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This is PART 16: Centers X(30001) -

PART 1: Introduction and Centers X(1) - X(1000)
PART 2: Centers X(1001) - X(3000)
PART 3: Centers X(3001) - X(5000)
PART 4: Centers X(5001) - X(7000)
PART 5: Centers X(7001) - X(10000)
PART 6: Centers X(10001) - X(12000)
PART 7: Centers X(12001) - X(14000)
PART 8: Centers X(14001) - X(16000)
PART 9: Centers X(16001) - X(18000)
PART 10: Centers X(18001) - X(20000)
PART 11: Centers X(20001) - X(22000)
PART 12: Centers X(22001) - X(24000)
PART 13: Centers X(24001) - X(26000)
PART 14: Centers X(26001) - X(28000)
PART 15: Centers X(28001) - X(30000)
PART 16: Centers X(30001) - X(32000)


X(30001) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    a^5 b^2 - a b^6 + 2 a^5 b c + a^4 b^2 c - a^3 b^3 c - 2 a^2 b^4 c + a b^5 c - b^6 c + a^5 c^2 + a^4 b c^2 - 2 a^3 b^2 c^2 - a^2 b^3 c^2 - a b^4 c^2 - a^3 b c^3 - a^2 b^2 c^3 - 2 a b^3 c^3 + b^4 c^3 - 2 a^2 b c^4 - a b^2 c^4 + b^3 c^4 + a b c^5 - a c^6 - b c^6 : :

X(30001) lies on these lines:


X(30002) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    -a^5 b^2 + a b^6 - a^4 b^2 c - a^2 b^4 c + a b^5 c + b^6 c - a^5 c^2 - a^4 b c^2 + 2 a^3 b^2 c^2 + a^2 b^3 c^2 - 2 a b^4 c^2 + a^2 b^2 c^3 - 2 a b^3 c^3 - b^4 c^3 - a^2 b c^4 - 2 a b^2 c^4 - b^3 c^4 + a b c^5 + a c^6 + b c^6 : :

X(30002) lies on these lines:


X(30003) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    -a^5 b^2 + a b^6 - a^4 b^2 c - 2 a^3 b^3 c + a^2 b^4 c + a b^5 c + b^6 c - a^5 c^2 - a^4 b c^2 + 2 a^3 b^2 c^2 + a^2 b^3 c^2 - 2 a^3 b c^3 + a^2 b^2 c^3 - 2 a b^3 c^3 - b^4 c^3 + a^2 b c^4 - b^3 c^4 + a b c^5 + a c^6 + b c^6 : :

X(30003) lies on these lines:


X(30004) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    a^3 b^3 - a^2 b^4 + a^3 b^2 c + a^2 b^3 c - a b^4 c + a^3 b c^2 - 2 a^2 b^2 c^2 + a b^3 c^2 + a^3 c^3 + a^2 b c^3 + a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 - a b c^4 : :

X(30004) lies on these lines:


X(30005) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    -a^5 b^2 + a b^6 - a^4 b^2 c - 3 a^3 b^3 c + 2 a^2 b^4 c + a b^5 c + b^6 c - a^5 c^2 - a^4 b c^2 + 2 a^3 b^2 c^2 + a^2 b^3 c^2 + a b^4 c^2 - 3 a^3 b c^3 + a^2 b^2 c^3 - 2 a b^3 c^3 - b^4 c^3 + 2 a^2 b c^4 + a b^2 c^4 - b^3 c^4 + a b c^5 + a c^6 + b c^6 : :

X(30005) lies on these lines:


X(30006) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    -a^4 b^2 - a^3 b^3 + a^2 b^4 + a b^5 - 2 a^3 b^2 c - a^2 b^3 c + 2 a b^4 c + b^5 c - a^4 c^2 - 2 a^3 b c^2 + 6 a^2 b^2 c^2 - a b^3 c^2 - a^3 c^3 - a^2 b c^3 - a b^2 c^3 - 2 b^3 c^3 + a^2 c^4 + 2 a b c^4 + a c^5 + b c^5 : :

X(30006) lies on these lines:


X(30007) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    -a^4 b^2 - a^3 b^3 + a^2 b^4 + a b^5 - 2 a^3 b^2 c - a^2 b^3 c + 2 a b^4 c + b^5 c - a^4 c^2 - 2 a^3 b c^2 + 2 a^2 b^2 c^2 - a b^3 c^2 - a^3 c^3 - a^2 b c^3 - a b^2 c^3 - 2 b^3 c^3 + a^2 c^4 + 2 a b c^4 + a c^5 + b c^5 : :

X(30007) lies on these lines:


X(30008) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    -2 a^4 b^3 + 2 a^2 b^5 - 2 a^4 b^2 c - a^3 b^3 c + 3 a b^5 c - 2 a^4 b c^2 + 3 a^2 b^3 c^2 + b^5 c^2 - 2 a^4 c^3 - a^3 b c^3 + 3 a^2 b^2 c^3 - 2 a b^3 c^3 - b^4 c^3 - b^3 c^4 + 2 a^2 c^5 + 3 a b c^5 + b^2 c^5 : :

X(30008) lies on these lines:


X(30009) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    -a^9 b^2 + a b^10 - a^9 b c - a^8 b^2 c + a^5 b^5 c + b^10 c - a^9 c^2 - a^8 b c^2 + a^5 b^4 c^2 + a^4 b^5 c^2 + a^5 b^2 c^4 - a b^6 c^4 + a^5 b c^5 + a^4 b^2 c^5 - b^6 c^5 - a b^4 c^6 - b^5 c^6 + a c^10 + b c^10 : :

X(30009) lies on these lines:


X(30010) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    -a^5 b^2 + a b^6 - a^4 b^2 c + 2 a^3 b^3 c + a b^5 c + b^6 c - a^5 c^2 - a^4 b c^2 + 2 a^3 b c^3 + 2 a b^3 c^3 + a b c^5 + a c^6 + b c^6 : :

X(30010) lies on these lines:


X(30011) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    b c (-2 a^4 b + a^3 b^2 + a b^4 - 2 a^4 c + 2 a^3 b c + a^2 b^2 c - 2 a b^3 c + b^4 c + a^3 c^2 + a^2 b c^2 + 2 a b^2 c^2 - b^3 c^2 - 2 a b c^3 - b^2 c^3 + a c^4 + b c^4) : :

X(30011) lies on these lines:


X(30012) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    -a^4 b^4 + a^3 b^5 + 2 a^4 b^3 c - a^2 b^5 c - 2 a^4 b^2 c^2 - a^3 b^3 c^2 - a^2 b^4 c^2 - a b^5 c^2 + 2 a^4 b c^3 - a^3 b^2 c^3 - 2 a^2 b^3 c^3 + 2 a b^4 c^3 + b^5 c^3 - a^4 c^4 - a^2 b^2 c^4 + 2 a b^3 c^4 + a^3 c^5 - a^2 b c^5 - a b^2 c^5 + b^3 c^5 : :

X(30012) lies on these lines:


X(30013) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    -a^4 b^2 - a^3 b^3 + a^2 b^4 + a b^5 - 2 a^3 b^2 c - 2 a^2 b^3 c + a b^4 c + b^5 c - a^4 c^2 - 2 a^3 b c^2 + 16 a^2 b^2 c^2 - 3 a b^3 c^2 - b^4 c^2 - a^3 c^3 - 2 a^2 b c^3 - 3 a b^2 c^3 - 4 b^3 c^3 + a^2 c^4 + a b c^4 - b^2 c^4 + a c^5 + b c^5 : :

X(30013) lies on these lines:


X(30014) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    -2 a^4 b^2 - 2 a^3 b^3 + 2 a^2 b^4 + 2 a b^5 - 4 a^3 b^2 c - a^2 b^3 c + 5 a b^4 c + 2 b^5 c - 2 a^4 c^2 - 4 a^3 b c^2 + 5 a^2 b^2 c^2 + b^4 c^2 - 2 a^3 c^3 - a^2 b c^3 - 2 b^3 c^3 + 2 a^2 c^4 + 5 a b c^4 + b^2 c^4 + 2 a c^5 + 2 b c^5 : :

X(30014) lies on these lines:


X(30015) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    -a^5 b^2 + a b^6 - a^4 b^2 c + a b^5 c + b^6 c - a^5 c^2 - a^4 b c^2 + 4 a^3 b^2 c^2 + 2 a^2 b^3 c^2 - 2 a b^4 c^2 + 2 a^2 b^2 c^3 - 2 a b^3 c^3 - 2 b^4 c^3 - 2 a b^2 c^4 - 2 b^3 c^4 + a b c^5 + a c^6 + b c^6 : :

X(30015) lies on these lines:


X(30016) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    -a^4 b^2 + a^3 b^3 - a^2 b^4 + a b^5 - a b^4 c + b^5 c - a^4 c^2 + 4 a^2 b^2 c^2 - a b^3 c^2 - b^4 c^2 + a^3 c^3 - a b^2 c^3 - a^2 c^4 - a b c^4 - b^2 c^4 + a c^5 + b c^5 : :

X(30016) lies on these lines:


X(30017) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    -a^5 b^2 + a^4 b^3 - a^2 b^5 + a b^6 - a b^5 c + b^6 c - a^5 c^2 + 2 a^3 b^2 c^2 - b^5 c^2 + a^4 c^3 - a^2 c^5 - a b c^5 - b^2 c^5 + a c^6 + b c^6 : :

X(30017) lies on these lines:


X(30018) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    (b + c) (-a^5 b + a b^5 - a^5 c - a^4 b c + 2 a^3 b^2 c - 2 a b^4 c + b^5 c + 2 a^3 b c^2 + 2 a^2 b^2 c^2 - b^4 c^2 - b^3 c^3 - 2 a b c^4 - b^2 c^4 + a c^5 + b c^5) : :

X(30018) lies on these lines:


X(30019) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    -a^3 b^2 + a b^4 + a^2 b^2 c - a b^3 c + b^4 c - a^3 c^2 + a^2 b c^2 + 2 a b^2 c^2 - 2 b^3 c^2 - a b c^3 - 2 b^2 c^3 + a c^4 + b c^4 : :

X(30019) lies on these lines:


X(30020) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    (b - c) (a^3 b - 4 a^2 b^2 + a b^3 + a^3 c - 3 a^2 b c + b^3 c - 4 a^2 c^2 + 5 b^2 c^2 + a c^3 + b c^3) : :

X(30020) lies on these lines:


X(30021) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    -2 a^5 b^2 + 2 a b^6 - 2 a^4 b^2 c - a^3 b^3 c + 2 a b^5 c + 2 b^6 c - 2 a^5 c^2 - 2 a^4 b c^2 + 2 a^3 b^2 c^2 + a^2 b^3 c^2 - a b^4 c^2 - a^3 b c^3 + a^2 b^2 c^3 - 2 a b^3 c^3 - b^4 c^3 - a b^2 c^4 - b^3 c^4 + 2 a b c^5 + 2 a c^6 + 2 b c^6 : :

X(30021) lies on these lines:


X(30022) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    b^2 c^2 (-2 a^3 - a^2 b + a b^2 - a^2 c + a b c + b^2 c + a c^2 + b c^2) : :

X(30022) lies on these lines:


X(30023) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    (b - c) (-a^3 b^2 - a^2 b^3 - 3 a^2 b^2 c - a^3 c^2 - 3 a^2 b c^2 + a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3) : :

X(30023) lies on these lines:


X(30024) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    b c (b - c) (2 a^3 - a^2 b + a b^2 - a^2 c + 3 a b c + b^2 c + a c^2 + b c^2) : :

X(30024) lies on these lines:


X(30025) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    (b - c) (-a^4 b^2 - a^2 b^4 - a^2 b^3 c - a^4 c^2 - a^2 b^2 c^2 + 2 a b^3 c^2 + b^4 c^2 - a^2 b c^3 + 2 a b^2 c^3 + b^3 c^3 - a^2 c^4 + b^2 c^4) : :

X(30025) lies on these lines:


X(30026) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    b c (a^3 b^2 - 2 a^2 b^3 + a^2 b^2 c - a b^3 c + a^3 c^2 + a^2 b c^2 + b^3 c^2 - 2 a^2 c^3 - a b c^3 + b^2 c^3) : :

X(30026) lies on these lines:


X(30027) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    -2 a^4 b^2 - a^3 b^3 + a^2 b^4 + 2 a b^5 - 3 a^3 b^2 c - a^2 b^3 c + 3 a b^4 c + 2 b^5 c - 2 a^4 c^2 - 3 a^3 b c^2 + 2 a^2 b^2 c^2 - a b^3 c^2 - a^3 c^3 - a^2 b c^3 - a b^2 c^3 - 2 b^3 c^3 + a^2 c^4 + 3 a b c^4 + 2 a c^5 + 2 b c^5 : :

X(30027) lies on these lines:


X(30028) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    a^3 b^3 - a^2 b^4 + a^3 b^2 c + a^2 b^3 c - a b^4 c + a^3 b c^2 + a b^3 c^2 + a^3 c^3 + a^2 b c^3 + a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 - a b c^4 : :

X(30028) lies on these lines:


X(30029) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    -a^4 b^2 - a^3 b^3 + a^2 b^4 + a b^5 - 4 a^3 b^2 c - 2 a^2 b^3 c + 3 a b^4 c + b^5 c - a^4 c^2 - 4 a^3 b c^2 - 2 a^2 b^2 c^2 - a b^3 c^2 + b^4 c^2 - a^3 c^3 - 2 a^2 b c^3 - a b^2 c^3 + a^2 c^4 + 3 a b c^4 + b^2 c^4 + a c^5 + b c^5 : :

X(30029) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 100: X(30030) - X(30099)  rightri

Extending the preambles just before X(24537), X(26153), and X(27378), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 100, as in centers X(30030)-X(30099). Then

m(X) = (a b^2 + a c^2 + b^2 c + b c^2) x - b (b c + a b - a c) y - c (b c - a b + a c) z : :

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, December 12, 2018)

underbar




X(30030) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^2 b^2 + a b^3 - 2 a b^2 c + b^3 c - a^2 c^2 - 2 a b c^2 + a c^3 + b c^3 : :

X(30030) lies on these lines:


X(30031) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^5 b^2 + a b^6 - a^4 b^2 c + a^3 b^3 c - a b^5 c + b^6 c - a^5 c^2 - a^4 b c^2 + 2 a^3 b^2 c^2 + a^2 b^3 c^2 - a b^4 c^2 + a^3 b c^3 + a^2 b^2 c^3 + 2 a b^3 c^3 - b^4 c^3 - a b^2 c^4 - b^3 c^4 - a b c^5 + a c^6 + b c^6 : :

X(30031) lies on these lines:


X(30032) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^5 b^2 + a b^6 + a^5 b c - a^4 b^2 c - a^3 b^3 c + b^6 c - a^5 c^2 - a^4 b c^2 + 2 a^3 b^2 c^2 + a^2 b^3 c^2 - a b^4 c^2 - a^3 b c^3 + a^2 b^2 c^3 - b^4 c^3 - a b^2 c^4 - b^3 c^4 + a c^6 + b c^6 : :

X(30032) lies on these lines:


X(30033) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^5 b^2 + a b^6 + 2 a^5 b c - a^4 b^2 c - 3 a^3 b^3 c + a b^5 c + b^6 c - a^5 c^2 - a^4 b c^2 + 2 a^3 b^2 c^2 + a^2 b^3 c^2 - a b^4 c^2 - 3 a^3 b c^3 + a^2 b^2 c^3 - 2 a b^3 c^3 - b^4 c^3 - a b^2 c^4 - b^3 c^4 + a b c^5 + a c^6 + b c^6 : :

X(30033) lies on these lines:


X(30034) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^3 b^2 + a b^4 - a^2 b^2 c - a b^3 c + b^4 c - a^3 c^2 - a^2 b c^2 - a b c^3 + a c^4 + b c^4 : :

X(30034) lies on these lines:


X(30035) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^3 b^2 + a b^4 + a^3 b c - a^2 b^2 c - a b^3 c + b^4 c - a^3 c^2 - a^2 b c^2 + 2 a b^2 c^2 - b^3 c^2 - a b c^3 - b^2 c^3 + a c^4 + b c^4 : :

X(30035) lies on these lines:


X(30036) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^2 b^2 + a b^3 + a^2 b c - a b^2 c + b^3 c - a^2 c^2 - a b c^2 + a c^3 + b c^3 : :

X(30036) lies on these lines:


X(30037) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^3 b^2 + a b^4 + a^2 b^2 c - 2 a b^3 c + b^4 c - a^3 c^2 + a^2 b c^2 + 4 a b^2 c^2 - b^3 c^2 - 2 a b c^3 - b^2 c^3 + a c^4 + b c^4 : :

X(30037) lies on these lines:


X(30038) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^2 b^2 + a b^3 + 2 a^2 b c + b^3 c - a^2 c^2 + a c^3 + b c^3 : :

X(30038) lies on these lines:

X(30038) = complement of X(17752)


X(30039) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^4 b^2 + a^3 b^3 - a^2 b^4 + a b^5 + 2 a^4 b c - 2 a^3 b^2 c - a b^4 c + b^5 c - a^4 c^2 - 2 a^3 b c^2 + 6 a^2 b^2 c^2 - a b^3 c^2 - b^4 c^2 + a^3 c^3 - a b^2 c^3 - a^2 c^4 - a b c^4 - b^2 c^4 + a c^5 + b c^5 : :

X(30039) lies on these lines:


X(30040) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^7 b^2 - a^5 b^4 + a^3 b^6 + a b^8 - a^6 b^2 c + a^5 b^3 c - a^4 b^4 c + a^2 b^6 c - a b^7 c + b^8 c - a^7 c^2 - a^6 b c^2 + a^3 b^4 c^2 + a^2 b^5 c^2 + a^5 b c^3 + a b^5 c^3 - a^5 c^4 - a^4 b c^4 + a^3 b^2 c^4 - 2 a b^4 c^4 - b^5 c^4 + a^2 b^2 c^5 + a b^3 c^5 - b^4 c^5 + a^3 c^6 + a^2 b c^6 - a b c^7 + a c^8 + b c^8 : :

X(30040) lies on these lines:


X(30041) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^7 b^2 - a^5 b^4 + a^3 b^6 + a b^8 - a^6 b^2 c + a^5 b^3 c - a^4 b^4 c + a^2 b^6 c - a b^7 c + b^8 c - a^7 c^2 - a^6 b c^2 + a^3 b^4 c^2 + a^2 b^5 c^2 + a^5 b c^3 - 2 a^3 b^3 c^3 + a b^5 c^3 - a^5 c^4 - a^4 b c^4 + a^3 b^2 c^4 - 2 a b^4 c^4 - b^5 c^4 + a^2 b^2 c^5 + a b^3 c^5 - b^4 c^5 + a^3 c^6 + a^2 b c^6 - a b c^7 + a c^8 + b c^8 : :

X(30041) lies on these lines:


X(30042) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^4 b^2 + a b^5 - a^3 b^2 c - a b^4 c + b^5 c - a^4 c^2 - a^3 b c^2 - a b c^4 + a c^5 + b c^5 : :

X(30042) lies on these lines:


X(30043) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^5 b^2 + a b^6 - a^4 b^2 c - a b^5 c + b^6 c - a^5 c^2 - a^4 b c^2 - a b c^5 + a c^6 + b c^6 : :

X(30043) lies on these lines:


X(30044) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    b c (-2 a^2 b + a b^2 - 2 a^2 c - 4 a b c + b^2 c + a c^2 + b c^2) : :

X(30044) lies on these lines:


X(30045) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    b c (-a^3 b^2 - 2 a^3 b c - a^2 b^2 c + a b^3 c - a^3 c^2 - a^2 b c^2 - 2 a b^2 c^2 + b^3 c^2 + a b c^3 + b^2 c^3) : :

X(30045) lies on these lines:


X(30046) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^5 b^2 + a^4 b^3 - a^2 b^5 + a b^6 + a^3 b^3 c + a^2 b^4 c - 3 a b^5 c + b^6 c - a^5 c^2 + 2 a^3 b^2 c^2 + a b^4 c^2 - b^5 c^2 + a^4 c^3 + a^3 b c^3 + a^2 b c^4 + a b^2 c^4 - a^2 c^5 - 3 a b c^5 - b^2 c^5 + a c^6 + b c^6 : :

X(30046) lies on these lines:


X(30047) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^3 b^3 + a^2 b^4 - a^3 b^2 c - 2 a^2 b^3 c + 2 a b^4 c - a^3 b c^2 - 2 a^2 b^2 c^2 - a b^3 c^2 + b^4 c^2 - a^3 c^3 - 2 a^2 b c^3 - a b^2 c^3 + a^2 c^4 + 2 a b c^4 + b^2 c^4 : :

X(30047) lies on these lines:


X(30048) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^3 b^3 + a^2 b^4 - a^3 b^2 c - 2 a^2 b^3 c + 2 a b^4 c - a^3 b c^2 - a b^3 c^2 + b^4 c^2 - a^3 c^3 - 2 a^2 b c^3 - a b^2 c^3 + a^2 c^4 + 2 a b c^4 + b^2 c^4 : :

X(30048) lies on these lines:


X(30049) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -2 a^3 b^2 + 2 a b^4 - 3 a b^3 c + 2 b^4 c - 2 a^3 c^2 + 4 a b^2 c^2 - b^3 c^2 - 3 a b c^3 - b^2 c^3 + 2 a c^4 + 2 b c^4 : :

X(30049) lies on these lines:


X(30050) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^3 b^2 + a b^4 + 3 a^2 b^2 c - 3 a b^3 c + b^4 c - a^3 c^2 + 3 a^2 b c^2 + 8 a b^2 c^2 - 2 b^3 c^2 - 3 a b c^3 - 2 b^2 c^3 + a c^4 + b c^4 : :

X(30050) lies on these lines:


X(30051) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^4 b^2 + a^3 b^3 - a^2 b^4 + a b^5 + 2 a^2 b^3 c - 3 a b^4 c + b^5 c - a^4 c^2 + 2 a^2 b^2 c^2 + a b^3 c^2 - b^4 c^2 + a^3 c^3 + 2 a^2 b c^3 + a b^2 c^3 - a^2 c^4 - 3 a b c^4 - b^2 c^4 + a c^5 + b c^5 : :

X(30051) lies on these lines:


X(30052) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^3 b^2 + a b^4 + a^3 b c - a^2 b^2 c + b^4 c - a^3 c^2 - a^2 b c^2 + a c^4 + b c^4 : :

X(30052) lies on these lines:


X(30053) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^5 b^2 + a b^6 + 2 a^5 b c - a^4 b^2 c - 5 a^3 b^3 c + 2 a^2 b^4 c + a b^5 c + b^6 c - a^5 c^2 - a^4 b c^2 + 2 a^3 b^2 c^2 - 3 a^2 b^3 c^2 + a b^4 c^2 - 5 a^3 b c^3 - 3 a^2 b^2 c^3 - 2 a b^3 c^3 - b^4 c^3 + 2 a^2 b c^4 + a b^2 c^4 - b^3 c^4 + a b c^5 + a c^6 + b c^6 : :

X(30053) lies on these lines:


X(30054) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    b c (a^3 b^2 - 2 a^3 b c - a^2 b^2 c + a b^3 c + a^3 c^2 - a^2 b c^2 + b^3 c^2 + a b c^3 + b^2 c^3) : :

X(30054) lies on these lines:


X(30055) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^4 b^2 - a^3 b^3 + a^2 b^4 + a b^5 - 2 a^3 b^2 c - 2 a^2 b^3 c + a b^4 c + b^5 c - a^4 c^2 - 2 a^3 b c^2 - 4 a^2 b^2 c^2 - a b^3 c^2 + b^4 c^2 - a^3 c^3 - 2 a^2 b c^3 - a b^2 c^3 + a^2 c^4 + a b c^4 + b^2 c^4 + a c^5 + b c^5 : :

X(30055) lies on these lines:


X(30056) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^3 b^2 + a b^4 - 3 a^2 b^2 c - a b^3 c + b^4 c - a^3 c^2 - 3 a^2 b c^2 - 2 a b^2 c^2 - a b c^3 + a c^4 + b c^4 : :

X(30056) lies on these lines:


X(30057) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -2 a^2 b^2 + 2 a b^3 + a^2 b c - 3 a b^2 c + 2 b^3 c - 2 a^2 c^2 - 3 a b c^2 + 2 a c^3 + 2 b c^3 : :

X(30057) lies on these lines:


X(30058) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^4 b^2 + 3 a^3 b^3 - 3 a^2 b^4 + a b^5 + 5 a^2 b^3 c - 6 a b^4 c + b^5 c - a^4 c^2 - 2 a^2 b^2 c^2 + a b^3 c^2 - 2 b^4 c^2 + 3 a^3 c^3 + 5 a^2 b c^3 + a b^2 c^3 + 2 b^3 c^3 - 3 a^2 c^4 - 6 a b c^4 - 2 b^2 c^4 + a c^5 + b c^5 : :

X(30058) lies on these lines:


X(30059) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^3 b^2 + a b^4 - a^2 b^2 c + b^4 c - a^3 c^2 - a^2 b c^2 - 2 a b^2 c^2 + b^3 c^2 + b^2 c^3 + a c^4 + b c^4 : :

X(30059) lies on these lines:


X(30060) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    (b - c) (-a^4 b^2 - a^2 b^4 - a^3 b^2 c - a^4 c^2 - a^3 b c^2 - a b^3 c^2 + b^4 c^2 - a b^2 c^3 + b^3 c^3 - a^2 c^4 + b^2 c^4) : :

X(30060) lies on these lines:


X(30061) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    b c (b - c) (2 a^3 - a^2 b + a b^2 - a^2 c + a b c + b^2 c + a c^2 + b c^2) : :

X(30061) lies on these lines:


X(30062) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^4 b^2 + a b^5 - a^3 b^2 c - a b^4 c + b^5 c - a^4 c^2 - a^3 b c^2 + 2 a^2 b^2 c^2 - a b c^4 + a c^5 + b c^5 : :

X(30062) lies on these lines:


X(30063) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    a^2 b^2 - a b^3 + 2 a^2 b c + 4 a b^2 c - b^3 c + a^2 c^2 + 4 a b c^2 - a c^3 - b c^3 : :

X(30063) lies on these lines:


X(30064) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -2 a^5 b^2 + 2 a b^6 + a^5 b c - 2 a^4 b^2 c - a b^5 c + 2 b^6 c - 2 a^5 c^2 - 2 a^4 b c^2 + 4 a^3 b^2 c^2 + 2 a^2 b^3 c^2 - 2 a b^4 c^2 + 2 a^2 b^2 c^3 + 2 a b^3 c^3 - 2 b^4 c^3 - 2 a b^2 c^4 - 2 b^3 c^4 - a b c^5 + 2 a c^6 + 2 b c^6 : :

X(30064) lies on these lines:


X(30065) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^5 b^2 + a b^6 - a^4 b^2 c + a^3 b^3 c - a b^5 c + b^6 c - a^5 c^2 - a^4 b c^2 + 4 a^3 b^2 c^2 + 3 a^2 b^3 c^2 - a b^4 c^2 + a^3 b c^3 + 3 a^2 b^2 c^3 + 2 a b^3 c^3 - b^4 c^3 - a b^2 c^4 - b^3 c^4 - a b c^5 + a c^6 + b c^6 : :

X(30065) lies on these lines:


X(30066) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^7 b^2 - a^5 b^4 + a^3 b^6 + a b^8 - a^6 b^2 c + a^5 b^3 c - a^4 b^4 c + a^2 b^6 c - a b^7 c + b^8 c - a^7 c^2 - a^6 b c^2 + a^3 b^4 c^2 + a^2 b^5 c^2 + a^5 b c^3 - 4 a^3 b^3 c^3 + a b^5 c^3 - a^5 c^4 - a^4 b c^4 + a^3 b^2 c^4 - 2 a b^4 c^4 - b^5 c^4 + a^2 b^2 c^5 + a b^3 c^5 - b^4 c^5 + a^3 c^6 + a^2 b c^6 - a b c^7 + a c^8 + b c^8 : :

X(30066) lies on these lines:


X(30067) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^7 b^2 - 2 a^6 b^3 - a^5 b^4 + a^3 b^6 + 2 a^2 b^7 + a b^8 - a^6 b^2 c - 3 a^5 b^3 c - 3 a^4 b^4 c + 3 a^2 b^6 c + 3 a b^7 c + b^8 c - a^7 c^2 - a^6 b c^2 - 2 a^5 b^2 c^2 + a^3 b^4 c^2 - a^2 b^5 c^2 + 2 a b^6 c^2 + 2 b^7 c^2 - 2 a^6 c^3 - 3 a^5 b c^3 + 4 a^3 b^3 c^3 - 3 a b^5 c^3 - a^5 c^4 - 3 a^4 b c^4 + a^3 b^2 c^4 - 6 a b^4 c^4 - 3 b^5 c^4 - a^2 b^2 c^5 - 3 a b^3 c^5 - 3 b^4 c^5 + a^3 c^6 + 3 a^2 b c^6 + 2 a b^2 c^6 + 2 a^2 c^7 + 3 a b c^7 + 2 b^2 c^7 + a c^8 + b c^8 : :

X(30067) lies on these lines:


X(30068) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^8 b^2 - a^7 b^3 - a^6 b^4 - a^5 b^5 + a^4 b^6 + a^3 b^7 + a^2 b^8 + a b^9 - 2 a^7 b^2 c - 2 a^6 b^3 c - a^5 b^4 c - a^4 b^5 c + 2 a^3 b^6 c + 2 a^2 b^7 c + a b^8 c + b^9 c - a^8 c^2 - 2 a^7 b c^2 - a^5 b^3 c^2 - 2 a^4 b^4 c^2 + 2 a^3 b^5 c^2 + 2 a^2 b^6 c^2 + a b^7 c^2 + b^8 c^2 - a^7 c^3 - 2 a^6 b c^3 - a^5 b^2 c^3 - a^3 b^4 c^3 + a b^6 c^3 - a^6 c^4 - a^5 b c^4 - 2 a^4 b^2 c^4 - a^3 b^3 c^4 - 2 a^2 b^4 c^4 - 4 a b^5 c^4 - b^6 c^4 - a^5 c^5 - a^4 b c^5 + 2 a^3 b^2 c^5 - 4 a b^4 c^5 - 2 b^5 c^5 + a^4 c^6 + 2 a^3 b c^6 + 2 a^2 b^2 c^6 + a b^3 c^6 - b^4 c^6 + a^3 c^7 + 2 a^2 b c^7 + a b^2 c^7 + a^2 c^8 + a b c^8 + b^2 c^8 + a c^9 + b c^9 : :

X(30068) lies on these lines:


X(30069) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^8 b^2 + a^7 b^3 + a^6 b^4 - a^5 b^5 + a^4 b^6 - a^3 b^7 - a^2 b^8 + a b^9 - 2 a^7 b^2 c + 2 a^6 b^3 c + 5 a^5 b^4 c + a^4 b^5 c - 4 a^2 b^7 c - 3 a b^8 c + b^9 c - a^8 c^2 - 2 a^7 b c^2 + 4 a^6 b^2 c^2 + 3 a^5 b^3 c^2 - 2 a^4 b^4 c^2 + 4 a^3 b^5 c^2 - 5 a b^7 c^2 - b^8 c^2 + a^7 c^3 + 2 a^6 b c^3 + 3 a^5 b^2 c^3 - 4 a^4 b^3 c^3 - 7 a^3 b^4 c^3 + 4 a^2 b^5 c^3 + 3 a b^6 c^3 - 2 b^7 c^3 + a^6 c^4 + 5 a^5 b c^4 - 2 a^4 b^2 c^4 - 7 a^3 b^3 c^4 + 2 a^2 b^4 c^4 + 4 a b^5 c^4 + b^6 c^4 - a^5 c^5 + a^4 b c^5 + 4 a^3 b^2 c^5 + 4 a^2 b^3 c^5 + 4 a b^4 c^5 + 2 b^5 c^5 + a^4 c^6 + 3 a b^3 c^6 + b^4 c^6 - a^3 c^7 - 4 a^2 b c^7 - 5 a b^2 c^7 - 2 b^3 c^7 - a^2 c^8 - 3 a b c^8 - b^2 c^8 + a c^9 + b c^9 : :

X(30069) lies on these lines:


X(30070) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -3 a^5 b^2 + 3 a b^6 + a^5 b c - 3 a^4 b^2 c + a^3 b^3 c - 2 a b^5 c + 3 b^6 c - 3 a^5 c^2 - 3 a^4 b c^2 + 6 a^3 b^2 c^2 + 3 a^2 b^3 c^2 - 3 a b^4 c^2 + a^3 b c^3 + 3 a^2 b^2 c^3 + 4 a b^3 c^3 - 3 b^4 c^3 - 3 a b^2 c^4 - 3 b^3 c^4 - 2 a b c^5 + 3 a c^6 + 3 b c^6 : :

X(30070) lies on these lines:


X(30071) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^5 b^2 + a b^6 + a^5 b c - a^4 b^2 c - a^3 b^3 c + b^6 c - a^5 c^2 - a^4 b c^2 - a^2 b^3 c^2 - a b^4 c^2 - a^3 b c^3 - a^2 b^2 c^3 - b^4 c^3 - a b^2 c^4 - b^3 c^4 + a c^6 + b c^6 : :

X(30071) lies on these lines:


X(30072) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^5 b^2 + a b^6 - a^4 b^2 c - a^3 b^3 c - a b^5 c + b^6 c - a^5 c^2 - a^4 b c^2 + 2 a^3 b^2 c^2 + a^2 b^3 c^2 - a b^4 c^2 - a^3 b c^3 + a^2 b^2 c^3 - b^4 c^3 - a b^2 c^4 - b^3 c^4 - a b c^5 + a c^6 + b c^6 : :

X(30072) lies on these lines:


X(30073) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^5 b^2 + a b^6 - a^4 b^2 c + a^2 b^4 c - a b^5 c + b^6 c - a^5 c^2 - a^4 b c^2 + 2 a^3 b^2 c^2 - a^2 b^3 c^2 - a^2 b^2 c^3 + 2 a b^3 c^3 - b^4 c^3 + a^2 b c^4 - b^3 c^4 - a b c^5 + a c^6 + b c^6 : :

X(30073) lies on these lines:


X(30074) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^3 b^3 + a^2 b^4 + a^3 b^2 c - a^2 b^3 c + a b^4 c + a^3 b c^2 + 2 a^2 b^2 c^2 + a b^3 c^2 - a^3 c^3 - a^2 b c^3 + a b^2 c^3 - 2 b^3 c^3 + a^2 c^4 + a b c^4 : :

X(30074) lies on these lines:


X(30075) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^5 b^2 + a b^6 - a^4 b^2 c - a^3 b^3 c + 2 a^2 b^4 c - a b^5 c + b^6 c - a^5 c^2 - a^4 b c^2 + 2 a^3 b^2 c^2 - 3 a^2 b^3 c^2 + a b^4 c^2 - a^3 b c^3 - 3 a^2 b^2 c^3 + 2 a b^3 c^3 - b^4 c^3 + 2 a^2 b c^4 + a b^2 c^4 - b^3 c^4 - a b c^5 + a c^6 + b c^6 : :

X(30075) lies on these lines:


X(30076) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^4 b^2 - a^3 b^3 + a^2 b^4 + a b^5 - a^2 b^3 c + b^5 c - a^4 c^2 - 2 a^2 b^2 c^2 + a b^3 c^2 - a^3 c^3 - a^2 b c^3 + a b^2 c^3 - 2 b^3 c^3 + a^2 c^4 + a c^5 + b c^5 : :

X(30076) lies on these lines:


X(30077) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^5 b^2 - a^4 b^3 + a^2 b^5 + a b^6 - 2 a^4 b^2 c - 2 a^3 b^3 c + a b^5 c + b^6 c - a^5 c^2 - 2 a^4 b c^2 - 2 a^3 b^2 c^2 - 2 a^2 b^3 c^2 + b^5 c^2 - a^4 c^3 - 2 a^3 b c^3 - 2 a^2 b^2 c^3 + a^2 c^5 + a b c^5 + b^2 c^5 + a c^6 + b c^6 : :

X(30077) lies on these lines:


X(30078) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^4 b^2 - a^3 b^3 + a^2 b^4 + a b^5 - a^2 b^3 c + b^5 c - a^4 c^2 + 2 a^2 b^2 c^2 + a b^3 c^2 - a^3 c^3 - a^2 b c^3 + a b^2 c^3 - 2 b^3 c^3 + a^2 c^4 + a c^5 + b c^5 : :

X(30078) lies on these lines:


X(30079) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -2 a^4 b^3 + 2 a^2 b^5 - a^3 b^3 c - 2 a^2 b^4 c + 3 a b^5 c - 4 a^3 b^2 c^2 + a^2 b^3 c^2 + b^5 c^2 - 2 a^4 c^3 - a^3 b c^3 + a^2 b^2 c^3 - 2 a b^3 c^3 - b^4 c^3 - 2 a^2 b c^4 - b^3 c^4 + 2 a^2 c^5 + 3 a b c^5 + b^2 c^5 : :

X(30079) lies on these lines:


X(30080) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^9 b^2 + a b^10 + a^9 b c - a^8 b^2 c - a^5 b^5 c + b^10 c - a^9 c^2 - a^8 b c^2 + a^5 b^4 c^2 + a^4 b^5 c^2 + a^5 b^2 c^4 - a b^6 c^4 - a^5 b c^5 + a^4 b^2 c^5 - b^6 c^5 - a b^4 c^6 - b^5 c^6 + a c^10 + b c^10 : :

X(30080) lies on these lines:


X(30081) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^5 b^2 + a b^6 - a^4 b^2 c - 2 a^3 b^3 c - a b^5 c + b^6 c - a^5 c^2 - a^4 b c^2 - 2 a^3 b c^3 - 2 a b^3 c^3 - a b c^5 + a c^6 + b c^6 : :

X(30081) lies on these lines:


X(30082) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    b c (-3 a^3 b^2 + 2 a^2 b^3 + a b^4 + 2 a^3 b c - a^2 b^2 c - 2 a b^3 c + b^4 c - 3 a^3 c^2 - a^2 b c^2 + 2 a b^2 c^2 - b^3 c^2 + 2 a^2 c^3 - 2 a b c^3 - b^2 c^3 + a c^4 + b c^4) : :

X(30082) lies on these lines:


X(30083) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^4 b^2 - a^3 b^3 + a^2 b^4 + a b^5 + 2 a^3 b^2 c - a b^4 c + b^5 c - a^4 c^2 + 2 a^3 b c^2 - 4 a^2 b^2 c^2 + 3 a b^3 c^2 - b^4 c^2 - a^3 c^3 + 3 a b^2 c^3 - 4 b^3 c^3 + a^2 c^4 - a b c^4 - b^2 c^4 + a c^5 + b c^5 : :

X(30083) lies on these lines:


X(30084) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -2 a^4 b^2 - 2 a^3 b^3 + 2 a^2 b^4 + 2 a b^5 - 2 a^3 b^2 c - 3 a^2 b^3 c + a b^4 c + 2 b^5 c - 2 a^4 c^2 - 2 a^3 b c^2 - 5 a^2 b^2 c^2 + b^4 c^2 - 2 a^3 c^3 - 3 a^2 b c^3 - 2 b^3 c^3 + 2 a^2 c^4 + a b c^4 + b^2 c^4 + 2 a c^5 + 2 b c^5 : :

X(30084) lies on these lines:


X(30085) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^5 b^2 + a b^6 - a^4 b^2 c - a b^5 c + b^6 c - a^5 c^2 - a^4 b c^2 + 4 a^3 b^2 c^2 + 2 a^2 b^3 c^2 - 2 a b^4 c^2 + 2 a^2 b^2 c^3 + 2 a b^3 c^3 - 2 b^4 c^3 - 2 a b^2 c^4 - 2 b^3 c^4 - a b c^5 + a c^6 + b c^6 : :

X(30085) lies on these lines:


X(30086) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^4 b^2 + a^3 b^3 - a^2 b^4 + a b^5 + 2 a^2 b^3 c - 3 a b^4 c + b^5 c - a^4 c^2 + a b^3 c^2 - b^4 c^2 + a^3 c^3 + 2 a^2 b c^3 + a b^2 c^3 - a^2 c^4 - 3 a b c^4 - b^2 c^4 + a c^5 + b c^5 : :

X(30086) lies on these lines:


X(30087) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^5 b^2 + a^4 b^3 - a^2 b^5 + a b^6 + 2 a^2 b^4 c - 3 a b^5 c + b^6 c - a^5 c^2 + 2 a^3 b^2 c^2 - 2 a^2 b^3 c^2 + 2 a b^4 c^2 - b^5 c^2 + a^4 c^3 - 2 a^2 b^2 c^3 + 2 a^2 b c^4 + 2 a b^2 c^4 - a^2 c^5 - 3 a b c^5 - b^2 c^5 + a c^6 + b c^6 : :

X(30087) lies on these lines:


X(30088) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^5 b^2 + a b^6 + 2 a^5 b c - a^4 b^2 c - 2 a^3 b^3 c + a b^5 c + b^6 c - a^5 c^2 - a^4 b c^2 + 4 a^3 b^2 c^2 + 2 a^2 b^3 c^2 - 2 a b^4 c^2 - 2 a^3 b c^3 + 2 a^2 b^2 c^3 - 2 b^4 c^3 - 2 a b^2 c^4 - 2 b^3 c^4 + a b c^5 + a c^6 + b c^6 : :

X(30088) lies on these lines:


X(30089) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^3 b^2 + a b^4 + a^2 b^2 c - 3 a b^3 c + b^4 c - a^3 c^2 + a^2 b c^2 + 6 a b^2 c^2 - 2 b^3 c^2 - 3 a b c^3 - 2 b^2 c^3 + a c^4 + b c^4 : :

X(30089) lies on these lines:


X(30090) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    b c (-a^2 b + a b^2 - a^2 c - 3 a b c + b^2 c + a c^2 + b c^2) : :

X(30090) lies on these lines:


X(30091) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    (b - c) (a^3 b - 4 a^2 b^2 + a b^3 + a^3 c + a^2 b c - 2 a b^2 c + b^3 c - 4 a^2 c^2 - 2 a b c^2 + 5 b^2 c^2 + a c^3 + b c^3) : :

X(30091) lies on these lines:


X(30092) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    b^2 c^2 (-2 a^3 - a^2 b + a b^2 - a^2 c - a b c + b^2 c + a c^2 + b c^2) : :

X(30092) lies on these lines:


X(30093) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    b c (b - c) (-a^5 b + a^3 b^3 - a^5 c - a^4 b c + 2 a^3 b^2 c + a^2 b^3 c + a b^4 c + 2 a^3 b c^2 + a^2 b^2 c^2 - a b^3 c^2 + b^4 c^2 + a^3 c^3 + a^2 b c^3 - a b^2 c^3 + 2 b^3 c^3 + a b c^4 + b^2 c^4) : :

X(30093) lies on these lines:


X(30094) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    (b - c) (-a^3 b^2 - a^2 b^3 - a^2 b^2 c - a^3 c^2 - a^2 b c^2 - a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3) : :

X(30094) lies on these lines:


X(30095) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    (b - c) (-a^4 b^2 - a^2 b^4 + a^2 b^3 c - a^4 c^2 - a^2 b^2 c^2 + b^4 c^2 + a^2 b c^3 + b^3 c^3 - a^2 c^4 + b^2 c^4) : :

X(30095) lies on these lines:


X(30096) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    (b - c) (-a^5 b^3 - a^3 b^5 - a^5 b^2 c - a^4 b^3 c - a^2 b^5 c - a^5 b c^2 - 2 a^4 b^2 c^2 - a^2 b^4 c^2 + a b^5 c^2 - a^5 c^3 - a^4 b c^3 - a^2 b^3 c^3 + b^5 c^3 - a^2 b^2 c^4 + b^4 c^4 - a^3 c^5 - a^2 b c^5 + a b^2 c^5 + b^3 c^5) : :

X(30096) lies on these lines:


X(30097) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    (a + b - c) (a - b + c) (a b^2 + b^2 c + a c^2 + b c^2) : :

X(30097) lies on these lines:


X(30098) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -2 a^4 b^2 - a^3 b^3 + a^2 b^4 + 2 a b^5 - a^3 b^2 c - a^2 b^3 c - a b^4 c + 2 b^5 c - 2 a^4 c^2 - a^3 b c^2 + 2 a^2 b^2 c^2 + a b^3 c^2 - a^3 c^3 - a^2 b c^3 + a b^2 c^3 - 2 b^3 c^3 + a^2 c^4 - a b c^4 + 2 a c^5 + 2 b c^5 : :

X(30098) lies on these lines:


X(30099) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 100

Barycentrics    -a^3 b^3 + a^2 b^4 + a^3 b^2 c - a^2 b^3 c + a b^4 c + a^3 b c^2 + 4 a^2 b^2 c^2 + a b^3 c^2 - a^3 c^3 - a^2 b c^3 + a b^2 c^3 - 2 b^3 c^3 + a^2 c^4 + a b c^4 : :

X(30099) lies on these lines:


X(30100) = EULER LINE INTERCEPT OF X(8537)X(14531)

Barycentrics    a^2 (a^2+b^2-c^2) (a^2-b^2+c^2) (a^10-3 a^8 b^2+2 a^6 b^4+2 a^4 b^6-3 a^2 b^8+b^10-3 a^8 c^2+7 a^6 b^2 c^2-a^4 b^4 c^2-7 a^2 b^6 c^2+4 b^8 c^2+2 a^6 c^4-a^4 b^2 c^4-4 a^2 b^4 c^4-5 b^6 c^4+2 a^4 c^6-7 a^2 b^2 c^6-5 b^4 c^6-3 a^2 c^8+4 b^2 c^8+c^10) : :
Barycentrics S^2 (8 R^4 - 6 R^2 SW + SW^2) + SB SC (-16 R^4 + 9 R^2 SW - SW^2) : :

See Tran Quang Hung and Ercole Suppa, Hyacinthos 28720.

X(30100) lies on these lines: {2,3}, {8537,14531}, {11425,15872}

X(30100) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4,3520,7512), (378,3541,3520), (427,12225,4), (1593,7503,4), (1885,5133,4), (9818,12084,3547)


X(30101) = X(950)X(3664)∩X(3739)X(5745)

Barycentrics    ((b+3*c)*a^3+(2*b^2+3*b*c-c^2)*a^2+(b^2-c^2)*(b+3*c)*a+(b^2-c^2)*c*(3*b-c))*((3*b+c)*a^3-(b^2-3*b*c-2*c^2)*a^2-(b^2-c^2)*(3*b+c)*a+(b^2-c^2)*b*(b-3*c)) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28722.

X(30101) lies on these lines: {950, 3664}, {3739, 5745}, {5088, 17175}


X(30102) = X(140)X(6709)∩X(6748)X(11245)

Barycentrics    SB*SC*(S^2-4*R^2*(2*SW-SC)+SC*(2*SA+2*SB-SC)+2*SW^2)*(S^2-4*R^2*(2*SW-SB)+SB*(2*SA+2*SC-SB)+2*SW^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28722.

X(30102) lies on these lines: {140, 6709}, {6748, 11245}






leftri  Points Celaeno(h,i,j,k,u,v,w): X(30103) - X(30178)  rightri

Definition: Point Celaeno(h,i,j,k,u,v,w,a,b,c) = f(h,i,j,k,u,v,w,a,b,c) : f(h,i,j,k,u,v,w,b,c,a) : f(h,i,j,k,u,v,w,c,a,b) (barycentrics), where

f(h,i,j,k,u,v,w,a,b,c) = h (a^4 + b^4 + c^4) + i (a^3 b + b^3 c + c^3 a + a^3 c + b^3 a + c^3 b) + j (b^2 c^2 + c^2 a^2 + a^2 b^2) + k (a^2 b c + a b^2 c + a b c^2) + a (u (a^3 + b^3 + c^3) + v (a^2 b + b^2 c + c^2 a + a^2 c + b^2 a + c^2 b) + w a b c)

where h, i, j, k, u, v, w are real numbers, not all zero. These points lie on the line X(1)X(2). (Clark Kimberling, December 14, 2018)

underbar




X(30103) = POINT CELAENO(1,0,0,0,0,0,1)

Barycentrics    a^4 + b^4 + a^2 b c + c^4 : :

X(30103) lies on these lines:


X(30104) = POINT CELAENO(1,0,0,0,0,0,-1)

Barycentrics    a^4 + b^4 - a^2 b c + c^4 : :

X(30104) lies on these lines:


X(30105) = POINT CELAENO(0,1,0,0,1,0,0)

Barycentrics    a^4 + a^3 b + 2 a b^3 + a^3 c + b^3 c + 2 a c^3 + b c^3 : :

X(30105) lies on these lines:


X(30106) = POINT CELAENO(0,1,0,0,0,1,0)

Barycentrics    2 a^3 b + a^2 b^2 + a b^3 + 2 a^3 c + a b^2 c + b^3 c + a^2 c^2 + a b c^2 + a c^3 + b c^3 : :

X(30106) lies on these lines:


X(30107) = POINT CELAENO(0,1,0,0,0,0,1)

Barycentrics    a^3 b + a b^3 + a^3 c + a^2 b c + b^3 c + a c^3 + b c^3 : :

X(30107) lies on these lines:


X(30108) = POINT CELAENO(0,1,0,0,-1,0,0)

Barycentrics    -a^4 + a^3 b + a^3 c + b^3 c + b c^3 : :

X(30108) lies on these lines:


X(30109) = POINT CELAENO(0,1,0,0,0,-1,0)

Barycentrics    -a^2 b^2 + a b^3 - a b^2 c + b^3 c - a^2 c^2 - a b c^2 + a c^3 + b c^3 : :

X(30109) lies on these lines:


X(30110) = POINT CELAENO(0,1,0,0,0,0,-1)

Barycentrics    a^3 b + a b^3 + a^3 c - a^2 b c + b^3 c + a c^3 + b c^3 : :

X(30110) lies on these lines:


X(30111) = POINT CELAENO(0,0,1,0,1,0,0)

Barycentrics    a^4 + a^2 b^2 + a b^3 + a^2 c^2 + b^2 c^2 + a c^3 : :

X(30111) lies on these lines:


X(30112) = POINT CELAENO(0,0,1,0,0,1,0)

Barycentrics    a^3 b + 2 a^2 b^2 + a^3 c + a b^2 c + 2 a^2 c^2 + a b c^2 + b^2 c^2 : :

X(30112) lies on these lines:


X(30113) = POINT CELAENO(0,0,1,0,-1,0,0)

Barycentrics    -a^4 + a^2 b^2 - a b^3 + a^2 c^2 + b^2 c^2 - a c^3 : :

X(30113) lies on these lines:


X(30114) = POINT CELAENO(0,0,1,0,0,-1,0)

Barycentrics    -a^3 b - a^3 c - a b^2 c - a b c^2 + b^2 c^2 : :

X(30114) lies on these lines:


X(30115) = POINT CELAENO(0,0,0,1,1,0,0)

Barycentrics    a (a^3 + b^3 + a b c + b^2 c + b c^2 + c^3) : :

X(30115) lies on these lines:


X(30116) = POINT CELAENO(0,0,0,1,0,1,0)

Barycentrics    a (a^2 b + a b^2 + a^2 c + a b c + 2 b^2 c + a c^2 + 2 b c^2) : :

X(30116) lies on these lines:


X(30117) = POINT CELAENO(0,0,0,1,-1,0,0)

Barycentrics    a (a^3 + b^3 - a b c - b^2 c - b c^2 + c^3) : :

X(30117) lies on these lines:


X(30118) = POINT CELAENO(1,0,0,0,1,1,1)

Barycentrics    2 a^4 + a^3 b + a^2 b^2 + a b^3 + b^4 + a^3 c + a^2 b c + a b^2 c + a^2 c^2 + a b c^2 + a c^3 + c^4 : :

X(30118) lies on these lines:


X(30119) = POINT CELAENO(1,0,0,0,-1,1,1)

Barycentrics    a^3 b + a^2 b^2 - a b^3 + b^4 + a^3 c + a^2 b c + a b^2 c + a^2 c^2 + a b c^2 - a c^3 + c^4 : :

X(30119) lies on these lines:


X(30120) = POINT CELAENO(1,0,0,0,1,-1,1)

Barycentrics    2 a^4 - a^3 b - a^2 b^2 + a b^3 + b^4 - a^3 c + a^2 b c - a b^2 c - a^2 c^2 - a b c^2 + a c^3 + c^4 : :

X(30120) lies on these lines:


X(30121) = POINT CELAENO(1,0,0,0,1,1,-1)

Barycentrics    2*a^4+(b+c)*a^3+(b^2-b*c+c^2)*a^2+(b+c)*(b^2+c^2)*a+c^4+b^4 : :

X(30121) lies on these lines:


X(30122) = POINT CELAENO(1,0,0,0,-1,-1,-1)

Barycentrics    -a^3 b - a^2 b^2 - a b^3 + b^4 - a^3 c - a^2 b c - a b^2 c - a^2 c^2 - a b c^2 - a c^3 + c^4 : :

X(30122) lies on these lines:


X(30123) = POINT CELAENO(1,0,0,0,1,-1,-1)

Barycentrics    2 a^4 - a^3 b - a^2 b^2 + a b^3 + b^4 - a^3 c - a^2 b c - a b^2 c - a^2 c^2 - a b c^2 + a c^3 + c^4 : :

X(30123) lies on these lines:


X(30124) = POINT CELAENO(1,0,0,0,-1,1,-1)

Barycentrics    a^3 b + a^2 b^2 - a b^3 + b^4 + a^3 c - a^2 b c + a b^2 c + a^2 c^2 + a b c^2 - a c^3 + c^4 : :

X(30124) lies on these lines:


X(30125) = POINT CELAENO(1,0,0,0,-1,-1,1)

Barycentrics    -a^3 b - a^2 b^2 - a b^3 + b^4 - a^3 c + a^2 b c - a b^2 c - a^2 c^2 - a b c^2 - a c^3 + c^4 : :

X(30125) lies on these lines:


X(30126) = POINT CELAENO(0,1,0,0,1,1,1)

Barycentrics    a^4 + 2 a^3 b + a^2 b^2 + 2 a b^3 + 2 a^3 c + a^2 b c + a b^2 c + b^3 c + a^2 c^2 + a b c^2 + 2 a c^3 + b c^3 : :

X(30126) lies on these lines:


X(30127) = POINT CELAENO(0,1,0,0,-1,1,1)

Barycentrics    -a^4 + 2 a^3 b + a^2 b^2 + 2 a^3 c + a^2 b c + a b^2 c + b^3 c + a^2 c^2 + a b c^2 + b c^3 : :

X(30127) lies on these lines:


X(30128) = POINT CELAENO(0,1,0,0,1,-1,1)

Barycentrics    a^4 - a^2 b^2 + 2 a b^3 + a^2 b c - a b^2 c + b^3 c - a^2 c^2 - a b c^2 + 2 a c^3 + b c^3 : :

X(30128) lies on these lines:


X(30129) = POINT CELAENO(0,1,0,0,1,1,-1)

Barycentrics    a^4 + 2 a^3 b + a^2 b^2 + 2 a b^3 + 2 a^3 c - a^2 b c + a b^2 c + b^3 c + a^2 c^2 + a b c^2 + 2 a c^3 + b c^3 : :

X(30129) lies on these lines:


X(30130) = POINT CELAENO(0,1,0,0,-1,-1,-1)

Barycentrics    -a^4 - a^2 b^2 - a^2 b c - a b^2 c + b^3 c - a^2 c^2 - a b c^2 + b c^3 : :

X(30130) lies on these lines:


X(30131) = POINT CELAENO(0,1,0,0,1,-1,-1)

Barycentrics    a^4 - a^2 b^2 + 2 a b^3 - a^2 b c - a b^2 c + b^3 c - a^2 c^2 - a b c^2 + 2 a c^3 + b c^3 : :

X(30131) lies on these lines:


X(30132) = POINT CELAENO(0,1,0,0,-1,1,-1)

Barycentrics    -a^4 + 2 a^3 b + a^2 b^2 + 2 a^3 c - a^2 b c + a b^2 c + b^3 c + a^2 c^2 + a b c^2 + b c^3 : :

X(30132) lies on these lines:


X(30133) = POINT CELAENO(0,1,0,0,-1,-1,1)

Barycentrics    -a^4 - a^2 b^2 + a^2 b c - a b^2 c + b^3 c - a^2 c^2 - a b c^2 + b c^3 : :

X(30133) lies on these lines:


X(30134) = POINT CELAENO(0,0,1,0,1,1,1)

Barycentrics    a^4 + a^3 b + 2 a^2 b^2 + a b^3 + a^3 c + a^2 b c + a b^2 c + 2 a^2 c^2 + a b c^2 + b^2 c^2 + a c^3 : :

X(30134) lies on these lines:


X(30135) = POINT CELAENO(0,0,1,0,-1,1,1)

Barycentrics    -a^4 + a^3 b + 2 a^2 b^2 - a b^3 + a^3 c + a^2 b c + a b^2 c + 2 a^2 c^2 + a b c^2 + b^2 c^2 - a c^3 : :

X(30135) lies on these lines:


X(30136) = POINT CELAENO(0,0,1,0,1,-1,1)

Barycentrics    a^4 - a^3 b + a b^3 - a^3 c + a^2 b c - a b^2 c - a b c^2 + b^2 c^2 + a c^3 : :

X(30136) lies on these lines:


X(30137) = POINT CELAENO(0,0,1,0,1,1,-1)

Barycentrics    a^4 + a^3 b + 2 a^2 b^2 + a b^3 + a^3 c - a^2 b c + a b^2 c + 2 a^2 c^2 + a b c^2 + b^2 c^2 + a c^3 : :

X(30137) lies on these lines:


X(30138) = POINT CELAENO(0,0,1,0,-1,-1,-1)

Barycentrics    -a^4 - a^3 b - a b^3 - a^3 c - a^2 b c - a b^2 c - a b c^2 + b^2 c^2 - a c^3 : :

X(30138) lies on these lines:


X(30139) = POINT CELAENO(0,0,1,0,1,-1,-1)

Barycentrics    a^4 - a^3 b + a b^3 - a^3 c - a^2 b c - a b^2 c - a b c^2 + b^2 c^2 + a c^3 : :

X(30139) lies on these lines:


X(30140) = POINT CELAENO(0,0,1,0,-1,1,-1)

Barycentrics    -a^4 + a^3 b + 2 a^2 b^2 - a b^3 + a^3 c - a^2 b c + a b^2 c + 2 a^2 c^2 + a b c^2 + b^2 c^2 - a c^3 : :

X(30140) lies on these lines:


X(30141) = POINT CELAENO(0,0,1,0,-1,-1,1)

Barycentrics    -a^4 - a^3 b - a b^3 - a^3 c + a^2 b c - a b^2 c - a b c^2 + b^2 c^2 - a c^3 : :

X(30141) lies on these lines:


X(30142) = POINT CELAENO(0,0,0,1,1,1,1)

Barycentrics    a (a^3 + a^2 b + a b^2 + b^3 + a^2 c + 2 a b c + 2 b^2 c + a c^2 + 2 b c^2 + c^3) : :

X(30142) lies on these lines:


X(30143) = POINT CELAENO(0,0,0,1,-1,1,1)

Barycentrics    a (a^3 - a^2 b - a b^2 + b^3 - a^2 c - 2 a b c - 2 b^2 c - a c^2 - 2 b c^2 + c^3) : :

X(30143) lies on these lines:


X(30144) = POINT CELAENO(0,0,0,1,1,-1,1)

Barycentrics    a (a^3 - a^2 b - a b^2 + b^3 - a^2 c + 2 a b c - a c^2 + c^3) : :

X(30144) lies on these lines:


X(30145) = POINT CELAENO(0,0,0,1,1,1,-1)

Barycentrics    a (a^3 + a^2 b + a b^2 + b^3 + a^2 c + 2 b^2 c + a c^2 + 2 b c^2 + c^3) : :

X(30145) lies on these lines:


X(30146) = POINT CELAENO(2,0,0,1,1,-1,-1)

Barycentrics    3 a^4 - a^3 b - a^2 b^2 + a b^3 + 2 b^4 - a^3 c - a^2 c^2 + a c^3 + 2 c^4 : :

X(30146) lies on these lines:


X(30147) = POINT CELAENO(0,0,0,1,-1,1,-1)

Barycentrics    a (a^3 - a^2 b - a b^2 + b^3 - a^2 c - 2 b^2 c - a c^2 - 2 b c^2 + c^3) : :

X(30147) lies on these lines:


X(30148) = POINT CELAENO(0,0,0,1,-1,-1,1)

Barycentrics    a (a^3 + a^2 b + a b^2 + b^3 + a^2 c - 2 a b c + a c^2 + c^3) : :

X(30148) lies on these lines:


X(30149) = POINT CELAENO(1,1,1,1,-1,-1,-1)

Barycentrics    b^4 + b^3 c + b^2 c^2 + b c^3 + c^4 : :

X(30149) lies on these lines:


X(30150) = POINT CELAENO(-1,1,1,1,1,-1,-1)

Barycentrics    2 a b^3 - b^4 + b^3 c + b^2 c^2 + 2 a c^3 + b c^3 - c^4 : :

X(30150) lies on these lines:


X(30151) = POINT CELAENO(1,-1,1,1,-1,1,-1)

Barycentrics    2 a^2 b^2 - 2 a b^3 + b^4 + 2 a b^2 c - b^3 c + 2 a^2 c^2 + 2 a b c^2 + b^2 c^2 - 2 a c^3 - b c^3 + c^4 : :

X(30151) lies on these lines:


X(30152) = POINT CELAENO(1,1,-1,1,-1,-1,-1)

Barycentrics    -2 a^2 b^2 + b^4 + b^3 c - 2 a^2 c^2 - b^2 c^2 + b c^3 + c^4: :

X(30152) lies on these lines:


X(30153) = POINT CELAENO(1, 1, 1, -1, -1, -1, 1)

Barycentrics    b^4 - 2 a b^2 c + b^3 c - 2 a b c^2 + b^2 c^2 + b c^3 + c^4 : :

X(30153) lies on these lines:


X(30154) = POINT CELAENO(1, 1, -1, -1, -1, -1, 1)

Barycentrics    -2 a^2 b^2 + b^4 - 2 a b^2 c + b^3 c - 2 a^2 c^2 - 2 a b c^2 - b^2 c^2 + b c^3 + c^4 : :

X(30154) lies on these lines:


X(30155) = POINT CELAENO(1, -1, 1, -1, -1, 1, 1)

Barycentrics    2*(b^2-b*c+c^2)*a^2-b^4+b^2*c^2-b*c^3-c^4-b^3*c : :

X(30155) lies on these lines:


X(30156) = POINT CELAENO(-1, 1, 1, -1, 1, -1, 1)

Barycentrics    2 a b^3 - b^4 - 2 a b^2 c + b^3 c - 2 a b c^2 + b^2 c^2 + 2 a c^3 + b c^3 - c^4 : :

X(30156) lies on these lines:


X(30157) = POINT CELAENO(-1, 1, -1, 1, 1, -1, -1)

Barycentrics    -2 a^2 b^2 + 2 a b^3 - b^4 + b^3 c - 2 a^2 c^2 - b^2 c^2 + 2 a c^3 + b c^3 - c^4 : :

X(30157) lies on these lines:


X(30158) = POINT CELAENO(1, -1, 1, -1, 1, -1, 1)

Barycentrics    2 a^4 - 2 a^3 b + b^4 - 2 a^3 c - 2 a b^2 c - b^3 c - 2 a b c^2 + b^2 c^2 - b c^3 + c^4 : :

X(30158) lies on these lines:


X(30159) = POINT CELAENO(1, 0, 1, -1, 1, -1, 1)

Barycentrics    2 a^4 - a^3 b + a b^3 + b^4 - a^3 c - 2 a b^2 c - 2 a b c^2 + b^2 c^2 + a c^3 + c^4 : :

X(30159) lies on these lines:


X(30160) = POINT CELAENO(1, -1, 0, -1, 1, -1, 1)

Barycentrics    2 a^4 - 2 a^3 b - a^2 b^2 + b^4 - 2 a^3 c - 2 a b^2 c - b^3 c - a^2 c^2 - 2 a b c^2 - b c^3 + c^4 : :

X(30160) lies on these lines:


X(30161) = POINT CELAENO(1, -1, 1, 0, 1, -1, 1)

Barycentrics    2 a^4 - 2 a^3 b + b^4 - 2 a^3 c + a^2 b c - a b^2 c - b^3 c - a b c^2 + b^2 c^2 - b c^3 + c^4 : :

X(30161) lies on these lines:


X(30162) = POINT CELAENO(-1, -1, -1, 1, 1, 1, 1)

Barycentrics    -b^4 + 2 a^2 b c + 2 a b^2 c - b^3 c + 2 a b c^2 - b^2 c^2 - b c^3 - c^4 : :

X(30162) lies on these lines:


X(30163) = POINT CELAENO(1, -1, 1, 1, -1, -1, -1)

Barycentrics    -2 a^3 b - 2 a b^3 + b^4 - 2 a^3 c - b^3 c + b^2 c^2 - 2 a c^3 - b c^3 + c^4 : :

X(30163) lies on these lines:


X(30164) = POINT CELAENO(1, 1, -1, 1, -1, 1, -1)

Barycentrics    2 a^3 b + b^4 + 2 a^3 c + 2 a b^2 c + b^3 c + 2 a b c^2 - b^2 c^2 + b c^3 + c^4 : :

X(30164) lies on these lines:


X(30165) = POINT CELAENO(1, 1, -1, -1, -1, 1, 1)

Barycentrics    2 a^3 b + b^4 + 2 a^3 c + b^3 c - b^2 c^2 + b c^3 + c^4 : :

X(30165) lies on these lines:


X(30166) = POINT CELAENO(1, -1, 1, -1, -1, -1, 1)

Barycentrics    -2 a^3 b - 2 a b^3 + b^4 - 2 a^3 c - 2 a b^2 c - b^3 c - 2 a b c^2 + b^2 c^2 - 2 a c^3 - b c^3 + c^4 : :

X(30166) lies on these lines:


X(30167) = POINT CELAENO(0, 1, 1, 1, -1, -1, -1)

Barycentrics    -a^4 + b^3 c + b^2 c^2 + b c^3 : :

X(30167) lies on these lines:


X(30168) = POINT CELAENO(0, 0, 1, 1, -1, -1, -1)

Barycentrics    -a^4 - a^3 b - a b^3 - a^3 c + b^2 c^2 - a c^3 : :

X(30168) lies on these lines:


X(30169) = POINT CELAENO(1, 0, 1, 1, -1, -1, -1)

Barycentrics    -a^3 b - a b^3 + b^4 - a^3 c + b^2 c^2 - a c^3 + c^4 : :

X(30169) lies on these lines:


X(30170) = POINT CELAENO(1, 0, 0, 1, -1, -1, -1)

Barycentrics    -a^3 b - a^2 b^2 - a b^3 + b^4 - a^3 c - a^2 c^2 - a c^3 + c^4 : :

X(30170) lies on these lines:


X(30171) = POINT CELAENO(1, 1, 0, 1, -1, -1, -1)

Barycentrics    -a^2 b^2 + b^4 + b^3 c - a^2 c^2 + b c^3 + c^4 : :

X(30171) lies on these lines:


X(30172) = POINT CELAENO(1, 1, 0, 0, -1, -1, -1)

Barycentrics    -a^2 b^2 + b^4 - a^2 b c - a b^2 c + b^3 c - a^2 c^2 - a b c^2 + b c^3 + c^4 : :

X(30172) lies on these lines:


X(30173) = POINT CELAENO(1, 1, 1, 0, -1, -1, -1)

Barycentrics    b^4 - a^2 b c - a b^2 c + b^3 c - a b c^2 + b^2 c^2 + b c^3 + c^4 : :

X(30173) lies on these lines:


X(30174) = POINT CELAENO(1, 1, 1, 0, 0, -1, -1)

Barycentrics    a^4 + a b^3 + b^4 - a^2 b c - a b^2 c + b^3 c - a b c^2 + b^2 c^2 + a c^3 + b c^3 + c^4 : :

X(30174) lies on these lines:


X(30175) = POINT CELAENO(1, 1, 1, 1, 0, -1, -1)

Barycentrics    a^4 + a b^3 + b^4 + b^3 c + b^2 c^2 + a c^3 + b c^3 + c^4 : :

X(30175) lies on these lines:


X(30176) = POINT CELAENO(1, 1, 1, 1, 0, 0, -1)

Barycentrics    a^4 + a^3 b + a^2 b^2 + a b^3 + b^4 + a^3 c + a b^2 c + b^3 c + a^2 c^2 + a b c^2 + b^2 c^2 + a c^3 + b c^3 + c^4 : :

X(30176) lies on these lines:


X(30177) = POINT CELAENO(1, 1, 1, 1, -1, 0, -1)

Barycentrics    a^3 b + a^2 b^2 + b^4 + a^3 c + a b^2 c + b^3 c + a^2 c^2 + a b c^2 + b^2 c^2 + b c^3 + c^4 : :

X(30177) lies on these lines:


X(30178) = POINT CELAENO(1, 1, 1, 1, -1, 0, 0)

Barycentrics    a^3 b + a^2 b^2 + b^4 + a^3 c + a^2 b c + a b^2 c + b^3 c + a^2 c^2 + a b c^2 + b^2 c^2 + b c^3 + c^4 : :

X(30178) lies on these lines:


X(30179) = POINT CELAENO(1, 1, 1, 1, -1, -1, 0)

Barycentrics    b^4 + a^2 b c + b^3 c + b^2 c^2 + b c^3 + c^4 : :

X(30179) lies on these lines:


X(30180) = POINT CELAENO(1, -1, 1, -1, 0, -1, 1)

Barycentrics    a^4 - 2 a^3 b - a b^3 + b^4 - 2 a^3 c - 2 a b^2 c - b^3 c - 2 a b c^2 + b^2 c^2 - a c^3 - b c^3 + c^4 : :

X(30180) lies on these lines:






leftri  Frègier points: X(30181) - X(30257)  rightri

This preamble and centers X(30181)-X(30257) were contributed by César Eliud Lozada, December 15, 2018.

Let Γ be a conic and P a fixed point on it. The sides of any right angle with vertex P cut Γ again in P' and P". Then all the lines P'P" intersect in a common point QΓ(P).
(See: Frègier's Theorem in MathWorld)

This section deals with some named conics in the plane of a triangle ABC. The point QΓ(P) is denoted here as the P-Frègier point of Γ.

It can be proved that if Γ is a rectangular hyperbola then QΓ(P) lies in the line at infinity. Isogonal conjugates of these points in the infinity are also included in this section.

The appearance of (i, j) in the following table means that QΓ(X(i)) = X(j) for the indicated conic Γ:

Γ = dual of Yff parabola:
(2, 4786), (7, 30181), (27, 30182), (75, 30183), (86, 30184), (273, 30185), (310, 30186), (335, 30187), (673, 30188), (675, 4025), (871, 30189), (903, 30190), (1088, 30191), (1268, 30192), (1659, 30193), (2296, 30194), (4373, 30195)
Γ = excentral-incentral ellipse:
(1768, 20), (5400, 30196), (16528, 30197)
Γ = Feuerbach hyperbola:
(1, 513), (4, 521), (7, 14077), (8, 30198), (9, 30199), (21, 30200), (79, 8702), (80, 513), (84, 30201), (90, 30202), (104, 30202), (256, 30203), (294, 30204), (314, 30205), (885, 30206), (941, 30207), (943, 30208)
Γ = Jerabek hyperbola:
(3, 924), (4, 520), (6, 30209), (54, 30210), (64, 30211), (65, 30212), (66, 30213), (67, 30209), (69, 20186)
Γ = Johnson circumconic:
(110, 5562), (265, 52), (1625, 30214)
Γ = Kiepert hyperbola:
(2, 1499), (4, 525), (10, 6002), (13, 523), (14, 523), (17, 30215), (18, 30216), (76, 30217), (83, 30218), (98, 525)
Γ = Kiepert parabola:
(669, 30219), (1649, 30220), (3233, 30221), (3733, 30222)
Γ = Mandart inellipse:
(11, 30223), (3271, 30224)
Γ = orthic inconic:
(125, 184)
Γ = Stammler hyperbola:
(1, 6003), (3, 523), (6, 1499), (399, 523)
Γ = Steiner circumellipse:
(99, 69), (190, 30225), (290, 30226), (648, 30227), (664, 30228)
Γ = Steiner inellipse:
(115, 6), (1015, 24289), (1084, 30229), (1086, 24281)
Γ = Thomson-Gibert-Moses hyperbola:
(2, 512), (3, 8675), (6, 30230), (110, 512), (154, 8673), (354, 30231), (392, 9013)
Γ = Yff hyperbola:
(2, 3830), (4, 3), (14163, 30232), (14164, 30233)
Γ = conic {A, B, C, X(1), X(2)}:
(1, 3669), (2, 30234), (57, 30235)

underbar

X(30181) = X(7)-FRÈGIER POINT OF DUAL OF YFF PARABOLA

Barycentrics    (a^2-2*(b+c)*a+b^2+4*b*c+c^2)*(b-c)*(a-c+b)*(a-b+c) : :
X(30181) = 2*X(3676)-3*X(24002)

X(30181) lies on these lines: {7,514}, {77,4449}, {522,693}, {663,7190}, {1086,30188}, {3663,21185}, {4040,4328}, {7178,28898}

X(30181) = reflection of X(30188) in X(1086)


X(30182) = X(27)-FRÈGIER POINT OF DUAL OF YFF PARABOLA

Barycentrics    (b^2+c^2-a^2)*(a^6-(b+c)*a^5+(b^2+3*b*c+c^2)*a^4-(3*b^2-2*b*c+3*c^2)*(b+c)^2*a^2+(b^2-c^2)^2*(b+c)*a+(b+c)*(b^2-c^2)*(b^3-c^3))*(b-c) : :

X(30182) lies on these lines: {2,4064}, {441,525}, {4467,8611}


X(30183) = X(75)-FRÈGIER POINT OF DUAL OF YFF PARABOLA

Barycentrics    (2*a^4+(b+c)*a^3+b*c*a^2+(b^2-c^2)*(b-c)*a-(b^2+c^2)*b*c)*(b-c) : :

X(30183) lies on these lines: {69,4498}, {75,28470}, {513,3004}, {1086,30187}, {4106,24560}, {4357,4401}, {4481,4785}, {8643,17321}

X(30183) = reflection of X(30187) in X(1086)


X(30184) = X(86)-FRÈGIER POINT OF DUAL OF YFF PARABOLA

Barycentrics    (a^4+3*(b+c)*a^3+(2*b^2+7*b*c+2*c^2)*a^2-(b+c)*(b^2+c^2)*a-b^4-(b-c)^2*b*c-c^4)*(b-c) : :

X(30184) lies on these lines: {523,4025}, {2786,4791}, {17159,21187}


X(30185) = X(273)-FRÈGIER POINT OF DUAL OF YFF PARABOLA

Barycentrics    (b^2+c^2-a^2)*(2*a^6-3*(b+c)*a^5+b*c*a^4+2*(b+c)^3*a^3-2*(b^2+c^2)*(b+c)^2*a^2+(b^2-c^2)^2*(b+c)*a-(b^2-c^2)^2*b*c)*(b-c)*(a+b-c)*(a-b+c) : :

X(30185) lies on the line {521,4025}


X(30186) = X(310)-FRÈGIER POINT OF DUAL OF YFF PARABOLA

Barycentrics
(2*(b+c)^2*a^6+(b+c)*(b^2+c^2)*a^5+(b^2+3*b*c+c^2)*b*c*a^4+(b+c)*(b^4-b^2*c^2+c^4)*a^3+(b^4+c^4-(b^2+b*c+c^2)*b*c)*b*c*a^2-(b+c)*(b^2+c^2)*b^2*c^2*a-(b^2+c^2)*b^3*c^3)*(b-c) : :

X(30186) lies on the line {512,4025}


X(30187) = X(335)-FRÈGIER POINT OF DUAL OF YFF PARABOLA

Barycentrics    (2*a^6-(b+c)*a^5+(b^2-3*b*c+c^2)*a^4+(b^4+c^4-(2*b^2-b*c+2*c^2)*b*c)*a^2+(b+c)*(b^4+c^4-2*(b-c)^2*b*c)*a-(b^4+c^4+(b-c)^2*b*c)*b*c)*(b-c) : :

X(30187) lies on these lines: {335,28470}, {812,3776}, {1086,30183}

X(30187) = reflection of X(30183) in X(1086)


X(30188) = X(673)-FRÈGIER POINT OF DUAL OF YFF PARABOLA

Barycentrics    (a^6-3*(b+c)*a^5+5*(b^2+b*c+c^2)*a^4-2*(b+c)*(3*b^2-2*b*c+3*c^2)*a^3+(3*b^4+3*c^4-2*(b-c)^2*b*c)*a^2+(b^2-c^2)^2*(b+c)*a-(b^3-c^3)*(b-c)*(b^2+c^2))*(b-c) : :

X(30188) lies on these lines: {514,673}, {522,3912}, {650,918}, {1086,30181}, {1155,4786}, {3935,27486}

X(30188) = reflection of X(30181) in X(1086)


X(30189) = X(871)-FRÈGIER POINT OF DUAL OF YFF PARABOLA

Barycentrics
(2*(b^2+b*c+c^2)^2*a^8+(b^3+c^3)*(b^2+b*c+c^2)*a^7+(b^4+c^4+(b^2+3*b*c+c^2)*b*c)*b*c*a^6+(b+c)*(b^2+c^2)*(b^4+c^4)*a^5+(b^6+c^6+(b^4+c^4-(b^2+b*c+c^2)*b*c)*b*c)*b*c*a^4+(b+c)*(b^4+c^4-(2*b^2-b*c+2*c^2)*b*c)*b^2*c^2*a^3-(b^4+c^4+(b^2+b*c+c^2)*b*c)*b^3*c^3*a^2-(b+c)*(b^2+c^2)*b^4*c^4*a-(b^2+c^2)*b^5*c^5)*(b-c) : :

X(30189) lies on the line {788,4025}


X(30190) = X(903)-FRÈGIER POINT OF DUAL OF YFF PARABOLA

Barycentrics    (5*a^4-5*(b+c)*a^3-(2*b^2-7*b*c+2*c^2)*a^2+(b+c)*(7*b^2-12*b*c+7*c^2)*a-b^4-5*(b-c)^2*b*c-c^4)*(b-c) : :

X(30190) lies on these lines: {320,514}, {900,4025}, {903,3667}, {1086,4786}

X(30190) = reflection of X(4786) in X(1086)


X(30191) = X(1088)-FRÈGIER POINT OF DUAL OF YFF PARABOLA

Barycentrics
(2*a^6-7*(b+c)*a^5+(10*b^2+21*b*c+10*c^2)*a^4-4*(b+c)*(2*b^2+3*b*c+2*c^2)*a^3+2*(b^2+b*c+c^2)*(2*b^2+b*c+2*c^2)*a^2-(b^2-c^2)^2*(b+c)*a+(b^2+c^2)*(b-c)^2*b*c)*(b-c)*(a-c+b)*(a-b+c) : :

X(30191) lies on these lines: {3900,4025}, {21195,21453}


X(30192) = X(1268)-FRÈGIER POINT OF DUAL OF YFF PARABOLA

Barycentrics    (13*a^4+19*(b+c)*a^3+3*(b+2*c)*(2*b+c)*a^2-(b+c)*(b^2+4*b*c+c^2)*a-(b^2+b*c+c^2)*(b^2+4*b*c+c^2))*(b-c) : :

X(30192) lies on the line {4025,4977}


X(30193) = X(1659)-FRÈGIER POINT OF DUAL OF YFF PARABOLA

Barycentrics    ((a^3-3*(b+c)*a^2+(b^2+c^2)*a+(b^2-c^2)*(b-c))*S+(a+b-c)*(a+c-b)*(a^3-(b+c)*a^2-(b+c)*b*c))*(b-c) : :

X(30193) lies on these lines: {}


X(30194) = X(2296)-FRÈGIER POINT OF DUAL OF YFF PARABOLA

Barycentrics
((b+c)^2*a^6+(b+c)*(3*b^2+4*b*c+3*c^2)*a^5+(2*b^4+2*c^4+(9*b^2+16*b*c+9*c^2)*b*c)*a^4-(b+c)*(b^4+c^4-(4*b^2+7*b*c+4*c^2)*b*c)*a^3-(b^6+c^6+(3*b^4-5*b^2*c^2+3*c^4)*b*c)*a^2-(2*b^2-3*b*c+2*c^2)*(b+c)^3*b*c*a-(b^4+c^4+(b-c)^2*b*c)*b^2*c^2)*(b-c) : :

X(30194) lies on the line {784,4025}


X(30195) = X(4373)-FRÈGIER POINT OF DUAL OF YFF PARABOLA

Barycentrics    (13*a^4-6*(b+c)*a^3-2*(2*b-c)*(b-2*c)*a^2+2*(b+c)*(7*b^2-12*b*c+7*c^2)*a-b^4-2*(5*b^2-7*b*c+5*c^2)*b*c-c^4)*(b-c) : :

X(30195) lies on the line {3667,4025}


X(30196) = X(5400)-FRÈGIER POINT OF EXCENTRAL-HEXYL ELLIPSE

Barycentrics
3*(b+c)*a^6-3*(b+c)^2*a^5-(b+c)*(6*b^2-11*b*c+6*c^2)*a^4+2*(3*b^4+3*c^4+(3*b^2-4*b*c+3*c^2)*b*c)*a^3+(b+c)*(3*b^4+3*c^4-2*(6*b^2-7*b*c+6*c^2)*b*c)*a^2-3*(b^4-c^4)*(b^2-c^2)*a+(b^2-c^2)^2*(b+c)*b*c : :

X(30196) lies on these lines: {3,30197}, {4,8}, {20,3667}, {40,17780}, {499,28774}, {2800,3952}, {4427,6326}, {5690,25030}, {5693,25253}, {9803,17777}, {17164,20117}, {26364,28826}

X(30196) = reflection of X(30197) in X(3)


X(30197) = X(16528)-FRÈGIER POINT OF EXCENTRAL-HEXYL ELLIPSE

Barycentrics
(b+c)*a^6-(5*b^2-6*b*c+5*c^2)*a^5-(b+c)*(4*b^2-9*b*c+4*c^2)*a^4+2*(4*b^2+5*b*c+4*c^2)*(b-c)^2*a^3+(b^2-c^2)*(b-c)*(3*b^2-4*b*c+3*c^2)*a^2-3*(b^4-c^4)*(b^2-c^2)*a+(b^2-c^2)^2*(b+c)*b*c : :

X(30197) lies on these lines: {3,30196}, {4,2457}, {20,145}

X(30197) = reflection of X(30196) in X(3)


X(30198) = X(8)-FRÈGIER POINT OF FEUERBACH HYPERBOLA

Barycentrics    a*(a^3-(b+c)*a^2-(b^2-8*b*c+c^2)*a+(b+c)*(b^2-4*b*c+c^2))*(b-c) : :

X(30198) lies on these lines: {4,23836}, {7,23819}, {30,511}, {84,23800}, {885,10307}, {905,6615}, {1052,9355}, {1339,4498}, {1357,16185}, {1769,14353}, {2254,13252}, {2401,12246}, {4905,30235}, {6705,23808}, {7659,29487}, {7971,14812}, {12114,24457}, {17410,21390}

X(30198) = circumnormal isogonal conjugate of X(8686)
X(30198) = isogonal conjugate of X(30236)


X(30199) = X(9)-FRÈGIER POINT OF FEUERBACH HYPERBOLA

Barycentrics    a*(a^5-3*(b+c)*a^4+2*(b+c)^2*a^3+2*(b+c)*(b^2+c^2)*a^2-(3*b^4+3*c^4+2*(2*b^2-3*b*c+2*c^2)*b*c)*a+(b^2-c^2)^2*(b+c))*(b-c) : :

X(30199) lies on these lines: {3,4394}, {4,4106}, {20,4380}, {30,511}, {84,23893}, {885,3427}, {3576,30234}, {6223,20297}, {6260,20314}

X(30199) = circumnormal isogonal conjugate of X(15728)
X(30199) = isogonal conjugate of X(30237)


X(30200) = X(21)-FRÈGIER POINT OF FEUERBACH HYPERBOLA

Barycentrics    a*(a^6-2*(b+c)*a^5-(b^2-b*c+c^2)*a^4+(b+c)*(4*b^2-b*c+4*c^2)*a^3-(b^4+c^4+(b-c)^2*b*c)*a^2-(b+c)*(2*b^4+2*c^4-(b^2+c^2)*b*c)*a+(b^4-c^4)*(b^2-c^2))*(b-c) : :

X(30200) lies on these lines: {30,511}, {13250,23035}, {13251,23036}

X(30200) = isogonal conjugate of X(30238)


X(30201) = X(84)-FRÈGIER POINT OF FEUERBACH HYPERBOLA

Barycentrics    a*(a^5-(b+c)*a^4-2*(b-c)^2*a^3+2*(b+c)*(b^2+c^2)*a^2+(b^4+c^4-2*(2*b^2+b*c+2*c^2)*b*c)*a-(b^2-c^2)^2*(b+c))*(b-c)*(-a+b+c) : :

X(30201) lies on these lines: {30,511}, {17896,20297}

X(30201) = isogonal conjugate of X(30239)


X(30202) = X(90)-FRÈGIER POINT OF FEUERBACH HYPERBOLA

Barycentrics    a*(a^6-2*(b+c)*a^5-(b-c)^2*a^4+4*(b+c)*(b^2+c^2)*a^3-(b^2+c^2)*(b^2+4*b*c+c^2)*a^2-2*(b^2-c^2)^2*(b+c)*a+(b^2-c^2)^2*(b+c)^2)*(b-c) : :

X(30202) lies on these lines: {30,511}, {4730,23224}

X(30202) = isogonal conjugate of X(30240)


X(30203) = X(256)-FRÈGIER POINT OF FEUERBACH HYPERBOLA

Barycentrics    a*((b^2+3*b*c+c^2)*a^3+(b+c)*(b^2+c^2)*a^2+(b^2+3*b*c+c^2)*b*c*a-(b+c)*b^2*c^2)*(b-c) : :

X(30203) lies on these lines: {1,7234}, {11,2680}, {30,511}, {649,10459}, {810,4879}, {4507,25128}, {5710,16874}

X(30203) = isogonal conjugate of X(30241)


X(30204) = X(294)-FRÈGIER POINT OF FEUERBACH HYPERBOLA

Barycentrics
a*(a^9-3*(b+c)*a^8+(4*b^2+7*b*c+4*c^2)*a^7-4*(b+c)*(b^2+c^2)*a^6+(2*b^4+2*c^4+(b^2+c^2)*b*c)*a^5+2*(b^3-c^3)*(b^2-c^2)*a^4-(b^2-c^2)^2*(4*b^2+7*b*c+4*c^2)*a^3+4*(b^4-c^4)*(b-c)*(b^2+3*b*c+c^2)*a^2-(3*b^6+3*c^6+(7*b^4+7*c^4+3*(b+c)^2*b*c)*b*c)*(b-c)^2*a+(b^4-c^4)*(b^2-c^2)^2*(b-c))*(b-c) : :

X(30204) lies on these lines: {30,511}, {1530,4106}

X(30204) = isogonal conjugate of X(30242)


X(30205) = X(314)-FRÈGIER POINT OF FEUERBACH HYPERBOLA

Barycentrics
a*((b^2+3*b*c+c^2)*a^6+(b+c)*b*c*a^5-(2*b^4-7*b^2*c^2+2*c^4)*a^4+2*(b+c)*(b^2+c^2)*b*c*a^3+(b^6+c^6+(b^4+c^4+(b^2+4*b*c+c^2)*b*c)*b*c)*a^2+(-4*b^2*c^2+(b^2-c^2)^2)*(b+c)*b*c*a-(b^2+c^2)*(b+c)^2*b^2*c^2)*(b-c) : :

X(30205) lies on the line {30,511}

X(30205) = isogonal conjugate of X(30243)


X(30206) = X(885)-FRÈGIER POINT OF FEUERBACH HYPERBOLA

Barycentrics
a*((b^2+6*b*c+c^2)*a^8-(3*b+c)*(b+3*c)*(b+c)*a^7+(b^4+c^4+(11*b^2+24*b*c+11*c^2)*b*c)*a^6+(b+c)*(5*b^4+5*c^4-2*(6*b^2+b*c+6*c^2)*b*c)*a^5-(5*b^4+5*c^4-(7*b^2-2*b*c+7*c^2)*b*c)*(b+c)^2*a^4-(b^2-c^2)*(b-c)*(b^4+c^4-2*(6*b^2+b*c+6*c^2)*b*c)*a^3+(3*b^6+3*c^6-(5*b^4+5*c^4+3*(b+c)^2*b*c)*b*c)*(b-c)^2*a^2-(b^2-c^2)*(b-c)^3*(b^4+c^4-2*(2*b^2+b*c+2*c^2)*b*c)*a-3*(b^4-c^4)*(b^2-c^2)*(b-c)^2*b*c) : :

X(30206) lies on the line {30,511}

X(30206) = isogonal conjugate of X(30244)


X(30207) = X(941)-FRÈGIER POINT OF FEUERBACH HYPERBOLA

Barycentrics    a^2*((4*b^2+9*b*c+4*c^2)*a^2+2*(b+2*c)*(2*b+c)*(b+c)*a+(5*b^2+12*b*c+5*c^2)*b*c)*(b-c) : :

X(30207) lies on the line {30,511}

X(30207) = isogonal conjugate of X(30245)


X(30208) = X(943)-FRÈGIER POINT OF FEUERBACH HYPERBOLA

Barycentrics
a*(a^6-2*(b+c)*a^5-(b^2+3*b*c+c^2)*a^4+(b+c)*(4*b^2+3*b*c+4*c^2)*a^3-(b^4+c^4-(b+3*c)*(3*b+c)*b*c)*a^2-(b+c)*(2*b^4+2*c^4+(3*b^2-4*b*c+3*c^2)*b*c)*a+(b^4-c^4)*(b^2-c^2))*(b-c) : :

X(30208) lies on the line {30,511}

X(30208) = isogonal conjugate of X(30246)


X(30209) = X(6)-FRÈGIER POINT OF JERABEK HYPERBOLA

Barycentrics    a^2*(a^4-b^4+4*b^2*c^2-c^4)*(b^2-c^2)*(-a^2+b^2+c^2) : :

X(30209) lies on these lines: {3,647}, {4,850}, {6,7652}, {30,511}, {110,7482}, {111,23701}, {879,4846}, {1147,21905}, {1296,15406}, {1636,3288}, {1640,9730}, {2081,3581}, {2435,3426}, {2519,22159}, {3569,14696}, {4549,15421}, {5652,5654}, {5926,8651}, {6643,28729}, {6699,22264}, {8574,13335}, {12038,14135}, {15451,22089}, {16194,23616}, {16229,18314}, {19543,24782}

X(30209) = complementary conjugate of X(14672)
X(30209) = isogonal conjugate of X(30247)


X(30210) = X(54)-FRÈGIER POINT OF JERABEK HYPERBOLA

Barycentrics    a^2*(a^8-2*(b^2+c^2)*a^6+b^2*c^2*a^4+(2*b^2-c^2)*(b^2-2*c^2)*(b^2+c^2)*a^2-(b^2-c^2)^4)*(b^2-c^2)*(-a^2+b^2+c^2) : :

X(30210) lies on these lines: {30,511}, {110,13863}, {185,6798}, {1291,15958}, {5943,20392}, {16106,16107}

X(30210) = isogonal conjugate of X(30248)


X(30211) = X(64)-FRÈGIER POINT OF JERABEK HYPERBOLA

Barycentrics    a^2*(a^8-2*(b^2+c^2)*a^6+10*b^2*c^2*a^4+2*(b^2+c^2)*(b^4-4*b^2*c^2+c^4)*a^2-(b^4+4*b^2*c^2+c^4)*(b^2-c^2)^2)*(b^2-c^2)*(-a^2+b^2+c^2) : :

X(30211) lies on these lines: {30,511}, {2435,6391}, {14380,15316}

X(30211) = isogonal conjugate of X(30249)


X(30212) = X(65)-FRÈGIER POINT OF JERABEK HYPERBOLA

Barycentrics    a*(a^4-(b+c)*a^3+b*c*a^2+(b^3+c^3)*a-(b^2-c^2)^2)*(b-c)*(-a^2+b^2+c^2) : :

X(30212) lies on these lines: {3,656}, {4,7253}, {5,8062}, {30,511}, {355,4086}, {661,23090}, {7629,11248}

X(30212) = isogonal conjugate of X(30250)


X(30213) = X(66)-FRÈGIER POINT OF JERABEK HYPERBOLA

Barycentrics    a^2*(a^10-(b^2+c^2)*a^8-2*(b^4+c^4)*a^6+2*(b^4-c^4)*(b^2-c^2)*a^4+(b^8+6*b^4*c^4+c^8)*a^2-(b^4-c^4)*(b^2-c^2)^3)*(b^2-c^2)*(-a^2+b^2+c^2) : :

X(30213) lies on these lines: {30,511}, {10097,15316}

X(30213) = isogonal conjugate of X(30251)


X(30214) = X(1625)-FRÈGIER POINT OF JOHNSON CIRCUMCONIC

Barycentrics
a^2*((b^2+c^2)^2*a^12-(b^2+c^2)*(3*b^4+2*b^2*c^2+3*c^4)*a^10+(2*b^8+2*c^8+(3*b^2+c^2)*(b^2+3*c^2)*b^2*c^2)*a^8+2*(b^2+c^2)*(b^8+c^8-(b^2+c^2)^2*b^2*c^2)*a^6-(b^2-c^2)^2*(3*b^8+3*c^8+4*(b^4+b^2*c^2+c^4)*b^2*c^2)*a^4+(b^2-c^2)^4*(b^4+c^4)*b^2*c^2+(b^8-c^8)*(b^2-c^2)^3*a^2)*(-a^2+b^2+c^2) : :

X(30214) lies on these lines: {3,6}, {525,5562}, {3150,3917}


X(30215) = X(17)-FRÈGIER POINT OF KIEPERT HYPERBOLA

Barycentrics    (2*sqrt(3)*S-a^2-c^2+7*b^2)*(2*sqrt(3)*S-a^2-b^2+7*c^2)*(12*S^2+4*sqrt(3)*(7*a^2-b^2-c^2)*S+(7*a^2-b^2-c^2)^2)*(b^2-c^2) : :

X(30215) lies on these lines: {30,511}, {6137,22934}

X(30215) = isogonal conjugate of X(30252)


X(30216) = X(18)-FRÈGIER POINT OF KIEPERT HYPERBOLA

Barycentrics    (-2*sqrt(3)*S-a^2-c^2+7*b^2)*(-2*sqrt(3)*S-a^2-b^2+7*c^2)*(12*S^2-4*sqrt(3)*(7*a^2-b^2-c^2)*S+(7*a^2-b^2-c^2)^2)*(b^2-c^2) : :

X(30216) lies on these lines: {30,511}, {6138,22889}

X(30216) = isogonal conjugate of X(30253)


X(30217) = X(76)-FRÈGIER POINT OF KIEPERT HYPERBOLA

Barycentrics    (2*a^6+(b^2+c^2)*a^4+(b^4-b^2*c^2+c^4)*a^2-(b^2+c^2)*b^2*c^2)*(b^2-c^2) : :

X(30217) lies on these lines: {3,9491}, {30,511}, {5027,23301}, {5926,9494}, {9147,9210}

X(30217) = isogonal conjugate of X(30254)


X(30218) = X(83)-FRÈGIER POINT OF KIEPERT HYPERBOLA

Barycentrics    (a^6+2*(b^2+c^2)*a^4+(b^2+2*c^2)*(2*b^2+c^2)*a^2-b^6-c^6)*(b^2-c^2) : :

X(30218) lies on these lines: {3,13511}, {30,511}, {2531,3095}, {3005,14316}, {9210,13309}, {14886,22159}

X(30218) = isogonal conjugate of X(30255)


X(30219) = X(669)-FRÈGIER POINT OF KIEPERT PARABOLA

Barycentrics    a^2*(a^8-3*(b^2+c^2)*a^6+(3*b^4+7*b^2*c^2+3*c^4)*a^4-(b^2+c^2)*(b^4+4*b^2*c^2+c^4)*a^2+2*(b^4+c^4)*b^2*c^2)*(b^2-c^2) : :

X(30219) lies on these lines: {6,1649}, {323,523}, {394,8029}, {399,30220}, {511,669}, {647,1570}, {684,8675}, {1993,11123}, {1994,10190}, {2451,3231}, {3288,3906}, {8371,15066}, {9168,11004}


X(30220) = X(1649)-FRÈGIER POINT OF KIEPERT PARABOLA

Barycentrics
(16*a^10-32*(b^2+c^2)*a^8+(19*b^4+58*b^2*c^2+19*c^4)*a^6-(b^2+c^2)*(5*b^4+22*b^2*c^2+5*c^4)*a^4+(b^8+c^8+7*(b^4+c^4)*b^2*c^2)*a^2+(b^4-c^4)*(b^2-c^2)*(b^4-4*b^2*c^2+c^4))*(b^2-c^2) : :

X(30220) lies on these lines: {110,8371}, {399,30219}, {523,9143}, {542,1649}, {669,2930}, {690,24981}, {9168,14683}

X(30220) = midpoint of X(9168) and X(14683)
X(30220) = reflection of X(8371) in X(110)


X(30221) = X(3233)-FRÈGIER POINT OF KIEPERT PARABOLA

Barycentrics    (4*a^8-4*(b^2+c^2)*a^6-(7*b^4-18*b^2*c^2+7*c^4)*a^4+10*(b^4-c^4)*(b^2-c^2)*a^2-3*(b^2-c^2)^4)*(c^2-a^2)*(a^2-b^2) : :
X(30221) = 7*X(110)-3*X(476) = 4*X(110)-3*X(3233) = 5*X(110)-3*X(7471) = X(110)+3*X(14480) = X(110)-3*X(14611) = 4*X(476)-7*X(3233) = 5*X(476)-7*X(7471) = X(476)+7*X(14480) = X(476)-7*X(14611) = 5*X(3233)-4*X(7471) = X(3233)+4*X(14480) = X(3233)-4*X(14611) = 4*X(6723)-3*X(12079) = X(7471)+5*X(14480) = X(7471)-5*X(14611) = X(10620)-3*X(14934)

X(30221) lies on these lines: {30,24981}, {110,476}, {2452,3066}, {6723,12079}, {10620,14934}

X(30221) = midpoint of X(14480) and X(14611)


X(30222) = X(3733)-FRÈGIER POINT OF KIEPERT PARABOLA

Barycentrics
a*(b-c)*(a^7-(3*b^2+4*b*c+3*c^2)*a^5+4*(b+c)*b*c*a^4+(3*b^4+3*c^4+(6*b^2+7*b*c+6*c^2)*b*c)*a^3-2*(b+c)*(3*b^2+4*b*c+3*c^2)*b*c*a^2-(-4*b^2*c^2+(b^2-c^2)^2)*(b+c)^2*a+2*(b^2-c^2)^2*(b+c)*b*c)*(a+b)*(a+c) : :

X(30222) lies on these lines: {517,3733}, {3737,8702}


X(30223) = X(11)-FRÈGIER POINT OF MANDART INELLIPSE

Barycentrics    a*(a^4-2*(b-c)^2*a^2+(b^2-c^2)^2)*(-a+b+c) : :

X(30223) lies on these lines: {1,90}, {9,55}, {11,57}, {19,1857}, {21,6514}, {31,33}, {34,774}, {36,7171}, {40,1728}, {44,7074}, {46,3583}, {56,84}, {63,497}, {65,12705}, {165,11502}, {171,9817}, {184,15503}, {212,4319}, {223,8758}, {238,1040}, {390,3219}, {405,12711}, {496,24467}, {516,1708}, {522,30224}, {610,15494}, {612,7069}, {614,7004}, {920,1479}, {950,12514}, {968,14547}, {971,1617}, {1001,10391}, {1104,1854}, {1108,2192}, {1118,1712}, {1155,10860}, {1158,1210}, {1182,4207}, {1376,15297}, {1397,10535}, {1406,2956}, {1420,10085}, {1445,3474}, {1454,10896}, {1467,7992}, {1478,18540}, {1490,1898}, {1519,3086}, {1621,10394}, {1697,3632}, {1698,10958}, {1707,1936}, {1723,2361}, {1736,8270}, {1737,3359}, {1839,1856}, {1851,2385}, {1852,7713}, {1859,12723}, {2098,3962}, {2175,11429}, {2182,20991}, {2187,2261}, {2308,4336}, {2950,12832}, {3022,11189}, {3056,5227}, {3057,12629}, {3058,3929}, {3065,7284}, {3100,17127}, {3218,5274}, {3220,10832}, {3271,11436}, {3305,5218}, {3306,10589}, {3333,3649}, {3338,18393}, {3467,7162}, {3475,8545}, {3486,5250}, {3554,19354}, {3587,4302}, {3685,3719}, {3899,7962}, {3928,11238}, {4383,9371}, {4423,17603}, {4428,15296}, {5119,5727}, {5204,9841}, {5223,10388}, {5225,7098}, {5248,10393}, {5281,27065}, {5285,10833}, {5426,13384}, {5432,7308}, {5534,11508}, {5720,8069}, {5853,20588}, {6210,10319}, {6763,10959}, {6765,26358}, {7091,7285}, {7174,24431}, {7289,12589}, {7677,11220}, {7741,17700}, {7951,17699}, {8543,11020}, {8583,22768}, {9614,12704}, {10389,15298}, {10578,29007}, {10947,24392}, {11435,21746}, {12185,24469}, {12686,18838}, {12717,24310}, {15171,26921}, {16541,24320}, {20992,23207}

X(30223) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 90, 7330), (55, 7082, 9), (3683, 14100, 55)


X(30224) = X(3271)-FRÈGIER POINT OF MANDART INELLIPSE

Barycentrics    a*(a^6-(b+c)*a^5-(b^2-3*b*c+c^2)*a^4+2*(b^2-c^2)*(b-c)*a^3-(b^2+c^2)*(b-c)^2*a^2-(b^2-c^2)*(b-c)^3*a+(b^2+3*b*c+c^2)*(b-c)^4)*(-a+b+c) : :

X(30224) lies on these lines: {9,4124}, {57,7336}, {522,30223}, {1738,3359}, {2808,3271}


X(30225) = X(190)-FRÈGIER POINT OF STEINER CIRCUMELLIPSE

Barycentrics    a^4-(b+c)*a^3-(2*b^2-3*b*c+2*c^2)*a^2+(b+c)*(3*b^2-4*b*c+3*c^2)*a-b^4-c^4-(b-c)^2*b*c : :

X(30225) lies on these lines: {2,24262}, {8,14947}, {10,19932}, {69,514}, {145,7760}, {190,952}, {239,17740}, {315,20535}, {346,1016}, {519,1992}, {1121,3699}, {1146,4561}, {3570,6790}, {3807,21290}, {3912,5219}, {4370,24807}, {4555,29616}, {4671,6542}, {6631,17233}, {16086,29331}, {24247,24282}

X(30225) = reflection of X(24807) in X(4370)
X(30225) = anticomplement of X(24281)


X(30226) = X(290)-FRÈGIER POINT OF STEINER CIRCUMELLIPSE

Barycentrics
(b^4+3*b^2*c^2+c^4)*a^10-3*(b^2+c^2)*(b^4+c^4)*a^8+3*(b^4+b^2*c^2+c^4)*(b^4-b^2*c^2+c^4)*a^6-(b^2+c^2)*(b^4-b^2*c^2+c^4)^2*a^4-(b^2-c^2)^2*(b^4-b^2*c^2+c^4)*b^2*c^2*a^2+(b^4-c^4)*(b^2-c^2)*b^4*c^4 : :
X(30226) = 5*X(3618)-4*X(5661)

X(30226) lies on these lines: {4,69}, {6,10684}, {99,25332}, {1992,23878}, {2549,30227}, {3618,5661}


X(30227) = X(648)-FRÈGIER POINT OF STEINER CIRCUMELLIPSE

Barycentrics    (a^8-(b^2+c^2)*a^6-(2*b^2+b*c-2*c^2)*(2*b^2-b*c-2*c^2)*a^4+3*(b^4-c^4)*(b^2-c^2)*a^2+(b^4-3*b^2*c^2+c^4)*(b^2-c^2)^2)*(b^2+c^2-a^2) : :

X(30227) lies on these lines: {4,9513}, {20,18338}, {30,1351}, {69,525}, {99,1562}, {125,671}, {148,3269}, {287,543}, {376,2966}, {648,2777}, {1249,23582}, {2549,30226}, {3164,15351}, {4235,6794}, {7738,10684}, {7748,9289}, {13172,17974}, {14568,21663}


X(30228) = X(664)-FRÈGIER POINT OF STEINER CIRCUMELLIPSE

Barycentrics    a^6-(b+c)*a^5-(3*b^2-5*b*c+3*c^2)*a^4+2*(b^3+c^3)*a^3+(3*b^4+3*c^4-4*(b^2+c^2)*b*c)*a^2-(b^2-c^2)*(b-c)*(b^2+4*b*c+c^2)*a-(b^4+c^4-(b+c)^2*b*c)*(b-c)^2 : :

X(30228) lies on these lines: {7,3675}, {69,522}, {144,666}, {347,1275}, {527,1992}, {2481,4440}, {4357,24411}, {16091,20254}


X(30229) = X(1084)-FRÈGIER POINT OF STEINER INELLIPSE

Barycentrics    a^2*(b^2*c^2*a^4+(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^2-(b^4-3*b^2*c^2+c^4)*b^2*c^2) : :

X(30229) lies on these lines: {6,512}, {538,599}, {543,694}, {574,3229}, {671,3124}, {695,7765}, {1613,5118}, {2086,14700}, {2088,3981}, {2936,20998}, {3978,7790}, {7757,18829}, {8591,9998}, {11152,11654}, {11171,12525}


X(30230) = X(6)-FRÈGIER POINT OF THOMSON-GIBERT-MOSES HYPERBOLA

Barycentrics    a^2*(b^2-c^2)*(a^6-(b^2+c^2)*a^4-(b^4-24*b^2*c^2+c^4)*a^2+(b^2+c^2)*(b^4-8*b^2*c^2+c^4)) : :

X(30230) lies on these lines: {3,8644}, {30,511}, {74,10102}, {1351,2444}, {1597,2489}, {3426,10097}, {3524,15724}, {4550,21905}, {7464,9137}, {8651,9126}, {11472,21733}

X(30230) = isogonal conjugate of X(30256)


X(30231) = X(354)-FRÈGIER POINT OF THOMSON-GIBERT-MOSES HYPERBOLA

Barycentrics    a*(b-c)*(a^5-2*(b+c)*a^4+(b^2+b*c+c^2)*a^3+(b+c)^3*a^2-(2*b^4+2*c^4+(b^2+c^2)*b*c)*a+(b^4-c^4)*(b-c)) : :

X(30231) lies on these lines: {30,511}, {2254,22160}, {4170,12699}, {4729,12702}, {7216,15934}, {9404,24290}

X(30231) = isogonal conjugate of X(30257)


X(30232) = X(14163)-FRÈGIER POINT OF YFF HYPERBOLA

Barycentrics
a^16-5*(b^2+c^2)*a^14+(3*b^4+32*b^2*c^2+3*c^4)*a^12+9*(-4*b^2*c^2+(b^2-c^2)^2)*(b^2+c^2)*a^10-(13*b^8+13*c^8-(8*b^4+83*b^2*c^2+8*c^4)*b^2*c^2)*a^8+(b^2+c^2)*(9*b^8+9*c^8+(2*b^4-41*b^2*c^2+2*c^4)*b^2*c^2)*a^6-(7*b^12+7*c^12-(12*b^8+12*c^8-(39*b^4-76*b^2*c^2+39*c^4)*b^2*c^2)*b^2*c^2)*a^4+(b^4-c^4)*(b^2-c^2)*(3*b^8+3*c^8-2*(3*b^4-7*b^2*c^2+3*c^4)*b^2*c^2)*a^2-4*(b^2-c^2)^6*b^2*c^2 : :
X(30232) = 3*X(381)-2*X(14164) = X(14164)-3*X(14215)

X(30232) lies on these lines: {3,8029}, {381,14164}, {523,30233}, {524,3830}

X(30232) = reflection of X(381) in X(14215)


X(30233) = X(14164)-FRÈGIER POINT OF YFF HYPERBOLA

Barycentrics    7*S^6-3*(6*R^2*(12*R^2+3*SA-5*SW)-4*SA^2+SB*SC+4*SW^2)*S^4-(9*R^2*SA*(27*R^2*(SA-SW)-SW*(-7*SW+9*SA))+SW^2*(7*R^2*SW+12*SA^2-8*SA*SW-SW^2))*S^2+(9*R^2-SW)*SB*SC*SW^3 : :
X(30233) = 3*X(381)-2*X(14163) = X(14163)-3*X(14214)

X(30233) lies on these lines: {3,8151}, {381,14163}, {523,30232}

X(30233) = reflection of X(381) in X(14214)


X(30234) = X(2)-FRÈGIER POINT OF CONIC {A, B, C, X(1), X(2)}

Barycentrics    a*(b-c)*(5*a^2-b^2-c^2) : :
X(30234) = X(1)+2*X(4394) = 2*X(667)+X(905) = 5*X(667)+X(2530) = 4*X(667)-X(3803) = 5*X(905)-2*X(2530) = 2*X(905)+X(3803) = 4*X(1125)-X(4106) = X(2254)+5*X(8656) = 4*X(2530)+5*X(3803) = X(2530)-5*X(14419) = 5*X(3616)+X(4380) = X(3669)+2*X(4401) = X(3803)+4*X(14419) = X(4367)+2*X(6050) = 2*X(4794)+X(7659)

X(30234) lies on these lines: {1,4394}, {2,28475}, {3,8642}, {28,6591}, {36,238}, {110,2746}, {650,4160}, {665,29350}, {1125,4106}, {1437,1980}, {1499,4786}, {1635,14077}, {2254,8656}, {2832,3669}, {3309,8643}, {3576,30199}, {3616,4380}, {4367,6050}, {4794,7659}, {9048,16475}

X(30234) = midpoint of X(667) and X(14419)
X(30234) = reflection of X(905) in X(14419)
X(30234) = {X(667), X(905)}-harmonic conjugate of X(3803)


X(30235) = X(57)-FRÈGIER POINT OF CONIC {A, B, C, X(1), X(2)}

Barycentrics    a*(b-c)*(a^5-(b+c)*a^4-2*(b^2-4*b*c+c^2)*a^3+2*(b+c)*(b^2-4*b*c+c^2)*a^2+(b^4+6*b^2*c^2+c^4)*a-(b^2-c^2)^2*(b+c)) : :

X(30235) lies on these lines: {1,650}, {56,11934}, {522,905}, {693,14986}, {999,8760}, {1125,25925}, {2826,3669}, {3086,4885}, {3616,24562}, {3622,26641}, {4294,8142}, {4905,30198}, {10529,26546}, {11019,29066}, {11193,22767}


X(30236) = ISOGONAL CONJUGATE OF X(30198)

Barycentrics    a*(c-a)*(a^3-(b+3*c)*a^2-(b^2-8*b*c+3*c^2)*a+(b^2-c^2)*(b-c))*(a-b)*(a^3-(3*b+c)*a^2-(3*b^2-8*b*c+c^2)*a+(b^2-c^2)*(b-c)) : :

X(30236) lies on the circumcircle and these lines: {3,8686}, {40,106}, {100,25737}, {104,5854}, {105,6244}, {108,23832}, {165,1477}, {953,13528}, {1292,8683}, {2077,2718}, {2291,3973}, {4571,6079}, {5537,22942}, {6282,28233}, {8059,23703}, {14110,28219}, {23981,30239}

X(30236) = reflection of X(8686) in X(3)
X(30236) = circumnormal isogonal conjugate of X(3880)
X(30236) = circumperp conjugate of X(8686)
X(30236) = isogonal conjugate of X(30198)
X(30236) = antipode of X(8686) in the circumcircle
X(30236) = trilinear pole of the line {6, 20323}


X(30237) = ISOGONAL CONJUGATE OF X(30199)

Barycentrics
a*(c-a)*(a^5-(3*b-c)*a^4+2*(b^2-2*b*c-c^2)*a^3+2*(b^3-c^3+b*c*(b+3*c))*a^2-(b^2-c^2)*(b-c)*(3*b-c)*a+(b^2-c^2)*(b-c)^3)*(a-b)*(a^5+(b-3*c)*a^4-2*(b^2+2*b*c-c^2)*a^3-2*(b^3-c^3-b*c*(3*b+c))*a^2+(b^2-c^2)*(b-c)*(b-3*c)*a+(b^2-c^2)*(b-c)^3) : :

X(30237) lies on the circumcircle and these lines: {3,15728}, {40,2291}, {103,6282}, {104,5759}, {105,3428}, {1477,3576}, {8059,23890}, {10310,15731}

X(30237) = reflection of X(15728) in X(3)
X(30237) = circumnormal isogonal conjugate of X(15733)
X(30237) = circumperp conjugate of X(15728)
X(30237) = isogonal conjugate of X(30199)
X(30237) = antipode of X(15728) in the circumcircle


X(30238) = ISOGONAL CONJUGATE OF X(30200)

Barycentrics
a*(c-a)*(a^6-2*b*a^5-(b^2+b*c+c^2)*a^4+(4*b^2-b*c+c^2)*b*a^3-(b^4+c^4-b*c*(3*b^2+2*b*c+c^2))*a^2-(b^2-c^2)*(b-c)*b*(2*b+c)*a+(b^2-c^2)^2*(b-c)^2)*(a-b)*(a^6-2*c*a^5-(b^2+b*c+c^2)*a^4+(b^2-b*c+4*c^2)*c*a^3-(b^4+c^4-b*c*(b^2+2*b*c+3*c^2))*a^2-(b^2-c^2)*(b-c)*c*(b+2*c)*a+(b^2-c^2)^2*(b-c)^2) : :

X(30238) lies on the circumcircle and these lines: {74,14110}, {104,12519}, {759,11012}, {5951,10310}

X(30238) = isogonal conjugate of X(30200)


X(30239) = ISOGONAL CONJUGATE OF X(30201)

Barycentrics
a*(c-a)*(a-b+c)*(a^5-(b-c)*a^4-2*(b-c)^2*a^3+2*(b-c)*(b^2+c^2)*a^2+(b^2-c^2)*(b^2-4*b*c-c^2)*a-(b^2-c^2)^2*(b-c))*(a-b)*(a-c+b)*(a^5+(b-c)*a^4-2*(b-c)^2*a^3-2*(b-c)*(b^2+c^2)*a^2+(b^2-c^2)*(b^2+4*b*c-c^2)*a+(b^2-c^2)^2*(b-c)) : :

X(30239) lies on the circumcircle and these lines: {56,1295}, {102,1420}, {104,10309}, {972,1617}, {1319,2745}, {2291,8602}, {2716,5193}, {23981,30236}

X(30239) = isogonal conjugate of X(30201)
X(30239) = trilinear pole of the line {6, 8602}


X(30240) = ISOGONAL CONJUGATE OF X(30202)

Barycentrics
a*(c-a)*(a^6-2*(b-c)*a^5-(b+c)^2*a^4+4*(b-c)*(b^2+c^2)*a^3-(b^2+c^2)*(b^2-4*b*c+c^2)*a^2-2*(b^2-c^2)^2*(b-c)*a+(b^2-c^2)^2*(b-c)^2)*(a-b)*(a^6+2*(b-c)*a^5-(b+c)^2*a^4-4*(b-c)*(b^2+c^2)*a^3-(b^2+c^2)*(b^2-4*b*c+c^2)*a^2+2*(b^2-c^2)^2*(b-c)*a+(b^2-c^2)^2*(b-c)^2) : :

X(30240) lies on the circumcircle and these lines: {104,11415}, {915,11249}

X(30240) = isogonal conjugate of X(30202)


X(30241) = ISOGONAL CONJUGATE OF X(30203)

Barycentrics
a*(c-a)*((b^2+b*c-c^2)*a^3+(b^3-c^3+b*c*(b+3*c))*a^2+(3*b^2+b*c+c^2)*b*c*a+(b+c)*b^2*c^2)*(a-b)*((b^2-b*c-c^2)*a^3+(b^3-c^3-b*c*(3*b+c))*a^2-(b^2+b*c+3*c^2)*b*c*a-(b+c)*b^2*c^2) : :

X(30241) lies on the circumcircle and these lines: {}

X(30241) = isogonal conjugate of X(30203)


X(30242) = ISOGONAL CONJUGATE OF X(30204)

Barycentrics
a*(c-a)*(a^9-(3*b+c)*a^8+(4*b^2-b*c-2*c^2)*a^7-2*(2*b^3-c^3-4*(b+c)*b*c)*a^6+(2*b^3-7*c^3-b*c*(7*b+8*c))*b*a^5+2*(b^4+3*c^4+2*b*c^2*(b-c))*b*a^4-(4*b^4-2*c^4+b*c*(7*b^2+12*b*c+3*c^2))*(b-c)^2*a^3+2*(2*b^5-c^5+(2*b^3+2*c^3+b*c*(2*b+c))*b*c)*(b-c)^2*a^2-(b^2-c^2)*(b-c)^3*(3*b^3+c^3+2*b*c*(b+2*c))*a+(b^4-c^4)*(b^2+c^2)*(b-c)^3)*(a-b)*(a^9-(b+3*c)*a^8-(2*b^2+b*c-4*c^2)*a^7+2*(b^3-2*c^3+4*(b+c)*b*c)*a^6-(7*b^3-2*c^3+b*c*(8*b+7*c))*c*a^5+2*(3*b^4+c^4-2*b^2*c*(b-c))*c*a^4+(2*b^4-4*c^4-b*c*(3*b^2+12*b*c+7*c^2))*(b-c)^2*a^3-2*(b^5-2*c^5-(2*b^3+2*c^3+b*c*(b+2*c))*b*c)*(b-c)^2*a^2-(b^2-c^2)*(b-c)^3*(b^3+3*c^3+2*b*c*(2*b+c))*a+(b^4-c^4)*(b^2+c^2)*(b-c)^3) : :

X(30242) lies on the circumcircle and these lines: {}

X(30242) = isogonal conjugate of X(30204)


X(30243) = ISOGONAL CONJUGATE OF X(30205)

Barycentrics
a*(c-a)*((b^2+b*c-c^2)*a^6+(b+2*c)*(b-c)*c*a^5-(2*b^4+2*c^4-b*c*(b+2*c)*(2*b-3*c))*a^4+2*(b^3-c^3+b*c*(2*b-3*c))*c^2*a^3+(b^6-c^6+(b^4+c^4+b*c*(7*b^2+2*b*c+c^2))*b*c)*a^2+(b+c)*(3*b^4+c^4-2*b^2*c*(b-c))*b*c*a+(b^2-c^2)^2*b^2*c^2)*(a-b)*((b^2-b*c-c^2)*a^6+(2*b+c)*(b-c)*b*a^5+(2*b^4+2*c^4+b*c*(2*b+c)*(3*b-2*c))*a^4+2*(b^3-c^3+b*c*(3*b-2*c))*b^2*a^3+(b^6-c^6-(b^4+c^4+b*c*(b^2+2*b*c+7*c^2))*b*c)*a^2-(b+c)*(b^4+3*c^4+2*b*c^2*(b-c))*b*c*a-(b^2-c^2)^2*b^2*c^2) : :

X(30243) lies on the circumcircle and these lines: {}

X(30243) = isogonal conjugate of X(30205)


X(30244) = ISOGONAL CONJUGATE OF X(30206)

Barycentrics
a*((b+3*c)*a^9-(3*b^2+7*b*c+6*c^2)*a^8+(b^2+11*b*c+12*c^2)*b*a^7+(5*b^4+6*c^4-b*c*(13*b^2+10*b*c+4*c^2))*a^6-(b+c)*(5*b^4+6*c^4-b*c*(8*b^2+b*c+4*c^2))*a^5-(b-c)*(b^5+6*c^5-(6*b^3-4*c^3-b*c*(b-2*c))*b*c)*a^4+(3*b^4-4*c^4-b*c*(5*b^2-b*c+3*c^2))*(b-c)^2*b*a^3-(b^5-6*c^5-(10*b^3+6*c^3+b*c*(9*b+10*c))*b*c)*(b-c)^3*a^2-(b^4-c^4)*c*(b-c)^2*(6*b^2-b*c+3*c^2)*a-(b^2-c^2)^2*(b-c)^3*b*c^2)*((3*b+c)*a^9-(6*b^2+7*b*c+3*c^2)*a^8+(12*b^2+11*b*c+c^2)*c*a^7+(6*b^4+5*c^4-b*c*(4*b^2+10*b*c+13*c^2))*a^6-(b+c)*(6*b^4+5*c^4-b*c*(4*b^2+b*c+8*c^2))*a^5+(b-c)*(6*b^5+c^5+(4*b^3-6*c^3-b*c*(2*b-c))*b*c)*a^4-(4*b^4-3*c^4+b*c*(3*b^2-b*c+5*c^2))*(b-c)^2*c*a^3-(6*b^5-c^5+(6*b^3+10*c^3+b*c*(10*b+9*c))*b*c)*(b-c)^3*a^2+(b^4-c^4)*b*(b-c)^2*(3*b^2-b*c+6*c^2)*a+(b^2-c^2)^2*(b-c)^3*b^2*c) : :

X(30244) lies on the circumcircle and these lines: {}

X(30244) = isogonal conjugate of X(30206)


X(30245) = ISOGONAL CONJUGATE OF X(30207)

Barycentrics    (c-a)*((4*b+5*c)*a^3+2*(b+2*c)*(2*b+3*c)*a^2+(b+c)*(9*b+5*c)*c*a+4*b*c^2*(b+c))*(a-b)*((5*b+4*c)*a^3+2*(3*b+2*c)*(2*b+c)*a^2+(5*b+9*c)*(b+c)*b*a+4*b^2*c*(b+c)) : :

X(30245) lies on the circumcircle and these lines: {}

X(30245) = isogonal conjugate of X(30207)


X(30246) = ISOGONAL CONJUGATE OF X(30208)

Barycentrics
a*(c-a)*(a^6-2*b*a^5-(b^2+5*b*c+c^2)*a^4+(4*b^2+3*b*c+c^2)*b*a^3-(b+c)*(b^3+c^3-2*b*c*(4*b+c))*a^2-(b^2-c^2)*(b-c)*b*(2*b+5*c)*a+(b^2-c^2)^2*(b-c)^2)*(a-b)*(a^6-2*c*a^5-(b^2+5*b*c+c^2)*a^4+(b^2+3*b*c+4*c^2)*c*a^3-(b+c)*(b^3+c^3-2*b*c*(b+4*c))*a^2-(b^2-c^2)*(b-c)*c*(5*b+2*c)*a+(b^2-c^2)^2*(b-c)^2) : :

X(30246) lies on the circumcircle and these lines: {}

X(30246) = isogonal conjugate of X(30208)


X(30247) = ISOGONAL CONJUGATE OF X(30209)

Barycentrics    (a^2-b^2+c^2)*(a^4-4*c^2*a^2-b^4+c^4)*(a^2+b^2-c^2)*(a^4-4*b^2*a^2+b^4-c^4)*(c^2-a^2)*(a^2-b^2) : :

X(30247) lies on the circumcircle and these lines: {2,14672}, {3,2373}, {4,111}, {24,2374}, {25,9084}, {28,9061}, {74,5486}, {98,378}, {105,4227}, {110,4235}, {186,2770}, {376,1297}, {468,10102}, {476,7482}, {523,10098}, {524,23701}, {648,691}, {675,7431}, {842,10295}, {925,11634}, {935,1632}, {1302,4230}, {1311,7436}, {1576,10423}, {2409,9064}, {2697,7464}, {3520,9076}, {3563,18533}, {4221,26703}, {4238,9058}, {4244,9107}, {4247,9083}, {4249,9057}, {5966,7576}, {7463,9056}, {7468,16167}, {7472,10420}, {7473,9060}, {11456,26717}

X(30247) = reflection of X(i) in X(j) for these (i,j): (4, 1560), (2373, 3)
X(30247) = circumnormal isogonal conjugate of X(2393)
X(30247) = circumperp conjugate of X(2373)
X(30247) = isogonal conjugate of X(30209)
X(30247) = anticomplement of X(14672)
X(30247) = antipode of X(2373) in the circumcircle
X(30247) = inverse of X(5512) in the polar circle
X(30247) = trilinear pole of the line {6, 468}


X(30248) = ISOGONAL CONJUGATE OF X(30210)

Barycentrics
(a^2-b^2+c^2)*(a^8-2*(b^2+2*c^2)*a^6+3*(b^2+2*c^2)*c^2*a^4+(b^2-c^2)*(2*b^4+b^2*c^2+4*c^4)*a^2-(b^4-c^4)*(b^2-c^2)^2)*(a^2+b^2-c^2)*(a^8-2*(2*b^2+c^2)*a^6+3*(2*b^2+c^2)*b^2*a^4-(b^2-c^2)*(4*b^4+b^2*c^2+2*c^4)*a^2+(b^4-c^4)*(b^2-c^2)^2)*(c^2-a^2)*(a^2-b^2) : :

X(30248) lies on the circumcircle and these lines: {74,12254}, {427,5966}, {523,13863}, {550,18401}, {1141,3520}, {2383,6240}, {7722,15907}, {13619,14979}

X(30248) = isogonal conjugate of X(30210)
X(30248) = trilinear pole of the line {6, 13418}


X(30249) = ISOGONAL CONJUGATE OF X(30211)

Barycentrics
(a^2-b^2+c^2)*(a^8-2*(b^2-c^2)*a^6+6*(b^2-c^2)*c^2*a^4+2*(b^2-c^2)*(b^4-4*b^2*c^2-c^4)*a^2-(b^4-c^4)*(b^2-c^2)^2)*(a^2+b^2-c^2)*(a^8+2*(b^2-c^2)*a^6-6*(b^2-c^2)*b^2*a^4+2*(b^2-c^2)*(b^4+4*b^2*c^2-c^4)*a^2+(b^4-c^4)*(b^2-c^2)^2)*(c^2-a^2)*(a^2-b^2) : :

X(30249) lies on the circumcircle and these lines: {4,5897}, {74,12250}, {5896,18213}, {6776,15324}

X(30249) = isogonal conjugate of X(30211)
X(30249) = polar conjugate of the anticomplement of X(20580)
X(30249) = trilinear pole of the line {6, 1885}


X(30250) = ISOGONAL CONJUGATE OF X(30212)

Barycentrics    a*(c-a)*(a^2-b^2+c^2)*(a^4-b*a^3-2*c^2*a^2+(b-c)*b^2*a-(b-c)*(b^3+c^3))*(a-b)*(a^2+b^2-c^2)*(a^4-c*a^3-2*b^2*a^2-(b-c)*c^2*a+(b-c)*(b^3+c^3)) : :

X(30250) lies on the circumcircle and these lines: {4,759}, {102,21740}, {104,7414}, {105,4231}, {110,4242}, {186,12030}, {925,13589}, {1295,3651}, {4220,26703}, {7438,9061}

X(30250) = isogonal conjugate of X(30212)


X(30251) = ISOGONAL CONJUGATE OF X(30213)

Barycentrics
(a^2-b^2+c^2)*(a^10-(b^2+3*c^2)*a^8-2*(b^4-c^4)*a^6+2*(b^2-c^2)*(b^4+2*b^2*c^2-c^4)*a^4+(b^4-c^4)*(b^4+3*c^4)*a^2-(b^4-c^4)^2*(b^2-c^2))*(a^2+b^2-c^2)*(a^10-(3*b^2+c^2)*a^8+2*(b^4-c^4)*a^6+2*(b^2-c^2)*(b^4-2*b^2*c^2-c^4)*a^4-(b^4-c^4)*(3*b^4+c^4)*a^2+(b^4-c^4)^2*(b^2-c^2))*(c^2-a^2)*(a^2-b^2) : :

X(30251) lies on the circumcircle and these lines: {24,2373}, {111,3542}, {1297,18533}, {4235,13398}

X(30251) = isogonal conjugate of X(30213)


X(30252) = ISOGONAL CONJUGATE OF X(30215)

Barycentrics    a^2*(2*sqrt(3)*S-a^2+7*b^2-c^2)*(2*sqrt(3)*S-a^2-b^2+7*c^2)*(c^2-a^2)*(a^2-b^2) : :

X(30252) lies on the circumcircle and these lines: {98,5487}, {107,14185}, {2380,10645}

X(30252) = isogonal conjugate of X(30215)


X(30253) = ISOGONAL CONJUGATE OF X(30216)

Barycentrics    a^2*(2*sqrt(3)*S+a^2-7*b^2+c^2)*(2*sqrt(3)*S+a^2+b^2-7*c^2)*(c^2-a^2)*(a^2-b^2) : :

X(30253) lies on the circumcircle and these lines: {98,5488}, {107,14187}, {2381,10646}

X(30253) = isogonal conjugate of X(30216)


X(30254) = ISOGONAL CONJUGATE OF X(30217)

Barycentrics    a^2*((b^2-c^2)*a^4+(b^4-b^2*c^2-c^4)*a^2+b^2*(2*b^4+b^2*c^2+c^4))*(c^2-a^2)*((b^2-c^2)*a^4+(b^4+b^2*c^2-c^4)*a^2-c^2*(b^4+b^2*c^2+2*c^4))*(a^2-b^2) : :

X(30254) lies on the circumcircle and these lines: {3,699}, {98,7751}, {729,3098}, {733,6234}

X(30254) = reflection of X(699) in X(3)
X(30254) = circumnormal isogonal conjugate of X(698)
X(30254) = circumperp conjugate of X(699)
X(30254) = isogonal conjugate of X(30217)
X(30254) = antipode of X(699) in the circumcircle


X(30255) = ISOGONAL CONJUGATE OF X(30218)

Barycentrics    a^2*(a^6-2*b^2*a^4-(2*b^2+5*c^2)*b^2*a^2-b^6+c^6-2*(b^2+c^2)*b^2*c^2)*(c^2-a^2)*(a^6-2*c^2*a^4-(5*b^2+2*c^2)*c^2*a^2+b^6-c^6-2*(b^2+c^2)*b^2*c^2)*(a^2-b^2) : :

X(30255) lies on the circumcircle and these lines: {4,13499}, {98,8150}, {733,3398}, {755,5092}, {5188,29011}

X(30255) = reflection of X(4) in X(13499)
X(30255) = isogonal conjugate of X(30218)


X(30256) = ISOGONAL CONJUGATE OF X(30230)

Barycentrics    (a^6-(b^2+7*c^2)*a^4-(b^4-24*b^2*c^2+7*c^4)*a^2+(b^4-c^4)*(b^2-c^2))*(a^6-(7*b^2+c^2)*a^4-(7*b^4-24*b^2*c^2+c^4)*a^2+(b^4-c^4)*(b^2-c^2))*(c^2-a^2)*(a^2-b^2) : :

X(30256) lies on the circumcircle and these lines: {3,9084}, {30,10102}, {111,376}, {378,2374}, {1302,11634}, {2373,21312}, {2770,7464}, {4221,9061}, {4235,9064}, {7472,9060}

X(30256) = reflection of X(9084) in X(3)
X(30256) = circumnormal isogonal conjugate of X(9027)
X(30256) = circumperp conjugate of X(9084)
X(30256) = isogonal conjugate of X(30230)
X(30256) = antipode of X(9084) in the circumcircle
X(30256) = trilinear pole of the line {6, 16317}


X(30257) = ISOGONAL CONJUGATE OF X(30231)

Barycentrics
a*(c-a)*(a^5-(2*b+c)*a^4+(b-c)*b*a^3+(b+3*c)*b^2*a^2-(b^2-c^2)*(b-c)*(2*b+c)*a+(b^3+c^3)*(b-c)^2)*(a-b)*(a^5-(b+2*c)*a^4-(b-c)*c*a^3+(3*b+c)*c^2*a^2-(b^2-c^2)*(b-c)*(b+2*c)*a+(b^3+c^3)*(b-c)^2) : :

X(30257) lies on the circumcircle and these lines: {105,3651}, {376,759}, {1302,13589}, {4220,9061}, {4242,9064}, {7414,15344}, {7464,12030}

X(30257) = isogonal conjugate of X(30231)


X(30258) = X(3)X(6)∩X(5)X(264)

Barycentrics    a^2*((b^4+b^2*c^2+c^4)*a^4-2*(b^4-c^4)*(b^2-c^2)*a^2+(b^4-b^2*c^2+c^4)*(b^2-c^2)^2)*(-a^2+b^2+c^2) : :
Barycentrics    (S^2-SB*SC)*(S^2+4*R^2*SW+2*SB*SC-SW^2) : :
X(30258) = 3*X(2)-4*X(10003), 5*X(1656)-4*X(14767)

See Dao Thanh Oai and César Lozada, Hyacinthos 28728.

X(30258) lies on these lines: {2, 1972}, {3, 6}, {4, 3164}, {5, 264}, {51, 6638}, {184, 13558}, {237, 6403}, {339, 7697}, {417, 15043}, {418, 3060}, {426, 5422}, {441, 18583}, {852, 5640}, {1073, 14489}, {1656, 14059}, {1942, 4846}, {1993, 6641}, {1994, 23606}, {5562, 17039}, {5889, 26897}, {5943, 6509}, {6375, 9243}, {6389, 14561}, {6776, 20975}, {10519, 20819}, {10796, 15013}, {11272, 28407}, {12161, 14152}, {15073, 20775}, {15526, 24206}, {20576, 28697}, {21969, 26907}

X(30258) = midpoint of X(4) and X(3164)
X(30258) = reflection of X(i) in X(j) for these (i,j): (3, 216), (264, 5)
X(30258) = anticomplement of the anticomplement of X(10003)
X(30258) = X(216)-of-X3-ABC reflections triangle
X(30258) = X(264)-of-Johnson triangle
X(30258) = X(3164)-of-Euler triangle
X(30258) = inverse-in-circle-{X(371),X(372),PU(1),PU(39)} of X(1970)
X(30258) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 5093, 15905), (3, 15851, 5050), (371, 372, 1970)


X(30259) = X(3)X(6)∩X(5012)X(14652)

Barycentrics    (SB+SC)*((5*R^2-2*SA-2*SW)*S^2-R^2*SA*SW) : :

See Dao Thanh Oai and César Lozada, Hyacinthos 28728.

X(30259) lies on these lines: {3, 6}, {5012, 14652}


X(30260) = X(3)X(6)∩X(13)X(9159)

Barycentrics    (SB+SC)*(S^2+sqrt(3)*(SA-3*R^2+SW)*S+(9*R^2-SW)*SA) : :

See Dao Thanh Oai and César Lozada, Hyacinthos 28728.

X(30260) lies on these lines: {3, 6}, {13, 9159}, {17, 10217}, {11658, 16241}


X(30261) = X(3)X(6)∩X(14)X(9159)

Barycentrics    (SB+SC)*(S^2-sqrt(3)*(SA-3*R^2+SW)*S+(9*R^2-SW)*SA) : :

See Dao Thanh Oai and César Lozada, Hyacinthos 28728.

X(30261) lies on these lines: {3, 6}, {14, 9159}, {18, 10218}, {11659, 16242}


X(30262) = X(3)X(6)∩X(20)X(3186)

Barycentrics    a^2 (-a^2+b^2+c^2)*(2*(b^2+c^2)*a^6-(3*b^4-b^2*c^2+3*c^4)*a^4+2*(b^2+c^2)*b^2*c^2*a^2+(b^6-c^6)*(b^2-c^2)) : :
Barycentrics    (S^2-SB*SC)*(S^2-16*R^2*SW-2*SB*SC+3*SW^2) : :

See Dao Thanh Oai and César Lozada, Hyacinthos 28728.

X(30262) lies on these lines: {3, 6}, {20, 3186}, {185, 20794}, {2790, 23240}, {7750, 14615}, {9306, 14673}, {11328, 12294}

X(30262) = midpoint of X(20) and X(3186)


X(30263) = X(20)X(2979)∩X(185)X(5667)

Barycentrics    a^2 (a^16 b^4-7 a^14 b^6+21 a^12 b^8-35 a^10 b^10+35 a^8 b^12-21 a^6 b^14+7 a^4 b^16-a^2 b^18-4 a^16 b^2 c^2+14 a^14 b^4 c^2-15 a^12 b^6 c^2+4 a^10 b^8 c^2-5 a^8 b^10 c^2+14 a^6 b^12 c^2-9 a^4 b^14 c^2+b^18 c^2+a^16 c^4+14 a^14 b^2 c^4-38 a^12 b^4 c^4+39 a^10 b^6 c^4-33 a^8 b^8 c^4+20 a^6 b^10 c^4-4 a^4 b^12 c^4+7 a^2 b^14 c^4-6 b^16 c^4-7 a^14 c^6-15 a^12 b^2 c^6+39 a^10 b^4 c^6+6 a^8 b^6 c^6-13 a^6 b^8 c^6-15 a^4 b^10 c^6-11 a^2 b^12 c^6+16 b^14 c^6+21 a^12 c^8+4 a^10 b^2 c^8-33 a^8 b^4 c^8-13 a^6 b^6 c^8+42 a^4 b^8 c^8+5 a^2 b^10 c^8-26 b^12 c^8-35 a^10 c^10-5 a^8 b^2 c^10+20 a^6 b^4 c^10-15 a^4 b^6 c^10+5 a^2 b^8 c^10+30 b^10 c^10+35 a^8 c^12+14 a^6 b^2 c^12-4 a^4 b^4 c^12-11 a^2 b^6 c^12-26 b^8 c^12-21 a^6 c^14-9 a^4 b^2 c^14+7 a^2 b^4 c^14+16 b^6 c^14+7 a^4 c^16-6 b^4 c^16-a^2 c^18+b^2 c^18) : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28731.

X(30263) lies on these lines: {20,2979}, {185,5667}, {577,6759}, {3087,10110}


X(30264) = REFLECTION OF X(12) IN X(3)

Barycentrics    4*a^7 - 4*a^6*b - 7*a^5*b^2 + 7*a^4*b^3 + 2*a^3*b^4 - 2*a^2*b^5 + a*b^6 - b^7 - 4*a^6*c + 6*a^5*b*c - 3*a^4*b^2*c - 4*a^3*b^3*c + 6*a^2*b^4*c - 2*a*b^5*c + b^6*c - 7*a^5*c^2 - 3*a^4*b*c^2 + 12*a^3*b^2*c^2 - 4*a^2*b^3*c^2 - a*b^4*c^2 + 3*b^5*c^2 + 7*a^4*c^3 - 4*a^3*b*c^3 - 4*a^2*b^2*c^3 + 4*a*b^3*c^3 - 3*b^4*c^3 + 2*a^3*c^4 + 6*a^2*b*c^4 - a*b^2*c^4 - 3*b^3*c^4 - 2*a^2*c^5 - 2*a*b*c^5 + 3*b^2*c^5 + a*c^6 + b*c^6 - c^7 : :
X(30264) = 4 X[3] - 3 X[21155],2 X[12] - 3 X[21155],3 X[376] - X[11491], 5 X[631] - 4 X[6668],5 X[3522] - X[20060],2 X[8068] - 3 X[21154]

X(30264) lies on these lines: {3, 12}, {4, 4999}, {11, 7491}, {20, 2894}, {30, 11012}, {40, 550}, {56, 6868}, {104, 12519}, {376, 529}, {411, 2829}, {515, 3916}, {517, 15338}, {548, 2077}, {631, 6668}, {758, 1071}, {944, 5855}, {958, 6934}, {1329, 6942}, {1350, 5849}, {1385, 3649}, {1483, 3894}, {3058, 10680}, {3522, 20060}, {3612, 5812}, {3869, 9964}, {4018, 5882}, {4189, 7680}, {4302, 22770}, {4311, 12709}, {4325, 15931}, {4428, 10597}, {5204, 6827}, {5267, 6831}, {5303, 6840}, {5426, 5901}, {5433, 6928}, {5434, 10267}, {5441, 5536}, {5584, 6948}, {5732, 5857}, {5759, 5852}, {5794, 21165}, {6244, 15696}, {6253, 22758}, {6284, 10959}, {6691, 6902}, {6825, 12943}, {6872, 22753}, {6875, 25466}, {6907, 10483}, {6910, 10894}, {6917, 24953}, {6922, 7280}, {6936, 25524}, {6954, 10895}, {6971, 7294}, {7489, 7958}, {7681, 11114}, {7965, 13743}, {8736, 22056}, {10386, 16200}, {10543, 24474}, {10902, 18990}, {11014, 28174}, {11194, 12116}, {12512, 13528}, {16370, 26332}, {17768, 21740}

X(30264) = midpoint of X(20) and X(2975)
X(30264) = reflection of X(i) and X(j) for these {i,j}: {4, 4999}, {12, 3}, {6831, 5267}, {15908, 11012}
X(30264) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 12, 21155}, {3, 10526, 5432}, {20, 3428, 11826}, {40, 550, 24466}, {6922, 7280, 21154}, {7491, 26286, 11}


X(30265) = REFLECTION OF X(19) IN X(3)

Barycentrics    a*(a^8 - 2*a^7*b + 2*a^5*b^3 - 2*a^4*b^4 + 2*a^3*b^5 - 2*a*b^7 + b^8 - 2*a^7*c + 6*a^5*b^2*c - 6*a^3*b^4*c + 2*a*b^6*c + 6*a^5*b*c^2 + 4*a^4*b^2*c^2 - 4*a^3*b^3*c^2 - 2*a*b^5*c^2 - 4*b^6*c^2 + 2*a^5*c^3 - 4*a^3*b^2*c^3 + 2*a*b^4*c^3 - 2*a^4*c^4 - 6*a^3*b*c^4 + 2*a*b^3*c^4 + 6*b^4*c^4 + 2*a^3*c^5 - 2*a*b^2*c^5 + 2*a*b*c^6 - 4*b^2*c^6 - 2*a*c^7 + c^8) : :
X(30265) = 4 X[3] - 3 X[21160],2 X[19] - 3 X[21160],5 X[3522] - X[20061]

X(30265) lies on these lines: {1, 7}, {3, 19}, {4, 18589}, {22, 15931}, {33, 1214}, {78, 25252}, {103, 13397}, {109, 7070}, {152, 2822}, {165, 3101}, {204, 22119}, {219, 971}, {278, 1040}, {376, 534}, {612, 25080}, {1096, 22057}, {1295, 28291}, {1297, 6011}, {1350, 3827}, {1486, 11414}, {1490, 2324}, {1709, 2328}, {1723, 13329}, {1838, 6836}, {3152, 19860}, {3428, 21312}, {3522, 20061}, {3870, 6360}, {4219, 10319}, {5706, 9943}, {7520, 7987}, {7538, 19861}, {7991, 15954}, {8583, 27402}, {8680, 18446}, {9746, 26260}, {11471, 15951}, {12522, 22770}

X(30265) = midpoint of X(20) and X(4329)
X(30265) = reflection of X(i) and X(j) for these {i,j}: {4, 18589}, {19, 3}
X(30265) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1721, 3332}, {3, 19, 21160}, {347, 3100, 1}


X(30266) = REFLECTION OF X(27) IN X(3)

Barycentrics    a^10 - 3*a^9*b - 2*a^8*b^2 + 8*a^7*b^3 - 6*a^5*b^5 + 2*a^4*b^6 - a^2*b^8 + a*b^9 - 3*a^9*c - 3*a^8*b*c + 8*a^7*b^2*c + 8*a^6*b^3*c - 6*a^5*b^4*c - 6*a^4*b^5*c + a*b^8*c + b^9*c - 2*a^8*c^2 + 8*a^7*b*c^2 + 16*a^6*b^2*c^2 - 4*a^5*b^3*c^2 - 10*a^4*b^4*c^2 - 4*a^2*b^6*c^2 - 4*a*b^7*c^2 + 8*a^7*c^3 + 8*a^6*b*c^3 - 4*a^5*b^2*c^3 - 4*a^4*b^3*c^3 - 4*a*b^6*c^3 - 4*b^7*c^3 - 6*a^5*b*c^4 - 10*a^4*b^2*c^4 + 10*a^2*b^4*c^4 + 6*a*b^5*c^4 - 6*a^5*c^5 - 6*a^4*b*c^5 + 6*a*b^4*c^5 + 6*b^5*c^5 + 2*a^4*c^6 - 4*a^2*b^2*c^6 - 4*a*b^3*c^6 - 4*a*b^2*c^7 - 4*b^3*c^7 - a^2*c^8 + a*b*c^8 + a*c^9 + b*c^9 : :

X(30266) lies on these lines: {2, 3}, {74, 1305}, {101, 1294}, {2690, 2693}, {5897, 26705}, {8680, 18446}, {16099, 29243}


X(30267) = REFLECTION OF X(28) IN X(3)

Barycentrics    a*(a^9 - 2*a^7*b^2 + 2*a^3*b^6 - a*b^8 - 3*a^7*b*c + 3*a^6*b^2*c + 5*a^5*b^3*c - 5*a^4*b^4*c - a^3*b^5*c + a^2*b^6*c - a*b^7*c + b^8*c - 2*a^7*c^2 + 3*a^6*b*c^2 + 10*a^5*b^2*c^2 + a^4*b^3*c^2 - 6*a^3*b^4*c^2 - 3*a^2*b^5*c^2 - 2*a*b^6*c^2 - b^7*c^2 + 5*a^5*b*c^3 + a^4*b^2*c^3 - 6*a^3*b^3*c^3 + 2*a^2*b^4*c^3 + a*b^5*c^3 - 3*b^6*c^3 - 5*a^4*b*c^4 - 6*a^3*b^2*c^4 + 2*a^2*b^3*c^4 + 6*a*b^4*c^4 + 3*b^5*c^4 - a^3*b*c^5 - 3*a^2*b^2*c^5 + a*b^3*c^5 + 3*b^4*c^5 + 2*a^3*c^6 + a^2*b*c^6 - 2*a*b^2*c^6 - 3*b^3*c^6 - a*b*c^7 - b^2*c^7 - a*c^8 + b*c^8) : :

X(30267) lies on these lines: {2, 3}, {74, 13397}, {100, 1294}, {108, 5897}, {347, 3295}, {942, 3100}, {1290, 2693}, {1292, 1297}, {1295, 6011}, {2691, 2697}, {3101, 3579}, {3182, 7070}, {3430, 6282}, {3871, 6360}, {4296, 24929}, {4329, 6361}, {9538, 15934}, {9643, 11518}, {12262, 14110}, {26703, 30257}


X(30268) = REFLECTION OF X(29) IN X(3)

Barycentrics    a^10 + 3*a^9*b - 2*a^8*b^2 - 8*a^7*b^3 + 6*a^5*b^5 + 2*a^4*b^6 - a^2*b^8 - a*b^9 + 3*a^9*c - 3*a^8*b*c - 2*a^7*b^2*c + 2*a^6*b^3*c - 4*a^5*b^4*c + 4*a^4*b^5*c + 2*a^3*b^6*c - 2*a^2*b^7*c + a*b^8*c - b^9*c - 2*a^8*c^2 - 2*a^7*b*c^2 + 4*a^6*b^2*c^2 + 6*a^5*b^3*c^2 - 2*a^4*b^4*c^2 - 6*a^3*b^5*c^2 + 2*a*b^7*c^2 - 8*a^7*c^3 + 2*a^6*b*c^3 + 6*a^5*b^2*c^3 - 8*a^4*b^3*c^3 + 4*a^3*b^4*c^3 + 2*a^2*b^5*c^3 - 2*a*b^6*c^3 + 4*b^7*c^3 - 4*a^5*b*c^4 - 2*a^4*b^2*c^4 + 4*a^3*b^3*c^4 + 2*a^2*b^4*c^4 + 6*a^5*c^5 + 4*a^4*b*c^5 - 6*a^3*b^2*c^5 + 2*a^2*b^3*c^5 - 6*b^5*c^5 + 2*a^4*c^6 + 2*a^3*b*c^6 - 2*a*b^3*c^6 - 2*a^2*b*c^7 + 2*a*b^2*c^7 + 4*b^3*c^7 - a^2*c^8 + a*b*c^8 - a*c^9 - b*c^9 : :

X(30268) lies on these lines: {2, 3}, {74, 28788}, {109, 1294}, {2689, 2693}, {3916, 10538}, {4296, 7100}, {5897, 26704}, {6360, 12702}


X(30269) = REFLECTION OF X(31) IN X(3)

Barycentrics    a^2*(a^5 + 2*a^2*b^3 - a*b^4 - 2*b^5 + 2*a*b^2*c^2 - 2*b^3*c^2 + 2*a^2*c^3 - 2*b^2*c^3 - a*c^4 - 2*c^5) : :
X(30269) = 3 X[549] - 2 X[20575],5 X[631] - 4 X[6679],5 X[3522] - X[20064]

X(30269) lies on these lines: {3, 31}, {4, 2887}, {20, 6327}, {40, 758}, {73, 8193}, {74, 6010}, {102, 1292}, {103, 28474}, {104, 28469}, {106, 28584}, {209, 5584}, {326, 1310}, {376, 752}, {515, 4680}, {517, 3938}, {549, 20575}, {573, 7688}, {631, 6679}, {674, 1350}, {734, 11257}, {953, 28520}, {1006, 6210}, {1066, 12410}, {1293, 28159}, {1496, 11573}, {1742, 4221}, {1973, 20727}, {2390, 10310}, {3522, 20064}, {3556, 3682}, {5603, 28885}, {6905, 20368}, {28145, 28524}, {28173, 28518}, {28291, 28900}, {28293, 28876}, {28299, 28873}, {28303, 28912}, {28467, 29372}

X(30269) = midpoint of X(20) and X(6327)
X(30269) = reflection of X(i) and X(j) for these {i,j}: {4, 2887}, {31, 3}


X(30270) = REFLECTION OF X(32) IN X(3)

Barycentrics    a^2*(a^6 + a^2*b^4 - 2*b^6 + 2*a^2*b^2*c^2 - 2*b^4*c^2 + a^2*c^4 - 2*b^2*c^4 - 2*c^6) : :
Trilinears    2 cos(A + 2ω) + cos(A - 2ω) + cos A : :
X(30270) = 4 X[5] - 5 X[7867],3 X[549] - 2 X[20576],5 X[631] - 4 X[6680],5 X[631] - 3 X[9753],5 X[3522] - X[20065],4 X[6680] - 3 X[9753],2 X[7805] - 3 X[9755],2 X[18806] - 3 X[22712]}

X(30270) lies on these lines: {3, 6}, {4, 626}, {5, 7822}, {20, 99}, {30, 7801}, {40, 760}, {76, 5999}, {98, 7751}, {103, 28469}, {114, 7888}, {194, 12203}, {262, 7808}, {376, 754}, {515, 4769}, {549, 20576}, {631, 6680}, {736, 6309}, {805, 2710}, {980, 4220}, {1078, 6194}, {1092, 8922}, {1296, 14388}, {1352, 7794}, {1503, 3933}, {1513, 3788}, {1974, 20819}, {2001, 2979}, {2353, 5562}, {2386, 21312}, {3117, 7467}, {3148, 3917}, {3425, 10323}, {3522, 13571}, {3552, 10334}, {3564, 7855}, {3934, 13860}, {5480, 7819}, {6248, 17130}, {6287, 11178}, {6660, 9306}, {6776, 7758}, {7488, 28710}, {7764, 9744}, {7768, 9863}, {7782, 22676}, {7789, 29181}, {7800, 10519}, {7805, 9755}, {7815, 18806}, {7826, 10991}, {7832, 13862}, {7889, 14561}, {7896, 9873}, {8671, 10310}, {9888, 14645}, {10358, 14881}, {14931, 20081}, {16187, 21513}, {18502, 22728}, {28295, 28563}

X(30270) = midpoint of X(20) and X(315)
X(30270) = midpoint of X(11824) and X(11825)
X(30270) = X(32)-of-circumcevian-triangle-of-X(511)
X(30270) = reflection of X(i) and X(j) for these {i,j}: {4, 626}, {32, 3}
X(30270) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3, 1350, 5188}, {3, 3095, 182}, {3, 5013, 21163}, {3, 5171, 5206}, {3, 5188, 8722}, {3, 9605, 5085}, {3, 9737, 574}, {3, 9821, 5171}, {3, 12054, 17508}, {20, 3926, 8721}, {98, 12251, 7751}, {182, 3095, 7772}, {631, 9753, 6680}, {1350, 18860, 8722}, {3098, 9737, 3}, {3926, 8721, 14981}, {5013, 5017, 13357}, {5017, 13357, 32}, {5188, 18860, 3}, {12305, 12306, 5085}, {13334, 14810, 3}, {14538, 14539, 182}


X(30271) = REFLECTION OF X(37) IN X(3)

Barycentrics    a*(3*a^4*b - 2*a^2*b^3 - b^5 + 3*a^4*c + 4*a^3*b*c - 2*a^2*b^2*c - 4*a*b^3*c - b^4*c - 2*a^2*b*c^2 + 2*b^3*c^2 - 2*a^2*c^3 - 4*a*b*c^3 + 2*b^2*c^3 - b*c^4 - c^5) : :
X(30271) = 3 X[3] - X[20430],3 X[37] - 2 X[20430],3 X[165] - X[984],X[192] - 5 X[3522],8 X[548] - X[4718],4 X[550] + X[4686],5 X[631] - 4 X[4698],5 X[3091] - 7 X[4751],X[3146] - 5 X[4699],7 X[3523] - 5 X[4687],3 X[3524] - 2 X[4755],7 X[3528] - 2 X[4681],X[3529] + 4 X[4739],3 X[3576] - 2 X[15569],2 X[3842] - 3 X[10164],X[4664] - 3 X[10304],5 X[4704] - 13 X[21734],2 X[4726] + 5 X[17538],7 X[4772] + X[5059],3 X[9778] + X[24349],11 X[15717] - 7 X[27268]

X(30271) lies on these lines: {3, 37}, {4, 3739}, {20, 75}, {30, 4688}, {40, 518}, {56, 11997}, {63, 3198}, {71, 5784}, {103, 6011}, {104, 1296}, {165, 984}, {192, 3522}, {376, 536}, {515, 3696}, {516, 24325}, {517, 991}, {548, 4718}, {550, 4686}, {573, 971}, {631, 4698}, {726, 5188}, {740, 4297}, {851, 25939}, {910, 24320}, {944, 28581}, {1001, 12717}, {1009, 25887}, {1108, 3286}, {1212, 20605}, {1284, 17635}, {1764, 10167}, {1818, 21871}, {1824, 22060}, {2223, 12721}, {2691, 28838}, {2831, 16728}, {2941, 15931}, {3091, 4751}, {3146, 4699}, {3428, 21312}, {3523, 4687}, {3524, 4755}, {3528, 4681}, {3529, 4739}, {3576, 15569}, {3601, 7201}, {3752, 4192}, {3781, 21872}, {3842, 10164}, {3916, 16551}, {4259, 21866}, {4664, 10304}, {4704, 21734}, {4726, 17538}, {4772, 5059}, {5728, 20367}, {5927, 21363}, {6210, 15726}, {9778, 24349}, {10178, 20368}, {1031+0, 15624}, {10391, 24310}, {12545, 25124}, {14110, 20718}, {14872, 22271}, {15717, 27268}, {16602, 19540}, {16610, 19647}, {18607, 20243}

X(30271) = midpoint of X(20) and X(75)
X(30271) = reflection of X(i) and X(j) for these {i,j}: {4, 3739}, {37, 3}, {14872, 22271}
{X(40),X(5732)}-harmonic conjugate of X(1350)}


X(30272) = REFLECTION OF X(38) IN X(3)

Barycentrics    a*(2*a^5*b + a^4*b^2 - 2*a^3*b^3 - b^6 + 2*a^5*c - 2*a*b^4*c + a^4*c^2 - 2*a*b^3*c^2 + b^4*c^2 - 2*a^3*c^3 - 2*a*b^2*c^3 - 2*a*b*c^4 + b^2*c^4 - c^6) : :
X(30272) = X[631] - 4 X[6682],5 X[3522] - X[20068]

X(30272) lies on these lines: {3, 38}, {4, 1215}, {20, 17165}, {40, 758}, {42, 517}, {102, 26712}, {104, 28486}, {244, 19550}, {376, 537}, {515, 4692}, {573, 3949}, {631, 6682}, {984, 19262}, {1006, 6211}, {1350, 9020}, {3428, 12329}, {3522, 20068}, {4362, 5767}, {5690, 20653}, {5886, 29647}, {17442, 22061}, {19260, 25024}

X(30272) = midpoint of X(20) and X(17165)
X(30272) = reflection of X(i) and X(j) for these {i,j}: {4, 1215}, {38, 3}


X(30273) = REFLECTION OF X(75) IN X(3)

Barycentrics    2 a^5 b-2 a^3 b^3+2 a^5 c+a^4 b c-2 a^3 b^2 c-b^5 c-2 a^3 b c^2-2 a^3 c^3+2 b^3 c^3-b c^5 : :
X(30273) = 4 X[5] - 5 X[4687],8 X[140] - 7 X[4751],8 X[548] - X[4764],4 X[550] + X[3644],5 X[631] - 4 X[3739],X[1278] - 5 X[3522],7 X[3090] - 8 X[4698],5 X[3091] - 7 X[27268],X[3146] - 5 X[4704],7 X[3523] - 5 X[4699],3 X[3524] - 2 X[4688],7 X[3528] - 2 X[4686],X[3529] + 4 X[4681],3 X[3545] - 4 X[4755],3 X[3576] - 2 X[24325],2 X[3696] - 3 X[5657],4 X[3842] - 3 X[5587],3 X[4664] - 2 X[20430],2 X[4718] + 5 X[17538],4 X[4726] - 11 X[21735],8 X[4739] - 13 X[10299],X[4740] - 3 X[10304],7 X[4772] - 11 X[15717],5 X[4821] - 13 X[21734],3 X[5603] - 4 X[15569],3 X[5731] - X[24349],2 X[5805] - 3 X[27475],2 X[21443] - 3 X[22712]

X(30273) lies on these lines: {1, 4032}, {3, 75}, {4, 37}, {5, 4687}, {20, 192}, {30, 4664}, {40, 740}, {55, 7009}, {72, 25252}, {92, 228}, {98, 6011}, {140, 4751}, {198, 242}, {312, 4192}, {335, 29243}, {376, 536}, {411, 20171}, {515, 984}, {516, 3993}, {518, 944}, {548, 4764}, {550, 3644}, {573, 29016}, {631, 3739}, {726, 4297}, {742, 1350}, {991, 29069}, {1278, 3522}, {1742, 29057}, {2223, 4008}, {2329, 3923}, {2724, 2730}, {2805, 13199}, {3090, 4698}, {3091, 27268}, {3146, 4704}, {3190, 22001}, {3523, 4699}, {3524, 4688}, {3528, 4686}, {3529, 4681}, {3545, 4755}, {3576, 24325}, {3696, 5657}, {3842, 5587}, {3868, 25241}, {4292, 7201}, {4294, 7718}, {4358, 19647}, {4718, 17538}, {4726, 21735}, {4739, 10299}, {4740, 10304}, {4772, 15717}, {4821, 21734}, {5603, 15569}, {5731, 24349}, {5732, 24813}, {5768, 27472}, {5805, 27475}, {6210, 28850}, {6360, 17441}, {7414, 11491}, {7580, 20173}, {8680, 18446}, {9441, 24257}, {12245, 28581}, {12512, 28522}, {17479, 20243}, {18137, 19543}, {18743, 19540}, {18750, 20760}, {19513, 20923}, {19514, 30090}, {21443, 22712}

X(30273) = midpoint of X(20) and X(192)
X(30273) = reflection of X(i) and X(j) for these {i,j}: {4, 37}, {75, 3}
X(30273) = {X(944),X(5759)}-harmonic conjugate of X(6776)






leftri  Endo-homothetic centers: X(30274) - X(30434)  rightri

This preamble and centers X(30274)-X(30434) were contributed by César Eliud Lozada, December 19, 2018.

This section comprises the endo-homothetic centers of the family of triangles homothetic with the orthic triangle of a reference triangle ABC. This family is composed by the following 37 triangles:

anti-Ascella, anti-Atik, 1st anti-circumperp, anti-Conway, 2nd anti-Conway, 3rd anti-Euler, 4th anti-Euler, anti-excenters-reflections, 2nd anti-extouch, anti-Honsberger, anti-Hutson intouch, anti-incircle-circles, anti-inverse-in-incircle, 6th anti-mixtilinear, 1st anti-Sharygin, anti-tangential-midarc, anti-Ursa minor, anti-Wasat, circumorthic, Ehrmann-side, Ehrmann-vertex, 2nd Ehrmann, 2nd Euler, 1st excosine, extangents, intangents, 1st Kenmotu diagonals, 2nd Kenmotu diagonals, Kosnita, Lucas antipodal tangential, Lucas(-1) antipodal tangential, orthic, submedial, tangential, inner tri-equilateral, outer tri-equilateral, Trinh.

For definitions and coordinates of these triangles, see the index of triangles referenced in ETC.

A table showing the endo-homothetic centers among these triangles can be seen here.

underbar

X(30274) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA AND EHRMANN-VERTEX

Barycentrics    a*((b+c)*a^5-(b^2-b*c+c^2)*a^4-(b+c)*(2*b^2-b*c+2*c^2)*a^3+(2*b^4+2*c^4-(3*b^2+2*b*c+3*c^2)*b*c)*a^2+(b^3-c^3)*(b^2-c^2)*a-(b^2-c^2)^2*(b-c)^2) : :
X(30274) = 2*X(942)+X(17603) = 4*X(942)+X(30282)

The homothetic center of these triangles is X(18386). X(30274) is their endo-homothetic center only when ABC is acute.

X(30274) lies on these lines: {1,3}, {2,18389}, {7,6840}, {80,6826}, {142,1737}, {226,6830}, {388,12005}, {443,10573}, {498,5904}, {499,26725}, {581,1393}, {631,15556}, {758,5744}, {912,5219}, {920,5259}, {938,2475}, {1056,5083}, {1210,2476}, {1439,4888}, {1478,5768}, {1858,8227}, {2800,6935}, {2801,10590}, {3085,3874}, {3485,5884}, {3487,6952}, {3583,5805}, {3585,5787}, {3586,10391}, {3754,6904}, {3868,13411}, {5432,5771}, {5439,25525}, {5443,6824}, {5445,6989}, {5691,9942}, {5692,5745}, {5693,11375}, {5728,6173}, {5883,9776}, {6224,12736}, {6245,12047}, {6879,12691}, {6972,11036}, {8068,12831}, {8261,25524}, {8727,18393}, {8728,10954}, {8729,18408}, {8731,30358}, {8732,30329}, {8733,18399}, {8734,18409}, {9579,13369}, {9613,12675}, {9614,12711}, {9655,26201}, {10039,24391}, {10449,20882}, {10527,20612}, {10855,30286}, {10896,17637}, {11020,17579}, {11219,11570}, {11237,17660}, {11571,13226}, {18410,30276}, {18411,30277}, {18422,30280}, {18423,30281}

X(30274) = reflection of X(30282) in X(17603)
X(30274) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3336, 11507), (3601, 24474, 5697), (11529, 18838, 5902)


X(30275) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA AND 2nd EHRMANN

Barycentrics    (a^3+(b+c)*a^2-(5*b^2+2*b*c+5*c^2)*a+3*(b^2-c^2)*(b-c))*(a+b-c)*(a-b+c) : :
X(30275) = X(7)+2*X(5219) = 4*X(142)-X(5744)

The homothetic center of these triangles is X(11405). X(30275) is their endo-homothetic center only when ABC is acute.

X(30275) lies on these lines: {2,7}, {3,8543}, {347,4648}, {388,25557}, {390,5603}, {443,5730}, {497,8255}, {516,30282}, {942,5261}, {948,4675}, {952,1056}, {954,6905}, {971,6844}, {997,12560}, {1441,4869}, {2095,8164}, {2099,2550}, {2801,10590}, {3090,5729}, {3091,10394}, {3485,5880}, {3601,30332}, {3753,7672}, {4552,29621}, {5177,5784}, {5220,10588}, {5274,7671}, {5308,22464}, {5542,18391}, {5686,7679}, {5698,11375}, {5714,9940}, {5728,6843}, {5735,5766}, {5762,6954}, {5779,6859}, {5780,6147}, {6987,21151}, {8544,8726}, {8727,30311}, {8728,30312}, {8729,30404}, {8731,30359}, {8733,30367}, {8734,30405}, {9814,11407}, {10569,10865}, {10707,18801}, {10855,30287}, {10857,30353}, {11023,13407}, {11025,17620}, {11518,30318}, {12573,13462}, {15726,17603}, {15803,30424}, {18443,18450}

X(30275) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 12848), (7, 142, 8732), (142, 6173, 9776)


X(30276) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA AND 1st KENMOTU DIAGONALS

Barycentrics    (-2*S*a+(-a+b+c)*((b+c)*a-b^2+2*b*c-c^2))*(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(5410). X(30276) is their endo-homothetic center only when ABC is acute.

X(30276) lies on these lines: {2,7}, {3,30296}, {277,16232}, {942,30341}, {3513,24154}, {3514,24155}, {3601,30333}, {4000,13388}, {4648,13389}, {6351,8243}, {7133,10858}, {8726,30400}, {8727,30306}, {8728,30313}, {8729,30406}, {8731,30360}, {8733,30368}, {8734,30418}, {10857,30354}, {11018,30346}, {11407,30396}, {11518,30319}, {13390,13940}, {15803,30425}, {17603,30375}, {18410,30274}, {18443,18458}, {30282,30431}

X(30276) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 30324), (57, 142, 30277)


X(30277) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA AND 2nd KENMOTU DIAGONALS

Barycentrics    (2*S*a+(-a+b+c)*((b+c)*a-b^2+2*b*c-c^2))*(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(5411). X(30277) is their endo-homothetic center only when ABC is acute.

X(30277) lies on these lines: {2,7}, {3,30297}, {277,2362}, {942,30342}, {1659,13887}, {3514,24154}, {3601,30334}, {4000,13389}, {4648,13388}, {8726,30401}, {8727,30307}, {8728,30314}, {8729,30407}, {8731,30361}, {8733,30369}, {8734,30419}, {10855,30289}, {10857,30355}, {11018,30347}, {11407,30397}, {11518,30320}, {15803,30426}, {17603,30376}, {18411,30274}, {18443,18460}, {30282,30432}

X(30277) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 30325), (57, 142, 30276)


X(30278) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA AND LUCAS ANTIPODAL TANGENTIAL

Barycentrics
a*(2*(a^4+4*(b+c)*a^3-2*(b^2+c^2)*a^2-4*(b+c)*(b^2+c^2)*a+(b^2-c^2)^2)*S+(5*a^5+3*(b+c)*a^4-6*(b^2+c^2)*a^3-2*(b+c)^3*a^2+(b^2-c^2)^2*a-(b^2-c^2)*(b-c)^3)*(-a+b+c))*(S+(a-b+c)*b)*(S+(a+b-c)*c) : :

The homothetic center of these triangles is X(19404). X(30278) is their endo-homothetic center only when ABC is acute.

X(30278) lies on these lines: {2,30302}, {57,7133}, {610,15892}, {3601,30335}, {5745,30416}, {10857,30279}, {11018,30348}, {11518,16213}


X(30279) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA AND LUCAS(-1) ANTIPODAL TANGENTIAL

Barycentrics
a*(-2*(a^4+4*(b+c)*a^3-2*(b^2+c^2)*a^2-4*(b+c)*(b^2+c^2)*a+(b^2-c^2)^2)*S+(5*a^5+3*(b+c)*a^4-6*(b^2+c^2)*a^3-2*(b+c)^3*a^2+(b^2-c^2)^2*a-(b^2-c^2)*(b-c)^3)*(-a+b+c))*(-S+(a-b+c)*b)*(-S+(a+b-c)*c) : :

The homothetic center of these triangles is X(19405). X(30279) is their endo-homothetic center only when ABC is acute.

X(30279) lies on these lines: {2,30303}, {57,30430}, {610,15891}, {3601,30336}, {5745,30417}, {10857,30278}, {11018,30349}, {11518,16214}


X(30280) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA AND INNER TRI-EQUILATERAL

Barycentrics    (2*sqrt(3)*S*a-((b+c)*a-b^2+2*b*c-c^2)*(-a+b+c))*(a+b-c)*(a+c-b) : :

The homothetic center of these triangles is X(11408). X(30280) is their endo-homothetic center only when ABC is acute.

X(30280) lies on these lines: {2,7}, {3,30300}, {277,2306}, {942,30344}, {3601,30338}, {8726,10649}, {8727,30309}, {8728,30316}, {8729,30409}, {8731,30364}, {8733,30372}, {8734,30421}, {10855,30292}, {10857,30356}, {11018,30351}, {11407,10655}, {11518,30321}, {15803,10651}, {17603,30377}, {18422,30274}, {18443,18469}, {30282,30433}

X(30280) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 30327), (57, 142, 30281)


X(30281) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA AND OUTER TRI-EQUILATERAL

Barycentrics    (-2*sqrt(3)*S*a-((b+c)*a-b^2+2*b*c-c^2)*(-a+b+c))*(a+b-c)*(a+c-b) : :

The homothetic center of these triangles is X(11409). X(30281) is their endo-homothetic center only when ABC is acute.

X(30281) lies on these lines: {2,7}, {3,30301}, {942,30345}, {3601,30339}, {8726,10650}, {8727,30310}, {8728,30317}, {8729,30410}, {8731,30365}, {8733,30373}, {8734,30422}, {10855,30293}, {10857,30357}, {11018,30352}, {11407,10656}, {11518,30322}, {15803,10652}, {17603,30378}, {18423,30274}, {18443,18471}, {30282,30434}

X(30281) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 30328), (57, 142, 30280)


X(30282) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ASCELLA AND TRINH

Barycentrics    a*(5*a^3-(b+c)*a^2-(5*b^2+2*b*c+5*c^2)*a+(b^2-c^2)*(b-c)) : :
X(30282) = 2*X(942)-5*X(17603) = 4*X(942)-5*X(30274) = (r+2R)*X(1) + 4r*X(3)

The homothetic center of these triangles is X(11410). X(30282) is their endo-homothetic center only when ABC is acute.

X(30282) lies on these lines: {1,3}, {2,3586}, {7,10304}, {9,5440}, {10,4305}, {20,5226}, {21,936}, {30,5219}, {33,7501}, {63,17549}, {72,19535}, {78,3219}, {80,6174}, {100,9623}, {103,15730}, {140,9581}, {142,25055}, {200,993}, {226,376}, {284,1743}, {405,5438}, {443,1479}, {452,6700}, {474,5436}, {497,10165}, {498,5691}, {515,5218}, {516,30275}, {519,5744}, {549,5722}, {550,9579}, {551,9776}, {553,19708}, {579,16667}, {581,22072}, {610,3465}, {631,950}, {938,15717}, {944,6705}, {956,3158}, {975,7520}, {991,22350}, {997,4512}, {1012,1750}, {1125,4294}, {1210,3523}, {1335,9615}, {1387,3653}, {1453,4255}, {1490,6906}, {1698,6857}, {1699,4302}, {1708,21161}, {1724,19764}, {1817,4653}, {1876,11410}, {1914,9592}, {2975,6765}, {2999,4256}, {3062,15175}, {3085,4297}, {3086,4314}, {3216,13726}, {3306,13587}, {3452,11111}, {3486,6684}, {3487,3528}, {3488,3524}, {3522,4292}, {3583,6826}, {3584,5726}, {3585,6851}, {3616,10624}, {3632,12437}, {3633,24391}, {3679,5745}, {3811,5267}, {3871,12629}, {3876,17574}, {3916,11523}, {3928,19704}, {3929,3940}, {4031,15710}, {4114,21735}, {4224,5268}, {4258,16572}, {4276,17194}, {4293,13405}, {4295,12512}, {4299,5290}, {4324,6869}, {4330,6885}, {4652,17548}, {4654,5719}, {4853,8715}, {4866,15446}, {4882,5258}, {4995,5252}, {5013,16780}, {5044,17571}, {5248,8583}, {5251,8580}, {5259,19520}, {5281,5731}, {5313,16469}, {5414,9583}, {5432,5587}, {5435,15692}, {5437,16371}, {5439,19537}, {5441,6675}, {5444,23708}, {5705,6910}, {5714,17538}, {5715,6934}, {5720,6914}, {5727,11545}, {5732,6909}, {5768,12647}, {5886,9580}, {6245,10039}, {6284,8227}, {6666,17561}, {6824,7989}, {6872,27385}, {6875,10393}, {6950,18446}, {7308,16418}, {7675,10398}, {7741,8728}, {7951,8727}, {7972,13226}, {8729,30411}, {8731,16569}, {8732,30331}, {8733,30374}, {8734,30423}, {9578,18481}, {9582,16232}, {9588,10573}, {9624,12701}, {9668,11230}, {9785,11023}, {10058,15015}, {10164,18391}, {10386,11373}, {10391,18397}, {10543,24914}, {10590,28164}, {10591,19862}, {10855,30294}, {11112,25525}, {11375,15338}, {11491,12650}, {12433,15712}, {12526,22836}, {12572,17576}, {15326,17718}, {15677,27131}, {15688,18541}, {15705,15933}, {15935,17504}, {16056,25502}, {16132,16140}, {16342,19859}, {17284,24609}, {18540,28444}, {21483,23511}, {24604,29571}, {28452,30308}, {30276,30431}, {30277,30432}, {30280,30433}, {30281,30434}

X(30282) = reflection of X(i) in X(j) for these (i,j): (1, 13384), (30274, 17603)
X(30282) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3, 15803), (1, 484, 18421), (1, 7280, 3361), (484, 18421, 2093)


X(30283) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND EHRMANN-SIDE

Barycentrics    a*(a^6-(b+c)*a^5-2*(b^2-7*b*c+c^2)*a^4+2*(b+c)*(b^2-5*b*c+c^2)*a^3+(b^2-8*b*c+c^2)*(b-c)^2*a^2-(b^2-c^2)*(b-c)*(b^2-8*b*c+c^2)*a-4*(b^2-c^2)^2*b*c) : :
X(30283) = 3*X(999)-2*X(22753) = X(3062)-3*X(10384) = 4*X(10269)-3*X(16417) = 3*X(19541)-4*X(22753)

The homothetic center of these triangles is X(18917). X(30283) is their endo-homothetic center only when ABC is acute.

X(30283) lies on these lines: {1,971}, {3,8}, {4,7373}, {20,8158}, {36,30286}, {40,11519}, {56,5727}, {84,9957}, {153,17556}, {355,8582}, {381,12115}, {382,12001}, {392,5779}, {496,12667}, {515,999}, {517,7171}, {519,6244}, {942,3577}, {958,12447}, {997,18227}, {1012,6767}, {1056,8727}, {1319,17604}, {1376,28236}, {1385,5720}, {1490,24928}, {1656,10785}, {1709,5919}, {2096,28174}, {2098,13253}, {2801,5289}, {2829,9668}, {3057,10085}, {3295,5882}, {3304,5691}, {3526,10786}, {3576,8580}, {3600,20420}, {3655,16418}, {3895,17613}, {3940,9954}, {4297,5853}, {4308,5809}, {4317,6253}, {4423,30291}, {4511,11678}, {5258,8273}, {5288,5584}, {5697,12767}, {5787,10106}, {5789,24987}, {5818,16863}, {5881,9709}, {5886,10863}, {5927,6913}, {6256,9669}, {6259,12053}, {6260,11373}, {6265,13227}, {6831,10805}, {6911,28224}, {6918,16203}, {6971,18545}, {7962,30304}, {8166,14986}, {9614,22792}, {9623,11227}, {9785,12246}, {9841,12629}, {9943,12448}, {9949,11496}, {9951,12737}, {10267,17571}, {10269,16417}, {10569,15934}, {10855,18443}, {10861,18444}, {10865,30284}, {10868,30285}, {11260,12520}, {11499,17573}, {11858,18448}, {11859,18456}, {11860,18454}, {12672,12684}, {12678,30384}, {15178,18761}, {16202,26321}, {17612,19860}, {18450,30287}, {18458,30288}, {18460,30289}, {18469,30292}, {18471,30293}, {18481,22770}

X(30283) = midpoint of X(i) and X(j) for these {i,j}: {944, 5768}, {7962, 30304}
X(30283) = reflection of X(i) in X(j) for these (i,j): (5720, 1385), (19541, 999)
X(30283) = X(5727)-of-2nd circumperp tangential triangle
X(30283) = X(6244)-of-inner-Garcia triangle
X(30283) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 10864, 9856), (956, 5731, 3)


X(30284) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-HONSBERGER AND EHRMANN-SIDE

Barycentrics    a*(a^4-2*(b+c)*a^3-3*b*c*a^2+2*(b^2-c^2)*(b-c)*a-(b^2-b*c+c^2)*(b-c)^2)*(-a+b+c) : :
X(30284) = 3*X(2646)-X(15837)

The homothetic center of these triangles is X(19129). X(30284) is their endo-homothetic center only when ABC is acute.

X(30284) lies on these lines: {1,7}, {3,7672}, {9,2320}, {33,29814}, {36,30329}, {40,11526}, {55,3218}, {78,5686}, {100,17603}, {104,2346}, {142,6224}, {355,7679}, {411,16193}, {497,29817}, {515,21617}, {517,7676}, {518,2330}, {971,8543}, {997,18230}, {999,11025}, {1001,10394}, {1040,17018}, {1156,6265}, {1319,5572}, {1385,5728}, {1445,3576}, {1467,18221}, {1482,7673}, {1602,22769}, {1617,11020}, {1621,10391}, {1864,5284}, {2094,10388}, {2099,11495}, {2801,29007}, {3057,15570}, {3243,3601}, {3616,5809}, {3826,10950}, {3870,5281}, {3935,5218}, {4420,24393}, {4666,5274}, {4861,5853}, {5173,7411}, {5223,22836}, {5563,20116}, {5727,20195}, {5775,6765}, {5886,7678}, {7191,14547}, {7671,10246}, {8232,18446}, {8238,30285}, {8387,18448}, {8388,18456}, {8389,18454}, {8732,18443}, {10865,30283}, {11496,12706}, {12114,12669}, {12730,12737}, {12740,14100}, {27542,29835}, {30330,30392}

X(30284) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 991, 4318), (1, 12520, 4323), (5731, 11037, 4293)


X(30285) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-SHARYGIN AND EHRMANN-SIDE

Barycentrics    a*(a^6+3*b*c*a^4+(b+c)*(b^2-3*b*c+c^2)*a^3-(2*b^4+2*c^4+3*b*c*(b^2+c^2))*a^2-(b+c)*(b^4+c^4-3*b*c*(b-c)^2)*a+(b^4-c^4)*(b^2-c^2)) : :

The homothetic center of these triangles is X(19176). X(30285) is their endo-homothetic center only when ABC is acute.

X(30285) lies on these lines: {1,256}, {3,2292}, {21,104}, {36,30358}, {40,11533}, {182,5692}, {355,5051}, {515,4425}, {517,3920}, {846,3576}, {944,26117}, {960,7193}, {997,18235}, {999,11031}, {1319,17611}, {3145,10267}, {4199,18446}, {4511,11688}, {5289,8424}, {5492,15952}, {5693,13323}, {5731,9791}, {5886,8229}, {8238,30284}, {8249,18448}, {8250,18456}, {8425,18454}, {8731,18443}, {10868,30283}, {11496,12713}, {12114,12683}, {12737,12746}, {18450,30359}, {18458,30360}, {18460,30361}, {18469,30364}, {18471,30365}, {30363,30392}

X(30285) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 8235, 9840), (1385, 9959, 21)


X(30286) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND EHRMANN-VERTEX

Barycentrics    3*a^4-7*(b+c)*a^3+(b^2+6*b*c+c^2)*a^2+7*(b^2-c^2)*(b-c)*a-4*(b^2-c^2)^2 : :

The homothetic center of these triangles is X(18918). X(30286) is their endo-homothetic center only when ABC is acute.

X(30286) lies on these lines: {1,2}, {11,11224}, {36,30283}, {46,10864}, {65,9656}, {80,2093}, {165,5727}, {355,3339}, {484,10860}, {517,17604}, {758,11678}, {952,13462}, {1111,4902}, {1146,1743}, {1728,11010}, {1837,7991}, {2099,7988}, {2801,30287}, {3337,7091}, {3340,7989}, {3361,5881}, {3419,10398}, {3474,4848}, {3486,9588}, {3586,5759}, {3753,15587}, {3894,12736}, {4731,11018}, {5119,10384}, {5252,10980}, {5435,28236}, {5587,11545}, {5603,16236}, {5692,18227}, {5697,10866}, {5722,9819}, {5726,5790}, {5763,9581}, {5902,8581}, {5903,9856}, {5927,18397}, {7743,7982}, {7987,10950}, {8164,14563}, {8256,12625}, {10175,11041}, {10855,30274}, {10863,18393}, {10865,30329}, {10868,30358}, {11035,18398}, {11375,30315}, {11571,13227}, {11858,18399}, {11859,18409}, {11860,18408}, {18410,30288}, {18411,30289}, {18422,30292}, {18423,30293}, {24914,30389}

X(30286) = reflection of X(30294) in X(17604)
X(30286) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1210, 3632, 1), (3086, 3633, 1), (3679, 18391, 1)


X(30287) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND 2nd EHRMANN

Barycentrics    a*((b+c)*a^4-(4*b^2+b*c+4*c^2)*a^3+3*(b+c)*(2*b^2-b*c+2*c^2)*a^2-(4*b^4+4*c^4+7*(b-c)^2*b*c)*a+(b^2-c^2)*(b-c)*(b^2+5*b*c+c^2))*(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(18919). X(30287) is their endo-homothetic center only when ABC is acute.

X(30287) lies on these lines: {7,8}, {516,30294}, {527,11678}, {1156,1445}, {2801,30286}, {4326,30389}, {5435,15726}, {5927,12848}, {6172,18227}, {7671,11019}, {7677,10384}, {8543,8583}, {8544,10864}, {8545,8580}, {8582,30312}, {8732,11575}, {9814,30291}, {10855,30275}, {10860,30295}, {10863,30311}, {10866,30332}, {10868,30359}, {11035,30340}, {11519,30318}, {11858,30367}, {11859,30405}, {11860,30404}, {14986,21151}, {17615,20059}, {18450,30283}, {25722,26015}, {30290,30424}

X(30287) = {X(7), X(15587)}-harmonic conjugate of X(10865)


X(30288) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND 1st KENMOTU DIAGONALS

Barycentrics    a*((2*(b+c)*a^2-4*(b^2+c^2)*a+2*(b+c)^3)*S+(-a+b+c)*((b+c)*a^3-3*(b^2+c^2)*a^2+(b+c)*(3*b^2-2*b*c+3*c^2)*a-(b^2+4*b*c+c^2)*(b-c)^2))*(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(18923). X(30288) is their endo-homothetic center only when ABC is acute.

X(30288) lies on these lines: {7,8}, {3062,7133}, {5927,30324}, {6203,8580}, {8582,30313}, {8583,30385}, {10860,30296}, {10863,30306}, {10864,30400}, {10866,30333}, {10868,30360}, {11019,30346}, {11035,30341}, {11519,30319}, {11858,30368}, {11859,30418}, {11860,30406}, {17604,30375}, {18227,30412}, {18410,30286}, {18458,30283}, {30290,30425}, {30291,30396}, {30294,30431}

X(30288) = {X(8581), X(15587)}-harmonic conjugate of X(30289)


X(30289) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND 2nd KENMOTU DIAGONALS

Barycentrics    a*(-(2*(b+c)*a^2-4*(b^2+c^2)*a+2*(b+c)^3)*S+(-a+b+c)*((b+c)*a^3-3*(b^2+c^2)*a^2+(b+c)*(3*b^2-2*b*c+3*c^2)*a-(b^2+4*b*c+c^2)*(b-c)^2))*(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(18924). X(30289) is their endo-homothetic center only when ABC is acute.

X(30289) lies on these lines: {7,8}, {3062,30355}, {5927,30325}, {6204,8580}, {8582,30314}, {8583,30386}, {10855,30277}, {10860,30297}, {10863,30307}, {10864,30401}, {10866,30334}, {11019,30347}, {11035,30342}, {11519,30320}, {11858,30369}, {11859,30419}, {11860,30407}, {17604,30376}, {18227,30413}, {18411,30286}, {18460,30283}, {30290,30426}, {30291,30397}, {30294,30432}

X(30289) = {X(8581), X(15587)}-harmonic conjugate of X(30288)


X(30290) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND KOSNITA

Barycentrics    a*((b+c)*a^5-(b^2-b*c+c^2)*a^4-(2*b-c)*(b-2*c)*(b+c)*a^3+(2*b^4+2*c^4+(b+c)^2*b*c)*a^2+(b^2-c^2)*(b-c)*(b^2-3*b*c+c^2)*a-(b^2-c^2)^2*(b+c)^2) : :

The homothetic center of these triangles is X(18925). X(30290) is their endo-homothetic center only when ABC is acute.

X(30290) lies on these lines: {1,971}, {8,79}, {10,11678}, {35,10860}, {36,8583}, {46,8580}, {55,16143}, {65,9656}, {72,4312}, {80,13227}, {90,5563}, {226,9948}, {516,10865}, {942,17604}, {946,12666}, {1698,18227}, {1858,4654}, {1898,11518}, {2093,9954}, {3339,5777}, {3633,12448}, {3671,12528}, {4292,5692}, {4303,27785}, {5223,18251}, {5290,6001}, {5691,12709}, {5694,18541}, {5696,11523}, {5697,9589}, {5714,5884}, {5784,9814}, {5902,5927}, {6765,17646}, {7741,10863}, {7951,8582}, {7972,9951}, {7991,17634}, {8545,12520}, {9949,13407}, {9952,11571}, {9961,13405}, {10392,10399}, {10394,12563}, {10624,28164}, {10855,15803}, {10868,30362}, {10883,11019}, {10948,18393}, {11519,25415}, {11522,17625}, {11858,30370}, {11859,30420}, {11860,30408}, {12565,15298}, {14872,18421}, {30287,30424}, {30288,30425}, {30289,30426}, {30292,10651}, {30293,10652}

X(30290) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 9856, 30294), (8581, 10866, 11035), (10866, 11035, 1)


X(30291) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND SUBMEDIAL

Barycentrics    a*(a^5-7*(b+c)*a^4+2*(5*b^2+6*b*c+5*c^2)*a^3+2*(b+c)*(b^2-10*b*c+c^2)*a^2-(11*b^2+26*b*c+11*c^2)*(b-c)^2*a+(b^2-c^2)*(b-c)*(5*b^2+22*b*c+5*c^2)) : :

The homothetic center of these triangles is X(18928). X(30291) is their endo-homothetic center only when ABC is acute.

X(30291) lies on these lines: {1,9947}, {8,1699}, {9,165}, {200,24644}, {210,7991}, {226,7989}, {1698,5658}, {3339,5777}, {4423,30283}, {5047,8583}, {5223,11678}, {5226,7988}, {5437,24645}, {5587,11545}, {5691,12447}, {7308,7987}, {7994,10241}, {7997,16209}, {8581,10980}, {8582,30315}, {9814,30287}, {9819,30294}, {10855,11407}, {10861,30304}, {10863,30308}, {10865,30330}, {10866,30337}, {10868,30363}, {11035,30343}, {11519,12635}, {11858,30371}, {11859,30395}, {11860,30394}, {30288,30396}, {30289,30397}, {30292,10655}, {30293,10656}

X(30291) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1750, 30393, 165), (3062, 8580, 165), (9947, 10157, 17604)


X(30292) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND INNER TRI-EQUILATERAL

Barycentrics    a*(2*sqrt(3)*((b+c)*a^2-2*(b^2+c^2)*a+(b+c)^3)*S+(-a+b+c)*((b+c)*a^3+(b+c)*a*b*c-3*(b^2+c^2)*a^2+3*(b^3+c^3)*a-(b^2+4*b*c+c^2)*(b-c)^2))*(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(18929). X(30292) is their endo-homothetic center only when ABC is acute.

X(30292) lies on these lines: {7,8}, {1653,8580}, {3062,30356}, {5927,30327}, {8523,28097}, {8582,30316}, {8583,10647}, {10855,30280}, {10860,30300}, {10863,30309}, {10864,10649}, {10866,30338}, {10868,30364}, {11019,30351}, {11035,30344}, {11519,30321}, {11859,30421}, {11860,30409}, {17604,30377}, {18227,30414}, {18422,30286}, {18469,30283}, {30290,10651}, {30291,10655}, {30294,30433}

X(30292) = {X(8581), X(15587)}-harmonic conjugate of X(30293)


X(30293) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND OUTER TRI-EQUILATERAL

Barycentrics    a*(-2*sqrt(3)*((b+c)*a^2-2*(b^2+c^2)*a+(b+c)^3)*S+(-a+b+c)*((b+c)*a^3+(b+c)*a*b*c-3*(b^2+c^2)*a^2+3*(b^3+c^3)*a-(b^2+4*b*c+c^2)*(b-c)^2))*(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(18930). X(30293) is their endo-homothetic center only when ABC is acute.

X(30293) lies on these lines: {7,8}, {1251,3062}, {1652,8580}, {5927,30328}, {8582,30317}, {8583,10648}, {10855,30281}, {10860,30301}, {10863,30310}, {10864,10650}, {10866,30339}, {11019,30352}, {11035,30345}, {11519,30322}, {11858,30373}, {11859,30422}, {11860,30410}, {17604,30378}, {18227,30415}, {18471,30283}, {30290,10652}, {30291,10656}, {30294,30434}

X(30293) = {X(8581), X(15587)}-harmonic conjugate of X(30292)


X(30294) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-ATIK AND TRINH

Barycentrics    a*((b+c)*a^5-(b^2+b*c+c^2)*a^4-(b+c)*(2*b^2-7*b*c+2*c^2)*a^3+(2*b^4+2*c^4+3*(b^2-6*b*c+c^2)*b*c)*a^2+(b^2-c^2)*(b-c)*(b^2-5*b*c+c^2)*a-(b^2-c^2)^2*(b+c)^2) : :

The homothetic center of these triangles is X(18931). X(30294) is their endo-homothetic center only when ABC is acute

X(30294) lies on these lines: {1,971}, {8,80}, {35,8583}, {36,10860}, {392,15587}, {516,30287}, {517,17604}, {519,11678}, {1864,11224}, {2098,7993}, {2801,4345}, {3586,5927}, {3632,12448}, {3679,18227}, {4294,12446}, {4301,7672}, {4304,10861}, {5083,15071}, {5119,8580}, {5289,5696}, {5441,16120}, {5902,11019}, {5903,9614}, {6797,9669}, {7091,7284}, {7741,8582}, {7951,10863}, {7972,13227}, {8275,18908}, {9671,25414}, {9819,30291}, {9948,12053}, {10392,18397}, {10624,12447}, {10855,30282}, {10865,30331}, {10868,30366}, {11519,30323}, {11858,30374}, {11859,30423}, {11860,30411}, {30288,30431}, {30289,30432}, {30292,30433}, {30293,30434}

X(30294) = reflection of X(30286) in X(17604)
X(30294) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 9856, 30290), (9856, 10866, 1)


X(30295) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP AND 2nd EHRMANN

Barycentrics    a*(a^4-3*(b+c)*a^3+(3*b^2-b*c+3*c^2)*a^2+3*(b-c)^2*b*c-(b^2-c^2)*(b-c)*a) : :
X(30295) = 4*X(36)-3*X(7677) = 4*X(1155)-X(1156) = 2*X(4316)+X(20119) = X(12730)-4*X(21578)

The homothetic center of these triangles is X(11416). X(30295) is their endo-homothetic center only when ABC is acute.

X(30295) lies on these lines: {2,30311}, {3,8543}, {4,30312}, {7,55}, {21,5880}, {35,30424}, {36,516}, {40,8544}, {46,10394}, {56,30332}, {57,7671}, {100,527}, {142,5284}, {165,8545}, {376,390}, {404,5698}, {484,2801}, {517,14151}, {518,5183}, {528,15326}, {651,3000}, {1001,17549}, {1155,1156}, {1376,6172}, {1445,2951}, {1621,6173}, {1633,11349}, {1770,6986}, {2093,5732}, {2550,17579}, {3218,15733}, {3295,30340}, {3826,6175}, {4038,4343}, {4220,30359}, {4312,5010}, {4316,20119}, {4319,17092}, {4321,9819}, {4326,10980}, {4480,4578}, {5274,8732}, {6600,20059}, {6767,11038}, {7580,12848}, {7589,30404}, {7673,7962}, {7675,11529}, {7678,10589}, {7679,10590}, {7991,30318}, {8075,30367}, {8076,30405}, {8257,9352}, {10177,27003}, {10860,30287}, {12730,21578}, {13587,28534}, {14189,24011}, {15254,17531}

X(30295) = reflection of X(14151) in X(18450)
X(30295) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 7676, 2346), (7, 11495, 7676), (8255, 11246, 7)


X(30296) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP AND 1st KENMOTU DIAGONALS

Barycentrics    a*(a^4-3*(b+c)*a^3+2*(-a+b+c)*S*a+3*(b^2+c^2)*a^2+2*(b-c)^2*b*c-(b^2-c^2)*(b-c)*a) : :

The homothetic center of these triangles is X(11417). X(30283) is their endo-homothetic center only when ABC is acute.

X(30296) lies on these lines: {2,30306}, {3,30276}, {4,30313}, {7,55}, {35,30425}, {36,30431}, {40,30400}, {56,30333}, {57,30346}, {165,6203}, {484,18410}, {516,30380}, {517,18458}, {1155,30375}, {1376,30412}, {1721,6204}, {1742,5414}, {2066,9441}, {3295,30341}

X(30296) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (55, 11495, 30297), (165, 30354, 6203)


X(30297) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP AND 2nd KENMOTU DIAGONALS

Barycentrics    a*(a^4-3*(b+c)*a^3-2*(-a+b+c)*S*a+3*(b^2+c^2)*a^2+2*(b-c)^2*b*c-(b^2-c^2)*(b-c)*a) : :

The homothetic center of these triangles is X(11418). X(30297) is their endo-homothetic center only when ABC is acute.

X(30297) lies on these lines: {2,30307}, {3,30277}, {4,30314}, {7,55}, {35,30426}, {36,30432}, {40,30401}, {56,30334}, {57,30347}, {165,6204}, {484,18411}, {516,30381}, {517,18460}, {1155,30376}, {1376,30413}, {1721,6203}, {1742,2066}, {3295,30342}, {4220,30361}, {4319,13389}, {5414,9441}, {7580,30325}, {7589,30407}, {7991,30320}, {8075,30369}, {8076,30419}, {10860,30289}

X(30297) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (55, 11495, 30296), (165, 30355, 6204)


X(30298) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP AND LUCAS ANTIPODAL TANGENTIAL

Barycentrics
a^2*((8*a^5+12*(b+c)*a^4-16*(3*b^2+5*b*c+3*c^2)*a^3+24*(b^2-c^2)*(b-c)*a^2+8*(b^2+12*b*c+c^2)*(b-c)^2*a-4*(b^2-c^2)*(b-c)^3)*S-(a+b-c)*(a-b+c)*(3*a^5-13*(b+c)*a^4+2*(3*b^2-34*b*c+3*c^2)*a^3+2*(b+c)*(7*b^2+38*b*c+7*c^2)*a^2-(9*b^2+22*b*c+9*c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)*(b^2+6*b*c+c^2)))*((a^3+(5*b-c)*a^2-(5*b^2+2*b*c+c^2)*a-(b-c)^3)*S-(a+b-c)*(a^4-(2*b+c)*a^3-(3*b+c)*a^2*c+(2*b^3+c^3+(b+4*c)*b*c)*a-(b-c)^3*b))*((a^3-(b-5*c)*a^2-(b^2+2*b*c+5*c^2)*a+(b-c)^3)*S-(a-b+c)*(a^4-(b+2*c)*a^3-(b+3*c)*a^2*b+(b^3+2*c^3+(4*b+c)*b*c)*a+(b-c)^3*c))*((a^3-(b+3*c)*a^2-(b+3*c)*(b-c)*a+(b-c)*(b^2+6*b*c+c^2))*S-(-a+b+c)*((b+c)*a^3-(b-3*c)*(b-c)*a^2-(b^2-c^2)*(b+3*c)*a+(b^2-c^2)*(b-c)^2))*((a^3-(3*b+c)*a^2+(3*b+c)*(b-c)*a-(b-c)*(b^2+6*b*c+c^2))*S-(-a+b+c)*((b+c)*a^3-(3*b-c)*(b-c)*a^2+(b^2-c^2)*(3*b+c)*a-(b^2-c^2)*(b-c)^2)) : :

The homothetic center of these triangles is X(19406). X(30298) is their endo-homothetic center only when ABC is acute.

X(30298) lies on these lines: {}


X(30299) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP AND LUCAS(-1) ANTIPODAL TANGENTIAL

Barycentrics
a^2*(-(8*a^5+12*(b+c)*a^4-16*(3*b^2+5*b*c+3*c^2)*a^3+24*(b^2-c^2)*(b-c)*a^2+8*(b^2+12*b*c+c^2)*(b-c)^2*a-4*(b^2-c^2)*(b-c)^3)*S-(a+b-c)*(a-b+c)*(3*a^5-13*(b+c)*a^4+2*(3*b^2-34*b*c+3*c^2)*a^3+2*(b+c)*(7*b^2+38*b*c+7*c^2)*a^2-(9*b^2+22*b*c+9*c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)*(b^2+6*b*c+c^2)))*(-(a^3+(5*b-c)*a^2-(5*b^2+2*b*c+c^2)*a-(b-c)^3)*S-(a+b-c)*(a^4-(2*b+c)*a^3-(3*b+c)*a^2*c+(2*b^3+c^3+(b+4*c)*b*c)*a-(b-c)^3*b))*(-(a^3-(b-5*c)*a^2-(b^2+2*b*c+5*c^2)*a+(b-c)^3)*S-(a-b+c)*(a^4-(b+2*c)*a^3-(b+3*c)*a^2*b+(b^3+2*c^3+(4*b+c)*b*c)*a+(b-c)^3*c))*(-(a^3-(b+3*c)*a^2-(b+3*c)*(b-c)*a+(b-c)*(b^2+6*b*c+c^2))*S-(-a+b+c)*((b+c)*a^3-(b-3*c)*(b-c)*a^2-(b^2-c^2)*(b+3*c)*a+(b^2-c^2)*(b-c)^2))*(-(a^3-(3*b+c)*a^2+(3*b+c)*(b-c)*a-(b-c)*(b^2+6*b*c+c^2))*S-(-a+b+c)*((b+c)*a^3-(3*b-c)*(b-c)*a^2+(b^2-c^2)*(3*b+c)*a-(b^2-c^2)*(b-c)^2)) : :

The homothetic center of these triangles is X(19407). X(30299) is their endo-homothetic center only when ABC is acute.

X(30299) lies on these lines: {}


X(30300) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP AND INNER TRI-EQUILATERAL

Barycentrics    a*(2*sqrt(3)*(-a+b+c)*S*a+a^4-3*(b+c)*a^3+3*(b^2+c^2)*a^2-(b^2-c^2)*(b-c)*a+2*(b-c)^2*b*c) : :

The homothetic center of these triangles is X(11420). X(30300) is their endo-homothetic center only when ABC is acute.

X(30300) lies on these lines: {2,30309}, {3,30280}, {4,30316}, {7,55}, {35,10651}, {36,30433}, {40,10649}, {56,30338}, {57,30351}, {165,1653}, {484,18422}, {516,30382}, {1082,4336}, {1155,30377}, {1250,1742}, {1376,30414}, {3295,30344}, {4220,30364}, {7580,30327}, {7589,30409}, {7991,30321}, {8075,30372}, {8076,30421}, {9441,10638}, {10860,30292}

X(30300) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (55, 11495, 30301), (165, 30356, 1653)


X(30301) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-CIRCUMPERP AND OUTER TRI-EQUILATERAL

Barycentrics    a*(-2*sqrt(3)*(-a+b+c)*S*a+a^4-3*(b+c)*a^3+3*(b^2+c^2)*a^2-(b^2-c^2)*(b-c)*a+2*(b-c)^2*b*c) : :

The homothetic center of these triangles is X(11421). X(30301) is their endo-homothetic center only when ABC is acute.

X(30301) lies on these lines: {2,30310}, {3,30281}, {4,30317}, {7,55}, {35,10652}, {36,30434}, {40,10650}, {56,30339}, {57,30352}, {165,1652}, {484,18423}, {516,30383}, {517,18471}, {559,4336}, {1155,30378}, {1250,9441}, {1251,1653}, {1376,30415}, {1742,10638}, {3295,30345}, {4220,30365}, {7580,30328}, {7589,30410}, {7991,30322}, {8075,30373}, {8076,30422}, {10860,30293}

X(30301) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (55, 11495, 30300), (165, 30357, 1652)


X(30302) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-CONWAY AND LUCAS ANTIPODAL TANGENTIAL

Barycentrics
-(5*a^5+7*(b+c)*a^4-2*(b+c)^2*a^3-2*(b+c)*(3*b^2-2*b*c+3*c^2)*a^2-(3*b^2+2*b*c+3*c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)*(b^2+6*b*c+c^2))*S+(a+b+c)*(a^6-4*(b+c)*a^5+4*(b+c)*a^3*b*c+5*(b-c)^2*a^4-(5*b^2+2*b*c+5*c^2)*(b-c)^2*a^2+4*(b^3-c^3)*(b^2-c^2)*a-(b^2-c^2)^2*(b-c)^2) : :

The homothetic center of these triangles is X(19408). X(30302) is their endo-homothetic center only when ABC is acute.

X(30302) lies on these lines: {2,30278}, {7,10134}, {21,30387}, {63,30429}, {4313,30335}, {5273,30416}, {8822,10430}, {11020,30348}


X(30303) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-CONWAY AND LUCAS(-1) ANTIPODAL TANGENTIAL

Barycentrics
(5*a^5+7*(b+c)*a^4-2*(b+c)^2*a^3-2*(b+c)*(3*b^2-2*b*c+3*c^2)*a^2-(3*b^2+2*b*c+3*c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)*(b^2+6*b*c+c^2))*S+(a+b+c)*(a^6-4*(b+c)*a^5+4*(b+c)*a^3*b*c+5*(b-c)^2*a^4-(5*b^2+2*b*c+5*c^2)*(b-c)^2*a^2+4*(b^3-c^3)*(b^2-c^2)*a-(b^2-c^2)^2*(b-c)^2) : :

The homothetic center of these triangles is X(19409). X(30303) is their endo-homothetic center only when ABC is acute.

X(30303) lies on these lines: {2,30279}, {7,10135}, {21,30388}, {63,30430}, {4313,30336}, {5273,30417}, {8822,10430}, {11020,30349}


X(30304) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-CONWAY AND SUBMEDIAL

Barycentrics    a*(a^5+(b+c)*a^4-2*(3*b^2-4*b*c+3*c^2)*a^3+2*(b^2-c^2)*(b-c)*a^2+(5*b^2+2*b*c+5*c^2)*(b-c)^2*a+(b^2-c^2)*(b-c)*(-3*b^2-2*b*c-3*c^2)) : :
X(30304) = 3*X(57)-2*X(19541) = 3*X(165)-2*X(200) = 4*X(997)-5*X(7987) = 3*X(1699)-4*X(11019) = 3*X(1750)-4*X(19541) = 2*X(5720)-3*X(21164) = 3*X(9778)-X(20015)

The homothetic center of these triangles is X(9306). X(30304) is their endo-homothetic center only when ABC is acute.

X(30304) lies on these lines: {1,84}, {2,11407}, {3,3929}, {4,553}, {7,1699}, {9,10167}, {20,519}, {21,30363}, {40,5918}, {46,12671}, {57,971}, {63,100}, {65,10864}, {72,9841}, {223,7004}, {354,11372}, {388,9948}, {515,2093}, {516,9965}, {518,7994}, {610,8558}, {912,6282}, {936,12528}, {942,12684}, {944,14646}, {950,12246}, {997,5267}, {1210,6223}, {1490,1708}, {1728,9942}, {1765,3731}, {1776,6261}, {2808,3784}, {2951,15733}, {3057,9845}, {3158,17613}, {3219,21153}, {3333,12688}, {3339,4292}, {3361,9960}, {3522,3951}, {3576,3683}, {3586,5768}, {3679,6916}, {3742,16112}, {3868,11531}, {3874,12651}, {3911,5658}, {3928,7580}, {4197,30315}, {4297,12526}, {4304,9819}, {4313,30337}, {4350,8835}, {4654,8727}, {4866,9588}, {5220,10178}, {5249,7988}, {5272,9355}, {5273,5785}, {5400,8056}, {5437,5927}, {5587,12678}, {5720,21164}, {5735,10431}, {5779,7308}, {5784,8580}, {5787,9579}, {5851,24703}, {5882,10385}, {6173,8226}, {6245,6844}, {6259,9581}, {6260,6969}, {6883,7330}, {6950,18446}, {6993,7989}, {7291,9572}, {7489,18443}, {7962,30283}, {7996,10444}, {9778,20015}, {9949,12577}, {10122,18219}, {10202,18540}, {10270,17857}, {10384,12915}, {10861,30291}, {10883,30308}, {11020,24644}, {11036,30343}, {11520,16189}, {11888,30371}, {11889,30395}, {11890,30394}, {18389,18421}, {18444,30392}, {20307,26932}

X(30304) = midpoint of X(i) and X(j) for these {i,j}: {7991, 18452}, {9965, 10430}
X(30304) = reflection of X(i) in X(j) for these (i,j): (1750, 57), (3586, 5768), (5691, 18391), (6282, 7171), (7962, 30283), (7994, 10860)
X(30304) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10085, 15071, 1), (12675, 12705, 1)


X(30305) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-CONWAY AND TRINH

Barycentrics    a^4-8*a^2*b*c+2*(b+c)*a^3-2*(b^2-c^2)*(b-c)*a-(b^2-c^2)^2 : :
X(30305) = 3*X(1)-2*X(4315) = 3*X(497)-2*X(5722) = 5*X(497)-4*X(18527) = 3*X(4293)-4*X(4315) = X(4293)-4*X(4342) = X(4315)-3*X(4342) = 3*X(5603)-2*X(22753) = 4*X(5722)-3*X(18391) = 5*X(5722)-6*X(18527) = 5*X(18391)-8*X(18527)

The homothetic center of these triangles is X(11438). X(30305) is their endo-homothetic center only when ABC is acute.

X(30305) lies on these lines: {1,7}, {2,5119}, {3,1387}, {4,1000}, {8,80}, {10,6919}, {11,5657}, {30,3476}, {35,3616}, {36,9778}, {40,3086}, {46,14986}, {55,5603}, {56,6361}, {57,28194}, {65,1058}, {145,10572}, {319,4673}, {329,519}, {355,5225}, {376,1319}, {377,3890}, {388,9957}, {392,2550}, {411,11508}, {484,5435}, {496,1788}, {497,517}, {498,6979}, {499,11010}, {515,7962}, {528,5289}, {535,3241}, {551,9776}, {631,11376}, {908,3895}, {938,5903}, {944,1317}, {946,1697}, {950,5758}, {952,9668}, {956,5698}, {958,13463}, {960,5082}, {997,17784}, {999,3474}, {1056,1836}, {1145,17556}, {1191,17366}, {1210,7991}, {1265,5100}, {1320,11114}, {1388,15338}, {1478,9812}, {1482,3486}, {1699,5726}, {1737,5274}, {1837,12245}, {2093,11019}, {2099,3058}, {2136,21075}, {2475,16155}, {2478,14923}, {2551,10914}, {2646,10595}, {2800,5768}, {3091,10039}, {3189,5730}, {3218,11240}, {3295,3485}, {3296,18490}, {3303,3487}, {3333,4031}, {3340,14563}, {3421,3880}, {3434,3877}, {3436,3885}, {3465,3938}, {3475,6767}, {3523,18220}, {3579,7288}, {3583,12647}, {3601,13464}, {3612,3622}, {3617,10826}, {3624,11024}, {3632,5815}, {3651,11510}, {3656,10385}, {3679,18228}, {3698,17559}, {3746,5703}, {3753,26105}, {3832,10827}, {3902,5739}, {3992,8055}, {4511,20075}, {4853,12572}, {4857,10573}, {4861,6872}, {5048,7967}, {5080,12648}, {5084,5836}, {5183,17728}, {5218,5886}, {5222,5315}, {5226,10056}, {5229,22793}, {5234,9874}, {5250,19843}, {5270,5561}, {5290,30337}, {5328,5541}, {5330,9963}, {5425,15933}, {5441,14450}, {5493,15803}, {5526,5838}, {5556,13602}, {5586,30343}, {5687,12732}, {5690,9669}, {5714,15888}, {5727,28234}, {5744,21630}, {5748,11813}, {5809,18397}, {5811,5881}, {5812,13600}, {5818,10896}, {5880,10179}, {5902,10580}, {5904,6764}, {6001,17642}, {6666,19855}, {6845,10957}, {6906,10966}, {6909,22767}, {6969,22835}, {7080,21616}, {7741,9780}, {7743,10589}, {7951,9779}, {7957,10866}, {7972,9809}, {8163,12246}, {8164,17605}, {8715,27383}, {9578,18483}, {9581,11362}, {9670,10950}, {9791,30366}, {9793,30374}, {9795,30423}, {9955,10588}, {10200,26062}, {10284,10526}, {10944,12953}, {11491,26358}, {11522,13411}, {11531,16236}, {11891,30411}, {12247,13274}, {12248,20586}, {12527,12629}, {12700,17622}, {12740,13199}, {14217,15558}, {15170,15934}, {15172,15935}, {21669,22759}, {24046,28016}, {26129,26364}, {26839,27253}

X(30305) = midpoint of X(7962) and X(9580)
X(30305) = reflection of X(i) in X(j) for these (i,j): (1, 4342), (2093, 11019), (3421, 24703), (3474, 999), (4293, 1), (17784, 997), (18391, 497), (30353, 5542)
X(30305) = X(200)-of-inner-Garcia-triangle
X(30305) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 4302, 5731), (4345, 5731, 1), (5731, 30332, 4302)


X(30306) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER AND 1st KENMOTU DIAGONALS

Barycentrics    (b^2-4*b*c+c^2)*a^3-3*(b^2-c^2)*(b-c)*a^2+2*(b-c)^2*(-a+b+c)*S+(3*b^2+4*b*c+3*c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)^3 : :

The homothetic center of these triangles is X(11447). X(30306) is their endo-homothetic center only when ABC is acute.

X(30306) lies on these lines: {2,30296}, {4,30385}, {5,30313}, {7,11}, {12,30333}, {226,30346}, {496,30341}, {1699,6203}, {2886,30412}, {3817,30380}, {5886,18458}, {7133,8228}, {7741,30425}, {7951,30431}, {7988,30354}, {8085,30368}, {8086,30418}, {8226,30324}, {8227,30400}, {8229,30360}, {8379,30406}, {8727,30276}, {10863,30288}, {11522,30319}, {17605,30375}, {18393,18410}, {30308,30396}


X(30307) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER AND 2nd KENMOTU DIAGONALS

Barycentrics    (b^2-4*b*c+c^2)*a^3-3*(b^2-c^2)*(b-c)*a^2-2*(b-c)^2*(-a+b+c)*S+(3*b^2+4*b*c+3*c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)^3 : :

The homothetic center of these triangles is X(11448). X(30307) is their endo-homothetic center only when ABC is acute.

X(30307) lies on these lines: {2,30297}, {4,30386}, {5,30314}, {7,11}, {12,30334}, {226,30347}, {496,30342}, {1699,6204}, {2886,30413}, {3817,30381}, {5886,18460}, {7741,30426}, {7951,30432}, {7988,30355}, {8085,30369}, {8086,30419}, {8226,30325}, {8227,30401}, {8229,30361}, {8379,30407}, {8727,30277}, {10863,30289}, {11522,30320}, {17605,30376}, {18393,18411}, {30308,30397}


X(30308) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER AND SUBMEDIAL

Barycentrics    a^3+2*(b+c)*a^2+5*(b-c)^2*a-8*(b^2-c^2)*(b-c) : :
X(30308) = X(1)+4*X(381) = X(1)-16*X(9955) = 7*X(1)+8*X(18480) = X(1)+2*X(18492) = X(1)-4*X(18493) = 11*X(1)+4*X(18525) = 19*X(1)-4*X(18526) = 8*X(2)-3*X(165) = 2*X(2)+3*X(1699) = X(2)-6*X(3817) = 4*X(2)-9*X(7988) = 13*X(2)-3*X(9778) = X(2)+9*X(9779) = 7*X(2)+3*X(9812) = 11*X(2)-6*X(10164) = 7*X(2)-12*X(10171) = X(165)+4*X(1699) = X(165)-16*X(3817) = X(165)-6*X(7988) = 13*X(165)-8*X(9778) = 7*X(165)+8*X(9812) = 11*X(165)-16*X(10164) = X(381)+4*X(9955) = 7*X(381)-2*X(18480) = 11*X(381)-X(18525) = 19*X(381)+X(18526) = X(1699)+4*X(3817) = 2*X(1699)+3*X(7988) = 13*X(1699)+2*X(9778) = X(1699)-6*X(9779) = 7*X(1699)-2*X(9812) = 14*X(9955)+X(18480) = 8*X(9955)+X(18492) = 4*X(9955)-X(18493) = 4*X(18480)-7*X(18492) = 2*X(18480)+7*X(18493) = 22*X(18480)-7*X(18525) = X(18492)+2*X(18493)

The homothetic center of these triangles is X(11451). X(30308) is their endo-homothetic center only when ABC is acute.

X(30308) lies on these lines: {1,381}, {2,165}, {4,25055}, {5,3654}, {11,4654}, {12,30337}, {20,19883}, {30,7987}, {40,5055}, {376,3624}, {496,30343}, {517,19709}, {519,3091}, {528,15017}, {546,3655}, {547,12699}, {551,3839}, {946,3545}, {962,3828}, {1125,3543}, {1385,14269}, {1572,18362}, {1656,28198}, {1698,5071}, {1743,17737}, {2886,30393}, {3058,5219}, {3062,3255}, {3090,9589}, {3241,19925}, {3339,7741}, {3361,3582}, {3534,11230}, {3544,11362}, {3576,3830}, {3579,15703}, {3653,15687}, {3656,4677}, {3829,8226}, {3843,28208}, {3845,5886}, {3850,5881}, {3851,7982}, {3855,13464}, {3929,5536}, {4301,5068}, {4312,10589}, {4857,6849}, {4995,9580}, {5054,16192}, {5056,9588}, {5087,8580}, {5223,11680}, {5231,17781}, {5298,9579}, {5493,7486}, {5531,10707}, {5550,15683}, {5690,14892}, {5726,30384}, {5901,23046}, {6175,8583}, {6361,19872}, {7280,28444}, {7678,30330}, {7951,9819}, {7993,10711}, {8085,30371}, {8086,30395}, {8229,30363}, {8379,30394}, {8727,11407}, {9166,9860}, {9592,11648}, {9612,10072}, {9614,10056}, {9814,30311}, {9875,14639}, {10109,26446}, {10129,10582}, {10165,11001}, {10863,30291}, {10883,30304}, {11019,30340}, {11235,17618}, {11737,22791}, {12512,15708}, {13174,23234}, {13462,23708}, {13624,15684}, {14893,18481}, {15685,17502}, {15692,19862}, {15693,28146}, {15694,28202}, {15695,28154}, {15697,28158}, {15711,28182}, {15713,28178}, {15721,19878}, {18393,18421}, {19708,28150}, {28452,30282}, {30306,30396}, {30307,30397}, {30309,10655}, {30310,10656}

X(30308) = midpoint of X(381) and X(18493)
X(30308) = reflection of X(i) in X(j) for these (i,j): (1698, 5071), (3679, 5818), (15692, 19862), (18492, 381)
X(30308) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1699, 3817, 7988), (1699, 7988, 165), (18492, 18493, 1)


X(30309) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER AND INNER TRI-EQUILATERAL

Barycentrics    2*sqrt(3)*(b-c)^2*(-a+b+c)*S+(b^2-4*b*c+c^2)*a^3-3*(b^2-c^2)*(b-c)*a^2+(3*b^2+4*b*c+3*c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)^3 : :

The homothetic center of these triangles is X(11452). X(30309) is their endo-homothetic center only when ABC is acute.

X(30309) lies on these lines: {2,30300}, {4,10647}, {5,30316}, {7,11}, {12,30338}, {226,30351}, {496,30344}, {1653,1699}, {2886,30414}, {3817,30382}, {5886,18469}, {7741,10651}, {7951,30433}, {7988,30356}, {8086,30421}, {8226,30327}, {8227,10649}, {8229,30364}, {8379,30409}, {8727,30280}, {10863,30292}, {11522,30321}, {17605,30377}, {18393,18422}, {30308,10655}


X(30310) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER AND OUTER TRI-EQUILATERAL

Barycentrics    -2*sqrt(3)*(b-c)^2*(-a+b+c)*S+(b^2-4*b*c+c^2)*a^3-3*(b^2-c^2)*(b-c)*a^2+(3*b^2+4*b*c+3*c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)^3 : :

The homothetic center of these triangles is X(11453). X(30310) is their endo-homothetic center only when ABC is acute.

X(30310) lies on these lines: {2,30301}, {4,10648}, {5,30317}, {7,11}, {12,30339}, {226,30352}, {496,30345}, {1652,1699}, {2886,30415}, {3817,30383}, {5886,18471}, {7741,10652}, {7951,30434}, {7988,30357}, {8085,30373}, {8086,30422}, {8226,30328}, {8227,10650}, {8229,30365}, {8379,30410}, {8727,30281}, {10863,30293}, {11522,30322}, {17605,30378}, {18393,18423}, {30308,10656}


X(30311) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 3rd ANTI-EULER AND 2nd EHRMANN

Barycentrics    (b^2-5*b*c+c^2)*a^3-3*(b^2-c^2)*(b-c)*a^2+(3*b^2+5*b*c+3*c^2)*(b-c)^2*a-(b^2-c^2)*(b-c)^3 : :
X(30311) = 3*X(7679)-4*X(7951)

The homothetic center of these triangles is X(11443). X(30311) is their endo-homothetic center only when ABC is acute.

X(30311) lies on these lines: {2,30295}, {4,390}, {5,30312}, {7,11}, {9,5057}, {12,10248}, {142,10129}, {226,7671}, {496,30340}, {516,6932}, {527,11680}, {920,1445}, {1001,11114}, {1699,8545}, {2476,3826}, {2550,17577}, {2801,18393}, {2886,6172}, {3817,30379}, {3869,24393}, {4193,5880}, {4197,15254}, {4293,6912}, {5218,7676}, {5226,7965}, {5542,10394}, {5603,14151}, {5886,18450}, {6600,20095}, {7672,18397}, {7741,30424}, {7988,30353}, {8085,30367}, {8086,30405}, {8226,12848}, {8227,8544}, {8229,30359}, {8257,20292}, {8379,30404}, {8727,30275}, {9814,30308}, {10863,30287}, {11522,30318}, {15726,17605}

X(30311) = {X(954), X(9668)}-harmonic conjugate of X(390)


X(30312) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER AND 2nd EHRMANN

Barycentrics    ((b^2+5*b*c+c^2)*a^2-(b+c)*(2*b^2+b*c+2*c^2)*a+(b^2-c^2)^2)*(a-b+c)*(a+b-c) : :
X(30312) = 3*X(7678)-4*X(7741)

The homothetic center of these triangles is X(11458). X(30312) is their endo-homothetic center only when ABC is acute.

X(30312) lies on these lines: {2,8543}, {4,30295}, {5,30311}, {7,12}, {8,14151}, {9,25010}, {10,30379}, {11,30332}, {46,6991}, {56,26060}, {142,4848}, {355,18450}, {390,496}, {404,2550}, {442,12848}, {443,956}, {495,30340}, {516,6943}, {527,11681}, {528,5433}, {1001,14882}, {1155,10883}, {1156,6932}, {1210,7671}, {1329,6172}, {1445,3841}, {1698,8545}, {1737,10394}, {2346,6989}, {2476,5880}, {2801,18395}, {3671,5692}, {3679,30318}, {3925,5435}, {4193,5698}, {4294,6986}, {4308,9710}, {4318,17278}, {4323,5289}, {4429,17077}, {5051,30359}, {5273,25973}, {5587,8544}, {5729,6937}, {6049,6067}, {6901,9655}, {7951,30424}, {7989,30353}, {8087,30367}, {8088,30405}, {8270,26724}, {8382,30404}, {8582,30287}, {8728,30275}, {9814,30315}, {11495,12953}, {15726,17606}

X(30312) = {X(7), X(3826)}-harmonic conjugate of X(7679)


X(30313) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER AND 1st KENMOTU DIAGONALS

Barycentrics    (2*(b+c)^2*S+(-a+b+c)*((b^2+4*b*c+c^2)*a-(b+c)*(b-c)^2))*(a-b+c)*(a+b-c) : :

The homothetic center of these triangles is X(11462). X(30313) is their endo-homothetic center only when ABC is acute.

X(30313) lies on these lines: {2,30385}, {4,30296}, {5,30306}, {7,12}, {10,30380}, {11,30333}, {355,18458}, {442,30324}, {495,30341}, {1210,30346}, {1329,30412}, {1698,6203}, {3679,30319}, {5051,30360}, {5587,30400}, {7133,8230}, {7741,30431}, {7951,30425}, {7989,30354}, {8088,30418}, {8382,30406}, {8582,30288}, {8728,30276}, {17606,30375}, {18395,18410}, {30315,30396}

X(30313) = {X(12), X(3826)}-harmonic conjugate of X(30314)


X(30314) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER AND 2nd KENMOTU DIAGONALS

Barycentrics    (-2*(b+c)^2*S+(-a+b+c)*((b^2+4*b*c+c^2)*a-(b+c)*(b-c)^2))*(a-b+c)*(a+b-c) : :

The homothetic center of these triangles is X(11463). X(30314) is their endo-homothetic center only when ABC is acute.

X(30314) lies on these lines: {2,30386}, {4,30297}, {5,30307}, {7,12}, {10,30381}, {11,30334}, {355,18460}, {442,30325}, {495,30342}, {1210,30347}, {1329,30413}, {1698,6204}, {3679,30320}, {5051,30361}, {5587,30401}, {7741,30432}, {7951,30426}, {7989,30355}, {8087,30369}, {8088,30419}, {8382,30407}, {8582,30289}, {8728,30277}, {17606,30376}, {18395,18411}, {30315,30397}

X(30314) = {X(12), X(3826)}-harmonic conjugate of X(30313)


X(30315) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER AND SUBMEDIAL

Barycentrics    a^4+3*(b+c)*a^3-3*(3*b^2+2*b*c+3*c^2)*a^2-3*(b^2-c^2)*(b-c)*a+8*(b^2-c^2)^2 : :
X(30315) = 3*X(1)-20*X(1656) = 5*X(1)+12*X(5790) = X(1)+16*X(9956) = 13*X(1)+4*X(12645) = 24*X(2)-7*X(30389) = 8*X(4)+9*X(165) = 2*X(4)+15*X(1698) = 5*X(4)+12*X(6684) = X(4)-18*X(10175) = 3*X(165)-20*X(1698) = 3*X(165)+14*X(7989) = X(165)+16*X(10175) = 5*X(1656)+12*X(9956) = 35*X(1656)-18*X(11230) = 25*X(1698)-8*X(6684) = 10*X(1698)+7*X(7989) = 5*X(1698)+12*X(10175) = 3*X(5790)-20*X(9956) = 7*X(5790)+10*X(11230) = 2*X(6684)+15*X(10175) = 14*X(9956)+3*X(11230)

The homothetic center of these triangles is X(11465). X(30315) is their endo-homothetic center only when ABC is acute.

X(30315) lies on these lines: {1,1656}, {2,30389}, {3,19876}, {4,165}, {5,3654}, {10,5056}, {11,30337}, {12,10980}, {40,3851}, {140,5587}, {355,30392}, {442,30326}, {495,30343}, {515,3533}, {516,3854}, {519,7486}, {547,4677}, {1210,30350}, {1329,30393}, {1657,11231}, {1699,5068}, {3062,3826}, {3090,3679}, {3091,3828}, {3339,7951}, {3361,5270}, {3522,19877}, {3523,3634}, {3544,28194}, {3545,9589}, {3584,6887}, {3617,10171}, {3624,5818}, {3628,5881}, {3656,12812}, {3850,26446}, {4197,30304}, {4301,15022}, {4668,5886}, {4745,5734}, {5051,30363}, {5055,7982}, {5059,10164}, {5067,25055}, {5071,11362}, {5223,11681}, {5425,5780}, {5531,6702}, {5559,23708}, {7679,30330}, {7741,9819}, {8087,30371}, {8088,30395}, {8227,11224}, {8382,30394}, {8582,30291}, {8728,11407}, {8960,19004}, {9814,30312}, {9851,17529}, {10827,13462}, {11375,30286}, {15178,15703}, {15720,18480}, {18395,18421}, {30313,30396}, {30314,30397}, {30316,10655}, {30317,10656}

X(30315) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1698, 7989, 165), (1698, 10175, 7989)


X(30316) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER AND INNER TRI-EQUILATERAL

Barycentrics    (a+b-c)*(a-b+c)*(2*sqrt(3)*(b+c)^2*S+(-a+b+c)*((b^2+4*b*c+c^2)*a-(b+c)*(b-c)^2)) : :

The homothetic center of these triangles is X(11466). X(30316) is their endo-homothetic center only when ABC is acute.

X(30316) lies on these lines: {2,10647}, {4,30300}, {5,30309}, {7,12}, {10,30382}, {11,30338}, {355,18469}, {442,30327}, {495,30344}, {1210,30351}, {1329,30414}, {1653,1698}, {3679,30321}, {5051,30364}, {5587,10649}, {7741,30433}, {7951,10651}, {7989,30356}, {8088,30421}, {8382,30409}, {8582,30292}, {8728,30280}, {17606,30377}, {18395,18422}, {30315,10655}

X(30316) = {X(12), X(3826)}-harmonic conjugate of X(30317)


X(30317) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 4th ANTI-EULER AND OUTER TRI-EQUILATERAL

Barycentrics    (a+b-c)*(a-b+c)*(-2*sqrt(3)*(b+c)^2*S+(-a+b+c)*((b^2+4*b*c+c^2)*a-(b+c)*(b-c)^2)) : :

The homothetic center of these triangles is X(11467). X(30317) is their endo-homothetic center only when ABC is acute.

X(30317) lies on these lines: {2,10648}, {4,30301}, {5,30310}, {7,12}, {10,30383}, {11,30339}, {355,18471}, {442,30328}, {1210,30352}, {1329,30415}, {1652,1698}, {3679,30322}, {5051,30365}, {5587,10650}, {7741,30434}, {7951,10652}, {7989,30357}, {8087,30373}, {8088,30422}, {8382,30410}, {8582,30293}, {8728,30281}, {17606,30378}, {18395,18423}, {30315,10656}

X(30317) = {X(12), X(3826)}-harmonic conjugate of X(30316)


X(30318) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS AND 2nd EHRMANN

Barycentrics    a*(a^3-5*(b+c)*a^2+(7*b^2+4*b*c+7*c^2)*a-3*(b+c)*(b^2+c^2))*(a+b-c)*(a-b+c) : :
X(30318) = 4*X(56)-3*X(1445)

The homothetic center of these triangles is X(11470). X(30318) is their endo-homothetic center only when ABC is acute.

X(30318) lies on these lines: {1,651}, {7,145}, {8,30379}, {9,1404}, {40,18450}, {56,78}, {57,3935}, {63,2078}, {77,3242}, {142,5554}, {390,5882}, {516,30323}, {517,8544}, {527,11682}, {938,5261}, {1319,5220}, {1388,15254}, {1420,3984}, {1998,3873}, {2098,15726}, {2951,7673}, {3306,5083}, {3339,3874}, {3679,30312}, {3870,17625}, {3988,5223}, {4308,11523}, {4312,11280}, {4326,30337}, {4864,6180}, {5252,25557}, {5728,7373}, {5729,24928}, {5880,10944}, {6049,6172}, {7962,30332}, {7991,30295}, {8232,10392}, {8581,11011}, {8732,24391}, {9814,16189}, {9846,12560}, {10388,11220}, {11025,30343}, {11518,30275}, {11519,30287}, {11522,30311}, {11529,30340}, {11531,30353}, {11533,30359}, {11534,30367}, {11535,30404}, {11899,30405}, {25415,30424}

X(30318) = reflection of X(5729) in X(24928)
X(30318) = {X(7), X(3243)}-harmonic conjugate of X(11526)


X(30319) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS AND 1st KENMOTU DIAGONALS

Barycentrics    a*(2*(a-3*b-3*c)*S+(-a+b+c)*(a^2-4*(b+c)*a+3*b^2-2*b*c+3*c^2))*(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(11473). X(30319) is their endo-homothetic center only when ABC is acute.

X(30319) lies on these lines: {1,372}, {7,145}, {8,30380}, {40,18458}, {517,30400}, {2098,30375}, {3641,6204}, {3679,30313}, {5605,13389}, {7962,30333}, {7991,30296}, {11518,30276}, {11519,30288}, {11522,30306}, {11523,30324}, {11529,30341}, {11531,30354}, {11533,30360}, {11534,30368}, {11535,30406}, {11899,30418}, {15829,30412}, {16189,30396}, {25415,30425}, {30323,30431}

X(30319) = {X(3243), X(3340)}-harmonic conjugate of X(30320)


X(30320) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS AND 2nd KENMOTU DIAGONALS

Barycentrics    a*(-2*(a-3*b-3*c)*S+(-a+b+c)*(a^2-4*(b+c)*a+3*b^2-2*b*c+3*c^2))*(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(11474). X(30320) is their endo-homothetic center only when ABC is acute.

X(30320) lies on these lines: {1,371}, {7,145}, {8,30381}, {40,18460}, {517,30401}, {2098,30376}, {3640,6203}, {3679,30314}, {5604,13388}, {7962,30334}, {7991,30297}, {11518,30277}, {11519,30289}, {11522,30307}, {11523,30325}, {11529,30342}, {11531,30355}, {11533,30361}, {11534,30369}, {11535,30407}, {11899,30419}, {15829,30413}, {16189,30397}, {25415,30426}, {30323,30432}

X(30320) = {X(3243), X(3340)}-harmonic conjugate of X(30319)


X(30321) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS AND INNER TRI-EQUILATERAL

Barycentrics    a*(2*sqrt(3)*(a-3*b-3*c)*S+(-a+b+c)*(a^2+3*b^2-2*b*c+3*c^2-4*(b+c)*a))*(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(11475). X(30321) is their endo-homothetic center only when ABC is acute.

X(30321) lies on these lines: {1,16}, {7,145}, {8,30382}, {40,18469}, {517,10649}, {2098,30377}, {3679,30316}, {7962,30338}, {7991,30300}, {11518,30280}, {11519,30292}, {11522,30309}, {11523,30327}, {11529,30344}, {11531,30356}, {11533,30364}, {11534,30372}, {11535,30409}, {11899,30421}, {15829,30414}, {16189,10655}, {25415,10651}, {30323,30433}

X(30321) = {X(3243), X(3340)}-harmonic conjugate of X(30322)


X(30322) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS AND OUTER TRI-EQUILATERAL

Barycentrics    a*(-2*sqrt(3)*(a-3*b-3*c)*S+(-a+b+c)*(a^2+3*b^2-2*b*c+3*c^2-4*(b+c)*a))*(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(11476). X(30322) is their endo-homothetic center only when ABC is acute.

X(30322) lies on these lines: {1,15}, {7,145}, {8,30383}, {40,18471}, {517,10650}, {2098,30378}, {3679,30317}, {7962,30339}, {7991,30301}, {11518,30281}, {11519,30293}, {11522,30310}, {11523,30328}, {11529,30345}, {11531,30357}, {11533,30365}, {11534,30373}, {11535,30410}, {11899,30422}, {15829,30415}, {16189,10656}, {25415,10652}, {30323,30434}

X(30322) = {X(3243), X(3340)}-harmonic conjugate of X(30321)


X(30323) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-EXCENTERS-REFLECTIONS AND TRINH

Barycentrics    a*(a^3-3*(b+c)*a^2-(b^2-8*b*c+c^2)*a+3*(b^2-c^2)*(b-c)) : :
X(30323) = 3*X(1)-2*X(56) = 5*X(1)-4*X(24928) = 3*X(46)-4*X(56) = X(46)-4*X(2098) = 5*X(46)-8*X(24928) = X(56)-3*X(2098) = 5*X(56)-6*X(24928) = 4*X(1329)-3*X(3679) = 5*X(1698)-4*X(8256) = 3*X(3241)-X(20076) = 2*X(4848)-3*X(10072) = 8*X(6691)-9*X(25055) = 4*X(7681)-5*X(11522)

The homothetic center of these triangles is X(24). X(30323) is their endo-homothetic center only when ABC is acute.

X(30323) lies on these lines: {1,3}, {8,5187}, {9,17444}, {10,6931}, {12,3656}, {63,22837}, {72,10912}, {78,2802}, {80,3632}, {90,1320}, {145,10572}, {498,13464}, {499,11362}, {516,30318}, {519,1479}, {550,12735}, {946,6968}, {952,12701}, {997,5330}, {998,1126}, {1000,3485}, {1145,25681}, {1317,18481}, {1329,3679}, {1387,24914}, {1389,7162}, {1404,1766}, {1405,17452}, {1478,4301}, {1537,12749}, {1572,7296}, {1698,8256}, {1720,10696}, {1737,12245}, {1770,3476}, {1837,5844}, {2800,10085}, {2829,7971}, {2841,13541}, {3085,5734}, {3086,4345}, {3241,4294}, {3243,5441}, {3244,10624}, {3419,13463}, {3544,7317}, {3577,5559}, {3583,5881}, {3586,3633}, {3623,4305}, {3625,3984}, {3654,5433}, {3655,15338}, {3811,3885}, {3877,5260}, {3880,5730}, {3884,19860}, {3893,3940}, {3895,22836}, {4299,28194}, {4302,5882}, {4304,11520}, {4311,28228}, {4333,28174}, {4338,18990}, {4668,11525}, {4674,11512}, {4816,11524}, {4848,10072}, {4853,5692}, {4857,5727}, {4861,12514}, {4867,6765}, {4919,17742}, {5252,22791}, {5261,12047}, {5288,12526}, {5289,10914}, {5603,10039}, {5665,13606}, {5690,11376}, {5904,12629}, {6261,10698}, {6361,21578}, {6691,25055}, {6796,10087}, {6833,15868}, {7171,11571}, {7294,26446}, {8543,15298}, {8666,10058}, {8679,15430}, {9578,18393}, {9589,10483}, {9897,12764}, {10051,10935}, {10526,10947}, {10573,12053}, {10944,12699}, {11519,30294}, {11526,30331}, {11533,30366}, {11534,30374}, {11535,30411}, {11899,30423}, {12648,21077}, {12737,24467}, {12953,28204}, {21271,24179}, {30319,30431}, {30320,30432}, {30321,30433}, {30322,30434}

X(30323) = midpoint of X(145) and X(11415)
X(30323) = reflection of X(i) in X(j) for these (i,j): (1, 2098), (8, 21616), (46, 1), (7991, 10310), (9897, 12764), (10573, 12053), (10680, 24680)
X(30323) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 7982, 25415), (1, 11531, 5903), (3340, 7982, 11280)


X(30324) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-EXTOUCH AND 1st KENMOTU DIAGONALS

Barycentrics    (2*(b+c)*S+(-a+b+c)^2*a)*(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(19355). X(30324) is their endo-homothetic center only when ABC is acute.

X(30324) lies on these lines: {1,1588}, {2,7}, {4,1123}, {5,8957}, {6,1659}, {12,14121}, {37,13390}, {65,7090}, {176,7586}, {281,13459}, {405,30385}, {442,30313}, {486,8953}, {950,30333}, {1490,30400}, {1750,30354}, {1864,30375}, {3069,13389}, {3085,6212}, {3487,30341}, {3586,30431}, {4000,8243}, {4199,30360}, {4295,6213}, {5728,30346}, {5927,30288}, {6351,13388}, {7580,30296}, {7593,30406}, {8079,30368}, {8080,30418}, {8226,30306}, {9612,30425}, {10910,18995}, {11523,30319}, {18397,18410}, {18446,18458}, {30326,30396}

X(30324) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 30276), (7, 30412, 6203), (9, 226, 30325)


X(30325) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-EXTOUCH AND 2nd KENMOTU DIAGONALS

Barycentrics    (-2*(b+c)*S+(-a+b+c)^2*a)*(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(19356). X(30325) is their endo-homothetic center only when ABC is acute.

X(30325) lies on these lines: {1,1587}, {2,7}, {4,1336}, {6,13390}, {12,7090}, {37,1659}, {65,14121}, {175,7585}, {281,13437}, {405,30386}, {442,30314}, {942,8957}, {950,30334}, {1490,30401}, {1750,30355}, {1864,30376}, {3068,13388}, {3085,6213}, {3586,30432}, {4199,30361}, {4295,6212}, {4419,8243}, {5728,30347}, {5927,30289}, {6352,13389}, {7580,30297}, {7593,30407}, {8079,30369}, {8080,30419}, {8226,30307}, {9612,30426}, {10911,18996}, {11523,30320}, {18397,18411}, {18446,18460}, {30326,30397}

X(30325) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 30277), (7, 30413, 6204), (9, 226, 30324)


X(30326) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-EXTOUCH AND SUBMEDIAL

Barycentrics    a*(a^5+(b+c)*a^4-6*(b^2+c^2)*a^3+2*(b+c)^3*a^2+5*(b^2-c^2)^2*a-(b^2-c^2)*(b-c)*(3*b+c)*(b+3*c)) : :

The homothetic center of these triangles is X(10601). X(30326) is their endo-homothetic center only when ABC is acute.

X(30326) lies on these lines: {1,1864}, {2,11407}, {4,3679}, {5,4654}, {9,165}, {40,3715}, {57,5779}, {72,4915}, {200,15064}, {210,7994}, {223,7069}, {226,5817}, {329,1699}, {355,9580}, {405,9851}, {442,30315}, {920,10045}, {936,6909}, {950,30337}, {971,7308}, {1006,1490}, {1697,9947}, {1698,6260}, {1728,3361}, {1737,3339}, {1754,3973}, {1836,2093}, {2801,10582}, {3058,5881}, {3305,5732}, {3487,30343}, {3586,9819}, {3646,12680}, {3678,12651}, {3829,8226}, {3832,3951}, {3929,19541}, {4199,30363}, {5268,9355}, {5691,12572}, {5720,6914}, {5728,30350}, {5735,17781}, {5785,18228}, {6282,18540}, {6667,11219}, {6846,10072}, {6907,19875}, {6911,7330}, {7593,30394}, {7701,10270}, {7996,10445}, {8001,11379}, {8079,30371}, {8080,30395}, {8232,30330}, {9814,12848}, {10863,24477}, {10864,25917}, {11523,16189}, {12526,19925}, {18397,18421}, {18446,30392}, {21153,27065}, {30324,30396}, {30325,30397}, {30327,10655}, {30328,10656}

X(30326) = reflection of X(10857) in X(7308)
X(30326) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 30304, 11407), (9, 1750, 165), (3062, 30393, 165)


X(30327) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-EXTOUCH AND INNER TRI-EQUILATERAL

Barycentrics    (2*sqrt(3)*(b+c)*S+(-a+b+c)^2*a)*(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(19363). X(30327) is their endo-homothetic center only when ABC is acute.

X(30327) lies on these lines: {1,5334}, {2,7}, {4,1833}, {388,5240}, {405,10647}, {442,30316}, {498,3179}, {950,30338}, {1276,3085}, {1277,4295}, {1490,10649}, {1750,30356}, {1864,30377}, {2306,5714}, {3340,5245}, {3485,5239}, {3487,30344}, {3586,30433}, {4199,30364}, {5246,9578}, {5728,30351}, {5927,30292}, {7580,30300}, {7593,30409}, {8079,30372}, {8080,30421}, {8226,30309}, {9612,10651}, {11523,30321}, {18397,18422}, {18446,18469}, {30326,10655}

X(30327) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 30280), (7, 30414, 1653), (9, 226, 30328)


X(30328) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd ANTI-EXTOUCH AND OUTER TRI-EQUILATERAL

Barycentrics    (-2*sqrt(3)*(b+c)*S+(-a+b+c)^2*a)*(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(19364). X(30328) is their endo-homothetic center only when ABC is acute.

X(30328) lies on these lines: {1,5335}, {2,7}, {4,1251}, {388,5239}, {405,10648}, {442,30317}, {950,30339}, {1276,4295}, {1277,3085}, {1490,10650}, {1750,30357}, {1864,30378}, {3340,5246}, {3485,5240}, {3586,30434}, {4199,30365}, {5245,9578}, {5728,30352}, {5927,30293}, {7580,30301}, {7593,30410}, {8079,30373}, {8080,30422}, {8226,30310}, {9612,10652}, {11523,30322}, {18397,18423}, {18446,18471}, {30326,10656}

X(30328) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 30281), (7, 30415, 1652), (9, 226, 30327)


X(30329) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-HONSBERGER AND EHRMANN-VERTEX

Barycentrics    a*((b+c)*a^4-2*(b^2+b*c+c^2)*a^3-(b+c)*b*c*a^2+2*(b^2-c^2)^2*a-(b^3+c^3)*(b-c)^2) : :
X(30329) = 3*X(1)-5*X(11025) = X(7)-3*X(5902) = 3*X(65)+X(14100) = 2*X(142)-3*X(5883) = X(3059)-3*X(3753) = 6*X(3833)-5*X(20195) = 2*X(4757)+X(5698) = 3*X(5686)-X(5904) = 3*X(5692)-5*X(18230) = X(5693)-3*X(5817) = X(5697)-3*X(8236) = 3*X(5728)-X(14100) = 3*X(5902)+X(18412) = 4*X(6666)-3*X(10176) = 3*X(7672)+5*X(11025) = X(7672)+2*X(20116) = 5*X(11025)-6*X(20116) = 3*X(11038)-5*X(18398) = 5*X(15016)-3*X(21151)

The homothetic center of these triangles is X(19130). X(30329) is their endo-homothetic center only when ABC is acute.

X(30329) lies on these lines: {1,1170}, {7,80}, {9,758}, {10,141}, {36,30284}, {40,12564}, {46,7675}, {65,516}, {72,12563}, {165,11020}, {200,3306}, {214,999}, {226,15064}, {354,3911}, {390,5903}, {484,7676}, {517,5572}, {519,15185}, {954,15556}, {971,5884}, {1001,3878}, {1156,11571}, {1159,2800}, {1737,21617}, {1837,13159}, {1858,10392}, {2093,4326}, {2099,15558}, {2550,3754}, {2802,11041}, {2807,29957}, {3059,3753}, {3085,5445}, {3243,3333}, {3339,5732}, {3487,3678}, {3555,6743}, {3833,20195}, {3868,5223}, {3919,15733}, {3946,22465}, {3956,8164}, {4295,5809}, {4312,10394}, {4343,4424}, {4757,5698}, {4860,5083}, {5045,10165}, {5173,11019}, {5493,12710}, {5686,5904}, {5690,15901}, {5691,12669}, {5692,18230}, {5693,5817}, {5696,20612}, {5697,8236}, {5708,11500}, {6666,10176}, {7678,18393}, {7679,18395}, {7680,10265}, {8232,18397}, {8238,30358}, {8387,18399}, {8388,18409}, {8389,18408}, {8732,30274}, {10090,14151}, {10164,11018}, {10398,12560}, {10578,15104}, {10865,30286}, {15006,28194}, {15016,21151}, {15570,24928}, {17706,24474}, {18421,30330}

X(30329) = midpoint of X(i) and X(j) for these {i,j}: {1, 7672}, {7, 18412}, {65, 5728}, {80, 12755}, {390, 5903}, {1156, 11571}, {3868, 5223}, {4312, 10394}, {5691, 12669}
X(30329) = reflection of X(i) in X(j) for these (i,j): (1, 20116), (2550, 3754), (3243, 3881), (3878, 1001), (5542, 942), (30331, 5572)
X(30329) = {X(5902), X(18412)}-harmonic conjugate of X(7)


X(30330) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-HONSBERGER AND SUBMEDIAL

Barycentrics    a*(a^4-2*(3*b^2-2*b*c+3*c^2)*a^2+8*(b^2-c^2)*(b-c)*a-(3*b^2+2*b*c+3*c^2)*(b-c)^2) : :
X(30330) = X(3062)+3*X(24645)

The homothetic center of these triangles is X(19137). X(30330) is their endo-homothetic center only when ABC is acute.

X(30330) lies on these lines: {1,6}, {3,15008}, {7,1699}, {57,2951}, {65,10384}, {144,10580}, {165,1445}, {269,21346}, {388,10392}, {390,6738}, {516,938}, {942,7992}, {971,3333}, {1125,5785}, {1156,18240}, {1479,4312}, {2093,15006}, {2310,4328}, {3059,8580}, {3174,8257}, {3303,9898}, {3306,25722}, {3361,5732}, {3600,9851}, {3829,6173}, {4321,10394}, {4907,5228}, {5045,5779}, {5437,15587}, {5542,14986}, {5691,5809}, {5817,21620}, {5833,10916}, {5850,21625}, {7672,11531}, {7675,7987}, {7677,30389}, {7678,30308}, {7679,30315}, {7988,21617}, {8056,9445}, {8236,30337}, {8238,30363}, {8255,20195}, {8387,30371}, {8388,30395}, {8389,30394}, {8545,11025}, {8732,11407}, {9819,16236}, {10389,15837}, {10582,11020}, {10865,30291}, {11038,30343}, {11526,16189}, {13405,18230}, {16112,17626}, {18421,30329}, {30284,30392}

X(30330) = reflection of X(i) in X(j) for these (i,j): (7, 15841), (15829, 1001)
X(30330) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3973, 9440), (1, 10398, 5223), (9, 5572, 1)


X(30331) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-HONSBERGER AND TRINH

Barycentrics    4*a^3-5*(b+c)*a^2+2*(b-c)^2*a-(b^2-c^2)*(b-c) : :
X(30331) = 3*X(1)-X(7) = 5*X(1)-X(4312) = X(1)-3*X(8236) = 5*X(1)-3*X(11038) = 5*X(1)+X(30332) = 11*X(1)-5*X(30340) = 4*X(1)-X(30424) = X(7)+3*X(390) = 5*X(7)-3*X(4312) = 2*X(7)-3*X(5542) = X(7)-9*X(8236) = 5*X(7)-9*X(11038) = 5*X(7)+3*X(30332) = 11*X(7)-15*X(30340) = 4*X(7)-3*X(30424) = 3*X(10)-4*X(6666) = 5*X(390)+X(4312) = 2*X(390)+X(5542) = X(390)+3*X(8236) = X(1000)+3*X(3488) = 3*X(1001)-2*X(6666)

The homothetic center of these triangles is X(5092). X(30331) is their endo-homothetic center only when ABC is acute.

X(30331) lies on these lines: {1,7}, {3,21625}, {8,25101}, {9,519}, {10,1001}, {35,7677}, {36,7676}, {40,6744}, {55,3911}, {57,10385}, {65,20116}, {80,2346}, {142,214}, {144,3241}, {145,5223}, {165,10580}, {226,3058}, {319,3883}, {354,4031}, {392,3059}, {496,3826}, {497,3817}, {515,6767}, {517,5572}, {518,3244}, {673,29571}, {758,15185}, {938,21153}, {942,5493}, {944,11372}, {946,5719}, {950,954}, {971,5882}, {997,3174}, {999,11495}, {1056,28164}, {1058,1125}, {1156,7972}, {1210,3746}, {1279,3755}, {1317,2801}, {1445,5119}, {1479,3947}, {1621,4847}, {1699,10578}, {1757,4924}, {1890,6198}, {2321,4702}, {2802,10177}, {3057,5728}, {3158,20103}, {3189,12447}, {3243,5698}, {3247,5819}, {3333,12512}, {3475,9580}, {3486,10384}, {3586,8232}, {3625,6541}, {3632,5686}, {3679,18230}, {3689,5316}, {3731,5838}, {3753,12732}, {3870,21060}, {3871,8582}, {3874,12710}, {3898,15733}, {3938,4656}, {3957,17484}, {4078,17765}, {4085,16593}, {4421,6692}, {4428,5745}, {4660,21255}, {4684,17361}, {4882,5129}, {5045,10386}, {5234,6764}, {5274,10171}, {5441,16133}, {5697,7672}, {5726,19925}, {5759,7982}, {5762,24680}, {5766,10398}, {5805,13464}, {5817,5881}, {5851,12735}, {5902,11025}, {5903,7673}, {5918,10569}, {6068,25416}, {6690,24386}, {6765,18250}, {7671,18412}, {7674,9623}, {7678,7951}, {7679,7741}, {8238,30366}, {8255,20330}, {8387,30374}, {8388,30423}, {8389,30411}, {8543,10572}, {8581,10543}, {8715,9843}, {8732,30282}, {9577,29815}, {9778,10980}, {9797,11106}, {9819,16236}, {10175,18527}, {10179,15587}, {10390,18490}, {10392,10950}, {10865,30294}, {11362,12433}, {11373,15808}, {11526,30323}, {11529,28228}, {12053,15950}, {12702,17706}, {13159,16137}, {13411,23708}, {15171,21620}, {15174,15570}, {15933,18421}, {15934,28194}, {16173,20119}, {17023,20533}, {17601,24216}, {17603,18240}, {17715,17725}, {18530,26446}, {21454,30350}, {21617,30384}, {24175,29820}

X(30331) = midpoint of X(i) and X(j) for these {i,j}: {1, 390}, {80, 12730}, {145, 5223}, {944, 11372}, {1156, 7972}, {3057, 5728}, {3243, 5698}, {3632, 12630}, {4294, 12560}, {4312, 30332}, {5441, 16133}, {5697, 7672}, {5759, 7982}, {5903, 7673}, {6068, 25416}, {10624, 12573}
X(30331) = reflection of X(i) in X(j) for these (i,j): (10, 1001), (65, 20116), (2550, 1125), (3625, 24393), (5542, 1), (5805, 13464), (13159, 16137), (14563, 15935), (24393, 15254), (30329, 5572), (30424, 5542)
X(30331) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 20, 12577), (1, 10624, 3671), (11038, 30332, 4312)


X(30332) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH AND 2nd EHRMANN

Barycentrics    7*a^3-5*(b+c)*a^2+(b-c)^2*a-3*(b^2-c^2)*(b-c) : :
X(30332) = 4*X(1)-3*X(7) = 2*X(1)-3*X(390) = 5*X(1)-3*X(4312) = 7*X(1)-6*X(5542) = 8*X(1)-9*X(8236) = 10*X(1)-9*X(11038) = 5*X(1)-6*X(30331) = 6*X(1)-5*X(30340) = 3*X(1)-2*X(30424) = 5*X(7)-4*X(4312) = 7*X(7)-8*X(5542) = 2*X(7)-3*X(8236) = 5*X(7)-6*X(11038) = 5*X(7)-8*X(30331) = 9*X(7)-10*X(30340) = 9*X(7)-8*X(30424) = 3*X(8)-4*X(5220) = 2*X(8)-3*X(6172) = 5*X(390)-2*X(4312) = 7*X(390)-4*X(5542) = 4*X(390)-3*X(8236) = 2*X(5220)-3*X(5698) = 8*X(5220)-9*X(6172) = 4*X(5698)-3*X(6172)

The homothetic center of these triangles is X(11477). X(30332) is their endo-homothetic center only when ABC is acute.

X(30332) lies on these lines: {1,7}, {2,9580}, {4,5766}, {8,190}, {9,3617}, {11,30312}, {12,10248}, {44,5838}, {45,5819}, {55,5226}, {56,30295}, {65,7671}, {100,5328}, {144,3621}, {145,527}, {149,5744}, {165,5274}, {329,2900}, {376,5126}, {404,1001}, {497,1155}, {517,10394}, {518,3644}, {938,6361}, {950,12848}, {954,5714}, {1000,28160}, {1004,1621}, {1056,28146}, {1159,3488}, {1445,5128}, {1479,5445}, {1697,3146}, {1699,5281}, {1788,9670}, {1836,10385}, {2098,14151}, {2346,5556}, {2478,2550}, {2801,5697}, {2898,3599}, {3057,15726}, {3058,3474}, {3149,15911}, {3241,28534}, {3243,20059}, {3245,18391}, {3434,5273}, {3487,10386}, {3522,12053}, {3523,9614}, {3524,7743}, {3528,11373}, {3529,9957}, {3579,5704}, {3601,30275}, {3614,7679}, {3616,5880}, {3622,6173}, {3625,5223}, {3626,5686}, {3826,7173}, {3869,12536}, {3877,5784}, {3878,5696}, {3912,4779}, {3923,5772}, {4440,15590}, {5046,6594}, {5059,7320}, {5204,7677}, {5218,9779}, {5265,12512}, {5528,20066}, {5657,9668}, {5703,12699}, {5729,5759}, {5762,8148}, {5817,18357}, {5825,21168}, {5851,12730}, {6006,21105}, {6767,28178}, {7672,14100}, {7962,30318}, {7987,18220}, {8240,30359}, {8241,30367}, {8242,30405}, {9578,17578}, {9814,30337}, {9819,28164}, {10005,25728}, {10866,30287}, {10896,19877}, {11041,28212}, {11220,17642}, {11372,29007}, {11529,28232}, {11924,30404}, {12527,12632}, {15933,28198}, {15934,28216}, {16020,24715}, {17314,28566}, {17538,24928}, {17781,20015}, {17784,18228}, {20533,29579}

X(30332) = reflection of X(i) in X(j) for these (i,j): (7, 390), (8, 5698), (4312, 30331), (5696, 3878), (7672, 14100), (20059, 3243)
X(30332) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (20, 9785, 4308), (20, 10624, 9785), (11200, 30334, 17805)
X(30332) = outer-Garcia-to-inner-Garcia similarity image of X(7)


X(30333) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH AND 1st KENMOTU DIAGONALS

Barycentrics    4*S*a+(-a+b+c)*(3*a^2+(b-c)^2) : :

The homothetic center of these triangles is X(1151). X(30333) is their endo-homothetic center only when ABC is acute.

X(30333) lies on these lines: {1,7}, {8,7090}, {11,30313}, {12,30306}, {55,16440}, {56,30296}, {65,30346}, {144,3641}, {145,3640}, {950,30324}, {1659,9812}, {1697,6203}, {2066,30413}, {3057,6405}, {3083,17784}, {3601,30276}, {4419,5605}, {5274,5393}, {5281,5405}, {5697,18410}, {7962,30319}, {8240,30360}, {8241,30368}, {8242,30418}, {9778,13389}, {10578,13390}, {10580,13388}, {10866,30288}, {11370,17014}, {11924,30406}, {12053,30380}, {13386,14004}, {14942,30335}, {30337,30396}

X(30333) = reflection of X(i) in X(j) for these (i,j): (8, 7090), (176, 1), (8986, 4297)
X(30333) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4344, 9785, 30334), (8236, 17802, 1), (17802, 30332, 11200)


X(30334) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH AND 2nd KENMOTU DIAGONALS

Barycentrics    -4*S*a+(-a+b+c)*(3*a^2+(b-c)^2) : :

The homothetic center of these triangles is X(1152). X(30334) is their endo-homothetic center only when ABC is acute.

X(30334) lies on these lines: {1,7}, {8,14121}, {11,30314}, {12,30307}, {55,16441}, {56,30297}, {65,30347}, {144,3640}, {145,3641}, {950,30325}, {1659,10578}, {1697,6204}, {3057,6283}, {3084,17784}, {3601,30277}, {4419,5604}, {5274,5405}, {5281,5393}, {5414,30412}, {5697,18411}, {7962,30320}, {8240,30361}, {8241,30369}, {8242,30419}, {9778,13388}, {9812,13390}, {10580,13389}, {10866,30289}, {11371,17014}, {11924,30407}, {12053,30381}, {13387,14004}, {14942,30336}, {30337,30397}

X(30334) = reflection of X(i) in X(j) for these (i,j): (8, 14121), (175, 1)
X(30334) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 390, 30333), (4344, 9785, 30333), (17805, 30332, 11200)


X(30335) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH AND LUCAS ANTIPODAL TANGENTIAL

Barycentrics    a^2*(-a+b+c)*(S+b*(a-b+c))*(S+c*(a+b-c)) : :
Trilinears    1/(1 + sec A/2 cos B/2 cos C/2) : :

The homothetic center of these triangles is X(13021). X(30335) is their endo-homothetic center only when ABC is acute.

X(30335) lies on the cubic K632, curve Q104 and these lines: {1,16213}, {8,30416}, {33,16232}, {55,1152}, {65,30348}, {175,21453}, {200,15892}, {220,2066}, {221,2293}, {371,4845}, {1697,30429}, {1806,2328}, {3601,30278}, {4313,30302}, {14942,30333}

X(30335) = isogonal conjugate of X(176)
X(30335) = X(56)-cross conjugate of X(30336)
X(30335) = X(3207)-vertex conjugate of X(30336)
X(30335) = anticomplement of the complementary conjugate of X(7090)
X(30335) = {X(2293), X(3303)}-harmonic conjugate of X(30336)


X(30336) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH AND LUCAS(-1) ANTIPODAL TANGENTIAL

Barycentrics    a^2*(-a+b+c)*(-S+b*(a-b+c))*(-S+c*(a+b-c)) : :
Trilinears    1/(-1 + sec A/2 cos B/2 cos C/2) : :

The homothetic center of these triangles is X(13022). X(30336) is their endo-homothetic center only when ABC is acute.

X(30336) lies on the cubic K632, curve Q104 and these lines: {1,16214}, {8,30417}, {33,2362}, {55,1151}, {65,30349}, {176,21453}, {200,15891}, {220,5414}, {221,2293}, {372,4845}, {1697,30430}, {1805,2328}, {3601,30279}, {4313,30303}, {14942,30334}

X(30336) = isogonal conjugate of X(175)
X(30336) = anticomplement of the complementary conjugate of X(14121)
X(30336) = {X(2293), X(3303)}-harmonic conjugate of X(30335)
X(30336) = X(56)-cross conjugate of X(30335)
X(30336) = X(3207)-vertex conjugate of X(30335)


X(30337) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH AND SUBMEDIAL

Barycentrics    a*(a^3+3*(b+c)*a^2-(b^2+22*b*c+c^2)*a-3*(b^2-c^2)*(b-c)) : :

The homothetic center of these triangles is X(11479). X(30337) is their endo-homothetic center only when ABC is acute.

X(30337) lies on these lines: {1,3}, {4,11379}, {8,30393}, {9,11519}, {10,12541}, {11,30315}, {12,30308}, {145,5223}, {200,3890}, {390,3062}, {392,4882}, {496,19875}, {936,3898}, {950,30326}, {1000,3632}, {1001,3680}, {1056,9589}, {1058,3679}, {1699,5261}, {1706,10179}, {2136,8580}, {2347,3731}, {3241,12526}, {3488,3633}, {3884,6765}, {3893,7308}, {3895,8583}, {4313,30304}, {4326,30318}, {4342,11522}, {4355,28194}, {4668,5722}, {4677,15170}, {4853,5260}, {4900,7160}, {5234,12629}, {5290,30305}, {5436,10912}, {5531,15558}, {5558,11034}, {5691,12575}, {5726,9614}, {5785,12536}, {5881,15172}, {5882,7992}, {5918,12128}, {7988,10588}, {7989,10591}, {8078,10968}, {8167,11530}, {8236,30330}, {8240,30363}, {8241,30371}, {8242,30395}, {9588,14986}, {9814,30332}, {10582,14923}, {10866,30291}, {11037,28228}, {11108,11525}, {11924,30394}, {12447,12632}, {12577,20070}, {12640,26105}, {12735,12767}, {13463,25525}, {16673,17451}, {30333,30396}, {30334,30397}, {30338,10655}, {30339,10656}

X(30337) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3339, 30343), (3339, 30343, 10980), (6767, 7982, 1)


X(30338) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH AND INNER TRI-EQUILATERAL

Barycentrics    4*sqrt(3)*a*S+(-a+b+c)*(3*a^2+(b-c)^2) : :

The homothetic center of these triangles is X(11480). X(30338) is their endo-homothetic center only when ABC is acute.

X(30338) lies on these lines: {1,7}, {8,5240}, {11,30316}, {12,30309}, {55,10647}, {56,30300}, {65,30351}, {559,9778}, {950,30327}, {1082,10580}, {1653,1697}, {3057,30377}, {3601,30280}, {5697,18422}, {7962,30321}, {8240,30364}, {8241,30372}, {8242,30421}, {10866,30292}, {11924,30409}, {12053,30382}, {30337,10655}

X(30338) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 390, 30339), (1, 3639, 11038), (4344, 4345, 30339)


X(30339) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-HUTSON INTOUCH AND OUTER TRI-EQUILATERAL

Barycentrics    -4*sqrt(3)*a*S+(-a+b+c)*(3*a^2+(b-c)^2) : :

The homothetic center of these triangles is X(11481). X(30339) is their endo-homothetic center only when ABC is acute.

X(30339) lies on these lines: {1,7}, {8,1251}, {11,30317}, {12,30310}, {55,10648}, {56,30301}, {65,30352}, {559,10580}, {950,30328}, {1082,9778}, {1652,1697}, {3057,30378}, {3601,30281}, {5697,18423}, {7962,30322}, {8240,30365}, {8241,30373}, {8242,30422}, {10866,30293}, {11924,30410}, {12053,30383}, {30337,10656}

X(30339) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 390, 30338), (1, 3638, 11038), (4344, 4345, 30338)


X(30340) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES AND 2nd EHRMANN

Barycentrics    a^3+7*(b+c)*a^2-5*(b-c)^2*a-3*(b^2-c^2)*(b-c) : :
X(30340) = 2*X(1)+3*X(7) = 8*X(1)-3*X(390) = 7*X(1)+3*X(4312) = X(1)-6*X(5542) = 14*X(1)-9*X(8236) = 4*X(1)-9*X(11038) = 11*X(1)-6*X(30331) = 6*X(1)-X(30332) = 3*X(1)+2*X(30424) = 4*X(7)+X(390) = 7*X(7)-2*X(4312) = X(7)+4*X(5542) = 7*X(7)+3*X(8236) = 2*X(7)+3*X(11038) = 11*X(7)+4*X(30331) = 9*X(7)+X(30332) = 9*X(7)-4*X(30424) = 7*X(390)+8*X(4312) = X(390)-16*X(5542) = 7*X(390)-12*X(8236)

The homothetic center of these triangles is X(11482). X(30340) is their endo-homothetic center only when ABC is acute.

X(30340) lies on these lines: {1,7}, {2,3715}, {8,6173}, {9,5550}, {142,5686}, {144,15254}, {145,5880}, {149,25558}, {354,5274}, {388,18221}, {495,30312}, {496,30311}, {518,3617}, {527,3616}, {528,3623}, {553,10578}, {938,18492}, {942,5261}, {971,11025}, {999,8543}, {1001,20059}, {1056,1159}, {1125,6172}, {1155,3475}, {1156,3296}, {2346,5551}, {2550,3621}, {2801,3091}, {3295,30295}, {3333,8545}, {3337,10303}, {3487,5265}, {3488,28168}, {3523,5557}, {3579,21151}, {3622,5698}, {3634,5223}, {3797,29583}, {3873,5784}, {3874,4208}, {3881,5696}, {3889,15733}, {3982,9812}, {4114,10389}, {4461,4966}, {4869,24349}, {5045,7671}, {5226,10980}, {5232,24325}, {5284,20214}, {5556,10390}, {5558,12053}, {5704,21617}, {5708,5771}, {5714,5728}, {5729,6832}, {5759,13624}, {5772,21255}, {5819,16666}, {5850,18230}, {7988,11034}, {8092,30404}, {8351,30405}, {9814,30343}, {10861,15185}, {11019,30308}, {11035,30287}, {11529,30318}, {13743,16133}, {15726,17609}, {15934,28186}, {17784,26842}, {21620,30379}

X(30340) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 30424, 30332), (7, 8236, 4312), (7, 30332, 30424)


X(30341) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES AND 1st KENMOTU DIAGONALS

Barycentrics    3*(b+c)*a^2+2*S*a-2*(b-c)^2*a-(b^2-c^2)*(b-c) : :

The homothetic center of these triangles is X(3311). X(30341) is their endo-homothetic center only when ABC is acute.

X(30341) lies on these lines: {1,7}, {142,3641}, {354,1659}, {495,30313}, {496,30306}, {942,30276}, {1125,30412}, {3008,5589}, {3295,30296}, {3296,7133}, {3333,6203}, {3475,13389}, {3487,30324}, {4667,11370}, {4675,5605}, {5045,30346}, {5393,10980}, {8092,30406}, {11035,30288}, {11043,30360}, {17609,30375}, {18398,18410}, {21620,30380}, {30343,30396}

X(30341) = midpoint of X(1) and X(1373)
X(30341) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 30333, 30425), (390, 21169, 30426), (4310, 11037, 30342)


X(30342) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES AND 2nd KENMOTU DIAGONALS

Barycentrics    3*(b+c)*a^2-2*S*a-2*(b-c)^2*a-(b^2-c^2)*(b-c) : :

The homothetic center of these triangles is X(3312). X(30342) is their endo-homothetic center only when ABC is acute.

X(30342) lies on these lines: {1,7}, {142,3640}, {354,13390}, {495,30314}, {496,30307}, {942,30277}, {999,30386}, {1125,30413}, {3008,5588}, {3295,30297}, {3333,6204}, {3475,13388}, {4667,11371}, {4675,5604}, {5045,30347}, {5405,10980}, {8092,30407}, {8351,30419}, {11035,30289}, {11043,30361}, {11044,30369}, {11529,30320}, {17609,30376}, {18398,18411}, {21620,30381}, {30343,30397}

X(30342) = midpoint of X(1) and X(1374)
X(30342) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 5542, 30341), (7, 30334, 30426), (4310, 11037, 30341)


X(30343) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES AND SUBMEDIAL

Barycentrics    a*(a^3+3*(b+c)*a^2-(b^2-26*b*c+c^2)*a-3*(b^2-c^2)*(b-c)) : :

The homothetic center of these triangles is X(11484). X(30343) is their endo-homothetic center only when ABC is acute.

X(30343) lies on these lines: {1,3}, {496,30308}, {738,7274}, {936,3892}, {1058,4355}, {1125,30393}, {1475,3731}, {1699,9851}, {3062,5542}, {3487,30326}, {3616,5223}, {3624,4866}, {3633,17706}, {3753,12127}, {3812,11519}, {3889,8583}, {4342,11034}, {4353,7996}, {4915,5439}, {5260,10582}, {5261,7989}, {5290,10591}, {5586,30305}, {5691,10580}, {7190,17106}, {7988,21620}, {7992,13464}, {7993,18240}, {7997,11376}, {8092,30394}, {9814,30340}, {10085,24644}, {10569,15071}, {11025,30318}, {11035,30291}, {11036,30304}, {11043,30363}, {11044,30371}, {16469,28011}, {30341,30396}, {30342,30397}, {30344,10655}, {30345,10656}

X(30343) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 999, 30389), (1, 3339, 30337), (3339, 30337, 7991)


X(30344) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES AND INNER TRI-EQUILATERAL

Barycentrics    2*sqrt(3)*S*a+3*(b+c)*a^2-2*(b-c)^2*a-(b^2-c^2)*(b-c) : :

The homothetic center of these triangles is X(11485). X(30344) is their endo-homothetic center only when ABC is acute.

X(30344) lies on these lines: {1,7}, {495,30316}, {496,30309}, {497,554}, {559,3475}, {942,30280}, {999,10647}, {1125,30414}, {1653,3333}, {3295,30300}, {3487,30327}, {5045,30351}, {8092,30409}, {11035,30292}, {11043,30364}, {11044,30372}, {11529,30321}, {17609,30377}, {18398,18422}, {21620,30382}, {30343,10655}

X(30344) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 5542, 30345), (1, 10652, 390), (7, 30338, 10651)


X(30345) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INCIRCLE-CIRCLES AND OUTER TRI-EQUILATERAL

Barycentrics    -2*sqrt(3)*S*a+3*(b+c)*a^2-2*(b-c)^2*a-(b^2-c^2)*(b-c) : :

The homothetic center of these triangles is X(11486). X(30345) is their endo-homothetic center only when ABC is acute.

X(30345) lies on these lines: {1,7}, {496,30310}, {497,1081}, {942,30281}, {999,10648}, {1082,3475}, {1125,30415}, {1251,3296}, {1652,3333}, {3295,30301}, {5045,30352}, {8092,30410}, {8351,30422}, {11035,30293}, {11043,30365}, {11044,30373}, {11529,30322}, {17609,30378}, {18398,18423}, {21620,30383}, {30343,10656}

X(30345) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 5542, 30344), (1, 10651, 390), (1, 10652, 30339)


X(30346) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE AND 1st KENMOTU DIAGONALS

Barycentrics    a*((b+c)*a^3-3*(b^2+c^2)*a^2+3*(b^2-c^2)*(b-c)*a+(-2*(b+c)*a+2*b^2-4*b*c+2*c^2)*S-(b^2+c^2)*(b-c)^2) : :

The homothetic center of these triangles is X(3068). X(30346) is their endo-homothetic center only when ABC is acute.

X(30346) lies on these lines: {1,372}, {7,354}, {57,30296}, {65,30333}, {226,30306}, {518,30412}, {999,18458}, {1210,30313}, {1827,13390}, {3333,30400}, {4319,13389}, {5045,30341}, {5902,30431}, {8083,30406}, {10980,30354}, {11018,30276}, {11019,30288}, {11031,30360}, {11032,30368}, {11033,30418}, {18398,30425}, {30350,30396}

X(30346) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (354, 5572, 30347), (354, 30375, 7)


X(30347) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE AND 2nd KENMOTU DIAGONALS

Barycentrics    a*((b+c)*a^3-3*(b^2+c^2)*a^2+3*(b^2-c^2)*(b-c)*a-(-2*(b+c)*a+2*b^2-4*b*c+2*c^2)*S-(b^2+c^2)*(b-c)^2) : :

The homothetic center of these triangles is X(3069). X(30347) is their endo-homothetic center only when ABC is acute.

X(30347) lies on these lines: {1,371}, {7,354}, {57,30297}, {65,30334}, {226,30307}, {518,30413}, {999,18460}, {1210,30314}, {1659,1827}, {3333,30401}, {4319,13388}, {5045,30342}, {5728,30325}, {5902,30432}, {8083,30407}, {10980,30355}, {11018,30277}, {11019,30289}, {11031,30361}, {11032,30369}, {11033,30419}, {18398,30426}, {30350,30397}

X(30347) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (354, 5572, 30346), (354, 30376, 7)


X(30348) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE AND LUCAS ANTIPODAL TANGENTIAL

Barycentrics
a*((b+c)*a^12-(3*b^2+2*b*c+3*c^2)*a^11-(b+c)*(8*b^2+21*b*c+8*c^2)*a^10+(23*b^2-30*b*c+23*c^2)*(b+c)^2*a^9+(b+c)*(13*b^4+13*c^4+(113*b^2+36*b*c+113*c^2)*b*c)*a^8-2*(b+c)*(101*b^4+101*c^4+2*(21*b^2+17*b*c+21*c^2)*b*c)*a^6*b*c-2*(27*b^6+27*c^6+(34*b^4+34*c^4-(59*b^2+68*b*c+59*c^2)*b*c)*b*c)*a^7+2*(27*b^6+27*c^6+(2*b^4+2*c^4-3*(49*b^2-36*b*c+49*c^2)*b*c)*b*c)*(b+c)^2*a^5-(b^2-c^2)*(b-c)*(13*b^6+13*c^6-(128*b^4+128*c^4+(361*b^2+360*b*c+361*c^2)*b*c)*b*c)*a^4-(b^2-c^2)^2*(23*b^6+23*c^6+(74*b^4+74*c^4-(135*b^2+116*b*c+135*c^2)*b*c)*b*c)*a^3+(b^2-c^2)^3*(b-c)*(8*b^4+8*c^4-(33*b^2+94*b*c+33*c^2)*b*c)*a^2-(a+b+c)*((b+c)*a^9+7*(b+c)^2*a^8-28*(b+c)*(b^2+c^2)*a^7+4*(3*b^4+3*c^4-(17*b^2+20*b*c+17*c^2)*b*c)*a^6+2*(b+c)*(23*b^4+23*c^4+10*(4*b^2-3*b*c+4*c^2)*b*c)*a^5-2*(b^2+c^2)*(23*b^4+23*c^4-2*(14*b^2+59*b*c+14*c^2)*b*c)*a^4-4*(b^2-c^2)*(b-c)*(3*b^4+3*c^4+2*(19*b^2+17*b*c+19*c^2)*b*c)*a^3+4*(b^2-c^2)^2*(7*b^4+7*c^4+(b^2-24*b*c+c^2)*b*c)*a^2-(b^2-c^2)^3*(b-c)*(7*b^2-34*b*c+7*c^2)*a-(b^2-c^2)^2*(b-c)^4*(b^2+10*b*c+c^2))*S+(b^2-c^2)^4*(b-c)^2*(3*b^2+22*b*c+3*c^2)*a+(b^2-c^2)^3*(b-c)^3*(-b^4-c^4+(b^2+8*b*c+c^2)*b*c)) : :

The homothetic center of these triangles is X(19420). X(30348) is their endo-homothetic center only when ABC is acute.

X(30348) lies on these lines: {1,30387}, {65,30335}, {354,4328}, {518,30416}, {11018,30278}, {11020,30302}, {16213,17609}


X(30349) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE AND LUCAS(-1) ANTIPODAL TANGENTIAL

Barycentrics
a*((b+c)*a^12-(3*b^2+2*b*c+3*c^2)*a^11-(b+c)*(8*b^2+21*b*c+8*c^2)*a^10+(23*b^2-30*b*c+23*c^2)*(b+c)^2*a^9+(b+c)*(13*b^4+13*c^4+(113*b^2+36*b*c+113*c^2)*b*c)*a^8-2*(b+c)*(101*b^4+101*c^4+2*(21*b^2+17*b*c+21*c^2)*b*c)*a^6*b*c-2*(27*b^6+27*c^6+(34*b^4+34*c^4-(59*b^2+68*b*c+59*c^2)*b*c)*b*c)*a^7+2*(27*b^6+27*c^6+(2*b^4+2*c^4-3*(49*b^2-36*b*c+49*c^2)*b*c)*b*c)*(b+c)^2*a^5-(b^2-c^2)*(b-c)*(13*b^6+13*c^6-(128*b^4+128*c^4+(361*b^2+360*b*c+361*c^2)*b*c)*b*c)*a^4-(b^2-c^2)^2*(23*b^6+23*c^6+(74*b^4+74*c^4-(135*b^2+116*b*c+135*c^2)*b*c)*b*c)*a^3+(b^2-c^2)^3*(b-c)*(8*b^4+8*c^4-(33*b^2+94*b*c+33*c^2)*b*c)*a^2+(a+b+c)*((b+c)*a^9+7*(b+c)^2*a^8-28*(b+c)*(b^2+c^2)*a^7+4*(3*b^4+3*c^4-(17*b^2+20*b*c+17*c^2)*b*c)*a^6+2*(b+c)*(23*b^4+23*c^4+10*(4*b^2-3*b*c+4*c^2)*b*c)*a^5-2*(b^2+c^2)*(23*b^4+23*c^4-2*(14*b^2+59*b*c+14*c^2)*b*c)*a^4-4*(b^2-c^2)*(b-c)*(3*b^4+3*c^4+2*(19*b^2+17*b*c+19*c^2)*b*c)*a^3+4*(b^2-c^2)^2*(7*b^4+7*c^4+(b^2-24*b*c+c^2)*b*c)*a^2-(b^2-c^2)^3*(b-c)*(7*b^2-34*b*c+7*c^2)*a-(b^2-c^2)^2*(b-c)^4*(b^2+10*b*c+c^2))*S+(b^2-c^2)^4*(b-c)^2*(3*b^2+22*b*c+3*c^2)*a+(b^2-c^2)^3*(b-c)^3*(-b^4-c^4+(b^2+8*b*c+c^2)*b*c)) : :

The homothetic center of these triangles is X(19421). X(30349) is their endo-homothetic center only when ABC is acute.

X(30349) lies on these lines: {1,30388}, {65,30336}, {354,4328}, {518,30417}, {11018,30279}, {11020,30303}, {16214,17609}


X(30350) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE AND SUBMEDIAL

Barycentrics    a*(a^2-6*(b+c)*a+5*(b-c)^2) : :

The homothetic center of these triangles is X(7392). X(30350) is their endo-homothetic center only when ABC is acute.

X(30350) lies on these lines: {1,3}, {105,28156}, {200,9342}, {390,11034}, {497,3982}, {518,30393}, {551,5273}, {672,16673}, {1002,26102}, {1210,30315}, {1282,3315}, {1449,4906}, {1699,5542}, {1709,24645}, {1743,29820}, {2124,11029}, {3062,5572}, {3158,15570}, {3219,4666}, {3243,3742}, {3296,4355}, {3305,3873}, {3632,17706}, {3892,9623}, {3945,10520}, {4031,10385}, {4114,4312}, {4315,15933}, {4512,29817}, {4882,5439}, {4900,14563}, {5226,7988}, {5531,14151}, {5543,9533}, {5558,12577}, {5586,10624}, {5691,6744}, {5728,30326}, {6602,9327}, {7671,9814}, {7989,21620}, {7992,12005}, {8083,8090}, {8423,11033}, {8545,11025}, {9949,12563}, {11020,24644}, {11031,30363}, {11032,30371}, {11036,11522}, {11379,12688}, {17022,17450}, {21454,30331}, {24392,25557}, {30346,30396}, {30347,30397}, {30351,10655}, {30352,10656}

X(30350) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 65, 30337), (1, 5902, 9819), (354, 5173, 18398)


X(30351) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE AND INNER TRI-EQUILATERAL

Barycentrics    a*(-2*sqrt(3)*((b+c)*a-(b-c)^2)*S+(b+c)*a^3-3*(b^2+c^2)*a^2+3*(b^2-c^2)*(b-c)*a-(b^2+c^2)*(b-c)^2) : :

The homothetic center of these triangles is X(11488). X(30351) is their endo-homothetic center only when ABC is acute.

X(30351) lies on these lines: {1,16}, {7,354}, {57,30300}, {65,30338}, {226,30309}, {518,30414}, {559,4336}, {999,18469}, {1210,30316}, {5045,30344}, {5902,30433}, {10980,30356}, {11018,30280}, {11019,30292}, {11031,30364}, {11032,30372}, {18398,10651}, {30350,10655}

X(30351) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (354, 5572, 30352), (354, 30377, 7)


X(30352) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-INVERSE-IN-INCIRCLE AND OUTER TRI-EQUILATERAL

Barycentrics    a*(2*sqrt(3)*((b+c)*a-(b-c)^2)*S+(b+c)*a^3-3*(b^2+c^2)*a^2+3*(b^2-c^2)*(b-c)*a-(b^2+c^2)*(b-c)^2) : :

The homothetic center of these triangles is X(11489). X(30352) is their endo-homothetic center only when ABC is acute.

X(30352) lies on these lines: {1,15}, {7,354}, {57,30301}, {65,30339}, {226,30310}, {518,30415}, {999,18471}, {1082,4336}, {1210,30317}, {3333,10650}, {5045,30345}, {5728,30328}, {5902,30434}, {8083,30410}, {10980,30357}, {11018,30281}, {11019,30293}, {11031,30365}, {11032,30373}, {11033,30422}, {18398,10652}, {30350,10656}

X(30352) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (354, 5572, 30351), (354, 30378, 7)


X(30353) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 6th ANTI-MIXTILINEAR AND 2nd EHRMANN

Barycentrics    a*(a^4-4*(b+c)*a^3+2*(3*b^2-4*b*c+3*c^2)*a^2-4*(b^2-c^2)*(b-c)*a+(b^2+10*b*c+c^2)*(b-c)^2) : :
X(30353) = 2*X(1)-3*X(4321)

The homothetic center of these triangles is X(11511). X(30353) is their endo-homothetic center only when ABC is acute.

X(30353) lies on these lines: {1,7}, {9,1155}, {57,15726}, {165,8545}, {200,527}, {535,9623}, {1156,1445}, {1418,4907}, {1699,30379}, {1750,12848}, {1836,6173}, {2093,2801}, {3174,5528}, {3339,10394}, {3935,20059}, {4654,8255}, {4853,7354}, {4860,14100}, {5128,5220}, {5698,8583}, {5766,12512}, {5784,12526}, {5880,9579}, {6172,8580}, {7671,10980}, {7677,24644}, {7987,8543}, {7988,30311}, {7989,30312}, {8089,30367}, {8090,30405}, {8245,30359}, {8423,30404}, {10857,30275}, {11224,14151}, {11372,22753}, {11531,30318}, {11662,17857}

X(30353) = reflection of X(i) in X(j) for these (i,j): (390, 4315), (11372, 22753), (30305, 5542)
X(30353) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 2951, 4326), (4312, 5732, 12560)


X(30354) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 6th ANTI-MIXTILINEAR AND 1st KENMOTU DIAGONALS

Barycentrics    a*(a^4-4*(b+c)*a^3+2*(3*b^2-2*b*c+3*c^2)*a^2-4*(b^2-c^2)*(b-c)*a-2*(a^2-2*(b+c)*a+(b-c)^2)*S+(b^2+6*b*c+c^2)*(b-c)^2) : :

The homothetic center of these triangles is X(11513). X(30354) is their endo-homothetic center only when ABC is acute.

X(30354) lies on these lines: {1,7}, {9,9616}, {57,30375}, {165,6203}, {1699,30380}, {1750,30324}, {2093,18410}, {3062,7133}, {7988,30306}, {7989,30313}, {8090,30418}, {8245,30360}, {8580,30412}, {10980,30346}, {11531,30319}

X(30354) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 2951, 30355), (1742, 4326, 30355), (4335, 5732, 30355)


X(30355) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 6th ANTI-MIXTILINEAR AND 2nd KENMOTU DIAGONALS

Barycentrics    a*(a^4-4*(b+c)*a^3+2*(3*b^2-2*b*c+3*c^2)*a^2-4*(b^2-c^2)*(b-c)*a+2*(a^2-2*(b+c)*a+(b-c)^2)*S+(b^2+6*b*c+c^2)*(b-c)^2) : :

The homothetic center of these triangles is X(11514). X(30355) is their endo-homothetic center only when ABC is acute.

X(30355) lies on these lines: {1,7}, {57,30376}, {165,6204}, {1699,30381}, {1750,30325}, {2093,18411}, {3062,30289}, {7987,30386}, {7988,30307}, {7989,30314}, {8089,30369}, {8090,30419}, {8245,30361}, {8423,30407}, {8580,30413}, {10857,30277}, {10980,30347}, {11531,30320}, {13389,14100}

X(30355) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 2951, 30354), (1742, 4326, 30354), (4335, 5732, 30354)


X(30356) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 6th ANTI-MIXTILINEAR AND INNER TRI-EQUILATERAL

Barycentrics    a*(-2*sqrt(3)*(a^2-2*(b+c)*a+(b-c)^2)*S+a^4-4*(b+c)*a^3+2*(3*b^2-2*b*c+3*c^2)*a^2-4*(b^2-c^2)*(b-c)*a+(b^2+6*b*c+c^2)*(b-c)^2) : :

The homothetic center of these triangles is X(11515). X(30356) is their endo-homothetic center only when ABC is acute.

X(30356) lies on these lines: {1,7}, {57,30377}, {165,1653}, {1082,14100}, {1699,30382}, {1750,30327}, {2093,18422}, {3062,30292}, {5240,15587}, {7987,10647}, {7988,30309}, {7989,30316}, {8089,30372}, {8090,30421}, {8245,30364}, {8423,30409}, {8580,30414}, {10857,30280}, {10980,30351}, {11531,30321}

X(30356) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 2951, 30357), (165, 10655, 1653), (1653, 30300, 165)


X(30357) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 6th ANTI-MIXTILINEAR AND OUTER TRI-EQUILATERAL

Barycentrics    a*(2*sqrt(3)*(a^2-2*(b+c)*a+(b-c)^2)*S+a^4-4*(b+c)*a^3+2*(3*b^2-2*b*c+3*c^2)*a^2-4*(b^2-c^2)*(b-c)*a+(b^2+6*b*c+c^2)*(b-c)^2) : :

The homothetic center of these triangles is X(11516). X(30357) is their endo-homothetic center only when ABC is acute.

X(30357) lies on these lines: {1,7}, {165,1652}, {559,14100}, {1251,3062}, {1699,30383}, {1750,30328}, {2093,18423}, {5239,15587}, {7987,10648}, {7988,30310}, {7989,30317}, {8089,30373}, {8090,30422}, {8245,30365}, {8580,30415}, {10857,30281}, {10980,30352}, {11531,30322}

X(30357) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 2951, 30356), (165, 10656, 1652), (1652, 30301, 165)


X(30358) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-SHARYGIN AND EHRMANN-VERTEX

Barycentrics    a*((b^2+b*c+c^2)*a^4-2*(b+c)*b*c*a^3-(2*b^4+2*c^4+(2*b^2+b*c+2*c^2)*b*c)*a^2+(b+c)*(b^2-3*b*c+c^2)*b*c*a+(b^4-c^4)*(b^2-c^2)) : :

The homothetic center of these triangles is X(19177). X(30358) is their endo-homothetic center only when ABC is acute.

X(30358) lies on these lines: {1,21}, {5,986}, {36,30285}, {46,8235}, {65,9959}, {80,256}, {484,4220}, {517,17611}, {982,15950}, {1284,5902}, {1737,4425}, {2093,8245}, {2801,30359}, {3670,5443}, {4199,18397}, {5051,18395}, {5492,24851}, {5691,12683}, {5692,18235}, {5697,8240}, {5903,9840}, {6839,24248}, {6905,17596}, {7504,24443}, {8229,18393}, {8238,30329}, {8249,18399}, {8250,18409}, {8425,18408}, {8731,30274}, {9791,18391}, {10573,26117}, {10868,30286}, {10950,24430}, {11043,18398}, {11571,13265}, {11813,24239}, {18410,30360}, {18411,30361}, {18422,30364}, {18423,30365}

X(30358) = reflection of X(30366) in X(17611)
X(30358) = {X(65), X(9959)}-harmonic conjugate of X(30362)


X(30359) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-SHARYGIN AND 2nd EHRMANN

Barycentrics    a*((b+c)*a^4-(2*b^2+b*c+2*c^2)*a^3+(b+c)*(b^2-3*b*c+c^2)*a^2+(2*b^2+7*b*c+2*c^2)*b*c*a+(2*b^2-b*c+2*c^2)*(b+c)*b*c)*(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(19178). X(30359) is their endo-homothetic center only when ABC is acute.

X(30359) lies on these lines: {7,21}, {256,1156}, {516,30366}, {527,11688}, {846,8545}, {2801,30358}, {4199,12848}, {4220,30295}, {4335,10394}, {4425,30379}, {5051,30312}, {5057,30097}, {6172,18235}, {6912,24248}, {7671,11031}, {8229,30311}, {8235,8544}, {8240,30332}, {8245,30353}, {8249,30367}, {8250,30405}, {8425,30404}, {8731,30275}, {9814,30363}, {10868,30287}, {11043,30340}, {11533,30318}, {15726,17611}, {18450,30285}, {30362,30424}


X(30360) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-SHARYGIN AND 1st KENMOTU DIAGONALS

Barycentrics    a*(2*(b+c)*(a^2-b*c)*S+(-a+b+c)*((b+c)*a^3-(b^2+c^2)*a^2-3*(b+c)*b*c*a-(b^2+c^2)*b*c))*(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(19183). X(30360) is their endo-homothetic center only when ABC is acute.

X(30360) lies on these lines: {7,21}, {256,7133}, {846,6203}, {4199,30324}, {4220,30296}, {4425,30380}, {5051,30313}, {8229,30306}, {8235,30400}, {8240,30333}, {8245,30354}, {8249,30368}, {8250,30418}, {8425,30406}, {8731,30276}, {10868,30288}, {11031,30346}, {11043,30341}, {11533,30319}, {17611,30375}, {18235,30412}, {18410,30358}, {18458,30285}, {30362,30425}, {30363,30396}, {30366,30431}


X(30361) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-SHARYGIN AND 2nd KENMOTU DIAGONALS

Barycentrics    a*(-2*(b+c)*(a^2-b*c)*S+(-a+b+c)*((b+c)*a^3-(b^2+c^2)*a^2-3*(b+c)*b*c*a-(b^2+c^2)*b*c))*(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(19184). X(30361) is their endo-homothetic center only when ABC is acute.

X(30361) lies on these lines: {7,21}, {846,6204}, {4199,30325}, {4220,30297}, {4425,30381}, {5051,30314}, {8229,30307}, {8235,30401}, {8240,30334}, {8245,30355}, {8249,30369}, {8250,30419}, {8425,30407}, {8731,30277}, {11031,30347}, {11043,30342}, {11533,30320}, {17611,30376}, {18235,30413}, {18411,30358}, {18460,30285}, {30362,30426}, {30363,30397}, {30366,30432}


X(30362) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-SHARYGIN AND KOSNITA

Barycentrics    a*((b+c)*a^5+(b^2+b*c+c^2)*a^4-(b^3+c^3)*a^3-(b^4+c^4+(b^2+b*c+c^2)*b*c)*a^2-(b+c)*b^2*c^2*a+(b^2-c^2)^2*b*c) : :

The homothetic center of these triangles is X(19185). X(30362) is their endo-homothetic center only when ABC is acute.

X(30362) lies on these lines: {1,256}, {3,3944}, {10,11688}, {12,5143}, {21,36}, {35,4220}, {46,846}, {65,9959}, {80,13265}, {404,25385}, {405,8424}, {516,8238}, {859,24161}, {942,17611}, {958,24697}, {1054,28238}, {1281,7283}, {1283,14798}, {1478,26117}, {1580,1724}, {1698,18235}, {2093,13097}, {2292,5903}, {2475,3724}, {2975,4683}, {3120,4225}, {3339,30363}, {3633,12642}, {4199,9612}, {4295,9791}, {4297,23821}, {5051,7951}, {5248,29634}, {5251,25906}, {5902,11203}, {7741,8229}, {7742,20834}, {7972,12746}, {8249,30370}, {8250,30420}, {8425,30408}, {8731,15803}, {9579,16778}, {10868,30290}, {11031,18398}, {11533,25415}, {11571,12770}, {12683,15071}, {13731,17596}, {13738,17889}, {15507,28265}, {22769,26731}, {30359,30424}, {30360,30425}, {30361,30426}, {30364,10651}, {30365,10652}

X(30362) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1284, 8240, 11043), (1284, 9840, 1), (8240, 11043, 1)


X(30363) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-SHARYGIN AND SUBMEDIAL

Barycentrics    a*(a^5-6*(b+c)*a^4-(3*b^2-7*b*c+3*c^2)*a^3+(b+c)*(9*b^2-7*b*c+9*c^2)*a^2+(2*b^4+2*c^4-(b^2-6*b*c+c^2)*b*c)*a+(b^2-c^2)*(b-c)*(-3*b^2-5*b*c-3*c^2)) : :

The homothetic center of these triangles is X(19188). X(30363) is their endo-homothetic center only when ABC is acute.

X(30363) lies on these lines: {1,9959}, {21,30304}, {165,846}, {1284,10980}, {1699,9791}, {2292,11531}, {3339,30362}, {4199,30326}, {4425,7988}, {5051,30315}, {5223,11688}, {5691,12579}, {7987,8235}, {7991,9840}, {8229,30308}, {8238,30330}, {8240,30337}, {8249,30371}, {8250,30395}, {8425,30394}, {8731,11407}, {9814,30359}, {9819,30366}, {10868,30291}, {11031,30350}, {11043,30343}, {11533,16189}, {18235,30393}, {18421,30358}, {30285,30392}, {30360,30396}, {30361,30397}, {30364,10655}, {30365,10656}

X(30363) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (846, 8245, 165), (846, 11203, 8245)


X(30364) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-SHARYGIN AND INNER TRI-EQUILATERAL

Barycentrics    a*(2*sqrt(3)*(b+c)*(a^2-b*c)*S+(-a+b+c)*((b+c)*a^3-(b^2+c^2)*a^2-3*(b+c)*b*c*a-(b^2+c^2)*b*c))*(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(19190). X(30364) is their endo-homothetic center only when ABC is acute.

X(30364) lies on these lines: {7,21}, {846,1653}, {4199,30327}, {4220,30300}, {4425,30382}, {5051,30316}, {8229,30309}, {8235,10649}, {8240,30338}, {8245,30356}, {8249,30372}, {8250,30421}, {8425,30409}, {8731,30280}, {10868,30292}, {11031,30351}, {11043,30344}, {11098,24248}, {11533,30321}, {17611,30377}, {18235,30414}, {18422,30358}, {18469,30285}, {30362,10651}, {30363,10655}, {30366,30433}


X(30365) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-SHARYGIN AND OUTER TRI-EQUILATERAL

Barycentrics    a*(-2*sqrt(3)*(b+c)*(a^2-b*c)*S+(-a+b+c)*((b+c)*a^3-(b^2+c^2)*a^2-3*(b+c)*b*c*a-(b^2+c^2)*b*c))*(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(19191). X(30365) is their endo-homothetic center only when ABC is acute.

X(30365) lies on these lines: {7,21}, {256,1251}, {846,1652}, {4199,30328}, {4220,30301}, {4425,30383}, {5051,30317}, {8229,30310}, {8235,10650}, {8240,30339}, {8245,30357}, {8249,30373}, {8250,30422}, {8425,30410}, {8731,30281}, {11031,30352}, {11043,30345}, {11097,24248}, {11533,30322}, {17611,30378}, {18235,30415}, {18423,30358}, {30362,10652}, {30363,10656}, {30366,30434}


X(30366) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st ANTI-SHARYGIN AND TRINH

Barycentrics    a*((b+c)*a^5+(b^2-b*c+c^2)*a^4-(b^3+c^3)*a^3-(b^4+c^4+(b^2+3*b*c+c^2)*b*c)*a^2-(b+c)*(2*b^2-b*c+2*c^2)*b*c*a+(b^2-c^2)^2*b*c) : :

The homothetic center of these triangles is X(19192). X(30366) is their endo-homothetic center only when ABC is acute.

X(30366) lies on these lines: {1,256}, {10,21}, {36,4220}, {56,17722}, {516,30359}, {517,17611}, {519,11688}, {846,855}, {956,8424}, {958,7295}, {978,3612}, {993,3705}, {2292,5697}, {3057,9959}, {3145,15654}, {3586,4199}, {3632,12642}, {3679,18235}, {3920,28377}, {4294,12567}, {4425,30384}, {5051,7741}, {5143,5724}, {5259,25906}, {5902,11031}, {7419,21674}, {7951,8229}, {7972,13265}, {8238,30331}, {8249,30374}, {8250,30423}, {8425,30411}, {8731,16569}, {9791,30305}, {10624,12579}, {10868,30294}, {11113,20545}, {11533,30323}, {16975,21745}, {30360,30431}, {30361,30432}, {30364,30433}, {30365,30434}

X(30366) = reflection of X(30358) in X(17611)
X(30366) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 9840, 30362), (8240, 9840, 1)


X(30367) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-TANGENTIAL-MIDARC AND 2nd EHRMANN

Barycentrics    a*(F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c)) : : : : , where
F(a,b,c) = 6*(a+b-c)*(a-b+c)*(-a+b+c)*b*c
G(a,b,c) = 2*(a+b-c)*(a-b+c)*c*(a^2+(b-2*c)*a-(2*b+c)*(b-c))
H(a,b,c) = -(a+b-c)*(a-b+c)*(-a+b+c)*(a^2-2*(b+c)*a+(b-c)^2)

The homothetic center of these triangles is X(19369). X(30367) is their endo-homothetic center only when ABC is acute.

X(30367) lies on these lines: {1,30405}, {7,1488}, {177,1156}, {188,6172}, {516,30374}, {527,11690}, {2801,18399}, {7670,8389}, {7671,11032}, {8075,30295}, {8077,8543}, {8078,8545}, {8079,12848}, {8081,8544}, {8085,30311}, {8087,30312}, {8089,30353}, {8241,30332}, {8249,30359}, {8733,30275}, {9814,30371}, {10503,15726}, {11044,30340}, {11858,30287}, {18448,18450}, {21622,30379}, {30370,30424}


X(30368) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-TANGENTIAL-MIDARC AND 1st KENMOTU DIAGONALS

Barycentrics    a*(F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c)): : : : , where
F(a,b,c) = 4*(a+b-c)*(a-b+c)*b*c*(-a+c+b)
G(a,b,c) = -2*(a+b-c)*(a-b+c)*(2*S+(-a+c+b)*(a+b-c))*c
H(a,b,c) = (a+b-c)*(a-b+c)*(-a+c+b)*(-a^2+2*(b+c)*a+2*S-(b-c)^2)

The homothetic center of these triangles is X(2067). X(30368) is their endo-homothetic center only when ABC is acute.

X(30368) lies on these lines: {1,30418}, {7,1488}, {177,7133}, {188,30412}, {6203,8078}, {8075,30296}, {8077,30385}, {8079,30324}, {8081,30400}, {8085,30306}, {8087,30313}, {8089,30354}, {8241,30333}, {8249,30360}, {8733,30276}, {10503,30375}, {11032,30346}, {11534,30319}, {11858,30288}, {18399,18410}, {18448,18458}, {21622,30380}, {30370,30425}, {30371,30396}, {30374,30431}


X(30369) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-TANGENTIAL-MIDARC AND 2nd KENMOTU DIAGONALS

Barycentrics    a*(F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c)): : : : , where
F(a,b,c) = 4*(a+b-c)*(a-b+c)*b*c*(-a+c+b)
G(a,b,c) = 2*(a+b-c)*(a-b+c)*(2*S-(-a+c+b)*(a+b-c))*c
H(a,b,c) = -(a+b-c)*(a-b+c)*(-a+c+b)*(2*S+a^2-2*(b+c)*a+(b-c)^2)

The homothetic center of these triangles is X(6502). X(30369) is their endo-homothetic center only when ABC is acute.

X(30369) lies on these lines: {1,30419}, {7,1488}, {188,30413}, {6204,8078}, {8075,30297}, {8077,30386}, {8079,30325}, {8081,30401}, {8085,30307}, {8087,30314}, {8089,30355}, {8241,30334}, {8249,30361}, {8733,30277}, {10503,30376}, {11032,30347}, {11044,30342}, {11534,30320}, {11858,30289}, {18399,18411}, {18448,18460}, {21622,30381}, {30370,30426}, {30371,30397}, {30374,30432}


X(30370) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-TANGENTIAL-MIDARC AND KOSNITA

Barycentrics    a*(F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+2*S^2): : : : , where
F(a,b,c) = b*c*a*(-a+b+c)
G(a,b,c) = -c*(a-b+c)*(a^2+a*b+b^2-c^2)
X(30370) = 4*X(13385)-X(30423)

The homothetic center of these triangles is X(36). X(30370) is their endo-homothetic center only when ABC is acute.

X(30370) lies on these lines: {1,167}, {10,11690}, {35,8075}, {36,8077}, {46,8078}, {65,8099}, {79,15997}, {80,8103}, {188,1698}, {516,8387}, {517,10506}, {942,10503}, {1699,9836}, {2093,8101}, {3339,30371}, {3633,12643}, {4292,11888}, {4295,9793}, {5902,11192}, {5903,8093}, {7951,8087}, {7972,8097}, {8079,9612}, {8095,15071}, {8249,30362}, {8733,15803}, {10500,12813}, {11009,11013}, {11032,18398}, {11534,25415}, {11571,12771}, {11858,30290}, {12047,21622}, {30367,30424}, {30368,30425}, {30369,30426}, {30372,10651}, {30373,10652}

X(30370) = inverse of X(12908) in the incircle
X(30370) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 177, 30408), (1, 8091, 30374), (2089, 8091, 1)


X(30371) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-TANGENTIAL-MIDARC AND SUBMEDIAL

Barycentrics    a*(F(a,b,c)*sin(A/2)+G(a,b,c)*sin(B/2)+G(a,c,b)*sin(C/2)+H(a,b,c)): : : : , where
F(a,b,c) = (-a+b+c)*(a^2+2*(b+c)*a-3*(b-c)^2)
G(a,b,c) = (a-b+c)*(a^2+2*(b-3*c)*a-(3*b+5*c)*(b-c))
H(a,b,c) = 4*(-a+b+c)*(a-b+c)*(a+b-c)

The homothetic center of these triangles is X(19372). X(30371) is their endo-homothetic center only when ABC is acute.

X(30371) lies on these lines: {1,8099}, {165,8075}, {177,30394}, {188,30393}, {1699,9793}, {3339,30370}, {5223,11690}, {5691,12580}, {7987,8081}, {7988,21622}, {7991,8091}, {8077,30389}, {8079,30326}, {8085,30308}, {8087,30315}, {8090,10967}, {8093,11531}, {8241,30337}, {8249,30363}, {8387,30330}, {8733,11407}, {9814,30367}, {9819,30374}, {11032,30350}, {11044,30343}, {11379,12908}, {11534,16189}, {11858,30291}, {11888,30304}, {18399,18421}, {18448,30392}, {30368,30396}, {30369,30397}, {30373,10656}

X(30371) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8078, 8089, 165), (8078, 11192, 8089)


X(30372) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-TANGENTIAL-MIDARC AND INNER TRI-EQUILATERAL

Barycentrics    a*(2*sqrt(3)*(2*c*sin(B/2)+2*b*sin(C/2)+a-b-c)*S+(-4*sin(A/2)*b*c+2*c*(a+b-c)*sin(B/2)+2*b*(a-b+c)*sin(C/2)+a^2-2*(b+c)*a+(b-c)^2)*(-a+b+c))*(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(7051). X(30372) is their endo-homothetic center only when ABC is acute.

X(30372) lies on these lines: {1,30421}, {7,1488}, {177,30409}, {188,30414}, {1653,8078}, {8075,30300}, {8077,10647}, {8079,30327}, {8081,10649}, {8087,30316}, {8089,30356}, {8241,30338}, {8249,30364}, {8733,30280}, {10503,30377}, {11032,30351}, {11044,30344}, {11534,30321}, {11858,30292}, {18399,18422}, {18448,18469}, {21622,30382}, {30370,10651}, {30371,10655}, {30374,30433}


X(30373) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-TANGENTIAL-MIDARC AND OUTER TRI-EQUILATERAL

Barycentrics    a*(-2*sqrt(3)*(2*c*sin(B/2)+2*b*sin(C/2)+a-b-c)*S+(-4*sin(A/2)*b*c+2*c*(a+b-c)*sin(B/2)+2*b*(a-b+c)*sin(C/2)+a^2-2*(b+c)*a+(b-c)^2)*(-a+b+c))*(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(19373). X(30373) is their endo-homothetic center only when ABC is acute.

X(30373) lies on these lines: {1,30422}, {7,1488}, {177,1251}, {188,30415}, {1652,8078}, {8075,30301}, {8077,10648}, {8079,30328}, {8081,10650}, {8085,30310}, {8087,30317}, {8089,30357}, {8241,30339}, {8249,30365}, {8733,30281}, {10503,30378}, {11032,30352}, {11044,30345}, {11534,30322}, {11858,30293}, {18399,18423}, {18448,18471}, {21622,30383}, {30370,10652}, {30371,10656}, {30374,30434}


X(30374) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-TANGENTIAL-MIDARC AND TRINH

Barycentrics    a*(3*(-a+b+c)*sin(A/2)*a*b*c-(a+b-c)*(a^2-a*c-b^2+c^2)*b*sin(C/2)-(a-b+c)*(a^2-a*b+b^2-c^2)*c*sin(B/2)+2*S^2) : :

The homothetic center of these triangles is X(35). X(30374) is their endo-homothetic center only when ABC is acute.

X(30374) lies on these lines: {1,167}, {35,8077}, {36,8075}, {80,1128}, {188,3679}, {516,30367}, {517,10503}, {519,11690}, {3057,8099}, {3586,8079}, {3632,12643}, {4294,12568}, {4304,11888}, {5119,8078}, {5441,16146}, {5691,9836}, {5697,8093}, {5902,11032}, {7741,8087}, {7951,8085}, {7972,8103}, {8249,30366}, {8387,30331}, {8733,30282}, {9793,30305}, {9819,30371}, {9957,10506}, {10500,18408}, {10624,12580}, {11534,30323}, {11858,30294}, {17641,18409}, {21622,30384}, {30368,30431}, {30369,30432}, {30372,30433}, {30373,30434}

X(30374) = reflection of X(18399) in X(10503)
X(30374) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 177, 30411), (1, 8091, 30370), (8091, 8241, 1)


X(30375) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR AND 1st KENMOTU DIAGONALS

Barycentrics    a*((2*(b+c)*a-2*b^2+4*b*c-2*c^2)*S+(-a+b+c)*((b+c)*a^2-2*(b-c)^2*a+(b^2-c^2)*(b-c))) : :

The homothetic center of these triangles is X(590). X(30375) is their endo-homothetic center only when ABC is acute.

X(30375) lies on these lines: {6,7133}, {7,354}, {11,30380}, {55,6203}, {56,30400}, {57,30354}, {210,13359}, {517,18410}, {942,30425}, {1155,30296}, {1319,18458}, {1721,13389}, {1864,30324}, {2098,30319}, {2646,30385}, {3057,6405}, {10501,30418}, {10502,30406}, {10503,30368}, {11030,21746}, {17603,30276}, {17604,30288}, {17605,30306}, {17606,30313}, {17609,30341}, {17611,30360}

X(30375) = midpoint of X(18410) and X(30431)
X(30375) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 30346, 354), (354, 14100, 30376)


X(30376) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR AND 2nd KENMOTU DIAGONALS

Barycentrics    a*(-(2*(b+c)*a-2*b^2+4*b*c-2*c^2)*S+(-a+b+c)*((b+c)*a^2-2*(b-c)^2*a+(b^2-c^2)*(b-c))) : :

The homothetic center of these triangles is X(615). X(30376) is their endo-homothetic center only when ABC is acute.

X(30376) lies on these lines: {1,30397}, {6,9043}, {7,354}, {11,30381}, {55,6204}, {56,30401}, {57,30355}, {210,13360}, {517,18411}, {942,30426}, {1155,30297}, {1319,18460}, {1721,13388}, {1864,30325}, {2098,30320}, {2646,30386}, {3057,6283}, {10501,30419}, {10502,30407}, {10503,30369}, {17603,30277}, {17604,30289}, {17605,30307}, {17606,30314}, {17609,30342}, {17611,30361}

X(30376) = midpoint of X(18411) and X(30432)
X(30376) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 30347, 354), (354, 14100, 30375)


X(30377) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR AND INNER TRI-EQUILATERAL

Barycentrics    a*(2*sqrt(3)*((b+c)*a-(b-c)^2)*S+(-a+b+c)*((b+c)*a^2-2*(b-c)^2*a+(b^2-c^2)*(b-c))) : :

The homothetic center of these triangles is X(23302). X(30377) is their endo-homothetic center only when ABC is acute.

X(30377) lies on these lines: {1,10655}, {7,354}, {11,30382}, {55,1653}, {56,10649}, {57,30356}, {210,30414}, {517,18422}, {1100,7127}, {1155,30300}, {1319,18469}, {1864,30327}, {2098,30321}, {2646,10647}, {3057,30338}, {10501,30421}, {10502,30409}, {10503,30372}, {17603,30280}, {17604,30292}, {17605,30309}, {17606,30316}, {17609,30344}, {17611,30364}

X(30377) = midpoint of X(18422) and X(30433)
X(30377) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 30351, 354), (354, 14100, 30378)


X(30378) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-URSA-MINOR AND OUTER TRI-EQUILATERAL

Barycentrics    a*(-2*sqrt(3)*((b+c)*a-(b-c)^2)*S+(-a+b+c)*((b+c)*a^2-2*(b-c)^2*a+(b^2-c^2)*(b-c))) : :

The homothetic center of these triangles is X(23303). X(30378) is their endo-homothetic center only when ABC is acute.

X(30378) lies on these lines: {1,10656}, {7,354}, {11,30383}, {55,1652}, {56,10650}, {210,30415}, {517,18423}, {942,10652}, {1100,1251}, {1155,30301}, {1319,18471}, {1864,30328}, {2098,30322}, {2646,10648}, {3057,30339}, {10501,30422}, {10502,30410}, {10503,30373}, {17603,30281}, {17604,30293}, {17605,30310}, {17606,30317}, {17609,30345}, {17611,30365}

X(30378) = midpoint of X(18423) and X(30434)
X(30378) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 30352, 354), (354, 14100, 30377)


X(30379) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-WASAT AND 2nd EHRMANN

Barycentrics    ((b+c)*a^2-2*(b^2-b*c+c^2)*a+(b^2-c^2)*(b-c))*(a+b-c)*(a+c-b) : :
X(30379) = X(7)+2*X(3911) = 4*X(142)-X(908) = 2*X(10427)+X(26015)

The homothetic center of these triangles is X(21639). X(30379) is their endo-homothetic center only when ABC is acute.

X(30379) lies on these lines: {2,7}, {4,8544}, {8,30318}, {10,30312}, {11,15726}, {36,516}, {56,5880}, {65,25557}, {77,3554}, {222,26723}, {241,1086}, {269,4859}, {277,4350}, {279,24181}, {354,8255}, {377,1467}, {390,3576}, {514,7216}, {515,18450}, {518,6735}, {519,14151}, {528,1319}, {651,3008}, {653,5236}, {673,909}, {946,5265}, {971,1532}, {1001,1470}, {1012,5805}, {1042,24178}, {1125,8543}, {1156,11219}, {1210,6932}, {1266,4552}, {1323,15727}, {1407,24789}, {1420,4190}, {1422,15474}, {1429,1813}, {1441,24199}, {1442,3946}, {1443,17067}, {1458,1738}, {1519,15325}, {1699,30353}, {1737,2801}, {2078,3254}, {2550,3476}, {3011,9364}, {3243,12648}, {3361,12609}, {3488,6916}, {3522,12053}, {3523,5766}, {3586,5732}, {3660,10427}, {3668,17092}, {3817,30311}, {3826,8581}, {3912,20881}, {4292,6912}, {4298,5258}, {4312,6974}, {4321,9623}, {4341,24779}, {4425,30359}, {4648,7190}, {4652,5698}, {4675,5228}, {5220,24914}, {5298,28534}, {5433,15254}, {5445,13407}, {5542,5902}, {5696,10916}, {5704,6260}, {5723,6610}, {5728,6907}, {5735,6966}, {5784,6734}, {6049,21627}, {6067,15587}, {6180,17278}, {6913,18541}, {7176,17050}, {7671,11019}, {7676,15931}, {7988,9814}, {8236,13384}, {9358,24198}, {9710,9850}, {11038,11526}, {12047,30424}, {12709,24564}, {12832,25558}, {14100,15845}, {14953,17197}, {16133,26725}, {17080,24177}, {17095,17305}, {17625,25006}, {21620,30340}, {21622,30367}, {21623,30405}, {21624,30404}, {24389,25722}, {26001,26932}

X(30379) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 8545), (7, 142, 21617), (142, 6173, 5249)


X(30380) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-WASAT AND 1st KENMOTU DIAGONALS

Barycentrics    (2*(b+c)*S+(-a+b+c)*((b+c)*a-(b-c)^2))*(a+c-b)*(a+b-c) : :

The homothetic center of these triangles is X(21640). X(30380) is their endo-homothetic center only when ABC is acute.

X(30380) lies on these lines: {2,7}, {4,30400}, {8,30319}, {10,30313}, {11,30375}, {482,2362}, {515,18458}, {516,30296}, {1086,8243}, {1125,30385}, {1659,4000}, {1699,30354}, {1737,18410}, {3817,30306}, {4425,30360}, {4648,13390}, {5393,7133}, {7988,30396}, {11019,30288}, {12047,30425}, {12053,30333}, {13388,18589}, {21620,30341}, {21622,30368}, {21623,30418}, {21624,30406}, {30384,30431}

X(30380) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 6203), (142, 226, 30381)


X(30381) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-WASAT AND 2nd KENMOTU DIAGONALS

Barycentrics    (-2*(b+c)*S+(-a+b+c)*((b+c)*a-(b-c)^2))*(a+c-b)*(a+b-c) : :

The homothetic center of these triangles is X(21641). X(30381) is their endo-homothetic center only when ABC is acute.

X(30381) lies on these lines: {2,7}, {4,30401}, {8,30320}, {10,30314}, {11,30376}, {37,8243}, {481,16232}, {515,18460}, {516,30297}, {1125,30386}, {1659,4648}, {1699,30355}, {1737,18411}, {3817,30307}, {4000,13390}, {4425,30361}, {5405,7595}, {7988,30397}, {10858,16432}, {11019,30289}, {12047,30426}, {12053,30334}, {13389,18589}, {21620,30342}, {21622,30369}, {21623,30419}, {21624,30407}, {30384,30432}

X(30381) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 6204), (142, 226, 30380)


X(30382) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-WASAT AND INNER TRI-EQUILATERAL

Barycentrics    (a+b-c)*(a-b+c)*(2*sqrt(3)*(b+c)*S+(-a+b+c)*((b+c)*a-(b-c)^2)) : :

The homothetic center of these triangles is X(21647). X(30382) is their endo-homothetic center only when ABC is acute.

X(30382) lies on these lines: {2,7}, {4,10649}, {8,30321}, {10,30316}, {11,30377}, {515,18469}, {516,30300}, {1125,10647}, {1699,30356}, {1737,18422}, {3639,5074}, {3817,30309}, {4292,11098}, {4425,30364}, {7988,10655}, {11019,30292}, {12047,10651}, {12053,30338}, {21620,30344}, {21622,30372}, {21623,30421}, {21624,30409}, {30384,30433}

X(30382) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 1653), (142, 226, 30383)


X(30383) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-WASAT AND OUTER TRI-EQUILATERAL

Barycentrics    (a+b-c)*(a-b+c)*(-2*sqrt(3)*(b+c)*S+(-a+b+c)*((b+c)*a-(b-c)^2)) : :

The homothetic center of these triangles is X(21648). X(30383) is their endo-homothetic center only when ABC is acute.

X(30383) lies on these lines: {2,7}, {4,10650}, {8,30322}, {10,30317}, {11,30378}, {515,18471}, {516,30301}, {1125,10648}, {1699,30357}, {1737,18423}, {3638,5074}, {3817,30310}, {4292,11097}, {4425,30365}, {7988,10656}, {11019,30293}, {12047,10652}, {12053,30339}, {21620,30345}, {21622,30373}, {21623,30422}, {21624,30410}, {30384,30434}

X(30383) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7, 1652), (142, 226, 30382)


X(30384) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: ANTI-WASAT AND TRINH

Barycentrics    (b+c)*a^3+(b^2-4*b*c+c^2)*a^2-(b^2-c^2)*(b-c)*a-(b^2-c^2)^2 : :
X(30384) = 3*X(36)-X(15228) = X(36)-3*X(16173) = X(484)-3*X(3582) = X(908)+2*X(21630) = 4*X(1387)-X(21578) = X(1512)-4*X(16174) = X(1737)-4*X(7743) = 3*X(3582)-2*X(3911) = 5*X(3616)-3*X(4881) = 2*X(5122)-3*X(5298) = X(15228)-9*X(16173) = 5*X(18493)-X(18524)

The homothetic center of these triangles is X(21663). X(30384) is their endo-homothetic center only when ABC is acute.

X(30384) lies on these lines: {1,4}, {2,5119}, {3,11376}, {5,3057}, {7,7284}, {8,5187}, {10,3877}, {11,517}, {12,9955}, {20,18220}, {21,16155}, {30,1319}, {35,404}, {36,516}, {40,499}, {46,962}, {55,5886}, {56,1770}, {57,10072}, {65,496}, {72,3813}, {79,1476}, {80,519}, {90,10529}, {142,25055}, {145,21077}, {149,4511}, {354,11551}, {355,2098}, {381,5252}, {392,2886}, {411,14798}, {442,20288}, {484,3582}, {495,5919}, {498,1697}, {528,5440}, {551,4304}, {553,11552}, {758,26015}, {912,18839}, {920,12704}, {942,17705}, {952,5048}, {956,24703}, {960,24390}, {995,3914}, {997,3434}, {999,1836}, {1000,3545}, {1012,22767}, {1145,5123}, {1149,3120}, {1155,15325}, {1158,10785}, {1201,23537}, {1210,4301}, {1279,16581}, {1317,12611}, {1329,10914}, {1385,6284}, {1388,12953}, {1420,4299}, {1447,5195}, {1482,1837}, {1484,14988}, {1512,8068}, {1532,22835}, {1537,5570}, {1702,13904}, {1703,13962}, {1706,25522}, {1727,3218}, {1728,5758}, {1739,5121}, {1743,21068}, {1858,10959}, {2077,10090}, {2099,3656}, {2170,5179}, {2446,23517}, {2447,23477}, {2476,3890}, {2646,5901}, {2800,5533}, {2802,3814}, {3058,15950}, {3061,21073}, {3085,6953}, {3091,10827}, {3149,11508}, {3245,28228}, {3295,11375}, {3303,11374}, {3338,4295}, {3419,5289}, {3452,3679}, {3560,10966}, {3576,4302}, {3579,5433}, {3601,4309}, {3612,3616}, {3622,4305}, {3624,17567}, {3632,21075}, {3636,5441}, {3649,5045}, {3667,4017}, {3671,18398}, {3674,7264}, {3698,17527}, {3702,3969}, {3746,5443}, {3748,4870}, {3753,3816}, {3755,5313}, {3817,4342}, {3822,3898}, {3825,24982}, {3838,10179}, {3841,24564}, {3869,10916}, {3878,6734}, {3880,5087}, {3884,24987}, {3885,10915}, {3899,24386}, {3902,5741}, {3912,4975}, {3918,25011}, {3936,4742}, {3940,4863}, {4187,5836}, {4292,5563}, {4293,9812}, {4297,21842}, {4310,15430}, {4311,10483}, {4316,28150}, {4317,9579}, {4329,24179}, {4424,24239}, {4425,30366}, {4653,17167}, {4679,9708}, {4847,5692}, {4858,23580}, {4861,5046}, {4872,24203}, {5010,10165}, {5074,17761}, {5086,5330}, {5122,5298}, {5126,15326}, {5131,28232}, {5183,28212}, {5219,10056}, {5248,24541}, {5250,26363}, {5253,14803}, {5258,12572}, {5274,18391}, {5288,12527}, {5316,19875}, {5432,11230}, {5542,7671}, {5587,6973}, {5657,10589}, {5687,25681}, {5690,10593}, {5726,30308}, {5727,16200}, {5806,13375}, {5844,12019}, {5887,10943}, {5902,11019}, {5905,11240}, {6147,17609}, {6238,12259}, {6265,13274}, {6361,7288}, {6684,11010}, {6767,17718}, {6841,10957}, {6844,10051}, {6886,7162}, {6941,7704}, {6985,11510}, {6988,16208}, {7173,9956}, {7354,22793}, {7373,10404}, {7680,15845}, {7681,10523}, {7727,13605}, {7965,12915}, {7972,21635}, {7982,9581}, {7988,9819}, {8069,22753}, {8071,11496}, {8196,26417}, {8203,26393}, {8256,17619}, {8715,27385}, {9589,15803}, {9598,9619}, {9599,9620}, {9668,10246}, {9709,24954}, {9779,10590}, {10050,11372}, {10052,10085}, {10073,10698}, {10176,25006}, {10310,12700}, {10527,12514}, {10543,14526}, {10679,11502}, {10680,22760}, {10738,12740}, {10742,20586}, {10944,18480}, {10948,12672}, {10950,24680}, {11020,21625}, {11035,13865}, {11362,18395}, {11499,26358}, {11715,23243}, {11720,12896}, {12678,30283}, {12702,24914}, {12735,28224}, {12737,12764}, {12743,19907}, {12848,15299}, {13374,13750}, {13624,15338}, {14100,20330}, {16153,21669}, {17188,17519}, {18650,24202}, {18651,26728}, {18976,22938}, {18990,20323}, {21617,30331}, {21622,30374}, {21623,30423}, {21624,30411}, {23340,26476}, {24046,28018}, {24159,28011}, {24160,28027}, {24474,26475}, {25405,28160}, {30380,30431}, {30381,30432}, {30382,30433}, {30383,30434}

X(30384) = midpoint of X(i) and X(j) for these {i,j}: {1, 3583}, {149, 4511}, {1320, 5176}, {2077, 14217}, {3218, 5180}, {11813, 21630}
X(30384) = reflection of X(i) in X(j) for these (i,j): (11, 7743), (484, 3911), (908, 11813), (1145, 5123), (1155, 15325), (1319, 1387), (1519, 946), (1532, 22835), (1737, 11), (6735, 3814), (11570, 5570), (15326, 5126), (17757, 5087), (21578, 1319)
X(30384) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1479, 10572), (1, 4857, 950), (1, 9614, 1479)


X(30385) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: CIRCUMORTHIC AND 1st KENMOTU DIAGONALS

Barycentrics    a*(2*S*a+(-a+b+c)*(a^2-(b+c)*a-2*b*c))*(a+b-c)*(a+c-b) : :

The homothetic center of these triangles is X(10880). X(30385) is their endo-homothetic center only when ABC is acute.

X(30385) lies on these lines: {1,372}, {2,30313}, {3,30276}, {4,30306}, {7,21}, {25,13390}, {35,30431}, {36,30425}, {55,16440}, {105,175}, {198,13940}, {238,2067}, {405,30324}, {614,13388}, {958,30412}, {968,13389}, {999,30341}, {1125,30380}, {1385,18458}, {2646,30375}, {3576,30400}, {7587,30406}, {7588,30418}, {7987,30354}, {8077,30368}, {8583,30288}, {22756,24328}, {30389,30396}

X(30385) = {X(56), X(1001)}-harmonic conjugate of X(30386)


X(30386) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: CIRCUMORTHIC AND 2nd KENMOTU DIAGONALS

Barycentrics    a*(-2*S*a+(-a+b+c)*(a^2-(b+c)*a-2*b*c))*(a+b-c)*(a+c-b) : :

The homothetic center of these triangles is X(10881). X(30386) is their endo-homothetic center only when ABC is acute.

X(30386) lies on these lines: {1,371}, {2,30314}, {3,30277}, {4,30307}, {7,21}, {25,1659}, {35,30432}, {36,8225}, {55,16441}, {105,176}, {198,13887}, {238,6502}, {405,30325}, {614,13389}, {958,30413}, {968,13388}, {999,30342}, {1125,30381}, {1385,18460}, {2646,30376}, {3576,30401}, {7587,30407}, {7588,30419}, {7987,30355}, {8077,30369}, {8583,30289}, {22757,24328}, {30389,30397}

X(30386) = {X(56), X(1001)}-harmonic conjugate of X(30385)


X(30387) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: CIRCUMORTHIC AND LUCAS ANTIPODAL TANGENTIAL

Barycentrics
a^2*(a^5-(b+c)*a^4-2*(b^2+4*b*c+c^2)*a^3+2*(b+c)*(b^2+c^2)*a^2+(b^4+c^4+2*(4*b^2-b*c+4*c^2)*b*c)*a-(a^3+5*(b+c)*a^2-(5*b^2-2*b*c+5*c^2)*a-(b+c)*(b^2+6*b*c+c^2))*S-(b^2-c^2)^2*(b+c)) : :

The homothetic center of these triangles is X(19424). X(30387) is their endo-homothetic center only when ABC is acute.

X(30387) lies on these lines: {1,30348}, {3,30278}, {21,30302}, {55,1152}, {958,30416}, {2360,8273}, {3304,16213}


X(30388) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: CIRCUMORTHIC AND LUCAS(-1) ANTIPODAL TANGENTIAL

Barycentrics
a^2*(a^5-(b+c)*a^4-2*(b^2+4*b*c+c^2)*a^3+2*(b+c)*(b^2+c^2)*a^2+(b^4+c^4+2*(4*b^2-b*c+4*c^2)*b*c)*a+(a^3+5*(b+c)*a^2-(5*b^2-2*b*c+5*c^2)*a-(b+c)*(b^2+6*b*c+c^2))*S-(b^2-c^2)^2*(b+c)) : :

The homothetic center of these triangles is X(19425). X(30388) is their endo-homothetic center only when ABC is acute.

X(30388) lies on these lines: {1,30349}, {3,30279}, {21,30303}, {55,1151}, {958,30417}, {2360,8273}, {3304,16214}


X(30389) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: CIRCUMORTHIC AND SUBMEDIAL

Barycentrics    a*(7*a^3-3*(b+c)*a^2-(7*b^2-6*b*c+7*c^2)*a+3*(b^2-c^2)*(b-c)) : :
X(30389) = 3*X(1)+4*X(3) = 5*X(1)+2*X(40) = 4*X(1)+3*X(165) = X(1)-8*X(1385) = 11*X(1)-4*X(1482) = X(1)+6*X(3576) = 9*X(1)-2*X(7982) = 2*X(1)+5*X(7987) = 6*X(1)+X(7991) = 5*X(1)-12*X(10246) = 19*X(1)-12*X(10247) = 10*X(1)-3*X(11224) = 8*X(1)-X(11531) = 9*X(1)-16*X(15178) = 12*X(1)-5*X(16189) = 23*X(1)-9*X(16191) = 13*X(1)-6*X(16200) = 15*X(1)-8*X(24680) = 2*X(1)-9*X(30392) = 24*X(2)-17*X(30315)

The homothetic center of these triangles is X(3090). X(30389) is their endo-homothetic center only when ABC is acute.

X(30389) lies on these lines: {1,3}, {2,30315}, {4,25055}, {5,3653}, {9,23073}, {10,10303}, {20,551}, {21,30304}, {30,9624}, {104,28148}, {140,3655}, {145,10164}, {214,936}, {355,632}, {376,9589}, {392,15071}, {405,9851}, {515,3090}, {516,3622}, {518,10541}, {519,3523}, {546,8227}, {548,3656}, {549,4677}, {572,3731}, {573,17474}, {610,22357}, {631,3679}, {944,1698}, {946,3529}, {952,14869}, {958,30393}, {960,22333}, {962,3636}, {991,1201}, {997,5234}, {1001,3062}, {1125,3091}, {1279,15839}, {1376,7990}, {1490,6920}, {1699,3146}, {1702,6453}, {1703,6454}, {1743,21748}, {1750,5436}, {1768,5250}, {2771,15039}, {2948,15034}, {2975,3984}, {3083,21565}, {3084,21568}, {3158,11260}, {3241,15717}, {3522,4301}, {3524,11362}, {3526,19876}, {3528,28194}, {3582,6825}, {3584,6891}, {3592,18992}, {3594,18991}, {3627,5886}, {3628,5587}, {3632,6684}, {3633,5657}, {3646,16860}, {3654,15712}, {3680,4421}, {3698,10156}, {3851,28208}, {3857,28186}, {3897,17531}, {3899,5884}, {3901,12005}, {3951,4511}, {4308,13405}, {4311,5290}, {4315,5703}, {4317,6987}, {4325,6868}, {4326,30287}, {4330,6948}, {4668,12108}, {4669,15708}, {4745,15721}, {4853,4855}, {4857,6850}, {4866,5258}, {4881,17572}, {4882,5440}, {5007,9619}, {5047,8583}, {5056,19883}, {5072,11230}, {5076,9955}, {5079,18480}, {5259,12114}, {5265,6738}, {5270,6827}, {5272,7963}, {5281,6049}, {5303,11682}, {5426,16143}, {5433,5727}, {5444,10827}, {5450,7992}, {5493,5734}, {5550,15022}, {5573,8572}, {5603,17538}, {5660,24954}, {5692,12675}, {5732,24644}, {5818,19872}, {5901,15704}, {6176,19646}, {6419,9583}, {6420,19004}, {6425,7968}, {6426,7969}, {6713,9897}, {6908,10072}, {6926,10056}, {6946,12650}, {6960,10199}, {6972,10197}, {6986,8666}, {7308,9845}, {7415,28619}, {7419,17194}, {7587,30394}, {7588,30395}, {7677,30330}, {7772,9592}, {7786,22650}, {7972,21154}, {7993,11715}, {7997,26321}, {8077,30371}, {8543,9814}, {9579,15950}, {9612,21578}, {9780,28236}, {9904,15021}, {10085,19526}, {11194,11523}, {11219,26066}, {11231,18526}, {11363,11403}, {11477,16475}, {11709,14094}, {11710,23235}, {11720,15054}, {12103,12699}, {12104,19919}, {12407,15027}, {13731,22392}, {15017,25522}, {15808,28164}, {15829,24645}, {16126,21161}, {16132,16138}, {16842,17614}, {16865,19861}, {18444,30144}, {19546,25502}, {19647,26102}, {24914,30286}, {30385,30396}, {30386,30397}, {10647,10655}, {10648,10656}

X(30389) = midpoint of X(1) and X(16192)
X(30389) = reflection of X(i) in X(j) for these (i,j): (7989, 3624), (9588, 3523)
X(30389) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3, 7991), (1, 7280, 2093), (3612, 21842, 1), (7982, 15178, 1)


X(30390) = PERSPECTOR OF THESE TRIANGLES: INNER-NAPOLEON AND VERTEX TRIANGLE OF THE INNER AND OUTER TRI-EQUILATERAL

Barycentrics    a^2*(2*sqrt(3)*(a^4-b^4-c^4)*S*a^2+a^8-3*(b^2+c^2)*a^6+3*(b^4+c^4)*a^4+2*(b^2-c^2)^2*b^2*c^2-(b^4-c^4)*(b^2-c^2)*a^2) : :

X(30390) lies on these lines: {5,182}, {154,3129}, {184,398}, {3490,11088}, {8919,14560}

X(30390) = {X(206), X(6759)}-harmonic conjugate of X(30391)


X(30391) = PERSPECTOR OF THESE TRIANGLES: OUTER-NAPOLEON AND VERTEX TRIANGLE OF THE INNER AND OUTER TRI-EQUILATERAL

Barycentrics    a^2*(-2*sqrt(3)*(a^4-b^4-c^4)*S*a^2+a^8-3*(b^2+c^2)*a^6+3*(b^4+c^4)*a^4+2*(b^2-c^2)^2*b^2*c^2-(b^4-c^4)*(b^2-c^2)*a^2) : :

X(30391) lies on these lines: {5,182}, {154,3130}, {184,397}, {3489,11083}, {8918,14560}

X(30391) = {X(206), X(6759)}-harmonic conjugate of X(30390)


X(30392) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EHRMANN-SIDE AND SUBMEDIAL

Barycentrics    a*(9*a^3-5*(b+c)*a^2-(9*b^2-10*b*c+9*c^2)*a+5*(b^2-c^2)*(b-c)) : :
Trilinears    5 r + 4 R cos A : :
X(30392) = 5*X(1)+4*X(3) = 7*X(1)+2*X(40) = 2*X(1)+X(165) = X(1)+8*X(1385) = 13*X(1)-4*X(1482) = X(1)+2*X(3576) = 11*X(1)-2*X(7982) = 4*X(1)+5*X(7987) = 8*X(1)+X(7991) = X(1)-4*X(10246) = 7*X(1)-4*X(10247) = 4*X(1)-X(11224) = 10*X(1)-X(11531) = 7*X(1)-16*X(15178) = 14*X(1)-5*X(16189) = 3*X(1)-X(16191) = 11*X(1)+7*X(16192) = 5*X(1)-2*X(16200) = 7*X(1)+8*X(17502) = 17*X(1)-8*X(24680)

The homothetic center of these triangles is X(5055). X(30392) is their endo-homothetic center only when ABC is acute.

X(30392) lies on these lines: {1,3}, {2,28236}, {20,3636}, {104,28170}, {145,9588}, {214,7993}, {355,30315}, {376,28232}, {515,3545}, {519,15708}, {547,3655}, {551,1699}, {572,16673}, {631,3632}, {944,3624}, {952,3653}, {991,1149}, {997,18452}, {1125,5056}, {1698,3533}, {1702,9585}, {2320,3062}, {2801,16858}, {3083,21564}, {3084,21569}, {3091,15808}, {3241,10164}, {3244,3523}, {3524,28234}, {3616,3817}, {3622,4297}, {3626,10303}, {3633,6684}, {3654,19711}, {3656,15690}, {3679,7967}, {3845,5886}, {3850,8227}, {3853,9624}, {3897,8583}, {4423,30283}, {4511,5223}, {4512,24645}, {4677,11812}, {4915,5440}, {5041,9619}, {5102,16475}, {5234,10176}, {5400,6176}, {5426,28461}, {5436,5927}, {5531,11715}, {5603,11001}, {5657,15719}, {5734,12512}, {5790,15723}, {5881,16239}, {6429,9615}, {6431,18992}, {6432,18991}, {6433,9616}, {7990,19860}, {9589,10595}, {9618,10137}, {9814,18450}, {10167,10179}, {10283,15686}, {10304,28228}, {10442,17394}, {11379,16143}, {11714,15735}, {12767,19907}, {15717,20057}, {15726,24644}, {16859,19861}, {16864,17614}, {18444,30304}, {18446,30326}, {18448,30371}, {18454,30394}, {18456,30395}, {18458,30396}, {18460,30397}, {18469,10655}, {18471,10656}, {30284,30330}, {30285,30363}

X(30392) = reflection of X(7988) in X(25055)
X(30392) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3, 11531), (7982, 13624, 16192), (11531, 16189, 11278), (11531, 16200, 11224)


X(30393) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st EXCOSINE AND SUBMEDIAL

Barycentrics    a*(a^2+2*(b+c)*a-(b+3*c)*(3*b+c)) : :

The homothetic center of these triangles is X(17825). X(30393) is their endo-homothetic center only when ABC is acute.

X(30393) lies on these lines: {1,210}, {2,5223}, {8,30337}, {9,165}, {10,962}, {40,10157}, {43,3731}, {57,3715}, {63,9342}, {171,3973}, {188,30371}, {200,1621}, {226,1698}, {236,30394}, {392,4915}, {497,3679}, {518,30350}, {612,16469}, {756,2999}, {936,993}, {960,11531}, {984,23511}, {997,18452}, {1125,30343}, {1215,16832}, {1329,30315}, {1697,3983}, {1743,5268}, {1961,16667}, {2093,3820}, {2886,30308}, {3097,16569}, {3158,15254}, {3243,8167}, {3452,7988}, {3475,3624}, {3587,18529}, {3681,10582}, {3697,4882}, {3711,10389}, {3729,26038}, {3928,15481}, {3929,4413}, {3930,16673}, {3951,19877}, {3958,10158}, {3971,17151}, {4005,11518}, {4015,6765}, {4104,17284}, {4312,26040}, {4355,17582}, {4384,27538}, {4512,27065}, {4533,16842}, {4668,4863}, {4679,10826}, {4682,16670}, {5129,6743}, {5220,5437}, {5231,10584}, {5256,9330}, {5273,5785}, {5302,5438}, {5506,12658}, {5658,6684}, {5686,11019}, {5691,18250}, {5692,18421}, {5732,15064}, {5745,11407}, {5795,7990}, {6172,9814}, {6666,25568}, {7028,30395}, {7994,24644}, {7996,17355}, {7997,25440}, {9623,10176}, {9780,12526}, {9898,12053}, {10268,18524}, {13405,18230}, {15829,16189}, {18235,30363}, {24393,26105}, {30396,30412}, {30397,30413}, {10655,30414}, {10656,30415}

X(30393) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (210, 7308, 1), (4383, 7322, 1)


X(30394) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EXTANGENTS AND SUBMEDIAL

Barycentrics    a*(16*b*c*sin(A/2)+a^2+2*(b+c)*a-3*(b-c)^2) : :

The homothetic center of these triangles is X(9816). X(30394) is their endo-homothetic center only when ABC is acute.

X(30394) lies on these lines: {1,10502}, {165,173}, {177,30371}, {236,30393}, {1699,11891}, {5223,8126}, {5691,12582}, {7587,30389}, {7590,7987}, {7593,30326}, {7988,21624}, {7991,8351}, {8083,8090}, {8092,30343}, {8379,30308}, {8382,30315}, {8389,30330}, {8425,30363}, {8729,11407}, {9814,30404}, {9819,30411}, {11531,12445}, {11535,16189}, {11860,30291}, {11924,30337}, {18408,18421}, {18454,30392}, {30396,30406}, {30397,30407}, {10655,30409}, {10656,30410}

X(30394) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (173, 8423, 165), (173, 11195, 8423)


X(30395) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INTANGENTS AND SUBMEDIAL

Barycentrics    a*(-16*b*c*sin(A/2)+a^2+2*(b+c)*a-3*(b-c)^2) : :

The homothetic center of these triangles is X(9817). X(30395) is their endo-homothetic center only when ABC is acute.

X(30395) lies on these lines: {1,8099}, {165,258}, {174,10980}, {1699,9795}, {3339,30420}, {5223,8125}, {5691,12581}, {7028,30393}, {7588,30389}, {7987,8082}, {7988,21623}, {7991,8092}, {8080,30326}, {8084,8089}, {8086,30308}, {8088,30315}, {8094,11531}, {8242,30337}, {8250,30363}, {8351,30343}, {8388,30330}, {8423,11033}, {8734,11407}, {9814,30405}, {9819,30423}, {11859,30291}, {11889,30304}, {11899,16189}, {18409,18421}, {18456,30392}, {30396,30418}, {30397,30419}, {10655,30421}, {10656,30422}

X(30395) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (258, 8090, 165), (258, 11217, 8090)


X(30396) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st KENMOTU DIAGONALS AND SUBMEDIAL

Barycentrics    a*(-2*(a^2+2*(b+c)*a-3*(b-c)^2)*S+(a^2+2*(b-c)*a-(3*b+c)*(b-c))*(a^2-2*(b-c)*a+(b+3*c)*(b-c))) : :

The homothetic center of these triangles is X(10961). X(30396) is their endo-homothetic center only when ABC is acute.

X(30396) lies on these lines: {1,30375}, {7,1699}, {165,6203}, {1743,8941}, {3339,30425}, {30306,30308}, {30313,30315}, {30393,30412}

X(30396) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3062, 10980, 30397), (6203, 30354, 165)


X(30397) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd KENMOTU DIAGONALS AND SUBMEDIAL

Barycentrics    a*(2*(a^2+2*(b+c)*a-3*(b-c)^2)*S+(a^2+2*(b-c)*a-(3*b+c)*(b-c))*(a^2-2*(b-c)*a+(b+3*c)*(b-c))) : :

The homothetic center of these triangles is X(10963). X(30397) is their endo-homothetic center only when ABC is acute.

X(30397) lies on these lines: {1,30376}, {7,1699}, {165,6204}, {1743,8945}, {3339,30426}, {7987,30401}, {7988,30381}, {9819,30432}, {11407,30277}, {16189,30320}, {18411,18421}, {18460,30392}, {30289,30291}, {30307,30308}, {30314,30315}, {30325,30326}, {30334,30337}, {30342,30343}, {30347,30350}, {30361,30363}, {30369,30371}, {30386,30389}, {30393,30413}, {30394,30407}, {30395,30419}

X(30397) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3062, 10980, 30396), (6204, 30355, 165)


X(30398) = PERSPECTOR OF THESE TRIANGLES: INNER-SQUARES AND VERTEX TRIANGLE OF THE INNER AND OUTER TRI-EQUILATERAL

Barycentrics    a^2*(2*(a^4-(b^2+c^2)*a^2-2*b^2*c^2)*S+a^2*(a^4-b^4-c^4-4*b^2*c^2)) : :

X(30398) lies on these lines: {6,3156}, {140,141}, {157,371}, {159,8276}, {184,590}, {206,8969}, {491,5012}, {1151,12977}, {1503,23313}


X(30399) = PERSPECTOR OF THESE TRIANGLES: OUTER-SQUARES AND VERTEX TRIANGLE OF THE INNER AND OUTER TRI-EQUILATERAL

Barycentrics    a^2*(-2*(a^4-(b^2+c^2)*a^2-2*b^2*c^2)*S+a^2*(a^4-b^4-c^4-4*b^2*c^2)) : :

X(30399) lies on these lines: {6,3155}, {140,141}, {157,372}, {159,8277}, {184,615}, {206,13972}, {492,5012}, {1152,13068}, {1503,23314}


X(30400) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd EULER AND 1st KENMOTU DIAGONALS

Barycentrics    a*(a^5-3*(b+c)*a^4+2*(b-c)^2*a^3+2*(b+c)*(b^2+c^2)*a^2-(3*b^2+2*b*c+3*c^2)*(b-c)^2*a-(2*a^3-2*(b+c)*a^2-2*(b+c)^2*a+2*(b^2-c^2)*(b-c))*S+(b^2-c^2)^2*(b+c)) : :

The homothetic center of these triangles is X(10897). X(30400) is their endo-homothetic center only when ABC is acute.

X(30400) lies on these lines: {1,7}, {3,6203}, {4,30380}, {40,30296}, {46,18410}, {56,30375}, {84,2067}, {517,30319}, {936,30412}, {1490,30324}, {3333,30346}, {3423,6213}, {3576,30385}, {5587,30313}, {7590,30406}, {7987,30396}, {8081,30368}, {8082,30418}, {8227,30306}, {8235,30360}, {8726,30276}, {10391,13388}, {10864,30288}

X(30400) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 5732, 30401), (990, 4292, 30401)


X(30401) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd EULER AND 2nd KENMOTU DIAGONALS

Barycentrics    a*(a^5-3*(b+c)*a^4+2*(b-c)^2*a^3+2*(b+c)*(b^2+c^2)*a^2-(3*b^2+2*b*c+3*c^2)*(b-c)^2*a+(2*a^3-2*(b+c)*a^2-2*(b+c)^2*a+2*(b^2-c^2)*(b-c))*S+(b^2-c^2)^2*(b+c)) : :

The homothetic center of these triangles is X(10898). X(30401) is their endo-homothetic center only when ABC is acute.

X(30401) lies on these lines: {1,7}, {3,6204}, {4,30381}, {40,30297}, {46,18411}, {56,30376}, {84,6502}, {517,30320}, {936,30413}, {1490,30325}, {3333,30347}, {3423,6212}, {3576,30386}, {5587,30314}, {7590,30407}, {7987,30397}, {8081,30369}, {8082,30419}, {8227,30307}, {8235,30361}, {8726,30277}, {10391,13389}, {10864,30289}

X(30401) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 5732, 30400), (990, 4292, 30400)


X(30402) = HOMOTHETIC CENTER OF THESE TRIANGLES: INNER TRI-EQUILATERAL AND VERTEX TRIANGLE OF THE INNER AND OUTER TRI-EQUILATERAL

Barycentrics    a^2*(-2*sqrt(3)*(3*a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2)*S+(-a^2+c^2+b^2)*(a^2+c^2-b^2)*(a^2-c^2+b^2)) : :

X(30402) lies on these lines: {3,10639}, {6,25}, {15,6759}, {16,10282}, {26,10661}, {110,10681}, {156,10662}, {182,10643}, {1498,11480}, {1503,23302}, {1614,10632}, {2917,15962}, {5321,16252}, {6000,10645}, {6353,18929}, {7051,26888}, {9306,11515}, {9833,18582}, {10192,23303}, {10535,10638}, {10536,10636}, {10539,10634}, {10540,18468}, {10633,11466}, {10646,11202}, {10657,13289}, {10664,20773}, {10682,15647}, {11206,11488}, {11421,11452}, {11475,26883}, {11476,13367}, {11481,17821}, {11485,14530}, {11486,14819}, {16808,18400}, {16966,18381}, {19190,26887}

X(30402) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8740, 21648, 6), (10533, 10534, 11243), (11408, 19364, 6)


X(30403) = HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER TRI-EQUILATERAL AND VERTEX TRIANGLE OF THE INNER AND OUTER TRI-EQUILATERAL

Barycentrics    a^2*(2*sqrt(3)*(3*a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2)*S+(c^2-a^2+b^2)*(c^2+a^2-b^2)*(-c^2+a^2+b^2)) : :

X(30403) lies on these lines: {3,10640}, {6,25}, {15,10282}, {16,6759}, {26,10662}, {110,10682}, {156,10661}, {182,10644}, {1250,10535}, {1498,11481}, {1503,23303}, {1614,10633}, {2917,15961}, {5318,16252}, {6000,10646}, {6353,18930}, {9306,11516}, {9833,18581}, {10192,23302}, {10536,10637}, {10539,10635}, {10540,18470}, {10632,11467}, {10645,11202}, {10658,13289}, {10663,20773}, {10681,15647}, {11206,11489}, {11420,11453}, {11475,13367}, {11476,26883}, {11480,17821}, {11485,14818}, {11486,14530}, {16809,18400}, {16967,18381}, {19191,26887}, {19373,26888}

X(30403) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 17827, 11244), (154, 17827, 6), (10533, 10534, 11244)


X(30404) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd EHRMANN AND EXTANGENTS

Barycentrics    -2*(a^2-2*(b+c)*a+(b-c)^2)*sin(A/2)+3*(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(8539). X(30404) is their endo-homothetic center only when ABC is acute.

X(30404) lies on these lines: {7,174}, {173,8545}, {177,1156}, {236,6172}, {516,30411}, {527,8126}, {2801,18408}, {6173,8125}, {7587,8543}, {7589,30295}, {7590,8544}, {7593,12848}, {7671,8083}, {8092,30340}, {8379,30311}, {8382,30312}, {8423,30353}, {8425,30359}, {8729,30275}, {9814,30394}, {10502,15726}, {11535,30318}, {11860,30287}, {11924,30332}, {18450,18454}, {21624,30379}, {30408,30424}

X(30404) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 8389, 8388), (174, 30405, 8388), (8389, 30405, 174)


X(30405) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd EHRMANN AND INTANGENTS

Barycentrics    2*(a^2-2*(b+c)*a+(b-c)^2)*sin(A/2)+3*(a+b-c)*(a-b+c) : :
X(30405) = 3*X(8387)-4*X(13385)

The homothetic center of these triangles is X(8540). X(30405) is their endo-homothetic center only when ABC is acute.

X(30405) lies on these lines: {1,30367}, {7,174}, {258,8545}, {516,30423}, {527,8125}, {2801,18409}, {5542,30411}, {6172,7028}, {6173,8126}, {7588,8543}, {7671,11033}, {8076,30295}, {8080,12848}, {8082,8544}, {8086,30311}, {8088,30312}, {8090,30353}, {8242,30332}, {8250,30359}, {8351,30340}, {8387,13385}, {8734,30275}, {9814,30395}, {10501,15726}, {11859,30287}, {11899,30318}, {18450,18456}, {21623,30379}, {30420,30424}

X(30405) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 174, 30404), (7, 8388, 8389), (174, 30404, 8389)


X(30406) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EXTANGENTS AND 1st KENMOTU DIAGONALS

Barycentrics    (-2*S+a^2-2*(b+c)*a+(b-c)^2)*sin(A/2)-(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(5415). X(30406) is their endo-homothetic center only when ABC is acute.

X(30406) lies on these lines: {7,174}, {173,6203}, {177,7133}, {236,30412}, {7014,11923}, {7587,30385}, {7590,30400}, {7593,30324}, {8083,30346}, {8092,30341}, {8379,30306}, {8382,30313}, {8425,30360}, {8729,30276}, {10502,30375}, {11535,30319}, {11860,30288}, {11924,30333}, {18408,18410}, {18454,18458}, {21624,30380}, {30394,30396}, {30408,30425}, {30411,30431}

X(30406) = {X(7), X(174)}-harmonic conjugate of X(30418)


X(30407) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EXTANGENTS AND 2nd KENMOTU DIAGONALS

Barycentrics    (2*S+a^2-2*(b+c)*a+(b-c)^2)*sin(A/2)-(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(5416). X(30407) is their endo-homothetic center only when ABC is acute.

X(30407) lies on these lines: {7,174}, {173,6204}, {236,30413}, {7587,30386}, {7589,30297}, {7590,30401}, {7593,30325}, {8083,30347}, {8092,30342}, {8379,30307}, {8382,30314}, {8423,30355}, {8729,30277}, {10502,30376}, {11535,30320}, {11860,30289}, {11924,30334}, {18408,18411}, {18454,18460}, {21624,30381}, {30394,30397}, {30408,30426}, {30411,30432}

X(30407) = {X(7), X(174)}-harmonic conjugate of X(30419)


X(30408) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EXTANGENTS AND KOSNITA

Barycentrics    2*(a+b+c)*sin(A/2)+a : :

The homothetic center of these triangles is X(10902). X(30408) is their endo-homothetic center only when ABC is acute.

X(30408) lies on these lines: {1,167}, {10,8126}, {36,7587}, {40,8130}, {46,173}, {65,12491}, {79,1127}, {80,13267}, {236,1698}, {258,3338}, {516,8389}, {942,10502}, {1125,8125}, {1130,6724}, {2093,13098}, {3576,8129}, {3624,7028}, {3633,12646}, {3746,8076}, {4292,11890}, {4295,11891}, {5045,10501}, {5542,8388}, {5563,7588}, {5691,9837}, {5902,8094}, {5903,12445}, {6684,8128}, {7593,9612}, {7741,8379}, {7951,8382}, {7972,12748}, {8104,16173}, {8127,10165}, {8425,30362}, {11535,25415}, {11571,12774}, {11860,30290}, {12047,21624}, {12685,15071}, {13407,21623}, {30404,30424}, {30406,30425}, {30407,30426}, {30409,10651}, {30410,10652}

X(30408) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8092, 8351, 11924), (8092, 11924, 1), (30411, 30420, 1)


X(30409) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EXTANGENTS AND INNER TRI-EQUILATERAL

Barycentrics    (-2*sqrt(3)*S+a^2-2*(b+c)*a+(b-c)^2)*sin(A/2)-(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(10636). X(30409) is their endo-homothetic center only when ABC is acute.

X(30409) lies on these lines: {7,174}, {173,1653}, {177,30372}, {236,30414}, {7587,10647}, {7589,30300}, {7590,10649}, {7593,30327}, {8083,30351}, {8092,30344}, {8379,30309}, {8382,30316}, {8423,30356}, {8425,30364}, {11535,30321}, {11860,30292}, {11924,30338}, {18408,18422}, {18454,18469}, {30394,10655}, {30408,10651}, {30411,30433}

X(30409) = {X(7), X(174)}-harmonic conjugate of X(30421)


X(30410) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EXTANGENTS AND OUTER TRI-EQUILATERAL

Barycentrics    (2*sqrt(3)*S+a^2-2*(b+c)*a+(b-c)^2)*sin(A/2)-(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(10637). X(30410) is their endo-homothetic center only when ABC is acute.

X(30410) lies on these lines: {7,174}, {173,1652}, {177,1251}, {236,30415}, {7589,30301}, {7590,10650}, {7593,30328}, {8083,30352}, {8092,30345}, {8379,30310}, {8382,30317}, {8425,30365}, {8729,30281}, {10502,30378}, {11535,30322}, {11860,30293}, {11924,30339}, {18408,18423}, {18454,18471}, {21624,30383}, {30394,10656}, {30408,10652}, {30411,30434}

X(30410) = {X(7), X(174)}-harmonic conjugate of X(30422)


X(30411) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: EXTANGENTS AND TRINH

Barycentrics    2*(a+b+c)*sin(A/2)+3*a : :

The homothetic center of these triangles is X(7688). X(30411) is their endo-homothetic center only when ABC is acute.

X(30411) lies on these lines: {1,167}, {35,7587}, {36,7589}, {80,7707}, {173,5119}, {236,3679}, {354,18409}, {516,30404}, {517,10502}, {519,8126}, {551,8125}, {1699,9837}, {3057,12491}, {3586,7593}, {3632,12646}, {4294,12570}, {4304,11890}, {5049,10501}, {5441,16151}, {5542,30405}, {5697,12445}, {5902,8083}, {7028,25055}, {7741,8382}, {7951,8379}, {7972,13267}, {7982,8130}, {8094,18398}, {8100,17609}, {8128,11362}, {8389,30331}, {8425,30366}, {8729,30282}, {9819,30394}, {10500,18399}, {10624,12582}, {11535,30323}, {11860,30294}, {11891,30305}, {21624,30384}, {30406,30431}, {30407,30432}, {30409,30433}, {30410,30434}

X(30411) = reflection of X(18408) in X(10502)
X(30411) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 174, 30423), (1, 177, 30374), (174, 30423, 30420)


X(30412) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st EXCOSINE AND 1st KENMOTU DIAGONALS

Barycentrics    a*(-a+b+c)+S : :

The homothetic center of these triangles is X(17819). X(30412) is their endo-homothetic center only when ABC is acute.

X(30412) lies on these lines: {1,7586}, {2,7}, {6,6351}, {8,7090}, {10,1132}, {37,3069}, {44,3068}, {45,615}, {169,8231}, {188,15892}, {190,1267}, {198,16440}, {210,13359}, {219,13387}, {236,30406}, {281,1586}, {344,492}, {346,30416}, {354,13360}, {518,30346}, {590,16885}, {591,17243}, {936,30400}, {958,30385}, {962,6213}, {997,18458}, {1100,19053}, {1123,1124}, {1125,30341}, {1270,3912}, {1271,4416}, {1329,30313}, {1376,30296}, {1600,15817}, {1698,30425}, {1743,5393}, {2324,3083}, {2886,30306}, {3300,13905}, {3593,25101}, {3679,30431}, {3731,5405}, {3973,8972}, {4643,5591}, {4851,5860}, {5391,17277}, {5414,30334}, {5590,17279}, {5692,18410}, {5704,8957}, {6347,20262}, {7028,30418}, {8580,30354}, {9780,14121}, {11292,25066}, {13759,29574}, {13847,16675}, {15829,30319}, {16669,19054}, {18227,30288}, {18235,30360}, {30393,30396}

X(30412) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (329, 27382, 30413), (5279, 27540, 30413), (5282, 27547, 30413)


X(30413) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st EXCOSINE AND 2nd KENMOTU DIAGONALS

Barycentrics    a*(-a+b+c)-S : :

The homothetic center of these triangles is X(17820). X(30413) is their endo-homothetic center only when ABC is acute.

X(30413) lies on these lines: {1,7585}, {2,7}, {6,6352}, {8,14121}, {10,1131}, {37,3068}, {44,3069}, {45,590}, {188,30369}, {190,5391}, {198,16441}, {210,13360}, {219,13386}, {236,30407}, {281,1585}, {344,491}, {346,15891}, {354,13359}, {518,30347}, {615,16885}, {936,30401}, {938,8957}, {958,30386}, {962,6212}, {997,18460}, {1100,19054}, {1123,9646}, {1125,30342}, {1267,17277}, {1270,4416}, {1271,3912}, {1329,30314}, {1335,1336}, {1376,30297}, {1599,15817}, {1698,30426}, {1743,5405}, {1991,17243}, {2066,30333}, {2324,3084}, {2886,30307}, {3302,13963}, {3595,25101}, {3679,30432}, {3731,5393}, {3973,13941}, {4643,5590}, {4851,5861}, {5591,17279}, {5692,18411}, {6348,20262}, {7028,30419}, {7090,9780}, {8580,30355}, {11291,25066}, {13639,29574}, {13846,16675}, {15829,30320}, {16669,19053}, {18227,30289}, {18235,30361}, {30393,30397}

X(30413) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (329, 27382, 30412), (5279, 27540, 30412), (5282, 27547, 30412)


X(30414) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st EXCOSINE AND INNER TRI-EQUILATERAL

Barycentrics    a*(-a+b+c)+sqrt(3)*S : :

The homothetic center of these triangles is X(17826). X(30414) is their endo-homothetic center only when ABC is acute.

X(30414) lies on these lines: {2,7}, {8,5240}, {10,22237}, {37,11489}, {44,11488}, {45,23303}, {145,5245}, {219,5367}, {236,30409}, {281,471}, {302,344}, {395,16777}, {518,30351}, {936,10649}, {958,10647}, {962,1277}, {997,18469}, {1125,30344}, {1329,30316}, {1376,30300}, {1698,10651}, {2886,30309}, {3241,7026}, {3616,5239}, {3617,5246}, {3639,5199}, {3679,30433}, {5692,18422}, {7028,30421}, {8580,30356}, {9761,17243}, {11790,16808}, {16645,16675}, {16885,23302}, {18227,30292}, {18235,30364}, {30393,10655}

X(30414) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9, 5242, 2), (3306, 27508, 30415), (5328, 5749, 30415)


X(30415) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st EXCOSINE AND OUTER TRI-EQUILATERAL

Barycentrics    a*(-a+b+c)-sqrt(3)*S : :

The homothetic center of these triangles is X(17827). X(30415) is their endo-homothetic center only when ABC is acute.

X(30415) lies on these lines: {2,7}, {8,1251}, {10,22235}, {37,11488}, {44,11489}, {45,23302}, {145,5246}, {188,30373}, {210,30378}, {219,5362}, {236,30410}, {281,470}, {303,344}, {396,16777}, {518,30352}, {936,10650}, {958,10648}, {962,1276}, {997,18471}, {1125,30345}, {1329,30317}, {1376,30301}, {1698,10652}, {2886,30310}, {3179,4295}, {3241,7043}, {3616,5240}, {3617,5245}, {3638,5199}, {3679,30434}, {5692,18423}, {7028,30422}, {8580,30357}, {9763,17243}, {11791,16809}, {15829,30322}, {16644,16675}, {16885,23303}, {18227,30293}, {18235,30365}, {30393,10656}

X(30415) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 9, 30414), (3306, 27508, 30414), (5328, 5749, 30414)


X(30416) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st EXCOSINE AND LUCAS ANTIPODAL TANGENTIAL

Barycentrics    (a*(a+b+c)+S)*((a+b-c)*c+S)*((a-b+c)*b+S)*(-a+b+c) : :

The homothetic center of these triangles is X(19430). X(30416) is their endo-homothetic center only when ABC is acute.

X(30416) lies on these lines: {7,7090}, {8,30335}, {9,30429}, {346,15892}, {518,30348}, {958,30387}, {3616,16213}, {5273,30302}, {5745,30278}


X(30417) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st EXCOSINE AND LUCAS(-1) ANTIPODAL TANGENTIAL

Barycentrics    (a*(a+b+c)-S)*((a+b-c)*c-S)*((a-b+c)*b-S)*(-a+b+c) : :

The homothetic center of these triangles is X(19431). X(30417) is their endo-homothetic center only when ABC is acute.

X(30417) lies on these lines: {7,14121}, {8,30336}, {9,30430}, {346,15891}, {518,30349}, {958,30388}, {3616,16214}, {5273,30303}, {5745,30279}


X(30418) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INTANGENTS AND 1st KENMOTU DIAGONALS

Barycentrics    (-2*S+a^2-2*(b+c)*a+(b-c)^2)*sin(A/2)+(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(2066). X(30418) is their endo-homothetic center only when ABC is acute.

X(30418) lies on these lines: {1,30368}, {7,174}, {258,6203}, {1489,7014}, {7028,30412}, {7133,8248}, {7588,30385}, {8076,30296}, {8080,30324}, {8082,30400}, {8086,30306}, {8088,30313}, {8090,30354}, {8242,30333}, {8250,30360}, {8734,30276}, {10501,30375}, {11033,30346}, {11859,30288}, {11899,30319}, {18409,18410}, {18456,18458}, {21623,30380}, {30395,30396}, {30420,30425}, {30423,30431}

X(30418) = {X(7), X(174)}-harmonic conjugate of X(30406)


X(30419) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INTANGENTS AND 2nd KENMOTU DIAGONALS

Barycentrics    (a^2-2*(b+c)*a+2*S+(b-c)^2)*sin(A/2)+(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(5414). X(30419) is their endo-homothetic center only when ABC is acute.

X(30419) lies on these lines: {1,30369}, {7,174}, {258,6204}, {7028,30413}, {7588,30386}, {8076,30297}, {8080,30325}, {8082,30401}, {8086,30307}, {8088,30314}, {8090,30355}, {8242,30334}, {8250,30361}, {8351,30342}, {8734,30277}, {10501,30376}, {11033,30347}, {11859,30289}, {11899,30320}, {18409,18411}, {18456,18460}, {21623,30381}, {30395,30397}, {30420,30426}, {30423,30432}

X(30419) = {X(7), X(174)}-harmonic conjugate of X(30407)


X(30420) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INTANGENTS AND KOSNITA

Barycentrics    2*(a+b+c)*sin(A/2)-a : :

The homothetic center of these triangles is X(35). X(30420) is their endo-homothetic center only when ABC is acute.

X(30420) lies on these lines: {1,167}, {10,8125}, {35,8076}, {36,7588}, {40,8129}, {46,258}, {65,8100}, {79,16147}, {80,8104}, {173,3338}, {236,3624}, {354,12491}, {516,8388}, {942,10501}, {1125,8126}, {1130,6732}, {1698,7028}, {2093,8102}, {3339,30395}, {3576,8130}, {3633,12644}, {3746,7589}, {4292,11889}, {4295,9795}, {5045,10502}, {5557,7707}, {5563,7587}, {5902,11217}, {5903,8094}, {6684,8127}, {6724,10231}, {7741,8086}, {7951,8088}, {7972,8098}, {8080,9612}, {8096,15071}, {8128,10165}, {8250,30362}, {8734,15803}, {10023,18291}, {11033,18398}, {11571,12772}, {11859,30290}, {11899,25415}, {12047,21623}, {13267,16173}, {13407,21624}, {30405,30424}, {30418,30425}, {30419,30426}, {30421,10651}, {30422,10652}

X(30420) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 8092, 30423), (174, 8092, 1), (8092, 8351, 8242)


X(30421) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INTANGENTS AND INNER TRI-EQUILATERAL

Barycentrics    (a^2-2*(b+c)*a+(b-c)^2-2*sqrt(3)*S)*sin(A/2)+(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(10638). X(30421) is their endo-homothetic center only when ABC is acute.

X(30421) lies on these lines: {1,30372}, {7,174}, {258,1653}, {7028,30414}, {7588,10647}, {8076,30300}, {8080,30327}, {8082,10649}, {8086,30309}, {8088,30316}, {8090,30356}, {8242,30338}, {8250,30364}, {8734,30280}, {10501,30377}, {11033,30351}, {11859,30292}, {11899,30321}, {18409,18422}, {18456,18469}, {21623,30382}, {30395,10655}, {30420,10651}, {30423,30433}

X(30421) = {X(7), X(174)}-harmonic conjugate of X(30409)


X(30422) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INTANGENTS AND OUTER TRI-EQUILATERAL

Barycentrics    (a^2-2*(b+c)*a+(b-c)^2+2*sqrt(3)*S)*sin(A/2)+(a+b-c)*(a-b+c) : :

The homothetic center of these triangles is X(1250). X(30422) is their endo-homothetic center only when ABC is acute.

X(30422) lies on these lines: {1,30373}, {7,174}, {258,1652}, {7028,30415}, {7588,10648}, {8076,30301}, {8080,30328}, {8082,10650}, {8086,30310}, {8088,30317}, {8090,30357}, {8242,30339}, {8250,30365}, {8351,30345}, {8734,30281}, {10501,30378}, {11033,30352}, {11859,30293}, {11899,30322}, {18409,18423}, {18456,18471}, {21623,30383}, {30395,10656}, {30420,10652}, {30423,30434}

X(30422) = {X(7), X(174)}-harmonic conjugate of X(30410)


X(30423) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INTANGENTS AND TRINH

Barycentrics    2*(a+b+c)*sin(A/2)-3*a : :
X(30423) = 3*X(1)-2*X(13385) = 4*X(13385)-3*X(30370)

The homothetic center of these triangles is X(36). X(30423) is their endo-homothetic center only when ABC is acute.

X(30423) lies on these lines: {1,167}, {35,7588}, {36,8076}, {80,8098}, {236,25055}, {258,5119}, {354,18408}, {516,30405}, {517,10501}, {519,8125}, {551,8126}, {3057,8100}, {3586,8080}, {3632,12644}, {3679,7028}, {4294,12569}, {4304,11889}, {5049,10502}, {5441,16147}, {5542,30404}, {5697,8094}, {6732,10231}, {7741,8088}, {7951,8086}, {7972,8104}, {7982,8129}, {8127,11362}, {8250,30366}, {8388,30331}, {8734,30282}, {9795,30305}, {9819,30395}, {10624,12581}, {11859,30294}, {11899,30323}, {12445,18398}, {12491,17609}, {17641,18399}, {21623,30384}, {30418,30431}, {30419,30432}, {30421,30433}, {30422,30434}

X(30423) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 8092, 30420), (8092, 8242, 1), (30411, 30420, 174)


X(30424) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd EHRMANN AND KOSNITA

Barycentrics    4*a^3+(b+c)*a^2-2*(b-c)^2*a-3*(b^2-c^2)*(b-c) : :
X(30424) = X(1)-3*X(7) = 5*X(1)-3*X(390) = X(1)+3*X(4312) = 2*X(1)-3*X(5542) = 11*X(1)-9*X(8236) = 7*X(1)-9*X(11038) = 4*X(1)-3*X(30331) = 3*X(1)-X(30332) = 3*X(1)-5*X(30340) = 5*X(7)-X(390) = 11*X(7)-3*X(8236) = 7*X(7)-3*X(11038) = 4*X(7)-X(30331) = 9*X(7)-X(30332) = 9*X(7)-5*X(30340) = 3*X(9)-4*X(3634) = 3*X(10)-2*X(5220) = X(390)+5*X(4312) = 2*X(390)-5*X(5542) = 11*X(390)-15*X(8236) = X(5220)-3*X(5880)

The homothetic center of these triangles is X(575). X(30424) is their endo-homothetic center only when ABC is acute.

X(30424) lies on these lines: {1,7}, {9,3634}, {10,527}, {11,4031}, {12,11662}, {35,30295}, {36,8543}, {46,3947}, {55,3982}, {57,1776}, {65,2801}, {79,1156}, {142,3647}, {144,9780}, {226,1155}, {354,4114}, {382,17706}, {515,1159}, {518,3625}, {528,3244}, {551,5126}, {553,1836}, {673,16477}, {758,5784}, {938,5586}, {942,15726}, {946,20418}, {954,5217}, {971,5884}, {1001,5267}, {1056,28228}, {1088,10136}, {1125,5698}, {1698,6172}, {1699,21454}, {1743,7613}, {2094,5231}, {2550,3626}, {3008,24695}, {3062,5556}, {3339,9814}, {3474,4654}, {3487,12512}, {3488,28158}, {3579,5762}, {3617,5223}, {3755,17365}, {3826,10592}, {3868,5696}, {3874,15733}, {3883,7321}, {3923,21255}, {3935,17483}, {3950,28526}, {4052,29649}, {4078,17767}, {4652,5550}, {4663,5845}, {4847,20292}, {4851,28557}, {5221,5729}, {5435,10171}, {5493,21620}, {5551,10390}, {5572,13369}, {5708,5805}, {5728,17637}, {5819,16670}, {5843,18357}, {5851,12019}, {5852,24393}, {5902,10394}, {5905,21060}, {6006,21201}, {6147,8255}, {6738,9579}, {6767,28232}, {7263,28570}, {7671,18398}, {7741,30311}, {7951,30312}, {9612,12848}, {9812,10980}, {11009,14151}, {11529,28164}, {12047,30379}, {12609,15823}, {12699,21625}, {12702,12872}, {14100,15009}, {14563,28160}, {15803,30275}, {15934,28150}, {15935,28154}, {16125,18482}, {17274,19868}, {17298,24280}, {17376,28530}, {18230,19872}, {18421,28236}, {20103,28609}, {20533,29601}, {21635,24465}, {25415,30318}, {30287,30290}, {30359,30362}, {30367,30370}, {30404,30408}, {30405,30420}

X(30424) = midpoint of X(i) and X(j) for these {i,j}: {7, 4312}, {3868, 5696}, {5223, 20059}
X(30424) = reflection of X(i) in X(j) for these (i,j): (10, 5880), (5542, 7), (5698, 1125), (14100, 20116), (30331, 5542)
X(30424) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 30332, 30340), (4307, 4862, 4353), (30332, 30340, 1)


X(30425) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st KENMOTU DIAGONALS AND KOSNITA

Barycentrics    -2*S*a+3*a^3-(b-c)^2*a-2*(b^2-c^2)*(b-c) : :

The homothetic center of these triangles is X(372). X(30425) is their endo-homothetic center only when ABC is acute.

X(30425) lies on these lines: {1,7}, {35,30296}, {36,30385}, {46,486}, {65,18410}, {79,7133}, {942,30375}, {1478,9907}, {1698,30412}, {1836,13388}, {3339,30396}, {3474,13390}, {7741,30306}, {7951,30313}, {9612,30324}, {11246,13389}, {12047,30380}, {13436,28849}, {15803,30276}, {18398,30346}, {25415,30319}, {30288,30290}, {30360,30362}, {30368,30370}, {30406,30408}, {30418,30420}

X(30425) = reflection of X(1) in X(481)
X(30425) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (390, 30342, 1), (4292, 24248, 30426), (30333, 30341, 1)


X(30426) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd KENMOTU DIAGONALS AND KOSNITA

Barycentrics    2*S*a+3*a^3-(b-c)^2*a-2*(b^2-c^2)*(b-c) : :

The homothetic center of these triangles is X(371). X(30426) is their endo-homothetic center only when ABC is acute.

X(30426) lies on these lines: {1,7}, {35,30297}, {36,8225}, {46,485}, {65,18411}, {942,30376}, {1478,9906}, {1659,3474}, {1698,30413}, {1836,13389}, {3339,30397}, {7741,30307}, {7951,30314}, {9612,30325}, {11246,13388}, {12047,30381}, {13453,28849}, {15803,30277}, {18398,30347}, {25415,30320}, {30289,30290}, {30361,30362}, {30369,30370}, {30407,30408}, {30419,30420}

X(30426) = reflection of X(1) in X(482)
X(30426) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (7, 30334, 30342), (390, 30341, 1), (30334, 30342, 1)


X(30427) = PERSPECTOR OF THESE TRIANGLES: INNER-VECTEN AND VERTEX TRIANGLE OF THE INNER AND OUTER TRI-EQUILATERAL

Barycentrics    a^2*(a^8-3*(b^2+c^2)*a^6+3*(b^4+c^4)*a^4+2*(b^2-c^2)^2*b^2*c^2+2*(a^4-b^4-c^4)*S*a^2-(b^4-c^4)*(b^2-c^2)*a^2) : :
X(30427) = 3*X(154)+X(12970)

X(30427) lies on these lines: {5,182}, {6,8946}, {110,489}, {154,1151}, {184,3071}, {486,50004}, {3594,17843}, {5409,13056}, {8964,9306}, {8967,30398}, {8996,10666}, {12229,19148}

X(30427) = midpoint of X(8996) and X(10666)
X(30427) = {X(206), X(6759)}-harmonic conjugate of X(30428)


X(30428) = PERSPECTOR OF THESE TRIANGLES: OUTER-VECTEN AND VERTEX TRIANGLE OF THE INNER AND OUTER TRI-EQUILATERAL

Barycentrics    a^2*(a^8-3*(b^2+c^2)*a^6+3*(b^4+c^4)*a^4+2*(b^2-c^2)^2*b^2*c^2-2*(a^4-b^4-c^4)*S*a^2-(b^4-c^4)*(b^2-c^2)*a^2) : :
X(30428) = 3*X(154)+X(12964)

X(30428) lies on these lines: {5,182}, {6,8948}, {110,490}, {154,1152}, {184,3070}, {485,30398}, {3592,17840}, {5408,13055}, {8968,23313}, {9306,13027}, {12230,19147}

X(30428) = {X(206), X(6759)}-harmonic conjugate of X(30427)


X(30429) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS ANTIPODAL TANGENTIAL AND ORTHIC

Barycentrics    a*((a+b+c)*(a^2-2*(b+c)*a+(b-c)^2)*(a^3+(b+c)*a^2-(b+c)^2*a-(b^2-c^2)*(b-c))-S*(a^4+6*(b+c)*a^3+8*b*c*a^2-2*(b+c)*(3*b^2+2*b*c+3*c^2)*a-(b^2+6*b*c+c^2)*(b-c)^2)) : :

The homothetic center of these triangles is X(19446). X(30429) is their endo-homothetic center only when ABC is acute.

X(30429) lies on these lines: {1,30348}, {9,30416}, {40,971}, {57,7133}, {63,30302}, {1697,30335}


X(30430) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: LUCAS(-1) ANTIPODAL TANGENTIAL AND ORTHIC

Barycentrics    a*((a+b+c)*(a^2-2*(b+c)*a+(b-c)^2)*(a^3+(b+c)*a^2-(b+c)^2*a-(b^2-c^2)*(b-c))+S*(a^4+6*(b+c)*a^3+8*b*c*a^2-2*(b+c)*(3*b^2+2*b*c+3*c^2)*a-(b^2+6*b*c+c^2)*(b-c)^2)) : :

The homothetic center of these triangles is X(19447). X(30430) is their endo-homothetic center only when ABC is acute.

X(30430) lies on these lines: {1,30349}, {9,30417}, {40,971}, {57,30279}, {63,30303}, {1697,30336}


X(30431) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 1st KENMOTU DIAGONALS AND TRINH

Barycentrics    -6*S*a+5*a^3-4*(b+c)*a^2+(b-c)^2*a-2*(b^2-c^2)*(b-c) : :

The homothetic center of these triangles is X(6200). X(30431) is their endo-homothetic center only when ABC is acute.

X(30431) lies on these lines: {1,7}, {36,30296}, {80,7133}, {517,18410}, {3586,30324}, {3679,30412}, {5119,6203}, {5902,30346}, {7741,30313}, {7951,30306}, {9819,30396}, {30276,30282}, {30288,30294}, {30319,30323}, {30360,30366}, {30368,30374}, {30380,30384}, {30406,30411}, {30418,30423}

X(30431) = reflection of X(18410) in X(30375)


X(30432) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: 2nd KENMOTU DIAGONALS AND TRINH

Barycentrics    6*S*a+5*a^3-4*(b+c)*a^2+(b-c)^2*a-2*(b^2-c^2)*(b-c) : :

The homothetic center of these triangles is X(6396). X(30432) is their endo-homothetic center only when ABC is acute.

X(30432) lies on these lines: {1,7}, {35,30386}, {36,30297}, {517,18411}, {3586,30325}, {3679,30413}, {5119,6204}, {5902,30347}, {7741,30314}, {7951,30307}, {9819,30397}, {30277,30282}, {30289,30294}, {30320,30323}, {30361,30366}, {30369,30374}, {30381,30384}, {30407,30411}, {30419,30423}

X(30432) = reflection of X(18411) in X(30376)


X(30433) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: INNER TRI-EQUILATERAL AND TRINH

Barycentrics    -6*sqrt(3)*S*a+5*a^3-4*(b+c)*a^2+(b-c)^2*a-2*(b^2-c^2)*(b-c) : :

The homothetic center of these triangles is X(10645). X(30433) is their endo-homothetic center only when ABC is acute.

X(30433) lies on these lines: {1,7}, {14,80}, {35,10647}, {36,30300}, {517,18422}, {1653,5119}, {3586,30327}, {3679,30414}, {5902,30351}, {7741,30316}, {7951,30309}, {9819,10655}, {30280,30282}, {30292,30294}, {30321,30323}, {30364,30366}, {30372,30374}, {30382,30384}, {30409,30411}, {30421,30423}

X(30433) = reflection of X(18422) in X(30377)
X(30433) = {X(1), X(4312)}-harmonic conjugate of X(16038)


X(30434) = ENDO-HOMOTHETIC CENTER OF THESE TRIANGLES: OUTER TRI-EQUILATERAL AND TRINH

Barycentrics    6*sqrt(3)*S*a+5*a^3-4*(b+c)*a^2+(b-c)^2*a-2*(b^2-c^2)*(b-c) : :

The homothetic center of these triangles is X(10646). X(30434) is their endo-homothetic center only when ABC is acute.

X(30434) lies on these lines: {1,7}, {13,80}, {35,10648}, {36,30301}, {517,18423}, {1652,5119}, {3586,30328}, {3679,30415}, {5902,30352}, {7741,30317}, {7951,30310}, {9819,10656}, {30281,30282}, {30293,30294}, {30322,30323}, {30365,30366}, {30373,30374}, {30383,30384}, {30410,30411}, {30422,30423}

X(30434) = reflection of X(18423) in X(30378)


X(30435) = ISOGONAL CONJUGATE X(18840)

Barycentrics    a^2 (3 a^2 + b^2 + c^2) : :
Barycentrics    S^2 - SB SC - 2 SB SW - 2 SC SW : :
Trilinears    2 sin A - cos A tan ω : :
Trilinears    cos A - 2 sin A cot ω : :
Trilinears    a - R cos A tan ω : :
X(30435) = X[7896]-2*X[7915]

See Tran Quang Hung and Ercole Suppa, Hyacinthos 28746.

X(30435) lies on these lines: {2,7762}, {3,6}, {4,3172}, {5,7735}, {9,5266}, {20,1285}, {25,251}, {30,5286}, {31,218}, {41,16466}, {55,5280}, {56,609}, {69,7819}, {76,11286}, {81,11343}, {83,183}, {99,7894}, {101,1191}, {112,1593}, {115,3843}, {140,7736}, {141,14023}, {159,15257}, {169,1104}, {172,999}, {193,3933}, {194,1003}, {198,16470}, {217,19347}, {220,595}, {230,1656}, {232,3517}, {237,11402}, {248,3527}, {315,7792}, {316,7851}, {378,8778}, {381,3767}, {382,5254}, {384,7754}, {385,7770}, {393,6756}, {405,5276}, {441,11433}, {524,7795}, {550,7738}, {598,15031}, {599,7822}, {754,7784}, {940,16783}, {942,16780}, {966,17698}, {980,21509}, {988,16667}, {995,3207}, {1078,7878}, {1181,8779}, {1184,3291}, {1186,3511}, {1194,9909}, {1249,7487}, {1385,9575}, {1472,1496}, {1482,1572}, {1506,5070}, {1595,3087}, {1597,1968}, {1598,2207}, {1617,4548}, {1627,7484}, {1657,2549}, {1724,19761}, {1743,3965}, {1914,3295}, {1915,8780}, {1971,14530}, {1975,3972}, {1992,3926}, {1995,5354}, {2070,16308}, {2138,17409}, {2229,16396}, {2241,6767}, {2242,7373}, {2300,20818}, {2493,7506}, {2896,7875}, {3051,3167}, {3052,3730}, {3148,9777}, {3224,3499}, {3303,16785}, {3304,16784}, {3314,10583}, {3329,7793}, {3407,12206}, {3509,16787}, {3523,14930}, {3526,3815}, {3528,14482}, {3534,7739}, {3552,7839}, {3567,9475}, {3579,9593}, {3589,7800}, {3618,3785}, {3629,7758}, {3734,7805}, {3744,17742}, {3749,3991}, {3763,7854}, {3788,7838}, {3830,5309}, {3849,7872}, {3851,5475}, {3934,8667}, {4383,5337}, {4386,9709}, {4426,9708}, {5025,20088}, {5032,11165}, {5054,9300}, {5055,7746}, {5073,5355}, {5077,7802}, {5275,11108}, {5277,16408}, {5283,16418}, {5523,12173}, {5710,16788}, {5938,12167}, {6090,9463}, {6144,7820}, {6392,14033}, {6655,7920}, {6656,16989}, {6660,20977}, {6680,7759}, {6792,15000}, {7375,8974}, {7376,13950}, {7388,13763}, {7389,13644}, {7574,16306}, {7581,21736}, {7585,11292}, {7586,11291}, {7750,7803}, {7751,7804}, {7756,15681}, {7761,7829}, {7763,11288}, {7765,17800}, {7768,7846}, {7769,11163}, {7773,7812}, {7774,7807}, {7775,7886}, {7779,7881}, {7780,7808}, {7785,7806}, {7788,7832}, {7797,7823}, {7798,7816}, {7801,7890}, {7809,7942}, {7811,7859}, {7817,7825}, {7818,7852}, {7835,7905}, {7836,7837}, {7840,7945}, {7842,7902}, {7843,7844}, {7845,7867}, {7848,7914}, {7850,7944}, {7857,7858}, {7860,7919}, {7864,14712}, {7869,7882}, {7873,7913}, {7874,7903}, {7880,7916}, {7883,7943}, {7884,7911}, {7885,7932}, {7891,13571}, {7896,7915}, {7897,14043}, {7898,7923}, {7899,7926}, {7900,7901}, {7909,7949}, {7912,16984}, {7917,7930}, {7928,9939}, {7929,7948}, {7931,7946}, {8364,14929}, {8550,8721}, {8744,10594}, {8879,15809}, {8882,19173}, {9310,16483}, {9327,16486}, {9490,18899}, {9592,13624}, {9607,15696}, {9609,13564}, {9620,12702}, {9715,22240}, {9969,20993}, {10306,10315}, {10313,11414}, {10314,11484}, {11313,13758}, {11314,13638}, {11321,16998}, {11335,20023}, {11610,11641}, {11648,15684}, {12164,23128}, {12174,13509}, {12188,12829}, {12203,14532}, {12308,14901}, {13735,27523}, {14003,26869}, {14269,14537}, {14581,18535}, {14602,20854}, {14974,21793}, {14996,21516}, {14997,21540}, {15270,19153}, {15589,16045}, {15720,21843}, {16042,21448}, {16060,17379}, {16061,17349}, {16394,26035}, {16589,16857}, {16918,16995}, {17001,17541}, {17002,17686}, {17597,17736}, {18494,27376}, {19118,27369}, {19125,20960}, {19767,21982}, {20897,26864}

X(30435) = isogonal conjugate of X(18840)
X(30435) = midpoint of X(8396) and X(8416)
X(30435) = reflection of X(i) in X(j) for these {i,j}: {7784,7834}, {7896,7915}
X(30435) = crossdifference of every pair of points on line X(523)X(2525)
X(30435) = intersection of tangents at PU(1) to hyperbola {{X(6),PU(1),PU(2)}}
X(30435) = inverse-in-1st-Brocard-circle of X(9605)
X(30435) = inverse-in-circle-{X(371),X(372),PU(1),PU(39)} of X(1350)
X(30435) = radical center of Lucas(-4 cot ω) circles
X(30435) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {2,7762,7776}, {2,7893,7879}, {3,6,9605}, {3,32,1384}, {3,5093,3095}, {3,9605,5024}, {3,21309,32}, {4,5304,5305}, {6,32,3}, {6,574,22246}, {6,800,15851}, {6,1333,5120}, {6,1384,5024}, {6,2220,4254}, {6,3053,39}, {6,4252,4253}, {6,4258,386}, {6,5008,21309}, {6,5013,7772}, {6,5052,5093}, {6,6423,3312}, {6,6424,3311}, {6,12963,6422}, {6,12968,6421}, {6,13345,8573}, {6,21309,1384}, {6,22331,5013}, {32,39,3053}, {32,187,22331}, {32,5007,6}, {32,5041,5023}, {32,7772,187}, {32,13356,2080}, {32,14075,7772}, {39,187,15515}, {39,3053,3}, {39,5206,15815}, {39,15515,5013}, {41,21764,16466}, {61,62,11477}, {83,6179,183}, {172,5332,16502}, {172,16502,999}, {187,5013,3}, {187,7772,5013}, {187,14075,6}, {193,14001,3933}, {230,2548,1656}, {251,5359,25}, {315,7792,7866}, {316,7856,7851}, {371,372,1350}, {384,7766,7754}, {385,7787,7770}, {574,5023,3}, {609,5299,56}, {1078,7878,11174}, {1384,5024,15655}, {1384,9605,3}, {1692,2031,2080}, {1975,7760,22253}, {2207,10311,1598}, {2242,16781,7373}, {3053,15815,5206}, {3329,7793,11285}, {3618,3785,8362}, {3629,7789,7758}, {3767,7745,381}, {3788,7838,9766}, {3793,8362,3785}, {3972,7760,1975}, {4264,5037,6}, {4383,5337,21526}, {5007,5008,32}, {5007,21309,9605}, {5013,22331,187}, {5032,19661,11165}, {5052,13357,3095}, {5085,5188,3}, {5093,13357,9605}, {5206,15815,3}, {5254,7737,382}, {5280,7031,55}, {5305,18907,4}, {5306,7745,3767}, {5319,7737,5254}, {5368,7747,5309}, {5475,7755,13881}, {5475,13881,3851}, {6421,12968,6398}, {6422,12963,6221}, {6680,7759,7778}, {7750,7803,11287}, {7768,7846,7868}, {7772,15515,39}, {7772,22331,3}, {7773,7828,11318}, {7779,7892,7881}, {7780,7808,15271}, {7785,7806,7887}, {7797,7823,7841}, {7812,7828,7773}, {7822,7826,599}, {7832,7877,7788}, {7854,7889,3763}, {8573,10317,1384}, {8743,10312,25}, {12150,14614,11286}, {16989,20065,6656}


X(30436) = X(5)X(113)∩X(10)X(27555)

Barycentrics    (b+c) (-2 a^4 b^2-3 a^3 b^3+a^2 b^4+3 a b^5+b^6-a^3 b^2 c-2 a^2 b^3 c+a b^4 c+2 b^5 c-2 a^4 c^2-a^3 b c^2-2 a^2 b^2 c^2-4 a b^3 c^2-b^4 c^2-3 a^3 c^3-2 a^2 b c^3-4 a b^2 c^3-4 b^3 c^3+a^2 c^4+a b c^4-b^2 c^4+3 a c^5+2 b c^5+c^6) : :
X(30436) = 5 X[1698] - X[2940]

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28747.

X(30436) lies on these lines: {5,113}, {10,27555}, {12,8287}, {115,24443}, {429,1861}, {442,22798}, {542,3615}, {857,29610}, {1495,9958}, {1698,2940}, {1737,14873}, {2392,3142}, {2899,27704}, {3013,8614}, {3136,25972}, {5044,22076}, {5221,8818}, {5587,27685}, {6723,24904}, {6739,18357}, {7173,8286}, {9780,27554}, {10175,27687}

X(30436) = X(26734)-complementary conjugate of X(3741)


X(30437) = X(7)X(2808)∩X(674)X(7671)

Barycentrics    a^2*((b^2+c^2)*a^4-2*(b^3+c^3)*a^3-b^2*c^2*a^2+2*(b^3-c^3)*(b^2-c^2)*a-(b^4+c^4+(2*b^2+b*c+2*c^2)*b*c)*(b-c)^2) : :
X(30437) = X(7)-4*X(29957)

See Dao Thanh Oai and César Lozada, Hyacinthos 28749.

X(30437) lies on these lines: {7, 2808}, {674, 7671}, {942, 15058}, {2772, 5902}, {2836, 11188}, {5889, 10399}, {8236, 9052}, {10122, 11444}


X(30438) = X(8)X(29958)∩X(392)X(23155)

Barycentrics    a^2*((b^2+c^2)*a^3+(b^2-c^2)*(b-c)*a^2-(b^4-b^2*c^2+c^4)*a-(b+c)*(b^4+c^4-(2*b^2-b*c+2*c^2)*b*c)) : :
X(30438) = X(8)-4*X(29958), 5*X(3616)-2*X(23154), 3*X(5640)-2*X(5902), 3*X(5640)-4*X(15049), 2*X(5693)+X(5889), 4*X(5694)-X(11412), 4*X(5883)-5*X(11451), 4*X(5884)-7*X(15043), 8*X(5885)-11*X(15024), 3*X(7998)-4*X(10176), 5*X(10574)-2*X(15071), 5*X(11444)-8*X(20117), 10*X(15016)-13*X(15028)

See Dao Thanh Oai and César Lozada, Hyacinthos 28749.

X(30438) lies on these lines: {8, 29958}, {392, 23155}, {511, 7985}, {513, 17579}, {758, 3060}, {1464, 19245}, {2392, 2979}, {2771, 5890}, {2779, 15305}, {2810, 3241}, {2836, 11188}, {2842, 5640}, {3616, 23154}, {3877, 8679}, {4511, 26892}, {5693, 5889}, {5694, 11412}, {5752, 11684}, {5883, 11451}, {5884, 15043}, {5885, 15024}, {6126, 10546}, {7998, 10176}, {10574, 15071}, {11346, 24482}, {11444, 20117}, {15016, 15028}

X(30438) = reflection of X(i) in X(j) for these (i,j): (2979, 5692), (5902, 15049), (23155, 392)
X(30438) = {X(5902), X(15049)}-harmonic conjugate of X(5640)


X(30439) = REFLECTION OF X(13) IN X(11624)

Barycentrics    (SB+SC)*(2*S^2+3*sqrt(3)*R^2*S+(9*R^2-2*SW)*SA) : :
X(30439) = 3*X(5640)-X(16259)

See Dao Thanh Oai and César Lozada, Hyacinthos 28749.

X(30439) lies on these lines: {3, 6}, {4, 11581}, {13, 5663}, {14, 5640}, {17, 11459}, {18, 13363}, {531, 25165}, {1154, 16962}, {3411, 12006}, {3412, 5889}, {5946, 11626}, {6104, 14170}, {6780, 16637}, {7998, 16241}, {8929, 15441}, {10654, 11002}, {11455, 12816}, {13754, 16267}, {15045, 16963}, {15072, 16965}, {16261, 16808}

X(30439) = reflection of X(13) in X(11624)


X(30440) = REFLECTION OF X(13) IN X(11626)

Barycentrics    (SB+SC)*(2*S^2-3*sqrt(3)*R^2*S+(9*R^2-2*SW)*SA) : :
X(30440) = 3*X(5640)-X(16260)

See Dao Thanh Oai and César Lozada, Hyacinthos 28749.

X(30440) lies on these lines: {3, 6}, {4, 11582}, {13, 5640}, {14, 5663}, {17, 13363}, {18, 11459}, {530, 25155}, {1154, 16963}, {3411, 5889}, {3412, 12006}, {5946, 11624}, {6105, 14169}, {6779, 16636}, {7998, 16242}, {8930, 15442}, {10653, 11002}, {11455, 12817}, {13754, 16268}, {15045, 16962}, {15072, 16964}, {16261, 16809}

X(30440) = reflection of X(13) in X(11626)


X(30441) = BARYCENTRIC PRODUCT X(99)*X(15318)

Barycentrics    (SA-SB)*(SA-SC)*(SB^2-4*R^2*SB+SC*SA)*(SC^2-4*R^2*SC+SA*SB) : :

See Tran Quang Hung and César Lozada, Hyacinthos 28750.

X(30441) lies on this line: {2404, 14570}

X(30441) = isogonal conjugate of X(30442)
X(30441) = barycentric product X(99)*X(15318)
X(30441) = barycentric quotient X(i)/X(j) for these (i,j): (99, 20477), (110, 6759)
X(30441) = trilinear product X(i)*X(j) for these {i,j}: {662, 15318}, {811, 18890}
X(30441) = trilinear quotient X(i)/X(j) for these (i,j): (662, 6759), (799, 20477)


X(30442) = X(6)X(2430)∩X(421)X(2501)

Barycentrics    (SB^2-SC^2)*(SA^2-4*R^2*SA+SB*SC) : :

See Tran Quang Hung and César Lozada, Hyacinthos 28750.

X(30442) lies on these lines: {6, 2430}, {421, 2501}, {647, 657}, {2485, 17434}, {3265, 10601}, {3288, 7927}, {6753, 14398}, {11792, 15140}, {15609, 23438}

X(30442) = isogonal conjugate of X(30441)


X(30443) = X(30)X(21651)∩X(51)X(5895)

Barycentrics    a^2 ( a^12 (b^2+c^2)-4 a^10 (b^2-c^2)^2+5 a^8 (b^2-c^2)^2 (b^2+c^2) -40 a^6 b^2 c^2 (b^2-c^2)^2-5 a^4 (b^2-c^2)^2 (b^6-9 b^4 c^2-9 b^2 c^4+c^6) +4 a^2 (b^2-c^2)^2 (b^8-2 b^6 c^2-14 b^4 c^4-2 b^2 c^6+c^8) -(b^2-c^2)^4 (b^6+7 b^4 c^2+7 b^2 c^4+c^6)) : :

See Tran Quang Hung and Angel Montesdeoca, Hyacinthos 28754.

X(30443) lies on these lines: {30,21651}, {51,5895}, {185,1885}, {373,5893}, {1498,6090}, {1906,6247}, {2777,21649}, {3357,15030}, {3917,5894}, {4319,7355}, {4320,6285}, {5650,8567}, {5878,6816}, {6225,7386}, {10575,12362}, {10990,15105}, {12163,13093}, {12315,14641}, {13417,16879}, {14642,17807}

X(30443) = the reflection of X(i) in X(j), for these {i, j}: {5562,20427}, {11381,64}, {12315,14641}


X(30444) = EULER LINE INTERCEPT OF X(12)X(3931)

Barycentrics    (b+c) (a^5 b+a^4 b^2-a b^5-b^6+a^5 c+2 a^3 b^2 c-3 a b^4 c+a^4 c^2+2 a^3 b c^2+4 a^2 b^2 c^2+4 a b^3 c^2+b^4 c^2+4 a b^2 c^3-3 a b c^4+b^2 c^4-a c^5-c^6) : :
X(30444) = 5 X[1698] - X[2941]

See Tran Quang Hung and Peter Moses, Hyacinthos 28751.

X(30444) lies on these lines: {2, 3}, {12, 3931}, {115, 119}, {120, 5512}, {127, 25640}, {321, 17757}, {339, 21664}, {355, 1834}, {496, 5716}, {517, 1211}, {946, 3454}, {952, 17015}, {1213, 1766}, {1245, 21935}, {1329, 5955}, {1698, 2941}, {2345, 3820}, {3936, 5603}, {5016, 24390}, {5019, 5475}, {5706, 5810}, {5752, 5799}, {5886, 17056}, {5887, 10974}, {7682, 17052}, {7951, 17594}, {10175, 12618}, {10381, 24474}, {14672, 20621}

X(30444) = complement of X(4221)
X(30444) = midpoint of X(4) and X(4220)
X(30444) = {X(429),X(442)}-harmonic conjugate of X(21530)
X(30444) = X(i)-complementary conjugate of X(j) for these (i,j): {3420, 1125}, {9107, 8062}
X(30444) = X(9058)-Ceva conjugate of X(523)


X(30445) = EULER LINE INTERCEPT OF X(12)X(18588)

Barycentrics    (b + c)*(-a^2 + b^2 + c^2)*(a^6*b - a^4*b^3 - a^2*b^5 + b^7 + a^6*c + 2*a^5*b*c + 3*a^4*b^2*c + 2*a^3*b^3*c + a^2*b^4*c - b^6*c + 3*a^4*b*c^2 - 3*b^5*c^2 - a^4*c^3 + 2*a^3*b*c^3 + 3*b^4*c^3 + a^2*b*c^4 + 3*b^3*c^4 - a^2*c^5 - 3*b^2*c^5 - b*c^6 + c^7) : :

See Tran Quang Hung and Peter Moses, Hyacinthos 28751.

X(30445) lies on these lines: {2, 3}, {12, 18588}, {115, 18591}, {119, 127}, {120, 14672}, {339, 1234}, {1060, 17720}, {3822, 18589}, {4463, 17757}, {7951, 10319}, {10202, 18635}

X(30445) = complement of X(4227)
X(30445) = X(i)-complementary conjugate of X(j) for these (i,j): {998, 942}, {9058, 8062}


X(30446) = EULER LINE INTERCEPT OF X(12)X(3743)

Barycentrics    (b + c)*(a^5*b + a^4*b^2 - a*b^5 - b^6 + a^5*c + a^3*b^2*c - 2*a*b^4*c + a^4*c^2 + a^3*b*c^2 + 2*a^2*b^2*c^2 + 3*a*b^3*c^2 + b^4*c^2 + 3*a*b^2*c^3 - 2*a*b*c^4 + b^2*c^4 - a*c^5 - c^6) : :

See Tran Quang Hung and Peter Moses, Hyacinthos 28751.

X(30446) lies on these lines: {2, 3}, {12, 3743}, {80, 1834}, {119, 137}, {1089, 3704}, {1211, 3878}, {1213, 16548}, {3454, 11813}, {5443, 17056}, {20625, 25640}

X(30446) = X(26711)-Ceva conjugate of X(523)
X(30446) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {5, 1904, 11113}


X(30447) = EULER LINE INTERCEPT OF X(12)X(502)

Barycentrics    (b+c) (a^5 b+a^4 b^2-a b^5-b^6+a^5 c-a^3 b^2 c+a^4 c^2-a^3 b c^2-2 a^2 b^2 c^2+a b^3 c^2+b^4 c^2+a b^2 c^3+b^2 c^4-a c^5-c^6) : :

See Tran Quang Hung and Peter Moses, Hyacinthos 28751.

X(30447) lies on the cubic K720 and these lines: {2, 3}, {12, 502}, {115, 3290}, {119, 3258}, {120, 5099}, {125, 517}, {515, 6739}, {523, 1577}, {758, 10693}, {1211, 10176}, {1213, 16547}, {1290, 5080}, {1737, 8287}, {3454, 25079}, {3814, 5520}, {4872, 23674}, {5074, 21253}, {8286, 30384}, {9956, 30436}, {16177, 25640}

X(30447) = complement of X(1325)
X(30447) = midpoint of X(1290) and X(5080)
X(30447) = reflection of X(5520) and X(3814)
X(30447) = circumcircle-inverse of X(2915)
X(30447) = nine point circle inverse of X(442)
X(30447) = polar circle inverse of X(28)
X(30447) = orthoptic circle of the Steiner inellipe inverse of X(4220)
X(30447) = complement of the isogonal of X(10693)
X(30447) = X(i)-complementary conjugate of X(j) for these (i,j): {2766, 8062}, {10693, 10}
X(30447) = X(i)-Ceva conjugate of X(j) for these (i,j): {1290, 523}, {5080, 758}
X(30447) = crosssum of X(184) and X(19622)
X(30447) = crossdifference of every pair of points on line {647, 1333}
X(30447) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {2, 16430, 7483}, {5, 25648, 4187}, {5, 27687, 442}, {429, 21530, 442}, {1113, 1114, 2915}, {1312, 1313, 442}, {5142, 24933, 5}, {27553, 27685, 3}, {27554, 27686, 4}, {27555, 27687, 5}, {27561, 27693, 22}, {27562, 27694, 23}, {27578, 27715, 20}, {27579, 27716, 21}, {27580, 27717, 25}, {27581, 27718, 27}, {27582, 27721, 376}, {27583, 27722, 377}, {27584, 27723, 384}


X(30448) = EULER LINE INTERCEPT OF X(12)X(986)

Barycentrics    a^5*b^2 - a^3*b^4 + a^2*b^5 - b^7 + 2*a^5*b*c - a^3*b^3*c - a*b^5*c + a^5*c^2 + 2*a^3*b^2*c^2 + a^2*b^3*c^2 + 2*b^5*c^2 - a^3*b*c^3 + a^2*b^2*c^3 + 2*a*b^3*c^3 - b^4*c^3 - a^3*c^4 - b^3*c^4 + a^2*c^5 - a*b*c^5 + 2*b^2*c^5 - c^7 : :
X(30448) = 5 X[1698] - X[21375]

See Tran Quang Hung and Peter Moses, Hyacinthos 28760.

X(30448) lies on these lines: {2, 3}, {12, 986}, {119, 5518}, {517, 2887}, {970, 3454}, {1698, 21375}, {3821, 3822}, {4417, 9567}, {5254, 22380}, {6211, 26446}, {7680, 29243}, {7951, 17596}, {13323, 20083}

X(30448) = {X(6881),X(30444)}-harmonic conjugate of X(5)


X(30449) = EULER LINE INTERCEPT OF X(12)X(4424)

Barycentrics    (b + c)*(-(a^5*b) + a^3*b^3 - a^2*b^4 + b^6 - a^5*c - a^3*b^2*c + a^2*b^3*c + 2*a*b^4*c - b^5*c - a^3*b*c^2 - 2*a^2*b^2*c^2 - 2*a*b^3*c^2 - b^4*c^2 + a^3*c^3 + a^2*b*c^3 - 2*a*b^2*c^3 + 2*b^3*c^3 - a^2*c^4 + 2*a*b*c^4 - b^2*c^4 - b*c^5 + c^6) : :

See Tran Quang Hung and Peter Moses, Hyacinthos 28751.

X(30449) lies on these lines: {2, 3}, {12, 4424}, {115, 22425}, {517, 3454}, {952, 1834}, {1211, 5690}, {1482, 3936}, {3073, 20575}, {3814, 24850}, {5510, 9955}, {5901, 17056}, {7680, 12621}, {7951, 24851}, {10974, 14988}, {12610, 21245}

X(30449) = X(15617)-complementary conjugate of X(1125)
X(30449) = {X(5),X(5499)}-harmonic conjugate of X(15973)


X(30450) = X(4)X(6754)∩X(107)X(925)

Barycentrics    a*b^3*c^3*(a^2-b^2)*(a^2-c^2)*(a^2-b^2+c^2)*(a^2+b^2-c^2)*(a^4-2*b^2*a^2+(b^2-c^2)^2)*(a^4-2*c^2*a^2+(b^2-c^2)^2) : :

See César Lozada, Hyacinthos 28756.

X(30450) lies on these lines: {4, 6754}, {107, 925}, {687, 4558}, {847, 17983}, {2165, 16081}, {5392, 15466}, {6330, 20563}, {6528, 16813}, {15352, 16237}, {18817, 18883}

X(30450) = isogonal conjugate of X(30451)


X(30451) = X(6)X(2501)∩X(184)X(669)

Barycentrics    a^4*(-a^2+b^2+c^2)*(a^4-2*(b^2+c^2)*a^2+b^4+c^4)*(b^2-c^2) : :
X(30451) = 2*X(6753)-3*X(14397)

See César Lozada, Hyacinthos 28756.

X(30451) lies on these lines: {6, 2501}, {184, 669}, {520, 647}, {523, 2623}, {526, 16040}, {826, 3288}, {924, 6753}, {1181, 1499}, {1409, 7180}, {1899, 23301}, {1993, 6563}, {2451, 12077}, {5926, 19357}, {9009, 19459}, {10601, 14341}, {11422, 11450}, {13366, 21646}

X(30451) = isogonal conjugate of X(30450)


X(30452) = BARYCENTRIC PRODUCT X(13)*X(115)

Barycentrics    (b - c)^2*(b + c)^2*(3*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) + 4*S*(Sqrt[3]*a^2 + S))::

X(30452) lies on the Simmons inconic with foci X(13) and X(15) and on these lines: {13, 531}, {298, 11118}, {300, 18896}, {1989, 3457}, {2381, 22510}, {3124, 30453}, {6531, 8737}, {8014, 18777}

X(30452) = X(13)-Ceva conjugate of X(20578)
X(30452) = crosspoint of X(13) and X(20578)
X(30452) = crosssum of X(15) and X(17402)
X(30452) = crossdifference of every pair of points on line {10411, 17402}
X(30452) = X(i)-isoconjugate of X(j) for these (i,j): {15, 24041}, {298, 1101}, {662, 17402}, {2151, 4590}
X(30452) = barycentric product X(i) X(j) for these {i,j}: {13, 115}, {125, 8737}, {300, 3124}, {338, 3457}, {523, 20578}, {1109, 2153}, {5995, 23105}, {6138, 10412}, {8029, 23895}, {15475, 23871}, {20579, 23283}
X(30452) = barycentric quotient X(i) / X(j) for these {i,j}: {13, 4590}, {115, 298}, {512, 17402}, {2153, 24041}, {2971, 8739}, {3124, 15}, {3457, 249}, {6138, 10411}, {8029, 23870}, {8737, 18020}, {8754, 470}, {15475, 23896}, {20578, 99}, {22260, 6137}


X(30453) = BARYCENTRIC PRODUCT X(14)*X(115)

Barycentrics    (b - c)^2*(b + c)^2*(3*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) - 4*S*(Sqrt[3]*a^2 - S))::

X(30453) lies on the Simmons inconic with foci X(14) and X(16) and on these lines: {14, 530}, {299, 11117}, {301, 18896}, {1989, 3458}, {2380, 22511}, {3124, 30452}, {6531, 8738}, {8015, 18776}

X(30453) = X(14)-Ceva conjugate of X(20579)
X(30453) = crosspoint of X(14) and X(20579)
X(30453) = crosssum of X(16) and X(17403)
X(30453) = crossdifference of every pair of points on line {10411, 17403}
X(30453) = X(i)-isoconjugate of X(j) for these (i,j): {16, 24041}, {299, 1101}, {662, 17403}, {2152, 4590}
X(30453) = barycentric product X(i) X(j) for these {i,j}: {14, 115}, {125, 8738}, {301, 3124}, {338, 3458}, {523, 20579}, {1109, 2154}, {5994, 23105}, {6137, 10412}, {8029, 23896}, {15475, 23870}, {20578, 23284}
X(30453) = barycentric quotient X(i) / X(j) for these {i,j}: {14, 4590}, {115, 299}, {512, 17403}, {2154, 24041}, {2971, 8740}, {3124, 16}, {3458, 249}, {6137, 10411}, {8029, 23871}, {8738, 18020}, {8754, 471}, {15475, 23895}, {20579, 99}, {22260, 6138}


X(30454) = BARYCENTRIC PRODUCT X(13)*X(2482)

Barycentrics    (2*a^2 - b^2 - c^2)^2*(3*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) + 4*S*(Sqrt[3]*a^2 + S))::
X(30454) = X[16256] - 3 X[16962]

X(30454) lies on the Simmons inconic with foci X(13) and X(15) and on these lines: {13, 531}, {396, 18777}, {690, 9117}, {3180, 21466}, {16256, 16962}

X(30454) = barycentric product X(i) X(j) for these {i,j}: {13, 2482}, {1649, 23895}, {2153, 24038}, {9205, 14559}
X(30454) = barycentric quotient X(i) / X(j) for these {i,j}: {1649, 23870}, {2482, 298}, {3457, 10630}, {5095, 470}


X(30455) = BARYCENTRIC PRODUCT X(14)*X(2482)

Barycentrics    (2*a^2 - b^2 - c^2)^2*(3*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) - 4*S*(Sqrt[3]*a^2 - S))::
X(30455) = X[16255] - 3 X[16963]

X(30455) lies on the Simmons inconic with foci X(14) and X(16) and on these lines: on lines {14, 530}, {395, 18776}, {690, 9115}, {3181, 21467}, {16255, 16963}

X(30455) = barycentric product X(i) X(j) for these {i,j}: {14, 2482}, {1649, 23896}, {2154, 24038}, {9204, 14559}
X(30455) = barycentric quotient X(i) / X(j) for these {i,j}: {1649, 23871}, {2482, 299}, {3458, 10630}, {5095, 471}


X(30456) = X(9)X(223)∩X(37)X(73)

Barycentrics    a*(a + b - c)*(a - b + c)*(b + c)*(3*a^4 - 2*a^2*b^2 - b^4 - 2*a^2*c^2 + 2*b^2*c^2 - c^4): :

X(30456) lies on the cubic K1090 and these lines: {1, 5776}, {6, 19}, {9, 223}, {37, 73}, {48, 1455}, {71, 227}, {154, 204}, {198, 1035}, {213, 8898}, {219, 21147}, {225, 1901}, {241, 27623}, {278, 5746}, {573, 15498}, {579, 1465}, {581, 12664}, {604, 1104}, {610, 1394}, {651, 5279}, {828, 18591}, {910, 2199}, {965, 1038}, {966, 10361}, {1060, 5778}, {1108, 1457}, {1400, 1427}, {1402, 7083}, {1441, 5749}, {1451, 5115}, {1765, 17102}, {1766, 7078}, {1838, 5798}, {2269, 15852}, {2287, 4296}, {3209, 3556}, {3694, 4551}, {3931, 12705}, {3990, 4559}, {4343, 14100}, {5227, 9370}, {5317, 26888}, {5712, 5928}, {5930, 8804}, {14110, 22134}, {18623, 18750}

X(30456) = X(i)-Ceva conjugate of X(j) for these (i,j): {9, 1400}, {223, 73}, {1214, 65}, {5930, 3198}, {18623, 5930}
X(30456) = X(i)-isoconjugate of X(j) for these (i,j): {6, 5931}, {21, 2184}, {29, 1073}, {64, 333}, {253, 284}, {283, 459}, {314, 2155}, {1172, 19611}, {1301, 6332}, {2287, 8809}, {6514, 6526}, {8748, 15394}
X(30456) = crosspoint of X(i) and X(j) for these (i,j): {9, 27382}, {610, 1249}, {1394, 18623}
X(30456) = crosssum of X(i) and X(j) for these (i,j): {1, 5776}, {1073, 2184}
X(30456) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {6, 1841, 2262}, {37, 3330, 1903}, {1409, 1880, 65}, {1456, 2264, 608}
X(30456) = barycentric product X(i) X(j) for these {i,j}: {1, 5930}, {7, 3198}, {10, 1394}, {20, 65}, {37, 18623}, {57, 8804}, {73, 1895}, {108, 8057}, {109, 17898}, {154, 1441}, {204, 307}, {226, 610}, {306, 3213}, {651, 6587}, {934, 14308}, {1020, 14331}, {1214, 1249}, {1231, 3172}, {1400, 18750}, {1402, 14615}, {1409, 15466}, {1427, 27382}, {3344, 8807}, {3668, 7070}, {4551, 21172}, {14249, 22341}
X(30456) = barycentric quotient X(i) / X(j) for these {i,j}: {1, 5931}, {20, 314}, {65, 253}, {73, 19611}, {154, 21}, {204, 29}, {610, 333}, {1042, 8809}, {1394, 86}, {1400, 2184}, {1402, 64}, {1409, 1073}, {1880, 459}, {3172, 1172}, {3198, 8}, {3213, 27}, {5930, 75}, {6525, 1896}, {6587, 4391}, {7070, 1043}, {7156, 2322}, {8804, 312}, {14308, 4397}, {15905, 1812}, {18623, 274}, {18750, 28660}, {21172, 18155}, {22341, 15394}


X(30457) = X(9)X(223)∩X(64)X(71)

Barycentrics    a^2*(a - b - c)*(a^4 - 2*a^2*b^2 + b^4 + 2*a^2*c^2 + 2*b^2*c^2 - 3*c^4)*(a^4 + 2*a^2*b^2 - 3*b^4 - 2*a^2*c^2 + 2*b^2*c^2 + c^4): :

The trilinear polar of X(30457) passes through X(4105).

Let A'B'C' and A"B"C" be the Hutson intouch and anti-Hutson intouch triangles, resp. Let A* be the barycentric product A'*A", and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(30457). (Randy Hutson, January 15, 2019)

X(30457) lies on the cubic K1090 and these lines: {6, 7367}, {9, 223}, {37, 2331}, {64, 71}, {253, 6559}, {345, 5931}, {393, 5514}, {480, 2318}, {579, 2338}, {728, 2324}, {2192, 7037}, {2911, 14642}, {5776, 20226}

X(30457) = isogonal conjugate of X(18623)
X(30457) = X(2184)-Ceva conjugate of X(64)
X(30457) = X(i)-cross conjugate of X(j) for these (i,j): {607, 55}, {1400, 9}
X(30457) = crosspoint of X(281) and X(8805)
X(30457) = crosssum of X(610) and X(1394)
X(30457) = X(i)-isoconjugate of X(j) for these (i,j): {1, 18623}, {2, 1394}, {7, 610}, {20, 57}, {56, 18750}, {69, 3213}, {77, 1249}, {81, 5930}, {85, 154}, {204, 348}, {222, 1895}, {269, 27382}, {273, 15905}, {279, 7070}, {603, 15466}, {604, 14615}, {651, 21172}, {934, 14331}, {1014, 8804}, {1414, 6587}, {1434, 3198}, {2184, 7338}, {3172, 7182}, {4565, 17898}, {4637, 14308}, {6525, 7183}, {7056, 7156}, {7125, 14249}
X(30457) = barycentric product X(i) X(j) for these {i,j}: {8, 64}, {9, 2184}, {33, 19611}, {42, 5931}, {55, 253}, {200, 8809}, {219, 459}, {281, 1073}, {312, 2155}, {318, 19614}, {1259, 6526}, {1857, 15394}, {3343, 8805}, {7017, 14642}, {7068, 15384}
X(30457) = barycentric quotient X(i) / X(j) for these {i,j}: {6, 18623}, {8, 14615}, {9, 18750}, {31, 1394}, {33, 1895}, {41, 610}, {42, 5930}, {55, 20}, {64, 7}, {154, 7338}, {220, 27382}, {253, 6063}, {281, 15466}, {459, 331}, {607, 1249}, {657, 14331}, {663, 21172}, {1073, 348}, {1253, 7070}, {1334, 8804}, {1857, 14249}, {1973, 3213}, {2155, 57}, {2175, 154}, {2184, 85}, {2212, 204}, {3709, 6587}, {4041, 17898}, {4524, 14308}, {5931, 310}, {6059, 6525}, {7070, 1097}, {8809, 1088}, {14379, 1804}, {14642, 222}, {15394, 7055}, {19611, 7182}, {19614, 77}


X(30458) = (name pending)

Barycentrics    35150 a^16-224435 a^14 b^2+621425 a^12 b^4-960475 a^10 b^6+875725 a^8 b^8-439825 a^6 b^10+77435 a^4 b^12+25375 a^2 b^14-10375 b^16-224435 a^14 c^2+868714 a^12 b^2 c^2-1207083 a^10 b^4 c^2+532540 a^8 b^6 c^2+253363 a^6 b^8 c^2-202074 a^4 b^10 c^2-91925 a^2 b^12 c^2+70900 b^14 c^2+621425 a^12 c^4-1207083 a^10 b^2 c^4+516660 a^8 b^4 c^4+87335 a^6 b^6 c^4+76038 a^4 b^8 c^4+123525 a^2 b^10 c^4-217900 b^12 c^4-960475 a^10 c^6+532540 a^8 b^2 c^6+87335 a^6 b^4 c^6+97202 a^4 b^6 c^6-56975 a^2 b^8 c^6+399500 b^10 c^6+875725 a^8 c^8+253363 a^6 b^2 c^8+76038 a^4 b^4 c^8-56975 a^2 b^6 c^8-484250 b^8 c^8-439825 a^6 c^10-202074 a^4 b^2 c^10+123525 a^2 b^4 c^10+399500 b^6 c^10+77435 a^4 c^12-91925 a^2 b^2 c^12-217900 b^4 c^12+25375 a^2 c^14+70900 b^2 c^14-10375 c^16 : :
Barycentrics    99960 S^4+S^2 (42201 R^4-158760 SB SC-38772 R^2 SW+860 SW^2)+SB SC (-2187 R^4-18468 R^2 SW+23340 SW^2) : :

As a point on the Euler line, X(30458) has Shinagawa coefficients {42201 R^4 + 99960 S^2 - 38772 R^2 SW + 860 SW^2, -3 (729 R^4 + 52920 S^2 + 6156 R^2 SW - 7780 SW^2).

See Kadir Altintas and Ercole Suppa, Hyacinthos 28759.

X(30458) lies on this line: {2,3}


X(30459) = REFLECTION OF X(30452) IN X(396)

Barycentrics    3*(12*R^2-5*SA-2*SW)*S^2+sqrt(3)*(14*S^2+3*SA^2-18*SB*SC-3*SW^2)*S+9*SB*SC*SW : :

Centers X(30459)-X(30479) were contributed by César Eliud Lozada (December 26, 2018).

X(30459) lies on the Simmons inconic with foci {X(13), X(15)} and these lines: {2,14}, {13,1338}, {30,30460}, {395,15778}, {396,30452}, {523,30461}, {533,14921}, {9117,13305}, {16529,22738}

X(30459) = reflection of X(30452) in X(396)
X(30459) = antipode of X(30452) in the Simmons inconic with foci {X(13), X(15)}


X(30460) = REFLECTION OF X(30452) IN THE LINE X(13)X(15)

Barycentrics    (SB-SC)^2*(S^2+sqrt(3)*(9*R^2-2*SW)*S-3*(3*R^2-SA)*SA) : :

X(30460) lies on the Simmons inconic with foci {X(13), X(15)} and these lines: {13,476}, {30,30459}, {115,12077}, {323,532}, {523,30452}, {3258,15610}, {11537,30454}, {12079,20578}

X(30460) = antipode of X(30461) in the Simmons inconic with foci {X(13), X(15)}


X(30461) = REFLECTION OF X(30460) IN X(396)

Barycentrics
-(2*(4*a^12-(8*(b^2+c^2))*a^10+(8*((b^2+c^2)^2-b^2*c^2))*a^8-(8*(b^6+c^6))*a^6+(2*(b^8+c^8+6*(b^2-c^2)^2*b^2*c^2))*a^4-(4*(b^2*c^2-(b^2-c^2)^2))*(b^4-c^4)*(b^2-c^2)*a^2-(2*b^4+b^2*c^2+2*c^4)*(b^2-c^2)^4))*S+sqrt(3)*(4*a^14-(12*(b^2+c^2))*a^12+(8*(b^4+5*b^2*c^2+c^4))*a^10-(8*(4*b^2*c^2-(b^2-c^2)^2))*(b^2+c^2)*a^8-(12*(-b^2*c^6-b^6*c^2+b^8+c^8-3*b^4*c^4))*a^6+(4*(b^2+c^2))*((b^4-c^4)^2-b^4*c^4)*a^4-(b^2-c^2)^2*(3*b^4+2*b^2*c^2+3*c^4)*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2)^3*b^2*c^2) : :

X(30461) lies on the Simmons inconic with foci {X(13), X(15)} and these lines: {15,30467}, {30,30452}, {395,30465}, {523,30459}

X(30461) = antipode of X(30460) in the Simmons inconic with foci {X(13), X(15)}


X(30462) = REFLECTION OF X(30453) IN X(395)

Barycentrics    3*(12*R^2-5*SA-2*SW)*S^2-sqrt(3)*(14*S^2+3*SA^2-18*SB*SC-3*SW^2)*S+9*SB*SC*SW : :

X(30462) lies on the Simmons inconic with foci {X(14), X(16)} and these lines: {2,13}, {14,1337}, {30,30463}, {395,30453}, {523,30464}, {532,14922}, {9115,13304}, {16530,22739}

X(30462) = reflection of X(30453) in X(395)
X(30462) = antipode of X(30453) in the Simmons inconic with foci {X(14), X(16)}


X(30463) = REFLECTION OF X(30453) IN THE LINE X(14)X(16)

Barycentrics    (SB-SC)^2*(S^2-sqrt(3)*(9*R^2-2*SW)*S-3*(3*R^2-SA)*SA) : :

X(30463) lies on the Simmons inconic with foci {X(14), X(16)} and these lines: {14,476}, {115,12077}, {323,533}, {523,30453}, {3258,15609}, {11549,30455}, {12079,20579}

X(30463) = antipode of X(30464) in the Simmons inconic with foci {X(14), X(16)}
X(30463) = reflection of X(30464) in X(395)


X(30464) = REFLECTION OF X(30463) IN X(395)

Barycentrics
2*(4*a^12-8*(b^2+c^2)*a^10+8*((b^2+c^2)^2-b^2*c^2)*a^8-8*(b^6+c^6)*a^6+2*(b^8+c^8+6*(b^2-c^2)^2*b^2*c^2)*a^4-4*(b^2*c^2-(b^2-c^2)^2)*(b^4-c^4)*(b^2-c^2)*a^2-(2*b^4+b^2*c^2+2*c^4)*(b^2-c^2)^4)*S+sqrt(3)*(4*a^14-12*(b^2+c^2)*a^12+8*(b^4+5*b^2*c^2+c^4)*a^10-8*(4*b^2*c^2-(b^2-c^2)^2)*(b^2+c^2)*a^8-12*(-b^2*c^6-b^6*c^2+b^8+c^8-3*b^4*c^4)*a^6+4*(b^2+c^2)*((b^4-c^4)^2-b^4*c^4)*a^4-(b^2-c^2)^2*(3*b^4+2*b^2*c^2+3*c^4)*b^2*c^2*a^2-(b^4-c^4)*(b^2-c^2)^3*b^2*c^2) : :

X(30464) lies on the Simmons inconic with foci {X(14), X(16)} and these lines: {16,30470}, {30,30453}, {395,30463}, {396,30468}, {523,30462}

X(30464) = antipode of X(30463) in the Simmons inconic with foci {X(14), X(16)}


X(30465) = REFLECTION OF X(30454) IN X(396)

Barycentrics    (S+sqrt(3)*SA)*(2*S^2+SA^2+2*SB*SC-SW^2) : :

X(30465) lies on the Simmons inconic with foci {X(13), X(15)} and these lines: {2,14}, {5,14816}, {13,5916}, {30,30466}, {110,6777}, {115,125}, {338,30453}, {395,30461}, {396,18777}, {523,30467}, {1316,22512}, {3258,15609}, {3448,6778}, {3457,22513}, {5318,8014}, {10545,16809}, {10653,11658}, {11078,23005}, {12079,20578}, {23283,30452}

X(30465) = midpoint of X(13) and X(16256)
X(30465) = reflection of X(30454) in X(396)
X(30465) = antipode of X(30454) in the Simmons inconic with foci {X(13), X(15)}
X(30465) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (115, 125, 30468), (868, 1648, 30468), (13636, 13722, 30468)


X(30466) = REFLECTION OF X(30454) IN THE LINE X(13)X(15)

Barycentrics    (27*R^2*(12*R^2-SA-6*SW)+6*SA^2+12*SB*SC+22*SW^2)*S^2-S*sqrt(3)*(2*S^2+SA^2+2*SB*SC-SW^2)*(9*R^2-2*SW)+3*(27*R^2-8*SW)*SB*SC*SW : :

X(30466) lies on the Simmons inconic with foci {X(13), X(15)} and these lines: {13,476}, {23,6104}, {30,30465}, {115,30469}, {396,30467}, {523,30454}, {11537,30452}, {11549,18777}

X(30466) = reflection of X(30467) in X(396)
X(30466) = antipode of X(30467) in the Simmons inconic with foci {X(13), X(15)}


X(30467) = REFLECTION OF X(30466) IN X(396)

Barycentrics
(b^2-c^2)^2*(2*(3*b^2*c^2*(b^4+c^4)+4*a^8-12*(b^2+c^2)*a^6+2*(5*b^4+8*b^2*c^2+5*c^4)*a^4-8*(b^2+c^2)*b^2*c^2*a^2-2*(b^4+c^4)^2+2*b^4*c^4)*sqrt(3)*S+12*a^10-24*(b^2+c^2)*a^8+36*b^2*c^2*a^6+12*(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^4-3*(-2*b^4*c^4+b^6*c^2+b^2*c^6+4*b^8+4*c^8)*a^2+3*(b^4-c^4)*(b^2-c^2)*b^2*c^2) : :

X(30467) lies on the Simmons inconic with foci {X(13), X(15)} and these lines: {15,30461}, {30,30454}, {396,30466}, {523,30465}, {23992,30470}

X(30467) = reflection of X(30466) in X(396)
X(30467) = antipode of X(30466) in the Simmons inconic with foci {X(13), X(15)}


X(30468) = REFLECTION OF X(30455) IN X(395)

Barycentrics    (-S+sqrt(3)*SA)*(2*S^2+SA^2+2*SB*SC-SW^2) : :

X(30468) lies on the Simmons inconic with foci {X(14), X(16)} and these lines: {2,13}, {5,14817}, {14,5917}, {30,30469}, {110,6778}, {115,125}, {338,30452}, {395,18776}, {396,30464}, {523,30470}, {1316,22513}, {3258,15610}, {3448,6777}, {3458,22512}, {5321,8015}, {10545,16808}, {10654,11659}, {11092,23004}, {12079,20579}, {23284,30453}

X(30468) = midpoint of X(14) and X(16255)
X(30468) = reflection of X(30455) in X(395)
X(30468) = antipode of X(30455) in the Simmons inconic with foci {X(14), X(16)}
X(30468) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (115, 125, 30465), (868, 1648, 30465), (13636, 13722, 30465)


X(30469) = REFLECTION OF X(30455) IN THE LINE X(14)X(16)

Barycentrics    (27*R^2*(12*R^2-SA-6*SW)+6*SA^2+12*SB*SC+22*SW^2)*S^2+S*sqrt(3)*(2*S^2+SA^2+2*SB*SC-SW^2)*(9*R^2-2*SW)+3*(27*R^2-8*SW)*SB*SC*SW : :

X(30469) lies on the Simmons inconic with foci {X(14), X(16)} and these lines: {14,476}, {23,6105}, {30,30468}, {115,30466}, {395,30470}, {523,30455}, {11537,18776}, {11549,30453}

X(30469) = reflection of X(30470) in X(395)
X(30469) = antipode of X(30470) in the Simmons inconic with foci {X(14), X(16)}


X(30470) = REFLECTION OF X(30469) IN X(395)

Barycentrics
(b^2-c^2)^2*(-2*(3*b^2*c^2*(b^4+c^4)+4*a^8-12*(b^2+c^2)*a^6+2*(5*b^4+8*b^2*c^2+5*c^4)*a^4-8*(b^2+c^2)*b^2*c^2*a^2-2*(b^4+c^4)^2+2*b^4*c^4)*sqrt(3)*S+12*a^10-24*(b^2+c^2)*a^8+36*b^2*c^2*a^6+12*(b^2+c^2)*(2*b^4-3*b^2*c^2+2*c^4)*a^4-3*(-2*b^4*c^4+b^6*c^2+b^2*c^6+4*b^8+4*c^8)*a^2+3*(b^4-c^4)*(b^2-c^2)*b^2*c^2) : :

X(30470) lies on the Simmons inconic with foci {X(14), X(16)} and these lines: {16,30464}, {30,30455}, {395,30469}, {523,30468}, {23992,30467}

X(30470) = reflection of X(30469) in X(395)
X(30470) = antipode of X(30469) in the Simmons inconic with foci {X(14), X(16)}


X(30471) = CENTER OF THE DUAL CONIC OF THE SIMMONS INCONIC WITH FOCI X(13) AND X(15)

Barycentrics    (2*S+(3*a^2-b^2-c^2)*sqrt(3))*(2*S+(-a^2+b^2+c^2)*sqrt(3)) : :
X(30471) = X(616)+3*X(628) = X(616)-3*X(14145) = 3*X(630)-2*X(6669) = 4*X(6669)-3*X(22846) = 3*X(21359)-X(22849)

The perspector of this dual conic is X(298). It is a circumconic passing through X(99) and X(9198).

X(30471) lies on the cubic K341a and these lines: {2,11121}, {3,299}, {6,14972}, {13,99}, {15,298}, {18,629}, {76,16241}, {114,1080}, {302,11301}, {532,22855}, {620,22848}, {630,6669}, {1975,16644}, {3180,19780}, {3200,10411}, {3643,11132}, {5238,7796}, {5352,7768}, {5464,6298}, {7763,10654}, {7788,11480}, {7811,10645}, {7837,19781}, {8299,10648}, {9885,14904}, {11127,14921}, {11131,14922}, {11299,22861}, {21359,22849}

X(30471) = midpoint of X(628) and X(14145)
X(30471) = reflection of X(i) in X(j) for these (i,j): (298, 11133), (11121, 22847), (22846, 630), (22848, 620)
X(30471) = isotomic conjugate of the isogonal conjugate of X(3170)
X(30471) = anticomplement of X(22847)
X(30471) = complement of X(11121)
X(30471) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11121, 22847), (15, 7799, 298), (618, 11129, 298)


X(30472) = CENTER OF THE DUAL CONIC OF THE SIMMONS INCONIC WITH FOCI X(14) AND X(16)

Barycentrics    (-2*S+(3*a^2-b^2-c^2)*sqrt(3))*(-2*S+(-a^2+b^2+c^2)*sqrt(3)) : :
X(30472) = X(617)+3*X(627) = X(617)-3*X(14144) = 3*X(629)-2*X(6670) = 4*X(6670)-3*X(22891) = 3*X(21360)-X(22895)

The perspector of this dual conic is X(299). It is a circumconic passing through X(99) and X(9199).

X(30472) lies on the cubic K341b and these lines: {2,11122}, {3,298}, {6,14972}, {14,99}, {16,299}, {17,630}, {76,16242}, {114,383}, {303,11302}, {533,22901}, {620,22892}, {629,6670}, {1975,16645}, {3181,19781}, {3201,10411}, {3642,11133}, {5237,7796}, {5351,7768}, {5463,6299}, {7763,10653}, {7788,11481}, {7811,10646}, {7837,19780}, {8299,10647}, {9886,14905}, {11126,14922}, {11130,14921}, {11300,22907}, {21360,22895}

X(30472) = midpoint of X(627) and X(14144)
X(30472) = reflection of X(i) in X(j) for these (i,j): (299, 11132), (11122, 22893), (22891, 629), (22892, 620)
X(30472) = isotomic conjugate of the isogonal conjugate of X(3171)
X(30472) = anticomplement of X(22893)
X(30472) = complement of X(11122)
X(30472) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 11122, 22893), (16, 7799, 299), (619, 11128, 299)


X(30473) = CENTER OF THE DUAL CONIC OF THE DE LONGCHAMPS ELLIPSE

Barycentrics    b*c*((b^2+b*c+c^2)*a-b*c*(b+c)) : :

The perspector of this dual conic is X(76).

X(30473) lies on these lines: {2,18040}, {6,668}, {8,22289}, {9,29381}, {10,4446}, {37,2998}, {69,17790}, {75,4377}, {76,594}, {141,3596}, {190,29695}, {192,4033}, {239,18044}, {312,17229}, {313,3661}, {314,4445}, {346,26757}, {350,17299}, {536,4110}, {579,29400}, {646,17262}, {984,28593}, {1100,24524}, {1575,17149}, {1909,17303}, {2321,6381}, {2345,3770}, {3161,4391}, {3204,3570}, {3210,18136}, {3264,3662}, {3679,20174}, {3739,20917}, {3759,25298}, {3760,4007}, {3765,17289}, {3834,30090}, {3948,17233}, {3963,5224}, {3975,17279}, {4261,26752}, {4358,17240}, {4361,29802}, {4384,18065}, {4393,18046}, {4437,21933}, {4494,17272}, {4506,17345}, {4643,17787}, {4699,18143}, {5069,21226}, {5839,25278}, {6335,9308}, {6374,6386}, {6542,18147}, {7148,25625}, {9780,25457}, {13466,21796}, {16574,29511}, {16696,26042}, {16777,18140}, {16816,18073}, {17053,27076}, {17148,27044}, {17227,20892}, {17228,20891}, {17230,18137}, {17231,20923}, {17241,29982}, {17275,25280}, {17314,18135}, {17316,25660}, {17350,29423}, {17443,18055}, {17490,18739}, {18067,21101}, {19804,28633}, {20654,30149}, {21904,25287}, {24004,25269}, {25107,28244}

X(30473) = trilinear pole of the line {20909, 21260}
X(30473) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4377, 17239, 75), (6376, 17786, 37)


X(30474) = CENTER OF THE DUAL CONIC OF THE EVANS CONIC

Barycentrics    (a^4-2*(b^2+c^2)*a^2+b^4+4*b^2*c^2+c^4)*(b^2-c^2) : :
X(30474) = X(850)+2*X(3265) = 2*X(850)+X(6563) = X(2525)+2*X(30476) = 4*X(3265)-X(6563) = X(5996)-3*X(9191)

The perspector of this dual conic is X(30475).

X(30474) lies on these lines: {2,525}, {22,22089}, {99,7471}, {325,523}, {647,7630}, {1499,5971}, {1637,2525}, {2373,2693}, {3566,4108}, {3906,6333}, {9146,9182}, {11163,18311}, {12079,23965}, {14417,23878}, {16063,18556}

X(30474) = midpoint of X(i) and X(j) for these {i,j}: {850, 3268}, {1637, 2525}
X(30474) = reflection of X(i) in X(j) for these (i,j): (1637, 30476), (3268, 3265), (6563, 3268)
X(30474) = isotomic conjugate of X(1302)
X(30474) = anticomplement of X(9209)
X(30474) = {X(850), X(3265)}-harmonic conjugate of X(6563)


X(30475) = PERSPECTOR OF THE DUAL CONIC OF THE EVANS CONIC

Barycentrics
(b^2-c^2)*(a^8-2*(b^2+2*c^2)*a^6-(b^4-2*b^2*c^2-6*c^4)*a^4+2*(2*b^6-2*b^4*c^2+b^2*c^4-2*c^6)*a^2-(2*b^4-c^4)*(b^2-c^2)^2)*(a^8-2*(2*b^2+c^2)*a^6+(6*b^4+2*b^2*c^2-c^4)*a^4-2*(2*b^6-b^4*c^2+2*b^2*c^4-2*c^6)*a^2+(b^4-2*c^4)*(b^2-c^2)^2) : :

The center of this dual conic is X(30474).

X(30475) lies on these lines: {}


X(30476) = CENTER OF THE DUAL CONIC OF THE JERABEK HYPERBOLA

Barycentrics    (a^4+2*b^2*c^2-(b^2+c^2)*a^2)*(b^2-c^2) : :
X(30476) = 3*X(2)+X(850) = X(669)+3*X(9148) = 3*X(1637)+X(2525) = X(2525)-3*X(30474) = X(3804)-3*X(4108) = X(6563)-3*X(14417) = X(12077)+3*X(14417)

The perspector of this dual conic is X(6331).

X(30476) lies on these lines: {2,647}, {5,30209}, {10,4524}, {125,15630}, {126,16188}, {127,16177}, {141,8675}, {306,21719}, {512,625}, {520,6130}, {523,4885}, {525,3239}, {620,22104}, {669,9148}, {804,8651}, {1637,2525}, {1649,15850}, {2451,17215}, {2485,23285}, {2489,3267}, {2501,2799}, {2528,8891}, {2793,6562}, {3589,9030}, {3739,17069}, {3766,27345}, {3767,7652}, {3788,23105}, {3804,4108}, {3906,3934}, {4077,24459}, {4139,4928}, {4369,8672}, {4374,27527}, {5907,9242}, {5972,11595}, {6363,23803}, {6563,12077}, {6723,22264}, {7777,10567}, {7886,8574}, {9404,19732}, {10097,11318}, {15143,16229}, {17478,21050}, {17899,17921}, {20907,25098}, {24353,24718}

X(30476) = midpoint of X(i) and X(j) for these {i,j}: {647, 850}, {1637, 30474}, {2485, 23285}, {2489, 3267}, {2501, 3265}, {6563, 12077}, {7624, 18312}, {16229, 22089}, {24353, 24718}
X(30476) = reflection of X(i) in X(j) for these (i,j): (6587, 14341), (22264, 6723)
X(30476) = complementary conjugate of X(15526)
X(30476) = isotomic conjugate of the isogonal conjugate of X(2451)
X(30476) = isotomic conjugate of the polar conjugate of X(16229)
X(30476) = polar conjugate of the isogonal conjugate of X(22089)
X(30476) = complement of X(647)
X(30476) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 850, 647), (2, 24622, 24782), (2, 25258, 25594)


X(30477) = PERSPECTOR OF THE DUAL CONIC OF THE YFF HYPERBOLA

Barycentrics
((2*b^2-c^2)*a^6-(4*b^4-b^2*c^2-c^4)*a^4+(2*b^2-c^2)*(b^4+b^2*c^2-c^4)*a^2-(b^4-c^4)*(b^2-c^2)*c^2)*((b^2-2*c^2)*a^6-(b^4+b^2*c^2-4*c^4)*a^4-(b^2-2*c^2)*(b^4-b^2*c^2-c^4)*a^2+(b^4-c^4)*(b^2-c^2)*b^2) : :

The center of this dual conic is X(524).

X(30477) lies on the line {511,13619}


X(30478) = CENTER OF THE DUAL CONIC OF THE YIU CONIC

Barycentrics    (3*a^3+3*(b+c)*a^2-(b^2-6*b*c+c^2)*a-(b^2-c^2)*(b-c))*(-a+b+c) : :

Let Ab, Ac be the touchpoints of the A-excircle and sides AB and AC, respectively, and define Bc, Ba, Ca, Cb cyclically. These six points lie on the Yiu conic, defined at X(478).
The perspector of this dual conic is X(30479).

X(30478) lies on these lines: {1,5745}, {2,12}, {3,1602}, {4,993}, {7,15823}, {8,2320}, {9,1125}, {10,631}, {11,452}, {20,2886}, {21,497}, {35,5082}, {36,443}, {40,6935}, {48,966}, {63,3485}, {65,5744}, {71,10476}, {72,16193}, {75,6337}, {104,6889}, {140,9708}, {142,3361}, {145,18231}, {154,20306}, {210,27383}, {220,24512}, {261,30479}, {281,7521}, {329,11375}, {354,960}, {355,6954}, {376,5267}, {377,19841}, {390,3813}, {404,26040}, {405,3086}, {442,4293}, {474,19855}, {496,16418}, {498,3421}, {499,5084}, {515,5705}, {517,6892}, {518,5703}, {549,9709}, {551,12559}, {936,10165}, {946,5698}, {950,5231}, {956,3085}, {962,4640}, {988,4000}, {997,6878}, {999,6675}, {1000,22837}, {1001,14986}, {1006,10785}, {1036,5324}, {1056,8666}, {1058,5248}, {1107,7735}, {1212,2275}, {1376,3523}, {1377,13935}, {1378,9540}, {1385,5791}, {1468,5712}, {1469,28275}, {1478,6856}, {1479,11111}, {1588,9678}, {1621,10529}, {1698,5795}, {1706,10164}, {1788,19860}, {1935,25885}, {2476,5229}, {2478,10589}, {3035,10303}, {3189,3601}, {3419,4305}, {3428,6847}, {3434,4189}, {3452,3624}, {3474,4652}, {3476,24987}, {3486,6734}, {3488,10916}, {3524,25440}, {3525,10805}, {3526,3820}, {3622,5289}, {3649,9965}, {3671,3928}, {3683,11376}, {3698,26062}, {3812,5435}, {3814,5067}, {3816,5129}, {3826,17580}, {3878,10595}, {3913,5281}, {3916,4295}, {3925,5204}, {4190,5303}, {4220,22654}, {4252,4307}, {4267,16713}, {4294,16370}, {4298,25525}, {4314,24392}, {4426,7736}, {4512,12053}, {4648,25500}, {4679,26129}, {4719,5222}, {4855,25006}, {5080,6933}, {5123,19877}, {5177,7354}, {5217,17784}, {5219,12527}, {5225,6872}, {5259,10072}, {5274,11106}, {5284,10586}, {5288,10056}, {5302,5550}, {5325,25055}, {5432,7080}, {5436,11019}, {5450,6916}, {5587,6927}, {5603,12514}, {5657,6977}, {5716,29639}, {5731,5794}, {5818,6880}, {5836,6966}, {6284,17576}, {6351,13902}, {6352,13959}, {6636,9712}, {6684,9623}, {6690,12513}, {6737,13384}, {6762,13405}, {6824,11249}, {6825,12667}, {6826,26286}, {6844,11827}, {6846,22753}, {6852,10532}, {6853,12115}, {6855,26332}, {6859,10526}, {6861,22765}, {6868,26470}, {6875,12116}, {6908,12114}, {6921,9780}, {6961,26446}, {6970,9956}, {6989,10269}, {8167,17554}, {8227,12572}, {9701,11003}, {9710,15717}, {10167,18251}, {10200,17559}, {10448,11269}, {10591,11113}, {10934,19528}, {11036,11281}, {11108,15325}, {11110,27509}, {11365,17560}, {13411,25568}, {14001,17030}, {15171,17571}, {15654,19262}, {16678,27621}, {17314,17733}, {18250,19862}, {23207,27407}, {24363,24432}, {24570,27339}

X(30478) = complement of X(5261)
X(30478) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 2975, 388), (2, 3436, 10588), (11194, 25466, 3600)


X(30479) = PERSPECTOR OF THE DUAL CONIC OF THE YIU CONIC

Barycentrics    (-a+b+c)*(b^2+(a+c)^2)*(c^2+(a+b)^2) : :

The center of this dual conic is X(30478).

X(30479) lies on the Feuerbach hyperbola and these lines: {1,69}, {2,2221}, {4,75}, {7,4388}, {8,3718}, {9,345}, {10,989}, {21,332}, {63,2354}, {79,15434}, {80,4986}, {84,6210}, {104,1310}, {238,987}, {256,7019}, {261,30478}, {294,391}, {307,1041}, {314,497}, {319,1000}, {320,3296}, {326,1064}, {333,1172}, {464,28287}, {885,4811}, {941,5739}, {960,1264}, {966,981}, {1479,10447}, {2551,3596}, {2975,8048}, {2997,4441}, {3672,26117}, {3717,4866}, {4329,20911}, {4514,6601}, {5016,24547}, {5558,21296}, {5665,6604}, {5738,17137}, {9534,26939}, {10436,26098}, {16043,27633}, {17097,17152}, {17306,18141}, {17361,18490}

X(30479) = isogonal conjugate of X(1460)
X(30479) = isotomic conjugate of X(388)
X(30479) = trilinear pole of the line {650, 3910}


X(30480) = X(5)X(128)∩X(14071)X(25149)

Barycentrics    (-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) (-a^6+2 a^4 b^2-a^2 b^4-a^2 b^3 c+b^5 c+2 a^4 c^2+a^2 b^2 c^2-a^2 b c^3-2 b^3 c^3-a^2 c^4+b c^5) (a^6-2 a^4 b^2+a^2 b^4-a^2 b^3 c+b^5 c-2 a^4 c^2-a^2 b^2 c^2-a^2 b c^3-2 b^3 c^3+a^2 c^4+b c^5) (-a^10 b^2+4 a^8 b^4-6 a^6 b^6+4 a^4 b^8-a^2 b^10-a^10 c^2+6 a^8 b^2 c^2-9 a^6 b^4 c^2+5 a^4 b^6 c^2-2 a^2 b^8 c^2+b^10 c^2+4 a^8 c^4-9 a^6 b^2 c^4+3 a^2 b^6 c^4-4 b^8 c^4-6 a^6 c^6+5 a^4 b^2 c^6+3 a^2 b^4 c^6+6 b^6 c^6+4 a^4 c^8-2 a^2 b^2 c^8-4 b^4 c^8-a^2 c^10+b^2 c^10) : :
Barycentrics    S^6+S^4 (-9 R^4-5 R^2 SB-5 R^2 SC-3 SB SC+3 R^2 SW+2 SB SW+2 SC SW)+SB SC (18 R^8-6 R^6 SW-7 R^4 SW^2+5 R^2 SW^3-SW^4)+S^2 (114 R^8-40 R^6 SB-40 R^6 SC-126 R^6 SW+46 R^4 SB SW+46 R^4 SC SW+41 R^4 SW^2-17 R^2 SB SW^2-17 R^2 SC SW^2-R^2 SW^3+2 SB SW^3+2 SC SW^3-SW^4+SB SC (-53 R^4+33 R^2 SW-4 SW^2)) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28764.

X(30480) lies on these lines: {5,128}, {14071,25149}


X(30481) = X(5)X(51)∩X(110)X(1157)

Barycentrics    a^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^14 b^2-6 a^12 b^4+15 a^10 b^6-20 a^8 b^8+15 a^6 b^10-6 a^4 b^12+a^2 b^14+a^14 c^2-9 a^12 b^2 c^2+25 a^10 b^4 c^2-30 a^8 b^6 c^2+15 a^6 b^8 c^2-a^4 b^10 c^2-a^2 b^12 c^2-6 a^12 c^4+25 a^10 b^2 c^4-30 a^8 b^4 c^4+9 a^6 b^6 c^4+3 a^4 b^8 c^4-2 a^2 b^10 c^4+b^12 c^4+15 a^10 c^6-30 a^8 b^2 c^6+9 a^6 b^4 c^6-a^4 b^6 c^6+2 a^2 b^8 c^6-4 b^10 c^6-20 a^8 c^8+15 a^6 b^2 c^8+3 a^4 b^4 c^8+2 a^2 b^6 c^8+6 b^8 c^8+15 a^6 c^10-a^4 b^2 c^10-2 a^2 b^4 c^10-4 b^6 c^10-6 a^4 c^12-a^2 b^2 c^12+b^4 c^12+a^2 c^14) : :
Barycentrics    S^4 (R^2-SB-SC+SW)+SB SC (15 R^6-18 R^4 SW+7 R^2 SW^2-SW^3)+S^2 (-17 R^6-8 R^4 SB-8 R^4 SC+SB SC (5 R^2-SW)+22 R^4 SW+6 R^2 SB SW+6 R^2 SC SW-9 R^2 SW^2-SB SW^2-SC SW^2+SW^3) : :

See Alexandr Skutin, Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28764.

X(30481) lies on these lines: {5,51}, {110,1157}, {5944,6150}


X(30482) = X(5)X(51)∩X(1510)X(6150)

Barycentrics    a^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^4-2 a^2 b^2+b^4-2 a^2 c^2-b^2 c^2+c^4) (a^10 b^2-4 a^8 b^4+6 a^6 b^6-4 a^4 b^8+a^2 b^10+a^10 c^2-6 a^8 b^2 c^2+9 a^6 b^4 c^2-5 a^4 b^6 c^2+2 a^2 b^8 c^2-b^10 c^2-4 a^8 c^4+9 a^6 b^2 c^4-3 a^2 b^6 c^4+4 b^8 c^4+6 a^6 c^6-5 a^4 b^2 c^6-3 a^2 b^4 c^6-6 b^6 c^6-4 a^4 c^8+2 a^2 b^2 c^8+4 b^4 c^8+a^2 c^10-b^2 c^10) : :
Barycentrics    S^4 (3 R^2-SB-SC+SW)+SB SC (6 R^6-10 R^4 SW+5 R^2 SW^2-SW^3)+S^2 (-26 R^6-8 R^4 SB-8 R^4 SC+SB SC (7 R^2-SW)+30 R^4 SW+6 R^2 SB SW+6 R^2 SC SW-11 R^2 SW^2-SB SW^2-SC SW^2+SW^3) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28764.

X(30482) lies on these lines: {5,51}, {1510,6150}, {12060,18350}


X(30483) = X(5)X(128)∩X(6343)X(25149)

Barycentrics    (-a^2 b^2+b^4-a^2 c^2-2 b^2 c^2+c^4) (a^22 b^2-8 a^20 b^4+28 a^18 b^6-56 a^16 b^8+70 a^14 b^10-56 a^12 b^12+28 a^10 b^14-8 a^8 b^16+a^6 b^18+a^22 c^2-13 a^20 b^2 c^2+56 a^18 b^4 c^2-119 a^16 b^6 c^2+140 a^14 b^8 c^2-91 a^12 b^10 c^2+28 a^10 b^12 c^2-a^8 b^14 c^2-a^6 b^16 c^2-8 a^20 c^4+56 a^18 b^2 c^4-141 a^16 b^4 c^4+163 a^14 b^6 c^4-79 a^12 b^8 c^4-7 a^10 b^10 c^4+30 a^8 b^12 c^4-23 a^6 b^14 c^4+13 a^4 b^16 c^4-5 a^2 b^18 c^4+b^20 c^4+28 a^18 c^6-119 a^16 b^2 c^6+163 a^14 b^4 c^6-79 a^12 b^6 c^6+5 a^10 b^8 c^6-11 a^8 b^10 c^6+35 a^6 b^12 c^6-39 a^4 b^14 c^6+25 a^2 b^16 c^6-8 b^18 c^6-56 a^16 c^8+140 a^14 b^2 c^8-79 a^12 b^4 c^8+5 a^10 b^6 c^8+7 a^8 b^8 c^8-12 a^6 b^10 c^8+39 a^4 b^12 c^8-45 a^2 b^14 c^8+28 b^16 c^8+70 a^14 c^10-91 a^12 b^2 c^10-7 a^10 b^4 c^10-11 a^8 b^6 c^10-12 a^6 b^8 c^10-26 a^4 b^10 c^10+25 a^2 b^12 c^10-56 b^14 c^10-56 a^12 c^12+28 a^10 b^2 c^12+30 a^8 b^4 c^12+35 a^6 b^6 c^12+39 a^4 b^8 c^12+25 a^2 b^10 c^12+70 b^12 c^12+28 a^10 c^14-a^8 b^2 c^14-23 a^6 b^4 c^14-39 a^4 b^6 c^14-45 a^2 b^8 c^14-56 b^10 c^14-8 a^8 c^16-a^6 b^2 c^16+13 a^4 b^4 c^16+25 a^2 b^6 c^16+28 b^8 c^16+a^6 c^18-5 a^2 b^4 c^18-8 b^6 c^18+b^4 c^20) : :
Barycentrics    S^6+S^4 (R^4-5 R^2 SB-5 R^2 SC-3 SB SC-R^2 SW+2 SB SW+2 SC SW)+SB SC (-27 R^8+52 R^6 SW-33 R^4 SW^2+9 R^2 SW^3-SW^4)+S^2 (69 R^8-40 R^6 SB-40 R^6 SC-68 R^6 SW+46 R^4 SB SW+46 R^4 SC SW+15 R^4 SW^2-17 R^2 SB SW^2-17 R^2 SC SW^2+3 R^2 SW^3+2 SB SW^3+2 SC SW^3-SW^4+SB SC (-43 R^4+29 R^2 SW-4 SW^2)) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28764.

X(30483) lies on these lines: {5,128}, {6343,25149}


X(30484) = X(3)X(128)∩X(4)X(252)

Barycentrics    5*S^4+(R^2*(4*R^2-5*SA)+2*SA^2-7*SB*SC-SW^2)*S^2+(R^2*(20*R^2-19*SW)+5*SW^2)*SB*SC : :
X(30484) = 3*X(5)-2*X(20414)

See Tran Quang Hung and César Lozada, Hyacinthos 28766.

X(30484) lies on these lines: {3, 128}, {4, 252}, {5, 6150}, {30, 14143}, {550, 6247}, {930, 2888}, {933, 3462}, {1510, 11591}, {3153, 14097}, {3574, 12060}, {15619, 15704}

X(30484) = midpoint of X(15619) and X(15704)
X(30484) = {X(3), X(1601)}-harmonic conjugate of X(23320)
X(30484) = Napoleon-Feuerbach isogonal conjugate of X(54)


X(30485) = X(3)X(24303)∩X(61)X(16641)

Barycentrics    (SB+SC)*(3*(6*R^2+2*SA-3*SW)*S^2-sqrt(3)*(S^2-36*R^2*SA+5*SA^2+4*SB*SC)*S+3*(6*R^2-3*SA+2*SW)*SA*SW) : :
X(30485) = 2*X(13350)-3*X(14170)

See Tran Quang Hung and César Lozada, Hyacinthos 28766.

X(30485) lies on these lines: {3, 24303}, {61, 16461}, {616, 10409}, {1495, 13350}, {5663, 13859}


X(30486) = X(3)X(24304)∩X(62)X(16462)

Barycentrics    (SB+SC)*(3*(6*R^2+2*SA-3*SW)*S^2+qrt(3)*(S^2-36*R^2*SA+5*SA^2+4*SB*SC)*S+3*(6*R^2-3*SA+2*SW)*SA*SW) : :

See Tran Quang Hung and César Lozada, Hyacinthos 28766.

X(30486) lies on these lines: {3, 24304}, {62, 16462}, {617, 10410}, {1495, 13349}, {5663, 13858}


X(30487) = X(3)X(695)∩X(192)X(815)

Barycentrics    a^2*(b*c*a^6-(b^3+c^3)*(b+c)*a^4+(b^3+c^3)*b*c*a^3+(b^4-c^4)*(b^2-c^2)*a^2-(b^3+c^3)*(b^2+c^2)*b*c*a+2*b^4*c^4) : :

See Tran Quang Hung and César Lozada, Hyacinthos 28766.

X(30487) lies on these lines: {3, 695}, {192, 815}, {6310, 19548}


X(30488) = ISOGONAL CONJUGATE OF X(6031)

Barycentrics    a^2*(2*(2*b^2+c^2)*a^4+(b^2+2*c^2)*a^2*c^2-4*(b^4-c^4)*b^2)*(2*(b^2+2*c^2)*a^4+(2*b^2+c^2)*a^2*b^2+4*(b^4-c^4)*c^2) : :

See Tran Quang Hung and César Lozada, Hyacinthos 28766.

X(30488) lies on these lines: {6, 6324}, {574, 12367}, {599, 8705}

X(30488) = reflection of X(6) in X(6324)
X(30488) = anticomplement of the complementary conjugate of X(6032)
X(30488) = antigonal conjugate of the isogonal conjugate of X(5971)
X(30488) = isogonal conjugate of X(6031)


X(30489) = ISOGONAL CONJUGATE OF X(10130)

Barycentrics    a^2*(b^2+c^2)*(2*a^2+2*b^2-c^2)*(2*a^2+2*c^2-b^2) : :

See Tran Quang Hung and César Lozada, Hyacinthos 28766.

X(30489) lies on these lines: {6, 23}, {39, 9019}, {76, 524}, {141, 23297}, {230, 25488}, {523, 18907}, {597, 13410}, {755, 11636}, {882, 9009}, {2353, 30435}, {2393, 27375}, {2854, 5052}, {3291, 16776}, {3629, 6664}, {6698, 15820}, {7737, 11594}, {8584, 20380}, {9465, 9971}

X(30489) = isogonal conjugate of X(10130)


X(30490) = X(54)X(143)∩X(195)X(25043)

Barycentrics    (SB+SC) *(2*R^2-SA-SW) *(2*SB+R^2)*(2*SC+R^2)*(S^2+SA*SB)*(S^2+SA*SC) : :

See Tran Quang Hung and César Lozada, Hyacinthos 28766.

X(30490) lies on these lines: {54, 143}, {195, 25043}


X(30491) = X(512)X(2030)∩X(523)X(18907)

Barycentrics    a^2*(b^2-c^2)*(-a^2+b^2+c^2)*(2*a^2-b^2+2*c^2)*(2*a^2+2*b^2-c^2) : :
X(30491) = 3*X(3049)-X(10097)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28769.

X(30491) lies on these lines: {512, 2030}, {523, 18907}, {647, 9517}, {1383, 2433}, {1384, 17414}, {1499, 23287}, {2501, 8599}, {2715, 11636}, {3049, 10097}, {17979, 30209}

X(30491) = isogonal conjugate of the polar conjugate of X(8599)
X(30491) = barycentric product X(i)*X(j) for these {i, j}: {3, 8599}, {125, 11636}, {525, 1383}, {598, 647}, {895, 23287}
X(30491) = barycentric quotient X(i)/X(j) for these (i, j): (3, 9146), (184, 9145), (228, 3908), (512, 5094), (525, 9464), (598, 6331), (647, 599), (669, 8541), (1383, 648)
X(30491) = trilinear product X(i)*X(j) for these {i, j}: {48, 8599}, {598, 810}, {656, 1383}
X(30491) = trilinear quotient X(i)/X(j) for these (i, j): (48, 9145), (63, 9146), (71, 3908), (598, 811), (656, 599), (661, 5094), (798, 8541), (810, 574), (1383, 162)


X(30492) = TRILINEAR POLE OF THE LINE {338, 2086}

Barycentrics    b^2*c^2*(b^2-c^2)*(a^6+a^4*c^2+c^6-(b^2-c^2)*a^2*c^2)*(a^6+a^4*b^2+b^6+(b^2-c^2)*a^2*b^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28769.

X(30492) lies on this line: {850, 5027}

X(30492) = trilinear pole of the line {338, 2086}


X(30493) = X(1)X(1361)∩X(7)X(286)

Barycentrics    a^2*(a + b - c)*(a - b + c)*(a^2 - b^2 - c^2)*(a^2*b^2 - b^4 + a^2*c^2 + 2*b^2*c^2 - c^4) : :

See Tran Quang Hung and Peter Moses, Hyacinthos 287771.

X(30493) lies on the cubic K714 and these lines: {1, 1361}, {7, 286}, {12, 26932}, {34, 26892}, {49, 23070}, {51, 1393}, {56, 58}, {57, 1745}, {63, 7066}, {65, 515}, {73, 22345}, {84, 7355}, {109, 23850}, {181, 1454}, {185, 7004}, {201, 3917}, {221, 22654}, {226, 14058}, {227, 8679}, {388, 26871}, {603, 1425}, {942, 1875}, {971, 1887}, {1038, 3784}, {1122, 1439}, {1214, 11573}, {1354, 1363}, {1355, 1367}, {1358, 20618}, {1400, 14597}, {1406, 18954}, {1433, 3304}, {1455, 23842}, {1469, 7289}, {1473, 19349}, {2003, 19365}, {3220, 26888}, {4014, 7702}, {4303, 22341}, {5399, 23981}, {5907, 24430}, {7352, 24467}, {9291, 18026}, {15524, 20323}, {17114, 18838}, {18915, 26929}, {26933, 26955}

X(30493) = X(18180)-Ceva conjugate of X(1393)
X(30493) = X(i)-isoconjugate of X(j) for these (i,j): {8, 2190}, {9, 275}, {33, 95}, {41, 276}, {54, 318}, {78, 8884}, {212, 8795}, {281, 2167}, {312, 8882}, {933, 4086}, {2148, 7017}, {2289, 8794}, {4041, 18831}, {8611, 16813}
X(30493) = crosspoint of X(7) and X(222)
X(30493) = crosssum of X(55) and X(281)
X(30493) = barycentric product X(i) X(j) for these {i,j}: {5, 222}, {7, 216}, {51, 348}, {53, 1804}, {56, 343}, {63, 1393}, {65, 16697}, {73, 17167}, {77, 1953}, {217, 6063}, {278, 5562}, {324, 7335}, {331, 418}, {603, 14213}, {604, 18695}, {1214, 18180}, {1397, 28706}, {1625, 17094}, {1813, 21102}, {2179, 7182}, {2181, 7183}, {3199, 7055}, {4565, 6368}, {4573, 15451}, {7069, 7177}, {7178, 23181}, {8798, 18623}, {17076, 27372}
X(30493) = barycentric quotient X(i) / X(j) for these {i,j}: {5, 7017}, {7, 276}, {51, 281}, {56, 275}, {216, 8}, {217, 55}, {222, 95}, {278, 8795}, {343, 3596}, {418, 219}, {603, 2167}, {604, 2190}, {608, 8884}, {1118, 8794}, {1393, 92}, {1397, 8882}, {1953, 318}, {2179, 33}, {3199, 1857}, {4565, 18831}, {5562, 345}, {7069, 7101}, {7335, 97}, {15451, 3700}, {16697, 314}, {18695, 28659}, {23181, 645}
X(30493) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {56, 222, 7335}, {1425, 3937, 603}


X(30494) = PERSPECTOR OF ADAMS CIRCLE

Barycentrics    (a-b+c)*((2*b+c)*a^3-2*(2*b^2+b*c+c^2)*a^2+2*(b-c)^2*b*c+(b^2-c^2)*(2*b-c)*a)*(a+b-c)*((b+2*c)*a^3-2*(b^2+b*c+2*c^2)*a^2+2*(b-c)^2*b*c+(b^2-c^2)*(b-2*c)*a) : :

Centers X(30494)-X(30503) were contributed by César Lozada, December 30, 2018. See Triangle circles at MathWorld for definitions of circles referred in these centers.

X(30494) lies on the Feuerbach hyperbola and these lines: {7,15658}, {9,6706}

X(30494) = isogonal conjugate of X(30502)


X(30495) = PERSPECTOR OF BROCARD CIRCLE

Barycentrics    a^2*(c^2*a^2+2*b^4)*(b^2*a^2+2*c^4) : :

X(30495) is the pole, wrt the Brocard circle, of line X(669)X(688), which is the isogonal conjugate of the isotomic conjugate of the Lemoine axis, and the trilinear polar of X(32). (Randy Hutson, January 15, 2019)

X(30495) lies on these lines: {538,599}, {574,3117}, {694,3734}, {695,7751}, {2387,5028}, {2549,20021}, {3721,3760}, {3981,18546}, {7748,27366}, {11648,20859}, {14820,17130}

X(30495) = reflection of X(6195) in X(3117)
X(30495) = isogonal conjugate of X(3972)
X(30495) = anticomplement of the complementary conjugate of X(7853)
X(30495) = complement of the anticomplementary conjugate of X(7898)
X(30495) = trilinear pole of the line {888, 17414}


X(30496) = PERSPECTOR OF 2nd BROCARD CIRCLE

Barycentrics    a^2*((b^2-c^2)*a^2-b^2*(2*b^2-c^2))*((b^2-c^2)*a^2-(b^2-2*c^2)*c^2) : :

X(30496) lies on the Jerabek hyperbola and these lines: {3,3229}, {69,698}, {5254,19222}

X(30496) = isogonal conjugate of X(3552)
X(30496) = anticomplement of the complementary conjugate of X(5025)


X(30497) = PERSPECTOR OF DAO-MOSES-TELV CIRCLE

Barycentrics
(S^4+3*(16*R^2*(9*R^2-4*SW)-SB^2+7*SW^2)*S^2-3*(4*R^2-SW)*(36*R^2-8*SW+3*SB)*SB*SW)*(S^4+3*(16*R^2*(9*R^2-4*SW)-SC^2+7*SW^2)*S^2-3*(4*R^2-SW)*(36*R^2-8*SW+3*SC)*SC*SW) : :

X(30497) lies on these lines: {542,1651}, {6070,17986}


X(30498) = PERSPECTOR OF EHRMANN CIRCLE

Barycentrics    a^2*(2*a^4-(8*b^2+5*c^2)*a^2+2*(2*b^2-c^2)^2)*(2*a^4-(5*b^2+8*c^2)*a^2+2*(b^2-2*c^2)^2) : :

X(30498) lies on these lines: {353,20251}, {574,20977}, {599,625}, {6323,11002}


X(30499) = PERSPECTOR OF GALLATLY CIRCLE

Barycentrics    (SB+SC)*(S^4+(3*SW-SC)*SW*S^2-SA*SB*SW^2)*(S^4+(3*SW-SB)*SW*S^2-SW^2*SC*SA) : :

X(30499) lies on these lines: {262,20021}, {325,14994}, {511,14096}


X(30500) = PERSPECTOR OF HEXYL CIRCLE

Barycentrics
a*(a^6-2*(b+c)*a^5-(b^2-6*b*c+c^2)*a^4+4*(b^3+c^3+(b-c)*b*c)*a^3-(b^2+6*b*c+c^2)*(b-c)^2*a^2-2*(b^2-c^2)*(b^3-c^3+(b+3*c)*b*c)*a+(b^2-c^2)^2*(b-c)^2)*(a^6-2*(b+c)*a^5-(b^2-6*b*c+c^2)*a^4+4*(b^3+c^3-(b-c)*b*c)*a^3-(b^2+6*b*c+c^2)*(b-c)^2*a^2-2*(b^2-c^2)*(b^3-c^3-(3*b+c)*b*c)*a+(b^2-c^2)^2*(b-c)^2) : :

X(30500) lies on the Feuerbach hyperbola and these lines: {8,6769}, {9,1012}, {1000,10388}, {1537,3254}, {2096,10307}, {4292,10309}, {7284,30304}, {12867,21669}

X(30500) = isogonal conjugate of X(30503)


X(30501) = PERSPECTOR OF LONGUET-HIGGINS CIRCLE

Barycentrics    (a^4+2*a^3*b+2*(3*b-c)*a^2*c-2*(b+3*c)*(b-c)*a*b-(b^2-c^2)*(b+c)^2)*(a^4+2*a^3*c-2*(b-3*c)*a^2*b+2*(3*b+c)*(b-c)*a*c+(b^2-c^2)*(b+c)^2) : :

X(30501) lies on these lines: {7,5342}, {63,391}, {69,962}, {77,3616}, {938,969}, {17011,20211}

X(30501) = isotomic conjugate of X(5815)
X(30501) = cyclocevian conjugate of the isotomic conjugate of X(9874)
X(30501) = trilinear pole of the line {905, 4765}


X(30502) = ISOGONAL CONJUGATE OF X(30494)

Barycentrics    a^2*(-a+b+c)*(2*(b+c)*a^3-(4*b^2+b*c+4*c^2)*a^2+2*(b^2-c^2)*(b-c)*a+(b-c)^2*b*c) : :

X(30502) lies on these lines: {1,3}, {41,10482}, {1253,4251}, {2293,4253}, {5022,16688}, {5248,28071}, {5259,28053}, {14942,17687}


X(30503) = ISOGONAL CONJUGATE OF X(30500)

Barycentrics    a*(a^6-2*(b+c)*a^5-(b+c)^2*a^4+4*(b^2-c^2)*(b-c)*a^3-(b^2-6*b*c+c^2)*(b+c)^2*a^2-2*(b^2-c^2)*(b-c)^3*a+(b^2-c^2)^2*(b-c)^2) : :
X(30503) = 3*X(165)-2*X(3587) = 3*X(165)+X(18421) = X(1056)-3*X(21151)

X(30503) lies on these lines: {1,3}, {4,12565}, {8,10884}, {9,3197}, {10,1490}, {19,2267}, {20,19860}, {30,2951}, {71,2324}, {73,1103}, {84,958}, {200,5657}, {207,7952}, {380,572}, {405,12705}, {515,2550}, {516,6987}, {519,3174}, {631,8583}, {912,5223}, {936,6261}, {944,4853}, {946,6865}, {952,4915}, {956,10167}, {960,7971}, {971,9708}, {990,30116}, {997,10164}, {1006,4512}, {1012,10860}, {1056,4321}, {1064,2999}, {1072,23681}, {1125,6926}, {1483,12127}, {1519,6947}, {1698,6825}, {1699,6827}, {1706,11500}, {1709,5251}, {1750,3925}, {1935,2956}, {2270,23840}, {2551,6260}, {2800,21153}, {3062,18540}, {3088,19784}, {3189,5882}, {3753,7580}, {3880,7966}, {4297,12650}, {4847,5768}, {4882,5534}, {5234,7330}, {5258,10085}, {5260,9961}, {5437,22753}, {5705,12616}, {5715,12609}, {5720,8580}, {5745,14647}, {5790,18528}, {5795,12667}, {5811,18250}, {5924,12572}, {6154,6264}, {6174,6326}, {6245,19843}, {6361,12651}, {6705,30478}, {6762,12675}, {6765,11362}, {6838,24982}, {6846,21628}, {6848,8582}, {6882,7988}, {6890,24541}, {6913,11372}, {6953,25011}, {7171,22758}, {7680,25525}, {9579,11827}, {9841,12114}, {9856,11108}, {12664,18251}, {16132,17857}

X(30503) = midpoint of X(i) and X(j) for these {i,j}: {40, 11529}, {5732, 9623}
X(30503) = reflection of X(i) in X(j) for these (i,j): (1, 18443), (3062, 18540), (6767, 1385), (11372, 6913)
X(30503) = isogonal conjugate of X(30500)
X(30503) = X(18420)-of-excentral-triangle
X(30503) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (40, 7982, 7957), (165, 2093, 40), (1040, 24806, 1)


X(30504) = X(54)X(21394)∩X(186)X(323)

Barycentrics    a^2 (a^2-b^2-b c-c^2) (a^2-b^2+b c-c^2) (a^16-4 a^14 b^2+5 a^12 b^4+a^10 b^6-10 a^8 b^8+14 a^6 b^10-11 a^4 b^12+5 a^2 b^14-b^16-4 a^14 c^2+8 a^12 b^2 c^2-a^10 b^4 c^2-3 a^8 b^6 c^2-12 a^6 b^8 c^2+28 a^4 b^10 c^2-23 a^2 b^12 c^2+7 b^14 c^2+5 a^12 c^4-a^10 b^2 c^4-a^8 b^4 c^4-2 a^6 b^6 c^4-18 a^4 b^8 c^4+39 a^2 b^10 c^4-22 b^12 c^4+a^10 c^6-3 a^8 b^2 c^6-2 a^6 b^4 c^6+2 a^4 b^6 c^6-21 a^2 b^8 c^6+41 b^10 c^6-10 a^8 c^8-12 a^6 b^2 c^8-18 a^4 b^4 c^8-21 a^2 b^6 c^8-50 b^8 c^8+14 a^6 c^10+28 a^4 b^2 c^10+39 a^2 b^4 c^10+41 b^6 c^10-11 a^4 c^12-23 a^2 b^2 c^12-22 b^4 c^12+5 a^2 c^14+7 b^2 c^14-c^16) : :
Barycentrics    (17 R^2+SB+SC-5 SW)S^4 + (-64 R^6+8 R^4 SB+8 R^4 SC+68 R^4 SW-6 R^2 SB SW-6 R^2 SC SW-25 R^2 SW^2+SB SW^2+SC SW^2+3 SW^3+SB SC (-15 R^2+5 SW))S^2 + SB SC (30 R^6-40 R^4 SW+19 R^2 SW^2-3 SW^3) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28775.

X(30504) lies on these lines: {54,21394}, {186,323}, {6150,12006}


X(30505) = X(2)X(3613)∩X(76)X(3060)

Barycentrics    (a^2+b^2)*(a^2+c^2)*((b^2+c^2)*a^2+b^2*c^2-c^4)*((b^2+c^2)*a^2-b^4+b^2*c^2) : :

See Jean-Louis Ayme and César Lozada, Hyacinthos 28776.

X(30505) lies on the Kiepert hyperbola and these lines: {2, 3613}, {76, 3060}, {83, 14957}, {98, 251}, {1916, 11794}, {2052, 10550}

X(30505) = isotomic conjugate of the anticomplement of X(20965)
X(30505) = barycentric product X(i)*X(j) for these {i,j}: {83, 3613}, {308, 27375}
X(30505) = barycentric quotient X(i)/X(j) for these (i,j): (32, 3203), (82, 18042), (83, 1078), (251, 5012)
X(30505) = trilinear product X(82)*X(3613)
X(30505) = trilinear quotient X(i)/X(j) for these (i,j): (31, 3203), (82, 5012), (83, 18042)


X(30506) = EULER LINE INTERCEPT OF X(51)X(324)

Barycentrics    (a^2+b^2-c^2)*(a^2-b^2+c^2)*(a^4-b^2*c^2-(b^2+c^2)*a^2)*((b^2+c^2)*a^2-b^4+2*b^2*c^2-c^4) : :

See Jean-Louis Ayme and César Lozada, Hyacinthos 28776.

X(30506) lies on these lines: {2, 3}, {51, 324}, {53, 17500}, {110, 275}, {143, 14978}, {251, 6531}, {264, 3060}, {1629, 5012}, {2052, 5640}, {3289, 6748}, {6747, 19130}, {6749, 25051}, {8884, 13434}, {11451, 15466}, {14389, 19174}

X(30506) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 458, 14957), (4, 6819, 1370), (5, 6755, 467)
X(30506) = barycentric product X(i)*X(j) for these {i,j}: {53, 1078}, {308, 27370}, {311, 10312}, {324, 5012}, {343, 1629}
X(30506) = barycentric quotient X(i)/X(j) for these (i,j): (53, 3613), (1629, 275)
X(30506) = trilinear product X(i)*X(j) for these {i,j}: {53, 18042}, {1078, 2181}
X(30506) = trilinear quotient X(i)/X(j) for these (i,j): (1629, 2190), (2181, 27375)


X(30507) = X(3)X(21357)∩X(5)X(10721)

Barycentrics    (4*SA-143*R^2+30*SW)*S^2+11*(11*R^2-2*SW)*SB*SC : :
X(30507) = X(15704)+2*X(17505)

See Jean-Louis Ayme and César Lozada, Hyacinthos 28776.

X(30507) lies on these lines: {3, 21357}, {5, 10721}, {368, 8421}, {549, 12162}, {550, 20191}, {15704, 17505}


X(30508) = ANTICOMPLEMENT OF X(13636)

Barycentrics    (-(2*a^2-b^2-c^2)*K+2*a^4-2*(b^2+c^2)*a^2-b^4+4*b^2*c^2-c^4)*(b^2-c^2-K)*(-a^2+b^2-K)*(c^2-a^2-K)*(-b^2+c^2-K) : :, where K=sqrt(SW^2-3*S^2)
Barycentrics    = (b^2-c^2) (5 a^4-5 a^2 b^2+2 b^4-5 a^2 c^2+b^2 c^2+2 c^4+2 (2 a^2-b^2-c^2) Sqrt[a^4-a^2 b^2+b^4-a^2 c^2-b^2 c^2+c^4]) : : (Peter Moses, January 3, 2019)

See Jean-Louis Ayme and César Lozada, Hyacinthos 28776.

X(30508) lies on the Kiepert parabola, the cubics K010, K015, K242, K408 and these lines: {2, 1340}, {99, 110}, {523, 6190}

X(30508) = isotomic conjugate of X(30509)
X(30508) = anticomplement of X(13636)
X(30508) = trilinear pole of the line {115, 2029}
X(30508) = reflection of X(2) in the line X(3413)X(9168)
X(30508) = Gibert-Simson transform of X(1379)
X(30508) = barycentric product X(i)*X(j) for these {i,j}: {670, 2029}, {2966, 14501}, {3414, 6190}
X(30508) = barycentric quotient X(i)/X(j) for these (i,j): (512, 2028), (1379, 1380), (2029, 512), (2799, 14502), (3414, 3413)
X(30508) = trilinear product X(799)*X(2029)
X(30508) = trilinear quotient X(i)/X(j) for these (i,j): (661, 2028), (2029, 798)


X(30509) = ANTICOMPLEMENT OF X(13722)

Barycentrics    ((2*a^2-b^2-c^2)*K+2*a^4-2*(b^2+c^2)*a^2-b^4+4*b^2*c^2-c^4)*(b^2-c^2+K)*(-a^2+b^2+K)*(c^2-a^2+K)*(-b^2+c^2+K) : :, where K=sqrt(SW^2-3*S^2)
Barycentrics    (b^2-c^2) (5 a^4-5 a^2 b^2+2 b^4-5 a^2 c^2+b^2 c^2+2 c^4-2 (2 a^2-b^2-c^2) Sqrt[a^4-a^2 b^2+b^4-a^2 c^2-b^2 c^2+c^4]) : : (Peter Moses, January 3, 2019) 30519

See Jean-Louis Ayme and César Lozada, Hyacinthos 28776.

X(30509) lies on the Kiepert parabola, the cubics K010, K015, K242, K408 and these lines: {2, 1341}, {99, 110}, {523, 6189}

X(30509) = anticomplement of X(13722)
X(30509) = isotomic conjugate of X(30508)
X(30509) = Gibert-Simson transform of X(1380)
X(30509) = isotomic conjugate of the anticomplement of X(13636)
X(30509) = trilinear pole of the line {115, 2028}
X(30509) = reflection of X(2) in the line X(3414)X(9168)
X(30509) = barycentric product X(i)*X(j) for these {i,j}: {670, 2028}, {2966, 14502}, {3413, 6189}
X(30509) = barycentric quotient X(i)/X(j) for these (i,j): (512, 2029), (1380, 1379), (2028, 512), (2799, 14501), (3413, 3414)
X(30509) = trilinear product X(799)*X(2028)
X(30509) = trilinear quotient X(i)/X(j) for these (i,j): (661, 2029), (2028, 798)


X(30510) = X(3)X(6030)∩X(23)X(16186)

Barycentrics    (SB+SC)*(SA-SB)*(SA-SC)*(SA-15*R^2+3*SW) : :

See Jean-Louis Ayme and César Lozada, Hyacinthos 28776.

X(30510) lies on these lines: {3, 6030}, {23, 16186}, {30, 14385}, {110, 351}, {476, 1304}, {1113, 10288}, {1114, 10287}, {2070, 14670}, {2071, 7740}, {6760, 12113}, {9717, 15107}, {10130, 11058}, {12270, 14703}, {13595, 18114}, {14685, 15080}


X(30511) = X(67)X(9003)∩X(476)X(2407)

Barycentrics    (SB-SC)*(4*S^2-3*R^2*(30*R^2+5*SA-17*SW)+3*SA^2-4*SB*SC-7*SW^2) : :

See Jean-Louis Ayme and César Lozada, Hyacinthos 28776.

X(30511) lies on these lines: {67, 9003}, {476, 2407}, {523, 550}, {924, 12162}, {2528, 3313}, {9033, 12901}


X(30512) = EULER LINE INTERCEPT OF X(107)X(13398)

Barycentrics    (SA-SB)*(SA-SC)*(2*S^2-(SB+SC)*(SA-6*R^2+2*SW)) : :

See Jean-Louis Ayme and César Lozada, Hyacinthos 28776.

X(30512) lies on these lines: {2, 3}, {107, 13398}, {110, 925}, {476, 2407}, {691, 16167}, {1302, 3565}, {3233, 5502}, {5468, 6563}

X(30512) = isogonal conjugate of orthocenter of X(3)X(4)X(68)


X(30513) = ISOGONAL CONJUGATE OF X(1470)

Barycentrics    (-a+b+c)*(a^3-(b-c)*a^2-(b^2-2*b*c-c^2)*a+(b^2-c^2)*(b-c))*(a^3+(b-c)*a^2+(b^2+2*b*c-c^2)*a+(b^2-c^2)*(b-c)) : :

See Jean-Louis Ayme and César Lozada, Hyacinthos 28776.

X(30513) lies on the Feuerbach hyperbola and these lines: {1, 908}, {2, 104}, {4, 5554}, {7, 5080}, {9, 6735}, {10, 90}, {21, 2551}, {65, 5555}, {80, 3434}, {84, 377}, {149, 24297}, {388, 1476}, {390, 13278}, {404, 12667}, {405, 10942}, {442, 18542}, {443, 7705}, {452, 943}, {497, 1320}, {958, 10958}, {1000, 3421}, {1001, 10956}, {1041, 1877}, {1156, 2550}, {1329, 22768}, {1389, 5046}, {1392, 4345}, {1478, 3306}, {1512, 3359}, {1519, 6957}, {1537, 6929}, {1837, 10522}, {2320, 5328}, {2346, 11239}, {2475, 10308}, {2481, 11185}, {3577, 26333}, {3680, 5727}, {3753, 18516}, {3897, 5084}, {4187, 16203}, {4190, 18491}, {5187, 10532}, {5250, 7162}, {5251, 6910}, {5559, 5692}, {6850, 25005}, {6872, 11248}, {6919, 10586}, {6930, 12775}, {6931, 7951}, {8068, 10584}, {10527, 26476}, {10596, 14497}, {10679, 11113}, {11108, 18545}, {12608, 19860}, {12647, 18254}, {20895, 30479}, {21301, 23836}

X(30513) = isogonal conjugate of X(1470)
X(30513) = trilinear pole of the line {650, 2804}
X(30513) = {X(6256), X(24982)}-harmonic conjugate of X(377)
X(30513) = barycentric product X(312)*X(998)
X(30513) = barycentric quotient X(i)/X(j) for these (i,j): (8, 17740), (9, 997), (21, 26637), (607, 11383), (650, 9001), (998, 57)
X(30513) = trilinear product X(i)*X(j) for these {i,j}: {8, 998}, {522, 9058}
X(30513) = trilinear quotient X(i)/X(j) for these (i,j): (8, 997), (29, 4227), (33, 11383), (312, 17740), (333, 26637), (522, 9001), (998, 56)






leftri  Centers associated with intriangles and extriangles: X(30514) - X(30516)  rightri

Contributed by Clark Kimberling and Peter Moses, January 1, 2019.

Following TCCT (page 196), the intriangle of a point P = p: q : r (barycentrics) is the central triangle having A-vertex

A' = 0 : b (c q + b r cos A) : c (b r + c q) cos A

and the extriangle of P is the central triangle having A-vertex

A'' = -a p (c q + b r cos A)(b r + c q cos A) : a q (c q + b r cos A)(a r + c p cos B) : a p (a r + c q cos A)(a q + b p cos C)

Properties of A'B'C' and A'''B'''C'':

1. A'B'C' is inscribed in ABC, which is inscribed in A''B''C''.

2. The locus of P for which A'B'C' is perspective to ABC is the Darboux cubic, K004; the locus of the perspector is the Thomson cubic, K002.

3. The locus of P for which A''B''C'' is perspective to ABC is the union of the circumcircle, the lines BC, CA, AB, the line at infinity, and the cubic K004. If P is on the line at infinity, then the perspector is on the cubic K162.

4. The locus of P for which A''B''C'' is perspective to the cevian triangle of P is the same as for property 3.

5. The locus of P for which A'B'C' is perspective to the anticevian triangle of P is the union of the lines BC, CA, AB, and the cubic K004.

6. A'B'C'and A''B''C'' are perspective with perspector X(6).

7. The intriangle of X(2) is perspective to the circumsymmedial triangle at X(3).

8. The intriangle of X(6) is perspective to the 1st Ehrmann triangle (see X(8537) at X(12027); to the Artzt triangle (see X(9742) at X(6776), and to the anti-Honsberger triangle at X(184).

underbar




X(30514) = ORTHOLOGIC CENTER: INTRIANGLE OF X(6) TO 4TH BROCARD TRIANGLE

Barycentrics    (4 a^6-a^4 b^2-4 a^2 b^4+b^6-a^4 c^2+6 a^2 b^2 c^2-b^4 c^2-4 a^2 c^4-b^2 c^4+c^6) (5 a^10-22 a^8 b^2+14 a^6 b^4+20 a^4 b^6-19 a^2 b^8+2 b^10-22 a^8 c^2+83 a^6 b^2 c^2-48 a^4 b^4 c^2-19 a^2 b^6 c^2+14 b^8 c^2+14 a^6 c^4-48 a^4 b^2 c^4+60 a^2 b^4 c^4-16 b^6 c^4+20 a^4 c^6-19 a^2 b^2 c^6-16 b^4 c^6-19 a^2 c^8+14 b^2 c^8+2 c^10) : :

X(30514) lies on these lines: {111,381}, {3849,10295}


X(30515) = ORTHOLOGIC CENTER: INTRIANGLE OF X(6) TO CIRCIUMSYMMEDIAL TRIANGLE

Barycentrics    a^2 (a^2 b^2-b^4+a^2 c^2+3 b^2 c^2-c^4) (2 a^8 b^2-2 a^6 b^4-2 a^4 b^6+2 a^2 b^8+2 a^8 c^2-29 a^6 b^2 c^2+38 a^4 b^4 c^2+a^2 b^6 c^2-4 b^8 c^2-2 a^6 c^4+38 a^4 b^2 c^4-22 a^2 b^4 c^4+4 b^6 c^4-2 a^4 c^6+a^2 b^2 c^6+4 b^4 c^6+2 a^2 c^8-4 b^2 c^8) : :

X(30515) lies on these lines: {353,10564}, {575,6235}, {8705,14867}, {9027,12505}


X(30516) = CENTROID OF INTRIANGLE OF X(6)

Barycentrics    (a^2-2 b^2-2 c^2) (4 a^4+a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) : :

X(30516) lies on these lines: {2, 11166}, {39, 647}, {182, 9745}, {353, 6032}, {373, 10418}, {524, 10160}, {574, 13857}, {575, 6792}, {597, 6791}, {1495, 5475}, {1499, 12506}, {2502, 7603}, {3589, 16317}, {3815, 5642}, {3849, 10166}, {5913, 10168}, {7619, 9127}, {7736, 8779}, {9830, 10162}, {11550, 15880}, {12036, 12040}, {13394, 18907}

X(30516) = midpoint of X(353) and X(6032)
X(30516) = reflection of X(i) and X(j) for these {i,j}: {10163, 10160}, {13378, 10162}
X(30516) = crossdifference of every pair of points on line {23, 9871}
X(30516) = barycentric product X(i)*X(j) for these {i,j}: {599, 18907}, {5094, 13394}
X(30516) = barycentric quotient X(18907) / X(598)


X(30517) = COMPLEMENT OF X(22049)

Barycentrics    SA*((-64*R^2+SA+15*SW)*S^2+4*(SB+SC)*(4*R^2-SW)*(4*SA-32*R^2+7*SW)) : :

See Tran Quang Hung and César Lozada, Hyacinthos 28777.

X(30517) lies on this line: {2, 3}

X(30517) = reflection of X(i) in X(j) for these (i,j): (16273, 15948), (18017, 3)
X(30517) = complement of X(22049)
X(30517) = {X(15948), X(16273)}-harmonic conjugate of X(376)


X(30518) = COMPLEMENT OF X(22050)

Barycentrics    216*S^4-3*(R^2*(113*R^2+44*SW)-168*SB*SC-4*SW^2)*S^2-(7*R^2*(R^2-20*SW)-92*SW^2)*SB*SC : :

See Tran Quang Hung and César Lozada, Hyacinthos 28777.

X(30518) lies on this line: {2, 3}

X(30518) = complement of X(22050)


X(30519) = (name pending)

Barycentrics    (b - c)(2 b^2 + 2 c^2 + b c - a b - a c) : :

X(30519) lies on these lines: {30,511}, {39,6586}, {76,3261}, {194,21225}, {693,4931}, {1459,5145}, {3239,21212}, {3700,3776}, {3762,20906}, {3835,4120}, {3960,21348}, {4024,21116}, {4025,26248}, {4079,22037}, {4122,24720}, {4391,20908}, {4468,21196}, {4474,9902}, {4500,21104}, {4928,4944}, {6546,27486}, {7265,22043}, {14425,17069}, {20907,21443}

X(30519) = isogonal conjugate of X(30554)


X(30520) = (name pending)

Barycentrics    (b - c)(a^2 + 2b^2 + 2c^2 - a b - a c) : :

X(30520) lies on these lines: {30,511}, {312,693}, {650,3752}, {2490,7658}, {2509,3669}, {2516,11068}, {2526,4088}, {3004,4468}, {3210,17494}, {3700,23813}, {3776,4885}, {3777,21349}, {4024,22034}, {4025,4394}, {4106,25259}, {4171,17458}, {4379,21115}, {4382,4820}, {4391,4408}, {4462,15413}, {4728,4944}, {4949,20295}, {6590,21104}, {18071,21438}

X(30520) = isogonal conjugate of X(30555)
>


X(30521) = X(2)X(11147)∩X(187)X(11258)

Barycentrics    (SB+SC)*(27*S^4-9*(3*(6*SA+SW)*R^2-3*SA^2-6*SB*SC+SW^2)*S^2+(6*SA+SW)*SW^3) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28781.

X(30521) lies on these lines: {2, 11147}, {187, 11258}, {2930, 8586}, {5210, 14262}


X(30522) = ISOGONAL CONJUGATE OF X(22751)

Barycentrics    (5*R^2+SA-2*SW)*S^2-5*(3*R^2-SW)*SB*SC : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28781.

X(30522) lies on these lines: {3, 12278}, {4, 49}, {5, 13367}, {26, 12293}, {30, 511}, {68, 15138}, {110, 18403}, {125, 15646}, {143, 3575}, {186, 265}, {343, 550}, {382, 1993}, {403, 10113}, {546, 13403}, {568, 18559}, {1147, 18377}, {1495, 11563}, {1511, 2072}, {1568, 18572}, {1658, 9927}, {2070, 12902}, {2071, 12121}, {3153, 12383}, {3448, 13619}, {3618, 18420}, {3763, 7514}, {3853, 12897}, {5448, 18567}, {5449, 15331}, {5480, 19155}, {5576, 22804}, {5609, 18323}, {5654, 18568}, {5876, 14516}, {5899, 12412}, {5944, 10024}, {5946, 12022}, {6101, 12225}, {6102, 6240}, {6146, 13630}, {6241, 18565}, {6644, 18396}, {7564, 11425}, {7574, 15132}, {7577, 18430}, {7689, 18356}, {10095, 12241}, {10149, 12896}, {10151, 12140}, {10224, 12038}, {10226, 20299}, {10254, 11464}, {10255, 11449}, {10264, 21663}, {10282, 13406}, {10296, 23236}, {10610, 13160}, {10733, 14157}, {11250, 18381}, {11459, 18564}, {11591, 12605}, {12106, 18390}, {12111, 18562}, {12112, 12419}, {12118, 15139}, {12161, 12173}, {13142, 16982}, {13399, 16111}, {13490, 16657}, {14852, 18324}, {15114, 15122}, {18474, 18570}, {18945, 18952}, {19205, 19211}

X(30522) = isogonal conjugate of X(22751)


X(30523) = X(1153)X(3054)∩X(8588)X(14650)

Barycentrics    (SB+SC)*(567*S^6-27*(6*(18*SA-SW)*R^2-21*SA^2+30*SB*SC+SW^2)*S^4-3*(9*SA^2+18*SB*SC+2*(9*R^2-5*SW)*SW)*SW^2*S^2+2*SW^6) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28781.

X(30523) lies on these lines: {1153, 3054}, {8588, 14650}

X(30523) = reflection of X(11841) in the line X(28585)X(29235)


X(30524) = CIRCUMCIRCLE INVERSE OF X(28447)

Barycentrics    3*(S^2-3*SB*SC)*a*b*c - 4*(SB+SC)*OH*SA*S : :
X(30524) = 3*X(2100)+X(7982), 3*X(2104)-5*X(11482), X(11477)+3*X(15162)

As a point on the Euler line, X(30524) has Shinagawa coefficients [-3*R+2*OH, 9*R-2*OH] .

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28781.

X(30524) lies on these lines: {2, 3}, {2100, 7982}, {2104, 11482}, {2574, 5609}, {11477, 15162}

X(30524) = midpoint of X(i) and X(j) for these {i,j}: {3, 15157}, {10751, 15160}
X(30524) = reflection of X(20409) in X(140)
X(30524) = circumperp conjugate of X(28448)
X(30524) = circumcircle-inverse-of X(28447)
X(30524) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 15154, 15157), (1113, 1114, 28447), (1113, 15157, 3)


X(30525) = CIRCUMCIRCLE INVERSE OF X(28448)

Barycentrics    3*(S^2-3*SB*SC)*a*b*c + 4*(SB+SC)*OH*SA*S : :

As a point on the Euler line, X(30525) has Shinagawa coefficients [-3*R-2*OH, 9*R+2*OH] .

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28781.

X(30525) lies on these lines: {2, 3}, {2101, 7982}, {2105, 11482}, {2575, 5609}, {11477, 15163}

X(30525) = midpoint of X(i) and X(j) for these {i,j}: {3, 15156}, {10750, 15161}
X(30525) = reflection of X(20408) in X(140)
X(30525) = circumperp conjugate of X(28447)
X(30525) = circumcircle-inverse-of X(28448)
X(30525) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 15155, 15156), (140, 20408, 13626), (1113, 1114, 28448)


X(30526) = TRILINEAR POLE OF THE LINE X(1510)X(10095)

Barycentrics    (2*SB+R^2) *(2*SC+R^2)*(2*SB-3*R^2)*(2*SC-3*R^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28783.

X(30526) lies on these lines: {1263, 27684}, {1994, 11538}

X(30526) = trilinear pole of the line {1510, 10095}
X(30526) = barycentric quotient X(1263)/X(21230)


X(30527) = TRILINEAR POLE OF THE LINE X(30)X(54)

Barycentrics    (SA-SB)*(SA-SC)*(S^2+SA*SB)*(S^2+SA*SC)*(S^2+5*SB^2)*(S^2+5*SC^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28783.

X(30527) lies on these lines: {933, 4240}, {2407, 18315}, {9214, 11815}

X(30527) = trilinear pole of the line {30, 54}
X(30527) = barycentric product X(99)*X(11815)
X(30527) = barycentric quotient X(i)/X(j) for these (i,j): (110, 11591), (933, 3520)
X(30527) = trilinear product X(662)*X(11815)
X(30527) = trilinear quotient X(662)/X(11591)


X(30528) = TRILINEAR POLE OF THE LINE X(30)X(110)

Barycentrics    (SA-SB)*(SA-SC)*(3*S^2-18*R^2*SB-2*SA*SC+5*SB^2)*(3*S^2-18*R^2*SC-2*SA*SB+5*SC^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28783.

X(30528) lies on these lines: {249, 2407}, {250, 4240}, {376, 477}, {648, 5664}, {687, 15466}, {2421, 2436}, {14590, 23582}

X(30528) = trilinear pole of the line {30, 110}


X(30529) = TRILINEAR POLE OF THE LINE X(143)X(1510)

Barycentrics    (3*S^2-SA^2)*(S^2-3*SB^2)*(S^2-3*SC^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28783.

X(30529) lies on these lines: {2, 94}, {61, 8838}, {62, 8836}, {265, 3091}, {476, 5966}, {1141, 1166}, {3542, 6344}, {3832, 18300}, {3839, 18316}, {4232, 18384}, {7110, 18359}, {7533, 14356}, {7578, 9220}, {7785, 11004}, {15226, 25044}

X(30529) = polar conjugate of X(562)
X(30529) = trilinear pole of line {143, 1510}
X(30529) = barycentric product X(i)*X(j) for these {i,j}: {49, 18817}, {94, 1994}, {328, 3518}
X(30529) = barycentric quotient X(i)/X(j) for these (i,j): (4, 562), (49, 22115), (94, 11140), (143, 1154), (265, 3519), (476, 930)
X(30529) = trilinear product X(94)*X(2964)
X(30529) = trilinear quotient X(i)/X(j) for these (i,j): (92, 562), (94, 2962), (143, 2290)
X(30529) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (94, 18883, 2), (1989, 18883, 94)


X(30530) = TRILINEAR POLE OF THE LINE X(511)X(6033)

Barycentrics    (SA-SB)*(SA-SC)*(5*S^4-(4*SA*SC-SB^2-SW^2)*S^2+(5*SB-4*SW)*SB*SW^2)*(5*S^4-(4*SA*SB-SC^2-SW^2)*S^2+(5*SC-4*SW)*SC*SW^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28783.

X(30530) lies on this line: {5968, 7778}

X(30530) = trilinear pole of the line {511, 6033}


X(30531) = MIDPOINT OF X(5) AND X(11803)

Barycentrics    2 a^10-13 a^8 b^2+24 a^6 b^4-14 a^4 b^6-2 a^2 b^8+3 b^10-13 a^8 c^2+14 a^6 b^2 c^2+11 a^4 b^4 c^2-3 a^2 b^6 c^2-9 b^8 c^2+24 a^6 c^4+11 a^4 b^2 c^4+10 a^2 b^4 c^4+6 b^6 c^4-14 a^4 c^6-3 a^2 b^2 c^6+6 b^4 c^6-2 a^2 c^8-9 b^2 c^8+3 c^10 : :
Barycentrics    (19*R^2+6*SA-10*SW)*S^2+(23*R^2-8*SW)*SB*SC : :
X(30531) = X(3)-3*X(8254), X(3)+3*X(20424), 5*X(5)-X(3519), 3*X(5)+X(15801), 3*X(54)+X(3627), 3*X(195)+5*X(3091), X(546)-3*X(3574), X(546)+3*X(22051), 5*X(546)-3*X(22804), 3*X(1209)-5*X(12812), X(1493)+3*X(3574), X(1493)-3*X(22051), 5*X(1493)+3*X(22804), 3*X(2888)-11*X(5072), 7*X(3090)-3*X(21230), X(3519)+5*X(11803), 3*X(3519)+5*X(15801), 5*X(3574)-X(22804), 3*X(11803)-X(15801), 5*X(22051)+X(22804)

See Antreas Hatzipolakis, Ercole Suppa and César Lozada, Hyacinthos 28785 and Hyacinthos 28786

X(30531) lies on these lines: {3, 8254}, {4, 17507}, {5, 1173}, {30, 12242}, {54, 3627}, {113, 137}, {143, 12010}, {195, 3091}, {539, 3850}, {1154, 3628}, {1209, 12812}, {2888, 5072}, {2914, 11801}, {2918, 17714}, {3090, 21230}, {3518, 15806}, {3525, 12307}, {3851, 11271}, {3853, 10619}, {3857, 6288}, {5076, 12254}, {5079, 12316}, {6152, 13451}, {6689, 12108}, {7691, 14869}, {10272, 25714}, {10610, 12103}, {12102, 18400}, {12325, 15022}, {12606, 14449}, {12811, 20584}, {13432, 19709}, {15425, 16337}, {24144, 27423}

X(30531) = midpoint of X(i) and X(j) for these {i,j}: {5, 11803}, {546, 1493}, {2914, 11801}, {3574, 22051}, {3853, 10619}, {8254, 20424}, {12606, 14449}
X(30531) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (546, 22051, 1493), (1493, 3574, 546)


X(30532) = X(549)X(29959)∩X(21248)X(22110)

Barycentrics    4 a^8-18 a^6 b^2+25 a^4 b^4-12 a^2 b^6+b^8-18 a^6 c^2+22 a^4 b^2 c^2+28 a^2 b^4 c^2-6 b^6 c^2+25 a^4 c^4+28 a^2 b^2 c^4+10 b^4 c^4-12 a^2 c^6-6 b^2 c^6+c^8 : :
Barycentrics    15 S^2-18 R^2 SB-18 R^2 SC-9 SB SC+24 R^2 SW+3 SB SW+3 SC SW+2 SW^2 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28792.

X(30532) lies on these lines: {549,29959}, {21248,22110}


X(30533) = X(5)X(10216)∩X(140)X(6153)

Barycentrics    -2 (a-b-c)^2 (a+b-c)^2 (a-b+c)^2 (a+b+c)^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^12-6 a^10 b^2+14 a^8 b^4-16 a^6 b^6+9 a^4 b^8-2 a^2 b^10-6 a^10 c^2+18 a^8 b^2 c^2-10 a^6 b^4 c^2-11 a^4 b^6 c^2+10 a^2 b^8 c^2-b^10 c^2+14 a^8 c^4-10 a^6 b^2 c^4-11 a^4 b^4 c^4-8 a^2 b^6 c^4+4 b^8 c^4-16 a^6 c^6-11 a^4 b^2 c^6-8 a^2 b^4 c^6-6 b^6 c^6+9 a^4 c^8+10 a^2 b^2 c^8+4 b^4 c^8-2 a^2 c^10-b^2 c^10) : :
Barycentrics    3 S^4 + (-5 R^4-9 R^2 SB-9 R^2 SC-SB SC+R^2 SW+2 SB SW+2 SC SW+SW^2)S^2 + (-R^4-R^2 SW+SW^2)SB SC : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28792.

X(30533) lies on these lines: {5,10216}, {140,6153}, {523,3628}, {1209,13856}, {9827,13467}


X(30534) = X(182)X(1992)∩X(184)X(7736)

Barycentrics    2 a^4 b^2 c^2 (a^8-5 a^6 b^2+5 a^4 b^4-a^2 b^6-5 a^6 c^2+7 a^4 b^2 c^2+15 a^2 b^4 c^2-3 b^6 c^2+5 a^4 c^4+15 a^2 b^2 c^4+6 b^4 c^4-a^2 c^6-3 b^2 c^6) : :
Barycentrics    3 S^4 + (-5 R^4-9 R^2 SB-9 R^2 SC-SB SC+R^2 SW+2 SB SW+2 SC SW+SW^2)S^2 + (-R^4-R^2 SW+SW^2)SB SC : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28792.

X(30534) lies on these lines: {182,1992}, {184,7736}, {575,4558}, {7709,15033}


X(30535) = ISOGONAL CONJUGATE OF X(3815)

Barycentrics    a^2 (a^4-2 a^2 b^2+b^4-3 a^2 c^2-3 b^2 c^2) (a^4-3 a^2 b^2-2 a^2 c^2-3 b^2 c^2+c^4) : :
Barycentrics    (2 SB+2 SC+SW)S^2 -SB SC SW + 2 R^2 SW^2 + SB SW^2 + SC SW^2 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28792.

X(30535) lies on these lines: {2,5034}, {25,5012}, {37,26639}, {39,2987}, {97,14965}, {111,15018}, {182,263}, {251,1692}, {575,1976}, {597,1989}, {694,5038}, {1994,3108}, {2165,3618}, {2456,11673}, {2963,3589}, {8770,10601}, {8791,14389}, {9178,21460}

X(30535) = isogonal conjugate of X(3815)
X(30535) = cevapoint of X(6) and X(182)
X(30535) = trilinear pole of line X(512)X(2080)


X(30536) = X(186)X(6152)∩X(523)X(8254)

Barycentrics    -a^2 (a-b-c)^2 (a+b-c)^2 (a-b+c)^2 (a+b+c)^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (a^8-4 a^6 b^2+6 a^4 b^4-4 a^2 b^6+b^8-3 a^6 c^2+3 a^4 b^2 c^2+3 a^2 b^4 c^2-3 b^6 c^2+2 a^4 c^4+3 a^2 b^2 c^4+2 b^4 c^4+a^2 c^6+b^2 c^6-c^8) (a^8-3 a^6 b^2+2 a^4 b^4+a^2 b^6-b^8-4 a^6 c^2+3 a^4 b^2 c^2+3 a^2 b^4 c^2+b^6 c^2+6 a^4 c^4+3 a^2 b^2 c^4+2 b^4 c^4-4 a^2 c^6-3 b^2 c^6+c^8) : :
Barycentrics    (9 R^2-SB-SC-3 SW)S^4 + (-13 R^6-43 R^4 SB-43 R^4 SC+21 R^4 SW+30 R^2 SB SW+30 R^2 SC SW-9 R^2 SW^2-5 SB SW^2-5 SC SW^2+SW^3+SB SC (-7 R^2+3 SW))S^2 + (-R^6-R^4 SW+3 R^2 SW^2-SW^3)SB SC : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28792.

X(30536) lies on these lines: {186,6152}, {523,8254}, {567,15620}, {1209,13856}, {6288,9221}, {13434,14979}


X(30537) = ISOGONAL CONJUGATE OF X(15018)

Barycentrics    a^4 b^4 c^4 (a^4-2 a^2 b^2+b^4-5 a^2 c^2-2 b^2 c^2+c^4) (a^4-5 a^2 b^2+b^4-2 a^2 c^2-2 b^2 c^2+c^4) : :
Barycentrics    7 S^2+9 R^2 SB+9 R^2 SC-3 SB SC+3 SB SW+3 SC SW : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28792.

X(30537) lies on these lines: {2,13337}, {6,5054}, {37,3582}, {39,1989}, {50,251}, {111,3815}, {308,7799}, {393,13351}, {549,13338}, {597,2987}, {1383,7736}, {2165,7739}, {2963,5421}, {3108,5306}, {6128,9698}, {8749,9606}

X(30537) = isogonal conjugate of X(15018)


X(30538) = MIDPOINT OF X(1) AND X(15446)

Barycentrics    a^3 b^2 (a-b-c) c^2 (a+b+c)^3 (2 a^5-a^4 b-4 a^3 b^2+2 a^2 b^3+2 a b^4-b^5-a^4 c+6 a^3 b c-2 a^2 b^2 c-5 a b^3 c+4 b^4 c-4 a^3 c^2-2 a^2 b c^2+6 a b^2 c^2-3 b^3 c^2+2 a^2 c^3-5 a b c^3-3 b^2 c^3+2 a c^4+4 b c^4-c^5) : :
Barycentrics    (2 SB+2 SC+SW)S^2 - SB SC SW+2 R^2 SW^2+SB SW^2+SC SW^2 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28792.

X(30538) lies on these lines: {1,1399}, {10,2646}, {11,1385}, {55,3885}, {214,17606}, {946,1319}, {1001,10394}, {1071,11715}, {1388,12114}, {1459,24457}, {1737,26287}, {1837,15079}, {2320,3486}, {3583,11376}, {3601,11525}, {3646,13384}, {3649,24928}, {3916,5048}, {5542,20323}, {10543,10959}, {12743,24387}, {12758,26087}

X(30538) = midpoint of X(1) and X(15446)


X(30539) = X(141)X(574)∩X(525)X(11168)

Barycentrics    4 a^10-11 a^8 b^2-2 a^6 b^4+22 a^4 b^6-14 a^2 b^8+b^10-11 a^8 c^2+16 a^6 b^2 c^2+4 a^2 b^6 c^2-5 b^8 c^2-2 a^6 c^4+36 a^2 b^4 c^4+4 b^6 c^4+22 a^4 c^6+4 a^2 b^2 c^6+4 b^4 c^6-14 a^2 c^8-5 b^2 c^8+c^10 : :
Barycentrics    (54 R^2+3 SB+3 SC-18 SW)S^2 + (-54 R^2+12 SW)SB SC + 18 R^2 SB SW+18 R^2 SC SW-3 SB SW^2-3 SC SW^2-2 SW^3 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28792.

X(30539) lies on these lines: {141,574}, {525,11168}


X(30540) = X(2)X(575)∩X(2395)X(11166)

Barycentrics    2 a^4 b^4 c^4 (4 a^10-12 a^8 b^2+14 a^6 b^4-6 a^4 b^6-12 a^8 c^2-a^6 b^2 c^2+16 a^4 b^4 c^2-11 a^2 b^6 c^2-2 b^8 c^2+14 a^6 c^4+16 a^4 b^2 c^4+22 a^2 b^4 c^4+2 b^6 c^4-6 a^4 c^6-11 a^2 b^2 c^6+2 b^4 c^6-2 b^2 c^8) : :
Barycentrics    (-9 SB-9 SC-7 SW)S^4 + (3 SB SC SW+12 R^2 SW^2+3 SB SW^2+3 SC SW^2-3 SW^3)S^2 -SB SC SW^3 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28792.

X(30540) lies on these lines: {2,575}, {2395,11166}


X(30541) = ISOGONAL CONJUGATE OF X(7737)

Barycentrics    a^2 (a^4-2 a^2 b^2+b^4-3 c^4) (a^4-3 b^4-2 a^2 c^2+c^4) : :
Barycentrics    (12 R^2+SB+SC-4 SW)S^2 + (-18 R^2+4 SW)SB SC + 6 R^2 SB SW+6 R^2 SC SW-2 R^2 SW^2-SB SW^2-SC SW^2 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28792.

X(30541) lies on these lines: {3,11653}, {23,14906}, {183,525}, {297,11185}, {378,511}, {574,15066}, {599,6393}, {1078,9289}, {3455,6800}, {20977,21399}

X(30541) = isogonal conjugate of X(7737)


X(30542) = ISOGONAL CONJUGATE OF X(11002)

Barycentrics    a^4 b^4 c^4 (2 a^4-3 a^2 b^2+2 b^4-2 a^2 c^2-2 b^2 c^2) (2 a^4-2 a^2 b^2-3 a^2 c^2-2 b^2 c^2+2 c^4) : :
Barycentrics    7 S^4 + (9 R^2 SB+9 R^2 SC-3 SB SC-12 R^2 SW-4 SB SW-4 SC SW+3 SW^2)S^2 + SB SC SW^2 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28792.

X(30542) lies on these lines: {50,5094}, {182,599}, {183,7496}, {186,2453}, {187,18575}, {7771,11643}, {7778,10130}

X(30542) = isogonal conjugate of X(11002)






leftri  Centers associated with duple triangles: X(30543) -X(30550)  rightri

Suppose that A'B'C' is a central triangle in the plane of a reference triangle ABC, and that barycentrics for A' are u : v : w. The duple of A'B'C' is here introduced as the central triangle A''B''C'' having A'' = u : w : v; for example, the duple of the excentral triangle has A'' = -a : c : b. It is easy to prove that if two triangles are perspective, then their duples are perspective. (Clark Kimberling, January 1, 2019)

Peter Moses found the following perspectivities for the duple of the excentral triangle. (January 1, 2019):

ABC (TCCT 6.1): X(76)
medial (TCCT 6.2): X(6)
excentral (TCCT 6.7): X(3509)
half altitude / mid-height (TCCT 6.38): X(6)
second Neuberg (MathWorld): X(511)
first Brocard (CTC): X(76)
anti-first-Brocard (see ETC X(5939)): X(8782)
inner inscribed squares (MathWorld): X(6)
outer inscribed squares (MathWorld): X(6)
submedial (see ETC X(9813)): X(6)
Gemini 40: X(30543)
Gemini 42: X(141)
Gemini 75: X(30544)

See also the lists at X(30545), X(30547), and X(30556).

underbar




X(30543) = PERSPECTOR OF THESE TRIANGLES: DUPLE-OF-EXCENTRAL AND GEMINI 40

Barycentrics    a^4-a^3 b-2 a^2 b^2+a b^3+b^4-a^3 c+a^2 b c-5 a b^2 c-b^3 c-2 a^2 c^2-5 a b c^2-4 b^2 c^2+a c^3-b c^3+c^4 : :

X(30543) lies on these lines: {1, 26051}, {2, 4642}, {8, 12}, {10, 26136}, {73, 4861}, {75, 17084}, {145, 26137}, {946, 16824}, {3120, 5484}, {3616, 4000}, {3878, 25446}, {4054, 9369}, {4384, 11522}, {4673, 28628}, {4714, 5443}, {5260, 17777}, {5835, 9780}, {5836, 28389}, {6533, 16173}, {11376, 19804}, {12047, 16821}, {12053, 16823}, {16817, 30384}

X(30543) = X(4673),X(28628)}-harmonic conjugate of X(29839)


X(30544) = PERSPECTOR OF THESE TRIANGLES: DUPLE-OF-EXCENTRAL AND GEMINI 75

Barycentrics    a^7 b^3-a^6 b^4-a^5 b^5+a^8 b c-a^7 b^2 c-a^6 b^3 c+2 a^5 b^4 c-2 a^4 b^5 c-a^3 b^6 c+a^2 b^7 c-a^7 b c^2+3 a^6 b^2 c^2-3 a^5 b^3 c^2-3 a^4 b^4 c^2+a^3 b^5 c^2+a^7 c^3-a^6 b c^3-3 a^5 b^2 c^3+2 a^4 b^3 c^3-2 a^3 b^4 c^3-2 a^2 b^5 c^3+b^7 c^3-a^6 c^4+2 a^5 b c^4-3 a^4 b^2 c^4-2 a^3 b^3 c^4+a^2 b^4 c^4-a^5 c^5-2 a^4 b c^5+a^3 b^2 c^5-2 a^2 b^3 c^5-b^5 c^5-a^3 b c^6+a^2 b c^7+b^3 c^7 : :

X(30544) lies on these lines: {75, 21008}, {239, 27995}


X(30545) = PERSPECTOR OF THESE TRIANGLES: DUPLE-OF-ANTICEVIAN-OF-X(75) AND INTOUCH

Barycentrics    b c (a - b + c) (a + b - c) (- b c + a c + a b) : :

The A-vertex of the duple-of-anticevian-triangle-of-X(75) is A'' = -b c : a b : a c. Among its perspectivities with other triangles are these, all with perspector X(1):

ABC, media, incentral, excentral, mid-arc, 2nd midarc, 2nd circumperp, inner mixtilinears, outer mixtilinear, Andromeda, Antila, Aquilla, Caelum, innear Malfatti, outer Malfatti, inverse-of-ABC in circle, inner Yff, outer Yff, anti-Aquilla, 4th Conway, 5th Conway, inner Yff tangents, outer Yff tangents, Gemini 15. In addition to those, Peter Moses found the following perspectivities for A''B''C'' (January 5, 2019):

intouch (TCCT 6.8): X(30545)
2nd extouch (ETC X(5927)): X(30546)
2nd Conway (ETC X(9776)): X(30547)
Gemini 60: X(30548)

X(30545) lies on these lines: {1,18299}, {2,10030}, {7,350}, {33,18026}, {57,4554}, {65,18832}, {75,325}, {76,85}, {181,18057}, {194,28391}, {331,1848}, {335,1088}, {348,4352}, {497,6604}, {518,20935}, {693,3873}, {982,3663}, {1432,3978}, {1463,17082}, {1469,17149}, {1502,17786}, {1699,2481}, {2171,20567}, {3212,6376}, {3673,3944}, {4008,5988}, {4052,10029}, {4110,6382}, {4872,10446}, {5219,7243}, {6649,9316}, {7033,24524}, {7201,7205}, {8055,18135}, {16593,18045}, {17095,19786}, {17181,20256}, {21404,29641}, {22015,22019}


X(30546) = PERSPECTOR OF THESE TRIANGLES: DUPLE-OF-ANTICEVIAN-OF-X(75) AND 2ND EXTOUCH

Barycentrics    2 a^5 b^3-2 a^3 b^5+a^5 b^2 c-2 a^3 b^4 c+a b^6 c+a^5 b c^2-3 a^4 b^2 c^2-2 a^3 b^3 c^2-a b^5 c^2-b^6 c^2+2 a^5 c^3-2 a^3 b^2 c^3-2 a^3 b c^4+2 b^4 c^4-2 a^3 c^5-a b^2 c^5+a b c^6-b^2 c^6 : :

X(30546) lies on these lines: {9,17793}, {72,19222}, {226,262}, {440,20254}, {1513,7179}, {5928,21334}, {7350,7413}, {20256,27184}


X(30547) = PERSPECTOR OF THESE TRIANGLES: DUPLE-OF-ANTICEVIAN-OF-X(75) AND 2ND CONWAY

Barycentrics    a^4 b^2-a^2 b^4-a^4 b c-a^3 b^2 c-a^2 b^3 c+a b^4 c+a^4 c^2-a^3 b c^2+a^2 b^2 c^2-a b^3 c^2-b^4 c^2-a^2 b c^3-a b^2 c^3+2 b^3 c^3-a^2 c^4+a b c^4-b^2 c^4 : :

X(30547) lies on these lines: {2,7167}, {7,256}, {8,3978}, {329,1655}, {3794,9309}, {4388,17137}, {4594,8033}


X(30548) = PERSPECTOR OF THESE TRIANGLES: DUPLE-OF-ANTICEVIAN-OF-X(75) AND GEMINI 60

Barycentrics    a^4 b^4-a^4 b^3 c+a^3 b^4 c-4 a^4 b^2 c^2-5 a^3 b^3 c^2-2 a^2 b^4 c^2-a^4 b c^3-5 a^3 b^2 c^3+a^2 b^3 c^3-a b^4 c^3+a^4 c^4+a^3 b c^4-2 a^2 b^2 c^4-a b^3 c^4+b^4 c^4 : :

X(30548) lies on these lines: {2,22167}, {192,6378}


X(30549) = PERSPECTOR OF THESE TRIANGLES: DUPLE-OF-X(3)-REFLECTION-OF-ABC AND TANGENTIAL

Barycentrics    3 a^8-3 a^6 b^2+5 a^4 b^4-5 a^2 b^6-3 a^6 c^2-6 a^4 b^2 c^2+5 a^2 b^4 c^2+4 b^6 c^2+5 a^4 c^4+5 a^2 b^2 c^4-8 b^4 c^4-5 a^2 c^6+4 b^2 c^6 : :

The A-vertex of the duple-of-X(3)-reflection-of-ABC is A'' = -(a^2+b^2-c^2) (a^2-b^2+c^2) : 2 c^2 (a^2+b^2-c^2) : -2 b^2 (-a^2+b^2-c^2). Among its perspectivities with other triangles are these (Peter Moses, January 5, 2019):

ABC: X(264)
medial: X(1249)
anticomplementary: X(193)
orthic: X(3193)
Macbeath: X(264)
Artzt (ETC X(9742)) : X(523)
tangential: X(30549)
anti-Ascella (ETC X(11363): X(30550)

X(30549) lies on these lines: {6,1632}, {20,64}, {154,3164}, {230,393}, {264,9747}, {1368,6389}, {2847,6716}, {3053,3186}, {3515,15653}, {8667,9909}, {8716,20794}


X(30550) = PERSPECTOR OF THESE TRIANGLES: DUPLE-OF-X(3)-REFLECTION-OF-ABC AND ANTI-ASCELLA

Barycentrics    3 a^8-9 a^6 b^2+13 a^4 b^4-7 a^2 b^6-9 a^6 c^2+6 a^4 b^2 c^2-a^2 b^4 c^2+8 b^6 c^2+13 a^4 c^4-a^2 b^2 c^4-16 b^4 c^4-7 a^2 c^6+8 b^2 c^6 : :

X(30550) lies on these lines: {2,27364}, {20,3564}, {427,1007}, {2165,3054}, {2974,11404}, {7494,18287}, {8266,8667}, {14570,19118}


X(30551) = EULER LINE INTERCEPT OF X(54)X(20193)

Barycentrics    4 a^10-9 a^8 b^2+2 a^6 b^4+8 a^4 b^6-6 a^2 b^8+b^10-9 a^8 c^2+4 a^6 b^2 c^2-9 a^4 b^4 c^2+17 a^2 b^6 c^2-3 b^8 c^2+2 a^6 c^4-9 a^4 b^2 c^4-22 a^2 b^4 c^4+2 b^6 c^4+8 a^4 c^6+17 a^2 b^2 c^6+2 b^4 c^6-6 a^2 c^8-3 b^2 c^8+c^10 : :
Barycentrics    (41 R^2-10 SW)S^2 + (R^2+6 SW)SB SC : :

As a point on the Euler line, X(30551) has Shinagawa coefficients {E - 40 F, 25 E + 24 F}.

See Kadir Altintas and Ercole Suppa, Hyacinthos 28797.

X(30551) lies on these lines: {2,3}, {54,20193}, {5642,16982}

X(30551) = reflection of X(5) in X(21451)
X(30551) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {2,23409,5}, {5,2070,550}, {140,21308,5}, {3845,6240,3627}, {10096,13621,5},{13595,18282,5}


X(30552) = EULER LINE INTERCEPT OF X(69)X(11440)

Barycentrics    -5 a^10+9 a^8 b^2+2 a^6 b^4-10 a^4 b^6+3 a^2 b^8+b^10+9 a^8 c^2-32 a^6 b^2 c^2+18 a^4 b^4 c^2+8 a^2 b^6 c^2-3 b^8 c^2+2 a^6 c^4+18 a^4 b^2 c^4-22 a^2 b^4 c^4+2 b^6 c^4-10 a^4 c^6+8 a^2 b^2 c^6+2 b^4 c^6+3 a^2 c^8-3 b^2 c^8+c^10 : :
Barycentrics    (10 R^2-2 SW)S^2 + (-16 R^2+3 SW)SB SC : :

As a point on the Euler line, X(30552) has Shinagawa coefficients {E - 4 F, -2 E + 6 F}.

See Kadir Altintas and Ercole Suppa, Hyacinthos 28797.

X(30552) lies on these lines: {2,3}, {69,11440}, {74,11411}, {110,6225}, {343,8567}, {394,5894}, {925,5897}, {1092,20427}, {1204,6515}, {1294,13398}, {5012,15740}, {5504,16111}, {5895,11064}, {5925,20725}, {9140,15077}, {9833,16163}, {11206,12279}, {11441,12250}, {12324,13445}, {14457,18911}, {15072,18925}

X(30552) = reflection of X(i) in X(j) for these {i,j}: {4,3548}, {3542,3}
X(30552) = X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,382,16238}, {3,1885,2}, {20,2071,4}, {20,3522,22}, {20,7396,5059}, {20,11413,1370}, {22,858,4232}, {550,21312,20}, {11413,16386,20}


X(30553) = (name pending)

Barycentrics    -10 a^10+21 a^8 b^2-2 a^6 b^4-20 a^4 b^6+12 a^2 b^8-b^10+21 a^8 c^2-18 a^6 b^2 c^2+47 a^4 b^4 c^2-53 a^2 b^6 c^2+3 b^8 c^2-2 a^6 c^4+47 a^4 b^2 c^4+82 a^2 b^4 c^4-2 b^6 c^4-20 a^4 c^6-53 a^2 b^2 c^6-2 b^4 c^6+12 a^2 c^8+3 b^2 c^8-c^10 : :
Barycentrics    (115 R^2-22 SW)S^2 + (-R^2+18 SW)SB SC : :

As a point on the Euler line, X(30553) has Shinagawa coefficients {27 E - 88 F, 71 E + 72 F}.

See Kadir Altintas and Ercole Suppa, Hyacinthos 28797.

X(30553) lies on this line: {2,3}

X(30553) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {140,5899,548}, {547,13163,546}


X(30554) = ISOGONAL CONJUGATE OF X(30519)

Barycentrics    a^2 (a-b) (a-c) (2 a^2+a b+2 b^2-a c-b c) (2 a^2-a b+a c-b c+2 c^2) : :

X(30554) lies on the circumcircle and these lines: {6,28563}, {32,106}, {103,182}, {105,10789}, {753,4279}, {1477,12835}, {1691,2712}, {2080,2700}, {12212,28485}

X(30554) = isogonal conjugate of X(30519)
X(30554) = X(i)-isoconjugate of X(j) for these (i,j): {1, 30519}, {513, 17230}, {9461, 20568}
X(30554) = cevapoint of X(649) and X(5332)
X(30554) = trilinear pole of line {6, 9459}
X(30554) = barycentric quotient X(i) / X(j) for these {i,j}: {6, 30519}, {101, 17230}, {9459, 9461}


X(30555) = ISOGONAL CONJUGATE OF X(30520)

Barycentrics    a^2 (a-b) (a-c) (2 a^2+2 b^2-a c-b c+c^2) (2 a^2-a b+b^2-b c+2 c^2) : :

X(30555) lies on the circumcircle and these lines: {604,1477}, {692,6078}, {907,5546}, {5549,8695}

X(30555) = isogonal conjugate of X(30520)
X(30555) = X(i)-isoconjugate of X(j) for these (i,j): {1, 30520}, {513, 17284}, {514, 3242}, {3669, 4901}
X(30555) = barycentric quotient X(i) / X(j) for these {i,j}: {6, 30520}, {101, 17284}, {692, 3242}, {3939, 4901}


X(30556) = PERSPECTOR OF THESE TRIANGLES: DUPLE-OF-HALF-ALTITUDE AND CEVIAN OF X(13386)

Barycentrics    a (a^2-b^2-2 b c-c^2-2 S) : :

Let A'B'C' be the duple of the half-altitude triangle, so that

A' = a^2 : a^2 - b^2 + c^2 : a^2 + b^2 - c^2.

The locus of a point P such that A'B'C' is perpsective to the cevian triangle of P is the cubic K170.
The locus of a point P such that A'B'C' is perpsective to the anticevian triangle of P is the cubic K707.
The locus of a point P such that A'B'C' is orthologic to the cevian triangle of P is the cubic K170.
The locus of a point P such that A'B'C' is orthologic to the anticevian triangle of P is the cubic K045.
The locus of a point P such that A'B'C' is paralogic to the cevian triangle of P is the cubic K211.
The locus of a point P such that A'B'C' is paralogic to the anticevian triangle of P is a cubic; see just below for an equation.
(Peter Moses, January 8, 2019)

Let f(a,b,c,x,y,z) = a^2 (b^2 - c^2) (-a^2 + b^2 + c^2) (y - z) y z. The cubic mentioned just above is given by the following equation:

f(a,b,c,x,y,z) + f(b,c,a,y,z,x) + f(c,a,b,z,x,y) - 2 a^2 b^2 c^2 x y z = 0.

The following perspectivities were contributed by Peter Moses, January 5, 2019:

A'B'C' is perspective to the following triangles, with perspector X(2): orthic, submedial (see (9813)), orthic-of-anticomplementary.

A'B'C' is perspective to the following triangles, with perspector X(3): medial, tangential, 1st circumperp, 2nd circumperp, outer Napoleon, inner Napoleon, outer Fermat, inner Fermat, outer Vecten, inner Vector, 1st Neuberg, 2nd Neuberg, Fuhrmann, 1st Brocard, Kosnita, McCay, Trinh, Carnot, 2nd Euler, Ara, 1st Ehrmann, Ascella, Ae, Ai, infinite altitude, anti-Hutson intouch (see X11363)), anti-incircle-circles (see X(11363)), Ehrmann side-triangle

The appearance of T,i in the following list means that A'B'C' is perspective to the triangle T, and the perspector is X(i):

ABC,69
reflection of ABC in X(3), 11821
reflection of X(3) in ABC, 11850
extangents, 10319
circum-orthic, 631
inner Garcia, 11512
4th extouch, 69
2nd Ehrmann, 11511
1st Kenmotu diagonal, (see X(31)), 11513
2nd Kenmotu diagonal, 11514
inner tri-equilateral (see X(10631)), 11515
outer tri-equilateral (see X(10631)), 11516
anti-Ascella (see X(11363)), 7484
anti-Conway (see X(11363)), 182 anti-3rd Euler (see X(11363), 7998
anti-4rd Euler (see X(11363), 7999
5th mixtilinear of orthic (see X(11363)), 20
tangential-of-anticomplementary (see X(11363)), 7386
aAOS (see X(15015)), 19378
1st excosine (see X(17807)), 17811
Ehrmann vertex-triangle, 18531
anti-Atik, 69
1st anti-Sharygin, 95

X(30556) lies on the cubics K168, K199, K332 and these lines: {1,6}, {2,175}, {3,6212}, {8,7090}, {10,486}, {21,1805}, {55,7348}, {56,6204}, {63,3083}, {65,6203}, {78,2066}, {142,481}, {144,176}, {169,8225}, {188,3082}, {200,15892}, {329,1659}, {345,13425}, {348,13453}, {371,997}, {372,12514}, {388,30324}, {482,527}, {485,21616}, {517,1377}, {590,25681}, {615,26066}, {936,1702}, {993,13333}, {1123,3421}, {1152,4640}, {1267,6337}, {1329,13911}, {1336,6857}, {1372,20195}, {1374,6173}, {1378,5044}, {1385,9678}, {1806,1812}, {2067,19861}, {2362,3869}, {2551,6351}, {3070,24703}, {3071,5794}, {3084,3305}, {3086,8957}, {3452,5393}, {3485,30325}, {3616,30413}, {3681,15890}, {4517,6405}, {5250,5414}, {5405,5745}, {5409,6513}, {5438,9616}, {5698,6460}, {5784,30400}, {5837,13936}, {5880,30425}, {6172,17805}, {6352,13959}, {6561,17647}, {6700,13912}, {8945,17594}, {13941,18231}, {15587,30354}, {15891,16214}, {17768,30426}, {17802,18230}, {19029,21677}, {20059,21169}

X(30556) = isogonal conjugate of X(2362)
X(30556) = X(7347)-complementary conjugate of X(141)
X(30556) = X(15890)-Ceva conjugate of X(9)
X(30556) = X(2066)-cross conjugate of X(13389)
X(30556) = cevapoint of X(3) and X(1124)
X(30556) = X(i)-isoconjugate of X(j) for these (i,j): {1, 2362}, {4, 2067}, {6, 1659}, {19, 13388}, {56, 7090}, {57, 7133}, {225, 1805}, {278, 5414}, {1123, 6502}, {2066, 13437}, {6213, 16232}
X(30556) = barycentric product X(i) X(j) for these {i,j}: {8, 13389}, {63, 14121}, {75, 2066}, {78, 13390}, {312, 6502}, {321, 1806}, {345, 16232}, {1267, 7133}, {1806, 321}, {2066, 75}, {2362, 13425}, {3083, 7090}, {6502, 312}, {7090, 3083}, {7133, 1267}, {13389, 8}, {13390, 78}, {13425, 2362}, {14121, 63}, {16232, 345}
X(30556) = barycentric quotient X(i) / X(j) for these {i,j}: {1, 1659}, {3, 13388}, {6, 2362}, {9, 7090}, {48, 2067}, {55, 7133}, {212, 5414}, {605, 6502}, {1124, 13389}, {1806, 81}, {2066, 1}, {2193, 1805}, {2362, 13437}, {5414, 6213}, {6212, 13390}, {6502, 57}, {7133, 1123}, {13389, 7}, {13390, 273}, {14121, 92}, {16232, 278}
X(30556) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1, 1743, 18991}, {1, 5223, 3641}, {1, 16469, 11371}, {1, 19003, 1449}, {8, 30412, 7090}, {37, 7968, 1}, {63, 3083, 13389}, {1279, 5604, 1}


X(30557) = PERSPECTOR OF THESE TRIANGLES: DUPLE-OF-HALF-ALTITUDE AND CEVIAN OF X(13387)

Barycentrics    a (a^2-b^2-2 b c-c^2+2 S) : :

X(30557) lies on the cubics K168, K199, K332 and these lines: {1,6}, {2,176}, {3,6213}, {8,14121}, {10,485}, {21,1806}, {55,7347}, {56,6203}, {63,2067}, {65,6204}, {78,5414}, {142,482}, {144,175}, {188,483}, {200,15891}, {329,13390}, {345,13458}, {348,13436}, {371,12514}, {372,997}, {388,30325}, {481,527}, {486,21616}, {517,1378}, {590,26066}, {615,25681}, {936,1703}, {993,13332}, {1123,6857}, {1151,4640}, {1329,13973}, {1336,3421}, {1371,20195}, {1373,6173}, {1377,5044}, {1805,1812}, {2066,5250}, {2551,6352}, {3070,5794}, {3071,24703}, {3083,3305}, {3452,5405}, {3485,30324}, {3579,9679}, {3616,30412}, {3681,15889}, {3781,7594}, {3869,16232}, {4517,6283}, {5391,6337}, {5393,5745}, {5408,6513}, {5698,6459}, {5784,30401}, {5837,13883}, {5880,30426}, {6172,17802}, {6351,13902}, {6502,19861}, {6560,17647}, {6700,13975}, {8941,17594}, {8957,18391}, {8972,18231}, {15587,30355}, {15892,16213}, {17768,30425}, {17805,18230}, {19030,21677}

X(30557) = isogonal conjugate of X(16232)
X(30557) = isotomic conjugate of the polar conjugate of X(7133)
X(30557) = X(7348)-complementary conjugate of X(141)
X(30557) = X(15889)-Ceva conjugate of X(9)
X(30557) = X(5414)-cross conjugate of X(13388)
X(30557) = cevapoint of X(3) and X(1335)
X(30557) = X(i)-isoconjugate of X(j) for these (i,j): {1, 16232}, {4, 6502}, {6, 13390}, {19, 13389}, {56, 14121}, {225, 1806}, {278, 2066}, {1336, 2067}, {2362, 6212}, {5414, 13459}
X(30557) = barycentric product X(i) X(j) for these {i,j}: {8, 13388}, {63, 7090}, {69, 7133}, {75, 5414}, {78, 1659}, {312, 2067}, {321, 1805}, {345, 2362}, {1659, 78}, {1805, 321}, {2067, 312}, {2362, 345}, {3084, 14121}, {5414, 75}, {7090, 63}, {7133, 69}, {13388, 8}, {13458, 16232}, {14121, 3084}, {16232, 13458}
X(30557) = barycentric quotient X(i) / X(j) for these {i,j}: {1, 13390}, {3, 13389}, {6, 16232}, {9, 14121}, {48, 6502}, {212, 2066}, {606, 2067}, {1335, 13388}, {1659, 273}, {1805, 81}, {2066, 6212}, {2067, 57}, {2193, 1806}, {2362, 278}, {5414, 1}, {6213, 1659}, {7090, 92}, {7133, 4}, {13388, 7}, {16232, 13459}


X(30558) = PERSPECTOR OF THESE TRIANGLES: DUPLE-OF-HALF-ALTITUDE AND ANTICEVIAN OF X(6339)

Barycentrics    (3 a^2-b^2-c^2) (a^4+2 a^2 b^2+b^4-6 a^2 c^2+2 b^2 c^2+c^4) (a^4-6 a^2 b^2+b^4+2 a^2 c^2+2 b^2 c^2+c^4) : :

X(30558) lies on the cubic K168 and these lines: {2,14248}, {3,15369}, {6,6337}, {69,6342}, {439,19118}, {3926,15525}, {5395,11059}

X(30558) = X(19214)-complementary conjugate of X(141)
X(30558) = X(3)-cross conjugate of X(6337)
X(30558) = X(i)-isoconjugate of X(j) for these (i,j): {1611, 8769}, {2128, 14248}
X(30558) = barycentric product X(i) X(j) for these {i,j}: {193, 6339}, {6339, 193}
X(30558) = barycentric quotient X(i) / X(j) for these {i,j}: {193, 6392}, {3053, 1611}, {3167, 19588}, {6337, 19583}, {6339, 2996}, {10607, 6461}, {15369, 14248}


X(30559) = MIDPOINT OF X(16) AND X(5238)

Barycentrics    a^2*(a^4-(b^2+c^2)*a^2-2*b^2*c^2-2*sqrt(3)*(3*a^2-2*b^2-2*c^2)*S) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28802.

X(30559) lies on these lines: {3, 6}, {17, 10616}, {30, 22891}, {533, 14144}, {617, 16530}, {2004, 3131}

X(30559) = midpoint of X(16) and X(5238)
X(30559) = reflection of X(17) in X(10616)
X(30559) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 19780, 3107), (16, 21159, 5351), (182, 5206, 30560)


X(30560) = MIDPOINT OF X(15) AND X(5237)

Barycentrics    a^2*(a^4-(b^2+c^2)*a^2-2*b^2*c^2+2*sqrt(3)*(3*a^2-2*b^2-2*c^2)*S) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28802.

X(30560) lies on these lines: {3, 6}, {18, 10617}, {30, 22846}, {532, 14145}, {616, 16529}, {2005, 3132}

X(30560) = midpoint of X(15) and X(5237)
X(30560) = reflection of X(18) in X(10617)
X(30560) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 19781, 3106), (15, 21158, 5352), (182, 5206, 30559)




leftri  Centers associated with the Gemini triangles 1-40: X(30561) - X(30713)  rightri

These centers were contributed by Randy Hutson, January 9, 2019. The Gemini triangles are introduced in the preamble just before X(24537).

underbar



X(30561) = X(2)X(319)∩X(149)X(214)

Barycentrics    (a + 2 b + 2 c) (4 a^2 + 4 a b + 4 a c + 3 b c) : :

Let A'1B'1C'1 be as at X(25417). Triangle A'1B'1C'1 is homothetic to the medial triangle at X(30561).

X(30561) lies on these lines: {2, 319}, {149, 214}, {4671, 16826}, {5333, 16777}, {25418, 30563}, {30587, 30589} et al

X(30561) = complement of X(30590)
X(30561) = {X(2),X(30562)}-harmonic conjugate of X(16777)
X(30561) = barycentric product X(30563)*X(30589)
X(30561) = barycentric quotient X(30563)/X(30590)


X(30562) = X(2)X(319)∩X(20)X(1385)

Barycentrics    4 a^3 + 9 a^2 (b + c) + a (5 b^2 + 12 b c + 5 c^2) + 3 b c (b + c) : :

Let A'1B'1C'1 be as at X(25417). Triangle A'1B'1C'1 is homothetic to the anticomplementary triangle at X(30562).

X(30562) lies on these lines: {1, 17163}, {2, 319}, {20, 1385}, {63, 8025}, {81, 25418}, {86, 17147}, {194, 3995}, {5625, 17135} et al

X(30562) = {X(16777),X(30561)}-harmonic conjugate of X(2)


X(30563) = X(2)X(44)∩X(45)X(4671)

Barycentrics    (a - 2 b - 2 c) (4 a^2 + 4 a b + 4 a c + 3 b c) : :

Let A'2B'2C'2 be as at X(25418). Triangle A'2B'2C'2 is homothetic to the medial triangle at X(30563).

X(30563) lies on these lines: {2, 44}, {45, 4671}, {2475, 3647}, {25418, 30561} et al

X(30563) = complement of X(30589)
X(30563) = {X(2),X(30564)}-harmonic conjugate of X(89)
X(30563) = barycentric product X(30561)*X(30590)
X(30563) = barycentric quotient X(30561)/X(30589)


X(30564) = X(2)X(44)∩X(20)X(355)

Barycentrics    4 a^3 - a^2 (b + c) - a (5 b^2 + 4 b c + 5 c^2) - 3 b c (b + c) : :

Let A'2B'2C'2 be as at X(25418). Triangle A'2B'2C'2 is homothetic to the anticomplementary triangle at X(30564).

X(30564) lies on these lines: {1, 16704}, {2, 44}, {20, 355}, {45, 1150}, {81, 25418}, {194, 16816}, {213, 14996}, {333, 17147}, {1764, 3219} et al

X(30564) = anticomplement of X(30588)
X(30564) = {X(89),X(30563)}-harmonic conjugate of X(2)


X(30565) = CENTROID OF SIDE-TRIANGLE OF GEMINI TRIANGLES 2 AND 30

Barycentrics    (b - c) (a^2 + b^2 + c^2 - 2 a b - 2 a c + b c) : :

Since Gemini triangles 2 and 30 are perspective (at X(63)), their side-triangle is degenerate, lying on the perspectrix, line X(514)X(661).

X(30565) lies on these lines: {2, 918}, {100, 190}, {514, 661}, {522, 14392}, {523, 4800}, {650, 4467}, {654, 3219}, {812, 4120}, {824, 4893}, {926, 3681}, {1121, 6366}, {1635, 2786} et al

X(30565) = anticomplement of X(1638)


X(30566) = CENTROID OF CROSS-TRIANGLE OF GEMINI TRIANGLES 2 AND 30

Barycentrics    a^2 (b + c) - 4 a b c - (b + c) (b^2 - 3 b c + c^2) : :

Since the vertices of Gemini triangles 2 and 30 lie on the same conic (the circumconic centered at X(9)), their cross-triangle is degenerate, lying on line X(514)X(661).

X(30566) lies on these lines: {2, 45}, {8, 4767}, {11, 3952}, {100, 15507}, {121, 4674}, {149, 3699}, {244, 11814}, {312, 3969}, {321, 3452}, {329, 6557}, {514, 661}, {528, 17780}, {537, 1647}, {645, 24624}, {899, 4442}, {1265, 5187} et al


X(30567) = X(1)X(2)∩X(57)X(312)

Barycentrics    a^3 - a b^2 - a c^2 + 4 a b c - 2 b^2 c - 2 b c^2 : :

X(30567) is the perspector of Gemini triangle 30 and the tangential triangle, wrt Gemini triangle 2, of the circumconic centered at X(9).

X(30567) lies on these lines: {1, 2}, {9, 14829}, {57, 312}, {63, 4358}, {69, 1997}, {75, 5437}, {76, 7196}, {144, 8055}, {165, 3685}, {190, 3928}, {320, 28609}, {321, 3306}, {333, 7308}, {341, 6762}, {344, 5745}, {345, 3911}, {346, 5435}, {908, 21621}, {1043, 5438}, {1089, 3338}, {1150, 3305}, {1155, 4387}, {1265, 24391}, {1376, 3886}, {1453, 13741}, {1699, 4645}, {1706, 4673}, {1707, 4011}, {2050, 9856} et al

X(30567) = {X(63),X(4358)}-harmonic conjugate of X(30568)


X(30568) = X(1)X(979)∩X(9)X(312)

Barycentrics    (a - b - c) (a^2 + a b + a c - 2 b c) : :

X(30568) is the perspector of Gemini triangle 2 and the tangential triangle, wrt Gemini triangle 30, of the circumconic centered at X(9).

X(30568) lies on these lines: {1, 979}, {2, 2415}, {8, 4082}, {9, 312}, {10, 2899}, {43, 4368}, {55, 4009}, {57, 190}, {63, 4358}, {75, 7308}, {86, 25430}, {165, 5205}, {192, 2999}, {200, 3685}, {210, 3886}, {226, 344}, {321, 3294}, {329, 3912}, {341, 1697}, {345, 2325}, {346, 3687}, {390, 5423}, {497, 3717}, {552, 4633}, {726, 5272}, {748, 3994}, {908, 17776}, {936, 7283}, {950, 1265}, {1001, 3967}, {1120, 3622}, {1698, 4425}, {1699, 17777}, {1743, 1999}, {1997, 3911}, {2321, 14555} et al

X(30568) = anticomplement of X(24175)
X(30568) = {X(63),X(4358)}-harmonic conjugate of X(30567)


X(30569) = X(1936)X(2342)∩X(4781)X(5744)

Barycentrics    a^7 - a^6 (b + c) - a^5 (b^2 - 5 b c + c^2) - a^4 (b + c) (b^2 + c^2) + a^3 (b^2 + c^2) (3 b^2 - 5 b c + 3 c^2) + a^2 (b^3 - c^3) (b^2 - c^2) - a (3 b^6 - 4 b^5 c + b^4 c^2 + b^2 c^4 - 4 b c^5 + 3 c^6) + (b - c)^2 (b^5 - 3 b^3 c^2 - 3 b^2 c^3 + c^5) : :

Let A2B2C2 and A30B30C30 be Gemini triangles 2 and 30, resp. X(30569) is the radical center of the circumcircles of triangles AA2A30, BB2B30 and CC2C30.

X(30569) lies on these lines: {1936, 2342}, {4781, 5744}


X(30570) = (name pending)

Barycentrics    a/((a^2 + 2 a b + 2 a c + b c) (2 a^3 (b + c) + a^2 (b^2 + 3 b c + c^2) + a b c (b + c) + b^2 c^2)) : :

Let A4B4C4 be Gemini triangle 4. Let A' be the center of conic {{A,B,C,B4,C4}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30570).

X(30570) lies on the line {3736, 30571}


X(30571) = ISOGONAL CONJUGATE OF X(4649)

Barycentrics    a/(a^2 + 2 a b + 2 a c + b c) : :

Let A4B4C4 be Gemini triangle 4. Let A' be the perspector of conic {{A,B,C,B4,C4}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30571).

X(30571) lies on these lines: {1, 1573}, {2, 740}, {28, 2201}, {37, 291}, {42, 1255}, {43, 25430}, {57, 846}, {81, 238}, {86, 4368}, {105, 8297}, {274, 350}, {277, 24161}, {278, 1874}, {330, 1655}, {538, 25055}, {551, 3227}, {984, 1002}, {985, 1001}, {1929, 8299}, {3736, 30570} et al

X(30571) = isogonal conjugate of X(4649)


X(30572) = X(523)X(656)∩X(900)X(1317)

Barycentrics    (b^2 - c^2) (2 a - b - c)/(a - b - c) : :

X(30572) is the intersection of perspectrices of every pair of {ABC, Gemini triangle 9, Gemini triangle 10}.

X(30572) lies on these lines: {65, 4145}, {522, 4318}, {523, 656}, {900, 1317}, {1365, 2611}, {1769, 21132} et al

X(30572) = barycentric product X(i)*X(j) for these {i,j}: {7, 4120}, {65, 3762}, {226, 900}, {519, 7178}, {523, 3911}, {1317, 4049}, {3676, 3943}
X(30572) = barycentric quotient X(i)/X(j) for these (i,j): (7, 4615), (10, 4582), (56, 4591), (57, 4622), (65, 3257), (226, 4555), (519, 645), (523, 4997), (900, 333), (3762, 314), (3911, 99), (3943, 3699), (4120, 8), (7178, 903)


X(30573) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 9

Barycentrics    (b - c) (2 a - b - c) (2 a^2 - b^2 - c^2 - a b - a c + 2 b c) : :

The side-triangle of ABC and Gemini triangle 9 is degenerate, lying on the perspectrix, line X(900)X(1317).

X(30573) lies on these lines: {1, 514}, {390, 6006}, {513, 5919}, {519, 4543}, {522, 3241}, {900, 1317}

X(30573) = tripolar centroid of X(3911)
X(30573) = barycentric product X(i)*X(j) for these {i,j}: {514, 6174}, {519, 1638}, {16704, 30574}
X(30573) = barycentric quotient X(i)/X(j) for these (i,j): (1638, 903), (6174, 190), (30574, 4080)


X(30574) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 10

Barycentrics    (b^2 - c^2) (2 a^2 - b^2 - c^2 - a b - a c + 2 b c) : :

The side-triangle of ABC and Gemini triangle 10 is degenerate, lying on the perspectrix, line X(523)X(656).

X(30574) lies on the line {523, 656}

X(30574) = tripolar centroid of X(226)
X(30574) = barycentric product X(i)*X(j) for these {i,j}: {10, 1638}, {4080, 30573}
X(30574) = barycentric quotient X(i)/X(j) for these (i,j): (1638, 86), (30573, 16704)


X(30575) = X(44)X(88)∩X(81)X(4638)

Barycentrics    a (b + c)/(2 a - b - c)^2 : :

Let A9B9C9 be Gemini triangle 9. Let A' be the center of conic {{A,B,C,B9,C9}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30575).

X(30575) lies on these lines: {44, 88}, {81, 4638}, {758, 4674}, {903, 17495}, {908, 6549}, {1318, 1870} et al

X(30575) = isotomic conjugate of X(16729)
X(30575) = barycentric product X(i)*X(j) for these {i,j}: {10, 679}, {88, 4080}, {903, 4674}
X(30575) = barycentric quotient X(i)/X(j) for these (i,j): (2, 16729), (10, 4738), (88, 16704), (679, 86), (4080, 4358), (4674, 519)


X(30576) = X(21)X(849)∩X(81)X(593)

Barycentrics    a (2 a - b - c)/(b + c)^2 : :

Let A10B10C10 be Gemini triangle 10. Let A' be the center of conic {{A,B,C,B10,C10}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30576).

X(30576) lies on these lines: {21, 849}, {58, 5303}, {60, 13624}, {81, 593}, {88, 4591}, {100, 1326}, {249, 1931}, {261, 5235}, {1019, 3960}, {1255, 2298} et al

X(30576) = barycentric product X(i)*X(j) for these {i,j}: {81, 16704}, {261, 1319}, {519, 757}
X(30576) = barycentric quotient X(i)/X(j) for these (i,j): (81, 4080), (519, 1089), (757, 903), (1319, 12), (16704, 321)


X(30577) = PERSPECTOR OF GEMINI TRIANGLE 10 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 10 AND 27

Barycentrics    3 a^3 - 2 a^2 (b + c) - a (4 b^2 - 7 b c + 4 c^2) + b^3 + c^3 : :

X(30577) lies on these lines: {2, 45}, {10, 1054}, {56, 100}, {57, 3882}, {244, 986}, {519, 9324}, {678, 3241}, {1145, 14193}, {1155, 5211}, {1266, 3911}, {1282, 5212} et al

X(30577) = anticomplement of X(4997)
X(30577) = {X(2),X(30579)}-harmonic conjugate of X(30578)


X(30578) = PERSPECTOR OF GEMINI TRIANGLE 27 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 10 AND 27

Barycentrics    a^3 + a^2 (b + c) - a (b^2 + 5 b c + c^2) - (b + c) (b^2 - 4 b c + c^2) : :

X(30578) lies on these lines: {2, 45}, {8, 80}, {11, 4756}, {312, 2895}, {320, 4358}, {528, 4152}, {908, 2325}, {1644, 9324}, {1647, 24821}, {1698, 3120}, {1997, 23958}, {2607, 4427}, {2796, 9458}, {3218, 4480}, {3699, 20095}, {3992, 5180}, {4009, 5057} et al

X(30578) = isotomic conjugate of X(8046)
X(30578) = complement of X(20092)
X(30578) = anticomplement of X(88)
X(30578) = {X(2),X(30579)}-harmonic conjugate of X(30577)


X(30579) = {X(30577),X(30578)}-HARMONIC CONJUGATE OF X(2)

Barycentrics    4 a^3 - a^2 (b + c) - a (5 b^2 - 2 b c + 5 c^2) + 3 b c (b + c) : :

X(30579) lies on these lines: {1, 4427}, {2, 45}, {20, 952}, {44, 17495}, {75, 16729}, {89, 192}, {537, 678}, {3977, 4887} et al

X(30579) = anticomplement of X(4080)
X(30579) = {X(30577),X(30578)}-harmonic conjugate of X(2)


X(30580) = CENTROID OF SIDE-TRIANGLE OF GEMINI TRIANGLES 10 AND 27

Barycentrics    (b - c) (3 a^3 - 2 a^2 (b + c) - a (b^2 - b c + c^2) + b^3 + c^3) : :

The side-triangle of Gemini triangles 10 and 27 is degenerate, lying on the perspectrix, line X(1)X(523).

X(30580) lies on these lines: {1, 523}, {99, 110}, {392, 513}, {512, 3877}, {514, 551}, {522, 3251}, {764, 4778}, {953, 2726}, {993, 4367}, {1022, 4977}, {1125, 4049} et al


X(30581) = X(58)X(5253)∩X(81)X(593)

Barycentrics    a (2 a + b + c)/(b + c)^2 : :

Let A11B11C11 be Gemini triangle 11. Let A' be the center of conic {{A,B,C,B11,C11}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30581).

X(30581) lies on these lines: {58, 5253}, {81, 593}, {261, 5333}, {662, 1171}, {1255, 1963} et al

X(30581) = barycentric product X(i)*X(j) for these {i,j}: {1, 30593}, {58, 16709}, {81, 8025}, {593, 4359}, {757, 1125}, {1100, 1509}
X(30581) = barycentric quotient X(i)/X(j) for these (i,j): (1, 6538), (81, 6539), (593, 1255), (757, 1268), (1100, 594), (1125, 1089), (4359, 28654), (8025, 321), (16709, 313), (30593, 75)


X(30582) = X(1255)X(3723)∩X(3743)X(4540)

Barycentrics    a (b + c)/(2 a + b + c)^2 : :

Let A12B12C12 be Gemini triangle 12. Let A' be the center of conic {{A,B,C,B12,C12}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30582).

X(30582) lies on these lines: {1255, 3723}, {3743, 4540}, {6539, 30594}

X(30582) = barycentric product X(i)*X(j) for these {i,j}: {1, 30594}, {1255, 6539}
X(30582) = barycentric quotient X(i)/X(j) for these (i,j): (1255, 8025), (6539, 4359), (30594, 75)


X(30583) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 14

Barycentrics    (b - c) (2 a - b - c) (a b + a c - 2 b c) : :

The side-triangle of ABC and Gemini triangle 14 is degenerate, lying on the perspectrix, line X(900)X(1145).

X(30583) lies on these lines: {1, 9260}, {2, 14421}, {8, 6161}, {10, 514}, {513, 3679}, {519, 3251}, {667, 956}, {891, 4728}, {900, 1145}, {984, 4777}, {1022, 19875}, {1647, 2087} et al

X(30583) = barycentric product X(i)*X(j) for these {i,j}: {519, 4728}, {536, 900}
X(30583) = barycentric quotient X(i)/X(j) for these (i,j): (519, 4607), (536, 4555), (900, 3227), (4728, 903)


X(30584) = X(522)X(650)∩X(3835)X(4083)

Barycentrics    (b - c) (a - b - c) (a b + a c - b c) (a^2 + b c) : :

X(30584) is the intersection of perspectrices of [ABC and Gemini triangle 15] and [ABC and Gemini triangle 16].

X(30584) lies on these lines: {522, 650}, {1215, 3805}, {3835, 4083} et al

X(30584) = barycentric product X(i)*X(j) for these {i,j}: {192, 3907}, {522, 17752}, {894, 4147}, {3835, 7081}, {4369, 27538}
X(30584) = barycentric quotient X(i)/X(j) for these (i,j): (522, 27447), (3835, 7049), (3907, 330), (4147, 257), (7081, 4598), (17752, 664), (27538, 27805)


X(30585) = X(58)X(750)∩X(191)X(3294)

Barycentrics    2 a^6 + 7 a^5 (b + c) + 3 a^4 (3 b^2 + 7 b c + 3 c^2) + a^3 (b + c) (5 b^2 + 18 b c + 5 c^2) + a^2 (3 b^4 + 17 b^3 c + 29 b^2 c^2 + 17 b c^3 + 3 c^4) + a (b + c)^3 (2 b^2 + 3 b c + 2 c^2) + 2 b c (b + c)^2 (b^2 + c^2) : :

Let A19B19C19 and A20B20C20 be Gemini triangles 19 and 20, resp. X(30585) is the radical center of the circumcircles of triangles AA19A20, BB19B20 and CC19C20.

X(30585) lies on these lines: {58, 750}, {191, 3294}


X(30586) = (name pending)

Barycentrics    (b + c)/((a^2 - b^2 - c^2 + a b + a c - b c) (a^4 + 4 a^3 (b + c) + a^2 (b^2 + 4 b c + c^2) - 2 a (b + c) (b^2 + c^2) - b^4 - 2 b^3 c - b^2 c^2 - 2 b c^3 - c^4)) : :

Let A19B19C19 be Gemini triangle 19. Let A' be the center of conic {{A,B,C,B19,C19}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30586).

X(30586) lies on these lines: (pending)


X(30587) = (name pending)

Barycentrics    (b + c) (a + 2 b + 2 c)/(a - 2 b - 2 c)^2 : :

Let A20B20C20 be Gemini triangle 20. Let A' be the center of conic {{A,B,C,B20,C20}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30587).

X(30587) lies on the line {30561, 30589}

X(30587) = barycentric product X(30588)*X(30589)
X(30587) = barycentric quotient X(30589)/X(5235)


X(30588) = ISOGONAL CONJUGATE OF X(4273)

Barycentrics    (b + c)/(a - 2 b - 2 c) : :

Let A20B20C20 be Gemini triangle 20. Let A' be the perspector of conic {{A,B,C,B20,C20}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30588).

X(30588) lies on these lines: {2, 44}, {4, 1385}, {10, 2650}, {37, 4080}, {76, 4358}, {86, 4604}, {94, 20565}, {98, 4588}, {142, 14554}, {321, 3943}, {661, 4049}, {671, 4597}, {908, 17758}, {1125, 14020}, {1751, 2364}, {2051, 5249}, {2163, 3624} et al

X(30588) = isogonal conjugate of X(4273)
X(30588) = isotomic conjugate of X(5235)
X(30588) = complement of X(30564)
X(30588) = barycentric product X(i)*X(j) for these {i,j}: {37, 20569}, {76, 28658}, {89, 321}, {226, 30608}, {30587, 30590}
X(30588) = barycentric quotient X(i)/X(j) for these (i,j): (2, 5235), (6, 4273), (10, 3679), (37, 45), (89, 81), (321, 4671), (20569, 274), (28658, 6), (30587, 30589), (30608, 333)


X(30589) = BARYCENTRIC QUOTIENT X(30587)/X(30588)

Barycentrics    (a + 2 b + 2 c)/(a - 2 b - 2 c) : :

X(30589) lies on these lines: {2, 44}, {79, 2320}, {1125, 2163}, {30561, 30587} et al

X(30589) = isotomic conjugate of X(30590)
X(30589) = anticomplement of X(30563)
X(30589) = barycentric product X(i)*X(j) for these {i,j}: {89, 28605}, {2163, 30596}, {5235, 30587}
X(30589) = barycentric quotient X(i)/X(j) for these (i,j): (2, 30590), (89, 25417), (1698, 3679), (28605, 4671), (30561, 30563), (30587, 30588), (30595, 30605)


X(30590) = BARYCENTRIC QUOTIENT X(30588)/X(30587)

Barycentrics    (a - 2 b - 2 c)/(a + 2 b + 2 c) : :

X(30590) lies on these lines: {2, 319}, {80, 4294}, {8652, 9093}

X(30590) = isotomic conjugate of X(30589)
X(30590) = anticomplement of X(30561)
X(30590) = barycentric product X(i)*X(j) for these {i,j}: {3679, 30598}
X(30590) = barycentric quotient X(i)/X(j) for these (i,j): (2, 30589), (3679, 1698), (30563, 30561), (30588, 30587), (30605, 30595)


X(30591) = X(320)X(350)∩X(523)X(1577)

Barycentrics    b c (b^2 - c^2) (2 a + b + c) : :

X(30591) is the intersection of perspectrices of every pair of {ABC, Gemini triangle 21, Gemini triangle 22}.

X(30591) lies on these lines: {320, 350}, {522, 4823}, {523, 1577}, {656, 4804}, {2517, 4777}, {2533, 4132}, {4024, 8061}, {4304, 15526}, {4977, 4983} et al

X(30591) = isotomic conjugate of X(4596)
X(30591) = anticomplement of X(8043)
X(30591) = barycentric product X(i)*X(j) for these {i,j}: {10, 4978}, {75, 4988}, {321, 4977}, {514, 4647}, {523, 4359}, {693, 1213}, {1125, 1577}
X(30591) = barycentric quotient X(i)/X(j) for these (i,j): (2, 4596), (75, 4632), (321, 6540), (523, 1255), (1125, 662), (1213, 100), (1577, 1268), (4359, 99), (4647, 190), (4977, 81), (4978, 86), (4988, 1)


X(30592) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 22

Barycentrics    (b - c) (2 a + b + c) (a b + a c - 2 b c) : :

The side-triangle of ABC and Gemini triangle 22 is degenerate, lying on the perspectrix, line X(4977)X(4983).

X(30592) lies on these lines: {522, 2530}, {690, 6545}, {764, 4010}, {812, 14419}, {891, 4728}, {2787, 14421}, {2832, 4800} et al

X(30592) = tripolar centroid of X(4359)
X(30592) = barycentric product X(i)*X(j) for these {i,j}: {536, 4977}, {1125, 4728}
X(30592) = barycentric quotient X(i)/X(j) for these (i,j): (536, 6540), (1125, 4607), (4728, 1268), (4977, 3227)


X(30593) = X(58)X(86)∩X(81)X(17495)

Barycentrics    (2 a + b + c)/(b + c)^2 : :

Let A21B21C21 be Gemini triangle 21. Let A' be the center of conic {{A,B,C,B21,C21}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30593).

X(30593) lies on these lines: {58, 86}, {81, 17495}, {190, 1963}, {1224, 1268}, {1434, 7341} et al

X(30593) = isotomic conjugate of X(6538)
X(30593) = barycentric product X(i)*X(j) for these {i,j}: {75, 30581}, {81, 16709}, {86, 8025}
X(30593) = barycentric quotient X(i)/X(j) for these (i,j): (2, 6538), (86, 6539), (1125, 594), (8025, 10), (16709, 321), (30581, 1)


X(30594) = X(1268)X(3634)∩X(6539)X(30582)

Barycentrics    (b + c)/(2 a + b + c)^2 : :

Let A22B22C22 be Gemini triangle 22. Let A' be the center of conic {{A,B,C,B22,C22}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30594).

X(30594) lies on these lines: {1268, 3634}, {6539, 30582}

X(30594) = barycentric product X(i)*X(j) for these {i,j}: {75, 30582}, {1268, 6539}
X(30594) = barycentric quotient X(i)/X(j) for these (i,j): (1268, 8025), (6539, 1125), (30582, 1)


X(30595) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 24

Barycentrics    (b - c) (a + 2 b + 2 c) (2 a^2 - b^2 - c^2) : :

The side-triangle of ABC and Gemini triangle 24 is degenerate, lying on the perspectrix, line X(4716)X(4802).

X(30595) lies on these lines: {351, 690}, {2786, 14431}, {4716, 4802} et al

X(30595) = tripolar centroid of X(5333)
X(30595) = barycentric product X(i)*X(j) for these {i,j}: {514, 4938}, {524, 4802}, {1698, 4750}, {30589, 30605}
X(30595) = barycentric quotient X(i)/X(j) for these (i,j): (4750, 30598), (4802, 671), (4938, 190), (30605, 30590)


X(30596) = X(10)X(75)∩X(69)X(5080)

Barycentrics    b^2 c^2 (a + 2 b + 2 c) : :

Let A23B23C23 be Gemini triangle 23. Let A' be the center of conic {{A,B,C,B23,C23}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30596).

X(30596) lies on these lines: {10, 75}, {69, 5080}, {86, 3761}, {192, 4377}, {264, 5342}, {310, 28650}, {311, 322}, {312, 1230}, {314, 5560}, {319, 5016}, {1964, 9902} et al

X(30596) = isotomic conjugate of isogonal conjugate of X(1698)
X(30596) = barycentric product X(i)*X(j) for these {i,j}: {75, 28605}, {76, 1698}, {30590, 30595}
X(30596) = barycentric quotient X(i)/X(j) for these (i,j): (75, 25417), (76, 30598), (1698, 6), (28605, 1), (30589, 2163), (30595, 30589)


X(30597) = TRILINEAR SQUARE OF X(25417)

Barycentrics    a/(a + 2 b + 2 c)^2 : :

Let A24B24C24 be Gemini triangle 24. Let A' be the center of conic {{A,B,C,B24,C24}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30597).

X(30597) lies on these lines: {5224, 19862}, {25417, 28606}

X(30597) = trilinear square of X(25417)
X(30597) = barycentric product X(25417)*X(30598)
X(30597) = barycentric quotient X(25417)/X(1698)


X(30598) = ISOTOMIC CONJUGATE OF X(1698)

Barycentrics    1/(a + 2 b + 2 c) : :
Trilinears    1/(1 - cos 2A + 4 (1 + cos A) (cos B + cos C)) : :

Let A24B24C24 be Gemini triangle 24. Let A' be the perspector of conic {{A,B,C,B24,C24}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30598).

X(30598) lies on these lines: {1, 1268}, {2, 319}, {7, 5550}, {10, 28650}, {69, 28626}, {75, 1125}, {86, 3624}, {310, 16709}, {335, 4422}, {551, 4464}, {675, 8652}, {748, 757}, {749, 756}, {903, 10436}, {3616, 4460}, {3622, 5564}, {3644, 4472}, {3879, 19878} et al

X(30598) = isotomic conjugate of X(1698)
X(30598) = barycentric product X(i)*X(j) for these {i,j}: {75, 25417}, {310, 28625}
X(30598) = barycentric quotient X(i)/X(j) for these (i,j): (2, 1698), (25417, 1), (28625, 42), (30590, 3679)


X(30599) = X(2)X(39)∩X(75)X(81)

Barycentrics    b c (a^2 + b^2 + c^2 + a b + a c + 2 b c)/(b + c) : :

Let A21B21C21 and A23B23C23 be Gemini triangles 21 and 23, resp. Let LA be the tangent at A to conic {{A,B21,C21,B23,C23}}, and define LB, LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30599).

Let A22B22C22 and A24B24C24 be Gemini triangles 22 and 24, resp. Let A' be the center of conic {{A,B22,C22,B24,C24}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30599).

X(30599) lies on these lines: {2, 39}, {27, 20880}, {75, 81}, {86, 321}, {92, 14014}, {312, 5333}, {314, 8025}, {333, 2160}, {1010, 3920}, {1043, 3957}, {1255, 4043}, {1412, 6358} et al

X(30599) = barycentric product X(i)*X(j) for these {i,j}: {75, 25526}, {314, 10404}
X(30599) = barycentric quotient X(i)/X(j) for these (i,j): (10404, 65), (25526, 1)


X(30600) = X(44)X(513)∩X(4467)X(7265)

Barycentrics    a (b - c) (a^2 - b^2 - c^2 - b c) (a^2 + b^2 + c^2 + a b + a c + 2 b c) : :

X(30600) is the intersection of perspectrices of [ABC and Gemini triangle 25] and [ABC and Gemini triangle 26].

X(30600) lies on these lines: {44, 513}, {4467, 7265}

X(30600) = barycentric product X(i)*X(j) for these {i,j}: {4467, 5311}, {14838, 17303}
X(30600) = barycentric quotient X(i)/X(j) for these (i,j): (5311, 6742), (17303, 15455)


X(30601) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 25

Barycentrics    (b - c) (a + 2 b + 2 c) (a^2 - b^2 - c^2 - b c) (4 a^2 - 2 b^2 - 2 c^2 + a b + a c - 2 b c) : :

The side-triangle of ABC and Gemini triangle 25 is degenerate, lying on the perspectrix, line X(4467)X(7265).

X(30601) lies on the line {4467, 7265}

X(30601) = tripolar centroid of X(319)
X(30601) = barycentric product X(319)*X(514)*X(1698)*X(4725)
X(30601) = barycentric product X(319)*X(4725)*X(4802)
X(30601) = barycentric product X(1698)*X(4467)*X(4725)


X(30602) = X(57)X(267)∩X(79)X(81)

Barycentrics    1/((a^2 - b^2 - c^2 - b c) (a^3 + a^2 (b + c) - a (b^2 + b c + c^2) - (b + c) (b^2 + c^2))) : :

Let A25B25C25 be Gemini triangle 25. Let A' be the center of conic {{A,B,C,B25,C25}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30602).

X(30602) lies on these lines: {1, 8818}, {2, 6757}, {57, 267}, {79, 81}, {274, 20565}, {502, 1255}, {1029, 1479} et al

X(30602) = barycentric product X(i)*X(j) for these {i,j}: {79, 1029}, {267, 30690}, {3444, 20565}
X(30602) = barycentric quotient X(i)/X(j) for these (i,j): (79, 2895), (267, 3219), (1029, 319), (3444, 35)


X(30603) = (name pending)

Barycentrics    b c (b + c)/(2 a^3 + 3 a^2 (b + c) + a (b + c)^2 - b c (b + c)) : :

Let A26B26C26 be Gemini triangle 26. Let A' be the center of conic {{A,B,C,B26,C26}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30603).

X(30603) lies on these lines: (pending)


X(30604) = X(523)X(1577)∩X(4693)X(4775)

Barycentrics    (b^2 - c^2) (a - 2 b - 2 c) (2 a^2 - b^2 - c^2 + a b + a c + 2 b c) : :

X(30604) is the intersection of perspectrices of every pair of {ABC, Gemini triangle 27, Gemini triangle 28}.

X(30604) lies on these lines: {523, 1577}, {4693, 4775}, {4983, 6089} et al

X(30604) = barycentric product X(i)*X(j) for these {i,j}: {3664, 4931}, {3679, 23755}, {4777, 17056}
X(30604) = barycentric quotient X(17056)/X(4597)


X(30605) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 28

Barycentrics    (b - c) (a - 2 b - 2 c) (2 a^2 - b^2 - c^2) : :

The side-triangle of ABC and Gemini triangle 28 is degenerate, lying on the perspectrix, line X(4693)X(4775).

X(30605) lies on these lines: {351, 690}, {514, 4010}, {900, 6161}, {918, 14421}, {2785, 14431}, {4693, 4775} et al

X(30605) = tripolar centroid of X(5235)
X(30605) = barycentric product X(i)*X(j) for these {i,j}: {514, 4933}, {524, 4777}, {3679, 4750}
X(30605) = barycentric quotient X(i)/X(j) for these (i,j): (524, 4597), (4777, 671), (4933, 190)


X(30606) = X(60)X(1043)∩X(261)X(284)

Barycentrics    (a - b - c) (2 a - b - c)/(b + c)^2 : :

Let A27B27C27 be Gemini triangle 27. Let A' be the center of conic {{A,B,C,B27,C27}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30606).

X(30606) lies on these lines: {60, 1043}, {81, 17495}, {261, 284}, {1509, 2325}, {3285, 4969} et al

X(30606) = barycentric product X(i)*X(j) for these {i,j}: {60, 3264}, {261, 519}, {333, 16704}, {3285, 28660}
X(30606) = barycentric quotient X(i)/X(j) for these (i,j): (60, 106), (261, 903), (333, 4080), (519, 12), (3285, 1400), (16704, 226)


X(30607) = X(89)X(4850)∩X(2320)X(3877)

Barycentrics    a (a - b - c)/(a - 2 b - 2 c)^2 : :

Let A28B28C28 be Gemini triangle 28. Let A' be the center of conic {{A,B,C,B28,C28}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30607).

X(30607) lies on these lines: {89, 4850}, {2320, 3877}, {3306, 4604}, {3707, 5233}

X(30607) = barycentric product X(89)*X(30608)
X(30607) = barycentric quotient X(i)/X(j) for these (i,j): (5219, 89), (30608, 4671)


X(30608) = ISOGONAL CONJUGATE OF X(1405)

Barycentrics    (a - b - c)/(a - 2 b - 2 c) : :

Let A28B28C28 be Gemini triangle 28. Let A' be the perspector of conic {{A,B,C,B28,C28}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30608).

X(30608) lies on these lines: {2, 44}, {8, 2320}, {9, 4997}, {75, 18359}, {85, 3911}, {92, 8756}, {312, 2325}, {333, 2364}, {1121, 4384}, {1220, 1698}, {1311, 4588}, {1732, 3306}, {2217, 5303}, {3707, 5233} et al

X(30608) = isogonal conjugate of X(1405)
X(30608) = isotomic conjugate of X(5219)
X(30608) = barycentric product X(i)*X(j) for these {i,j}: {9, 20569}, {75, 2320}, {76, 2364}, {89, 312}, {333, 30588}, {4671, 30607}
X(30608) = barycentric quotient X(i)/X(j) for these (i,j): (2, 5219), (6, 1405), (9, 45), (89, 57), (312, 4671), (333, 5235), (2320, 1), (2364, 6), (20569, 85), (30588, 226), (30607, 89)


X(30609) = CENTROID OF CROSS-TRIANGLE OF GEMINI TRIANGLES 29 AND 30

Barycentrics    a (a b + a c - b^2 - c^2) (a^6 - 2 a^5 (b + c) + a^4 (3 b^2 - 2 b c + 3 c^2) - 4 a^3 (b - c)^2 (b + c) + a^2 (3 b^4 - 2 b^3 c - 3 b^2 c^2 - 2 b c^3 + 3 c^4) - 2 a (b - c)^2 (b^3 + c^3) + (b - c)^2 (b^4 - b^2 c^2 + c^4)) : :

X(30609) lies on these lines: {2, 30612}, {144, 145}, {294, 1642}

X(30609) = anticomplement of X(30612)


X(30610) = TRILINEAR POLE OF LINE X(144)X(145)

Barycentrics 1/((b - c) (a^2 - a b - a c + 2 b c)) : :

Line X(144)X(145) is the anticomplement of line X(7)X(8), and the line of the degenerate cross-triangle of Gemini triangles 29 and 30.

X(30610) lies on these lines: {2, 14936}, {100, 8641}, {650, 4554}, {1025, 4763}, {1026, 4595} et al

X(30610) = isogonal conjugate of X(20980)
X(30610) = isotomic conjugate of X(4885)
X(30610) = trilinear pole of line X(144)X(145)
X(30610) = barycentric product X(i)*X(j) for these {i,j}: {190, 9311}, {4595, 27498}
X(30610) = barycentric quotient X(i)/X(j) for these (i,j): (2, 4885), (6, 20980), (100, 1376), (190, 3729), (9311, 514)


X(30611) = CENTROID OF CROSS-TRIANGLE OF EXTOUCH AND INTOUCH TRIANGLES

Barycentrics    a (a b + a c - b^2 - c^2) (a^6 - 2 a^5 (b + c) + a^4 (3 b^2 + b c + 3 c^2) - a^3 (4 b^3 - b^2 c - b c^2 + 4 c^3) + a^2 (3 b^4 + b^3 c - 6 b^2 c^2 + b c^3 + 3 c^4) - a (b - c)^2 (2 b^3 + 3 b^2 c + 3 b c^2 + 2 c^3) + (b - c)^2 (b^4 - 4 b^2 c^2 + c^4)) : :

The cross-triangle of the extouch and intouch triangles is degenerate, lying on line X(7)X(8).

X(30611) lies on these lines: {2, 30609}, {7, 8}

X(30611) = complement of X(30609)
X(30611) = anticomplement of X(30612)


X(30612) = CENTROID OF CROSS-TRIANGLE OF 1st AND 2nd ZANIAH TRIANGLES

Barycentrics    a (a b + a c - b^2 - c^2) (a^6 - 2 a^5 (b + c) + a^4 (3 b^2 - 5 b c + 3 c^2) - a^3 (b + c) (4 b^2 - 11 b c + 4 c^2) + a^2 (3 b^4 - 5 b^3 c - 5 b c^3 + 3 c^4) - a (b - 2 c) (2 b - c) (b - c)^2 (b + c) + (b - c)^2 (b^2 + c^2)^2) : :

The cross triangle of the 1st and 2nd Zaniah triangles is degenerate, lying on line X(1)X(6).

X(30612) lies on these lines: {2, 30609}, {1, 6}

X(30612) = complement of X(30611)


X(30613) = CENTROID OF SIDE-TRIANGLE OF GEMINI TRIANGLES 29 AND 30

Barycentrics    a (b - c) (a^4 - 2 a^3 (b + c) + 2 a^2 (b + c)^2 - 2 a (b + c) (b^2 + c^2) + b^4 + b^2 c^2 + c^4) : :

X(30613) lies on these lines: {2, 11193}, {31, 1734}, {100, 11124}, {513, 3681}, {693, 3434}, {1621, 3126} et al

X(30613) = anticomplement of X(11193)


X(30614) = X(1)X(3710)∩X(2)X(4906)

Barycentrics    3 a^3 - 3 a^2 (b + c) + a (5 b^2 - 2 b c + 5 c^2) - (b + c) (b^2 + c^2) : :

X(30614) is the perspector of Gemini triangle 29 and the tangential triangle, wrt Gemini triangle 29, of the {Gemini 29, Gemini 30}-circumconic.

X(30614) lies on these lines: {1, 3710}, {2, 4906}, {65, 145}, {81, 3241}, {278, 1280}, {354, 20020}, {1211, 3242}, {1357, 5222} et al

X(30614) = anticomplement of X(30615)


X(30615) = ANTICOMPLEMENT OF X(4906)

Barycentrics    (a - b - c) (a^2 + b^2 + c^2 - a b - a c) : :

X(30615) is the perspector of the extouch triangle and the tangential triangle, wrt the extouch triangle, of the {extouch, intouch}-circumconic (the Privalov conic).

X(30615) lies on these lines: {2, 4906}, {8, 210}, {9, 4030}, {55, 3717}, {200, 1040}, {344, 3748}, {345, 3689}, {346, 2348}, {518, 10327}, {519, 4383}, {612, 6703}, {614, 9053}, {1211, 3679}, {1386, 20020} et al

X(30615) = complement of X(30614)
X(30615) = anticomplement of X(4906)
X(30615) = barycentric quotient X(30617)/X(279)


X(30616) = X(144)X(145)∩X(218)X(329)

Barycentrics    3 a^4 - 4 a^3 (b + c) + 2 a^2 (b + c)^2 - 2 a b c (b + c) - (b - c)^2 (b^2 + c^2) : :

X(30616) is the perspector of Gemini triangle 30 and the tangential triangle, wrt Gemini triangle 29, of the {Gemini 29, Gemini 30}-circumconic.

X(30616) lies on these lines: {2, 30617}, {8, 3732}, {63, 3730}, {144, 145}, {218, 329}, {348, 4564}, {3189, 20071} et al

X(30616) = anticomplement of X(30617)
X(30616) = barycentric product X(190)*X(28590)
X(30616) = barycentric quotient X(28590)/X(514)


X(30617) = X(7)X(8)∩X(218)X(226)

Barycentrics    (a^2 + b^2 + c^2 - a b - a c)/(a - b - c) : :

X(30617) is the perspector of the intouch triangle and the tangential triangle, wrt the extouch triangle, of the {extouch, intouch}-circumconic (the Privalov conic).

X(30617) lies on these lines: {1, 1565}, {2, 30616}, {7, 8}, {57, 16549}, {150, 1837}, {169, 4904}, {218, 226}, {279, 3476}, {348, 1319}, {355, 1111}, {651, 14078}, {664, 7185}, {950, 3663}, {1358, 9312}, {1420, 7181}, {1788, 3598}, {1836, 4911}, {2082, 5845}, {2099, 3674}, {2218, 3423}, {2295, 4675} et al

X(30617) = complement of X(30616)
X(30617) = anticomplement of X(30618)
X(30617) = {X(7),X(8)}-harmonic conjugate of X(7195)
X(30617) = barycentric product X(i)*X(j) for these {i,j}: {7, 17279}, {279, 30615}, {348, 5101}, {651, 4373}
X(30617) = barycentric quotient X(i)/X(j) for these (i,j): (651, 145), (4373, 4391), (5101, 281), (17279, 8), (30615, 346)


X(30618) = X(1)X(6)∩X(8)X(2348)

Barycentrics    a (a - b - c) (2 a^2 + b^2 + c^2 - a b - a c) : :

X(30618) is the perspector of the 2nd Zaniah triangle and the tangential triangle, wrt the 1st Zaniah triangle, of the {1st Zaniah, 2nd Zaniah}-circumconic.

X(30618) lies on these lines: {1, 6}, {2, 30616}, {8, 2348}, {21, 7259}, {41, 3693}, {169, 5836}, {346, 2264}, {644, 3057}, {728, 3913}, {910, 3501}, {1265, 3161}, {1376, 15876}, {1385, 24036} et al

X(30618) = barycentric product X(i)*X(j) for these {i,j}: {8, 3744}, {9, 17353}
X(30618) = barycentric quotient X(i)/X(j) for these (i,j): (3744, 7), (17353, 85)


X(30619) = X(144)X(145)∩X(344)X(765)

Barycentrics    3 a^5 - 5 a^4 (b + c) + 2 a^3 (b^2 + 4 b c + c^2) - 2 a^2 (b^3 + c^3) + a (b - c)^2 (3 b^2 + 2 b c + 3 c^2) - (b - c) (b^4 - c^4) : :

X(30619) is the perspector of Gemini triangle 29 and the tangential triangle, wrt Gemini triangle 30, of the {Gemini 29, Gemini 30}-circumconic.

X(30619) lies on these lines: {2, 30620}, {77, 3870}, {100, 1037}, {144, 145}, {344, 765}, {2398, 4000} et al

X(30619) = anticomplement of X(30620)


X(30620) = X(7)X(8)∩X(200)X(8271)

Barycentrics    (a - b - c) (a^4 - a^3 (b + c) + 2 a^2 b c - a (b - c)^2 (b + c) + (b - c)^2 (b^2 + c^2)) : :

X(30620) is the perspector of the extouch triangle and the tangential triangle, wrt the intouch triangle, of the {extouch, intouch}-circumconic (the Privalov conic).

X(30620) lies on these lines: {2, 30619}, {7, 8}, {200, 8271}, {219, 3686}, {480, 3912}, {1037, 1376} et al

X(30620) = complement of X(30619)
X(30620) = anticomplement of X(30621)
X(30620) = {X(7),X(8)}-harmonic conjugate of X(4012)
X(30620) = barycentric product X(346)*X(30623)
X(30620) = barycentric quotient X(30623)/X(279)


X(30621) = X(1)X(6)∩X(55)X(77)

Barycentrics    a (2 a^4 - 3 a^3 (b + c) + a^2 (b^2 + 4 b c + c^2) - a (b - c)^2 (b + c) + (b - c)^2 (b^2 + c^2)) : :
Trilinears    2 a^4 - 3 a^3 (b + c) + a^2 (b^2 + 4 b c + c^2) - a (b - c)^2 (b + c) + (b - c)^2 (b^2 + c^2) : :

X(30621) is the perspector of the 1st Zaniah triangle and the tangential triangle, wrt the 2nd Zaniah triangle, of the {1st Zaniah, 2nd Zaniah}-circumconic.

X(30621) lies on these lines: {1, 6}, {2, 30619}, {55, 77}, {241, 1253}, {269, 11495}, {390, 1456}, {651, 14100}, {916, 14520}, {1407, 10178}, {1418, 9441}, {1419, 4326}, {1442, 2346}, {1443, 7676}, {1697, 15832}, {1742, 6610} et al

X(30621) = complement of X(30620)
X(30621) = {X(1),X(6)}-harmonic conjugate of X(5572)


X(30622) = X(144)X(3059)∩X(6172)X(6605)

Barycentrics    3 a^6 - 6 a^5 (b + c) + 5 a^4 (b + c)^2 - 2 a^3 (b + c) (4 b^2 - 5 b c + 4 c^2) + 3 a^2 (b - c)^2 (3 b^2 + 4 b c + 3 c^2) - 2 a (b - c)^2 (b + c) (b^2 + 3 b c + c^2) - (b - c)^4 (b^2 + c^2) : :

X(30622) is the perspector of Gemini triangle 30 and the tangential triangle, wrt Gemini triangle 30, of the {Gemini 29, Gemini 30}-circumconic.

X(30622) lies on these lines: {2, 30623}, {144, 3059}, {6172, 6605}, {17732, 17781}

X(30622) = anticomplement of X(30623)


X(30623) = X(7)X(354)∩X(222)X(553)

Barycentrics    (a^4 - a^3 (b + c) + 2 a^2 b c - a (b - c)^2 (b + c) + (b - c)^2 (b^2 + c^2))/(a - b - c) : :

X(30623) is the perspector of the intouch triangle and the tangential triangle, wrt the intouch triangle, of the {extouch, intouch}-circumconic (the Privalov conic).

X(30623) lies on these lines: {2, 30622}, {7, 354}, {222, 553}, {269, 1040}, {279, 3474}, {348, 3683}, {658, 17728}, {1434, 2194}, {1565, 1709} et al

X(30623) = complement of X(30622)
X(30623) = anticomplement of X(30624)
X(30623) = barycentric product X(279)*X(30620)
X(30623) = barycentric quotient X(30620)/X(346)


X(30624) = X(9)X(165)∩X(210)X(6605)

Barycentrics    a (a - b - c) (2 a^4 - 3 a^3 (b + c) + 3 a^2 (b^2 + c^2) - 5 a (b - c)^2 (b + c) + (b - c)^2 (3 b^2 + 4 b c + 3 c^2)) : :

X(30624) is the perspector of the 2nd Zaniah triangle and the tangential triangle, wrt the 2nd Zaniah triangle, of the {1st Zaniah, 2nd Zaniah}-circumconic.

X(30624) lies on these lines: {2, 30622}, {9, 165}, {210, 6605}, {3967, 6559}, {7308, 21446}, {8012, 15481}

X(30624) = complement of X(30623)


X(30625) = X(8)X(144)∩X(9)X(85)

Barycentrics    a^4 - 3 a^3 (b + c) + 3 a^2 (b^2 + c^2) - (a b + a c - 2 b c) (b - c)^2 : :

X(30625) is the perspector, wrt Gemini triangle 30, of the {ABC, Gemini 30}-circumconic.

X(30625) lies on these lines: {1, 3177}, {2, 10481}, {8, 144}, {9, 85}, {10, 30694}, {40, 3732}, {63, 169}, {190, 728}, {220, 9312}, {329, 3912}, {519, 20111}, {527, 6604}, {2481, 21384}, {17732, 17781} et al

X(30625) = anticomplement of X(10481)


X(30626) = (name pending)

Barycentrics    a (a^6 - 2 a^5 (b + c) + a^4 (b^2 + 4 b c + c^2) - 2 a^3 b c (b + c) - a^2 (b - c)^2 (b^2 + c^2) + 2 a (b - c)^2 (b^3 + c^3) - (b - c)^2 (b^2 + c^2)^2)/(b - c) : :

Let A29B29C29 and A30B30C30 be Gemini triangles 29 and 30, resp. Let E* be the {Gemini 29, Gemini 30}-circumconic. Let A' be the intersection of the tangents to E* at A29 and A30. Define B' and C' cyclically. The lines AA', BB', CC' concur in X(13577). Let E29 and E30 be the {ABC, Gemini 29}-circumconic and {ABC, Gemini 30}-circumconic, resp. Let A" be the intersection of the tangent to E29 at A29 and the tangent to E30 at A30. Define B" and C" cyclically. The lines AA", BB", CC" concur in X(693). The lines A'A", B'B", C'C" concur in X(30626).

X(30626) lies on the line {2, 1814}


X(30627) = EIGENCENTER OF GEMINI TRIANGLE 29

Barycentrics    a (a - b - c)/(a^2 (b^2 + c^2) - a (b^3 + c^3) + b c (b - c)^2) : :

X(30627) lies on these lines: {6, 664}, {41, 100}, {220, 3699}, {607, 1897}, {644, 1253}, {2287, 7257} et al

X(30627) = trilinear pole of line X(9)X(8641)


X(30628) = PERSPECTOR OF THESE TRIANGLES: GEMINI 29 AND HONSBERGER

Barycentrics    a*((b+c)*a^3-(3*b^2+b*c+3*c^2)*a^2+(b+c)*(3*b^2-4*b*c+3*c^2)*a-(b^3-c^3)*(b-c)) : :

X(30628) lies on these lines: {1, 5785}, {2, 3059}, {7, 3434}, {8, 5728}, {9, 1174}, {38, 4335}, {63, 4326}, {65, 20008}, {100, 1445}, {142, 11025}, {144, 145}, {226, 10865}, {354, 15587}, {480, 3935}, {516, 3868}, {519, 18412}, {528, 12755}, {651, 8271}, {962, 971}, {1320, 2801} et al

X(30628) = anticomplement of X(3059)


X(30629) = CENTROID OF GEMINI TRIANGLE 31

Barycentrics    2 a^3 (b^3 + c^3) + 9 a^2 b^2 c^2 + 3 a b c (b^3 + c^3) + 8 b^3 c^3 : :

X(30629) lies on these lines: {2, 4495}, {3912, 26738}


X(30630) = CENTROID OF GEMINI TRIANGLE 32

Barycentrics    2 a^3 (b^3 + c^3) - 7 a^2 b^2 c^2 - 3 a b c (b^3 + c^3) + 8 b^3 c^3 : :

X(30630) lies on the line {2, 30636}


X(30631) = PERSPECTOR OF GEMINI TRIANGLE 32 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 31 AND 32

Barycentrics    b c (a^2 b c - a (b^3 + c^3) - b^2 c^2) : :

X(30631) lies on these lines: {2, 893}, {75, 3703}, {334, 561}, {350, 3914}, {1909, 5249}, {1920, 3836}, {1921, 2887}, {1965, 4645} et al

X(30631) = {X(2),X(17493)}-harmonic conjugate of X(30646)
X(30631) = {X(2),X(30632)}-harmonic conjugate of X(7018)
X(30631) = barycentric product X(76)*X(24575)
X(30631) = barycentric quotient X(24575)/X(6)


X(30632) = {X(7018),X(30631)}-HARMONIC CONJUGATE OF X(2)

Barycentrics    b c (a b^3 + a c^3 + b^2 c^2) : :

X(30632) lies on these lines: {2, 893}, {75, 3006}, {76, 3120}, {334, 30635}, {561, 2887}, {799, 4655}, {1920, 25957}, {1921, 25760}, {1965, 6327} et al

X(30632) = {X(7018),X(30631)}-harmonic conjugate of X(2)
X(30632) = barycentric product X(76)*X(4443)
X(30632) = barycentric quotient X(4443)/X(6)


X(30633) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 31 AND 32

Barycentrics    1/(a (a^2 - b c) (a^3 b^3 + a^3 c^3 - a^2 b^2 c^2 - b^3 c^3)) : :

X(30633) lies on these lines: {292, 1966}, {334, 14603}, {698, 3862}, {726, 24576}, {1581, 1921}, {1920, 30663}, {1965, 30648} et al

X(30633) = isogonal conjugate of X(30634)


X(30634) = PERSPECTOR OF UNARY COFACTOR TRIANGLES OF GEMINI TRIANGLES 31 AND 32

Barycentrics    a^3 (a^2 - b c) (a^3 b^3 + a^3 c^3 - a^2 b^2 c^2 - b^3 c^3) : :

X(30634) lies on these lines: {32, 18756}, {101, 699}, {110, 727}, {239, 1281}, {825, 18893}, {1922, 1967}, {1933, 14599} et al

X(30634) = isogonal conjugate of X(30633)
X(30634) = barycentric product X(i)*X(j) for these {i,j}: {31, 19579}, {32, 19581}, {238, 18278}, {1914, 3510}
X(30634) = barycentric quotient X(i)/X(j) for these (i,j): (6, 30633), (32, 24576), (3510, 18895), (18278, 334), (19579, 561), (19581, 1502)


X(30635) = ISOTOMIC CONJUGATE OF X(17126)

Barycentrics    1/(2 a^3 + a b c) : :
Barycentrics    1/(2 a^2 sin A + S) : :

Let A31B31C31 be Gemini triangle 31. Let LA be the line through A31 parallel to BC, and define LB and LC cyclically. Let A'31 = LB∩LC, and define B'31 and C'31 cyclically. Triangle A'31B'31C'31 is homothetic to ABC at X(30635).

X(30635) lies on these lines: {2, 4495}, {334, 30632}, {2887, 30636}, {3006, 7179}, {3661, 4044}, {4388, 7357}, {4389, 4441}, {6327, 7224}, {25760, 30638} et al

X(30635) = isotomic conjugate of X(17126)
X(30635) = {X(2887),X(30637)}-harmonic conjugate of X(30636)


X(30636) = ISOTOMIC CONJUGATE OF X(17127)

Barycentrics    1/(2 a^3 - a b c) : :
Barycentrics    1/(2 a^2 sin A - S) : :

Let A32B32C32 be Gemini triangle 32. Let LA be the line through A32 parallel to BC, and define LB and LC cyclically. Let A'32 = LB∩LC, and define B'32 and C'32 cyclically. Triangle A'32B'32C'32 is homothetic to ABC at X(30636).

X(30636) lies on these lines: {2, 30630}, {2887, 30635}, {3661, 20888}, {4645, 7357}, {6327, 7261}, {25957, 30638}

X(30636) = isotomic conjugate of X(17127)
X(30636) = {X(2887),X(30637)}-harmonic conjugate of X(30635)


X(30637) = {X(30635),X(30636)}-HARMONIC CONJUGATE OF X(2887)

Barycentrics    b^2 c^2 (a^2 + 4 b c) : :

X(30637) lies on these lines: {2, 561}, {1278, 6385}, {2887, 30635} et al

X(30637) = {X(2),X(561)}-harmonic conjugate of X(30638)
X(30637) = {X(30635),X(30636)}-harmonic conjugate of X(2887)
X(30637) = barycentric product X(i)*X(j) for these {i,j}: {76, 17118}, {561, 17124}
X(30637) = barycentric quotient X(i)/X(j) for these (i,j): (17118, 6), (17124, 31)


X(30638) = {X(2),X(561)}-HARMONIC CONJUGATE OF X(30637)

Barycentrics    b^2 c^2 (a^2 - 4 b c) : :

X(30638) lies on these lines: {2, 561}, {76, 27797}, {3596, 9464}, {25760, 30635}, {25957, 30636} et al

X(30638) = {X(2),X(561)}-harmonic conjugate of X(30637)
X(30638) = barycentric product X(i)*X(j) for these {i,j}: {76, 17119}, {561, 17125}
X(30638) = barycentric quotient X(i)/X(j) for these (i,j): (17119, 6), (17125, 31)


X(30639) = X(824)X(4391)∩X(2533)X(3805)

Barycentrics    b c (b^3 - c^3) (a^4 - b^2 c^2) : :

X(30639) is the intersection of perspectrices of every pair of {ABC, Gemini triangle 31, Gemini triangle 32}.

X(30639) lies on these lines: {788, 21301}, {824, 4391}, {2533, 3805}, {3766, 4010} et al

X(30639) = barycentric product X(i)*X(j) for these {i,j}: {561, 30654}, {788, 14603}, {824, 1966}, {1491, 3978}, {1920, 30665}, {1921, 3805}, {3661, 14296}
X(30639) = barycentric quotient X(i)/X(j) for these (i,j): (239, 30670), (385, 1492), (788, 9468), (824, 1581), (894, 30664), (1491, 694), (1966, 4586), (3805, 292), (3978, 789), (14296, 14621), (30654, 31), (30665, 893)


X(30640) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 31

Barycentrics    a (b^3 - c^3) (a^2 - b c) (a^3 b^3 + a^3 c^3 - 2 b^3 c^3) : :

The side-triangle of ABC and Gemini triangle 31 is degenerate, lying on the perspectrix, line X(3766)X(4010).

X(30640) lies on the line {3766, 4010}

X(30640) = tripolar centroid of X(1921)
X(30640) = barycentric product X(i)*X(j) for these {i,j}: {716, 30665}, {17493, 30641}
X(30640) = barycentric quotient X(30641)/X(30669)


X(30641) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 32

Barycentrics    a (b^3 - c^3) (a^2 + b c) (a^3 b^3 + a^3 c^3 - 2 b^3 c^3) : :

The side-triangle of ABC and Gemini triangle 32 is degenerate, lying on the perspectrix, line X(2533)X(3805).

X(30641) lies on the line {2533, 3805}

X(30641) = tripolar centroid of X(1920)
X(30641) = barycentric product X(i)*X(j) for these {i,j}: {716, 3805}, {30640, 30669}
X(30641) = barycentric quotient X(30640)/X(17493)


X(30642) = X(334)X(1921)∩X(1966)X(30657)

Barycentrics    b c (a^2 + b c)/(a^2 - b c)^2 : :

Let A31B31C31 be Gemini triangle 31. Let A' be the center of conic {{A,B,C,B31,C31}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30642).

X(30642) lies on these lines: {334, 1921}, {1966, 30657}

X(30642) = barycentric product X(i)*X(j) for these {i,j}: {334, 30669}, {561, 30657}, {1920, 30663}
X(30642) = barycentric quotient X(i)/X(j) for these (i,j): (334, 17493), (1581, 30658), (1909, 4366), (18896, 30643), (30657, 31), (30663, 893), (30669, 238)


X(30643) = X(1921)X(3846)∩X(17493)X(30658)

Barycentrics    b c (a^2 - b c)/(a^2 + b c)^2 : :

Let A32B32C32 be Gemini triangle 32. Let A' be the center of conic {{A,B,C,B32,C32}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30643).

X(30643) lies on these lines: {1921, 3846}, {17493, 30658}

X(30643) = barycentric product X(i)*X(j) for these {i,j}: {561, 30658}, {7018, 17493}
X(30643) = barycentric quotient X(i)/X(j) for these (i,j): (350, 6645), (1581, 30657), (7018, 30669), (17493, 171), (18896, 30642), (30658, 31)


X(30644) = CENTROID OF GEMINI TRIANGLE 33

Barycentrics    a (8 a^3 b c + 3 a^2 (b^3 + c^3) + 9 a b^2 c^2 + 2 b c (b^3 + c^3)) : :
Trilinears    8 a^3 b c + 3 a^2 (b^3 + c^3) + 9 a b^2 c^2 + 2 b c (b^3 + c^3) : :

X(30644) lies on these lines: {2, 1908}, {1015, 4850}, {1573, 25057}


X(30645) = CENTROID OF GEMINI TRIANGLE 34

Barycentrics    a (8 a^3 b c - 3 a^2 (b^3 + c^3) - 7 a b^2 c^2 + 2 b c (b^3 + c^3)) : :
Trilinears    8 a^3 b c - 3 a^2 (b^3 + c^3) - 7 a b^2 c^2 + 2 b c (b^3 + c^3) : :

X(30645) lies on the line {2, 30651}


X(30646) = PERSPECTOR OF GEMINI TRIANGLE 34 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 33 AND 34

Barycentrics    a^2 (a^2 b c + a b^3 + a c^3 - b^2 c^2) : :

X(30646) lies on these lines: {2, 893}, {6, 3725}, {31, 292}, {37, 3757}, {39, 846}, {63, 2275}, {228, 1914}, {238, 9285}, {968, 2276}, {1621, 21814}, {1908, 17122}, {2176, 5364} et al

X(30646) = {X(2),X(17493)}-harmonic conjugate of X(30631)
X(30646) = {X(2),X(30647)}-harmonic conjugate of X(893)
X(30646) = barycentric product X(1)*X(24478)
X(30646) = barycentric quotient X(24478)/X(75)


X(30647) = {X(893),X(30646)}-HARMONIC CONJUGATE OF X(2)

Barycentrics    a^3 (b^3 + a b c + c^3) : :

X(30647) lies on these lines: {1, 8620}, {2, 893}, {6, 3121}, {31, 1501}, {32, 3724}, {39, 4414}, {55, 21814}, {292, 17126}, {750, 1908}, {1185, 3725} et al

X(30647) = {X(893),X(30646)}-harmonic conjugate of X(2)
X(30647) = barycentric product X(i)*X(j) for these {i,j}: {1, 3764}, {6, 3735}, {31, 25760}
X(30647) = barycentric quotient X(i)/X(j) for these (i,j): (3764, 75), (3735, 76), (25760, 561)


X(30648) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 33 AND 34

Barycentrics    a^2/((a^2 - b c) (b^3 + c^3 - a^3 - a b c)) : :

X(30648) lies on these lines: {238, 1581}, {292, 1691}, {295, 7281}, {334, 1966}, {511, 1757}, {518, 7061}, {1326, 2223}, {1755, 2076}, {1965, 30633} et al

X(30648) = isogonal conjugate of X(1281)
X(30648) = trilinear pole of line X(5029)X(30654)
X(30648) = barycentric product X(i)*X(j) for these {i,j}: {1, 24479}, {292, 7261}, {694, 7061}
X(30648) = barycentric quotient X(i)/X(j) for these (i,j): (1, 18037), (6, 1281), (292, 4645), (24479, 75), (7061, 3978), (7261, 1921)


X(30649) = EIGNENCENTER OF GEMINI TRIANGLE 34

Barycentrics    a (a^4 (b - c)^2 - a^3 (b - c)^2 (b + c) + a^2 (b^4 + b^3 c - b^2 c^2 + b c^3 + c^4) + a b^2 c^2 (b + c) - b^2 c^2 (b^2 - b c + c^2)) : :
Trilinears    a^4 (b - c)^2 - a^3 (b - c)^2 (b + c) + a^2 (b^4 + b^3 c - b^2 c^2 + b c^3 + c^4) + a b^2 c^2 (b + c) - b^2 c^2 (b^2 - b c + c^2) : :

X(30649) lies on these lines: {1, 1281}, {2, 3494}, {21, 3551}, {41, 43}, {87, 256}, {404, 2108}, {846, 3229}, {1740, 8424} et al

X(30649) = eigencenter of 1st Sharygin triangle


X(30650) = ISOGONAL CONJUGATE OF X(4363)

Barycentrics    a^2/(a^2 + 2 b c) : :

Let A33B33C33 be Gemini triangle 33. Let LA be the line through A33 parallel to BC, and define LB, LC cyclically. Let A'33 = LB∩LC, and define B'33, C'33 cyclically. Triangle A'33B'33C'33 is homothetic to ABC at X(30650).

X(30650) lies on these lines: {2, 1908}, {44, 751}, {89, 1015}, {292, 17126}, {649, 995}, {869, 902}, {3285, 21793}, {16584, 30651} et al

X(30650) = isogonal conjugate of X(4363)
X(30650) = {X(16584),X(30652)}-harmonic conjugate of X(30651)
X(30650) = barycentric product X(1)*X(751)
X(30650) = barycentric quotient X(i)/X(j) for these (i,j): (1, 3761), (6, 4363), (751, 75)


X(30651) = ISOGONAL CONJUGATE OF X(4361)

Barycentrics    a^2/(a^2 - 2 b c) : :

Let A34B34C34 be Gemini triangle 34. Let LA be the line through A34 parallel to BC, and define LB, LC cyclically. Let A'34 = LB∩LC, and define B'34, C'34 cyclically. Triangle A'34B'34C'34 is homothetic to ABC at X(30651).

X(30651) lies on these lines: {2, 30645}, {292, 17127}, {749, 1100}, {869, 2308}, {893, 17126}, {1500, 25417}, {16584, 30650} et al

X(30651) = isogonal conjugate of X(4361)
X(30651) = {X(16584),X(30652)}-harmonic conjugate of X(30650)
X(30651) = barycentric product X(1)*X(749)
X(30651) = barycentric quotient X(i)/X(j) for these (i,j): (1, 3760), (6, 4361), (749, 75)


X(30652) = {X(30650),X(30651)}-HARMONIC CONJUGATE OF X(16584)

Barycentrics    a (4 a^2 + b c) : :

X(30652) lies on these lines: {2, 31}, {23, 1460}, {43, 21747}, {58, 145}, {81, 3052}, {89, 354}, {109, 9105}, {181, 11002}, {346, 4275}, {387, 20066}, {595, 3622}, {601, 3522}, {602, 15717}, {982, 9340}, {1054, 3892}, {1185, 1979}, {1376, 14997}, {1397, 11003}, {1399, 3600}, {1414, 7268}, {1468, 3623}, {1621, 14996}, {1707, 3920}, {16584, 30650} et al

X(30652) = anticomplement of X(25958)
X(30652) = {X(2),X(31)}-harmonic conjugate of X(30653)
X(30652) = {X(30650),X(30651)}-harmonic conjugate of X(16584)


X(30653) = {X(2),X(31)}-HARMONIC CONJUGATE OF X(30652)

Barycentrics    a (4 a^2 - b c) : :

X(30653) lies on these lines: {1, 21747}, {2, 31}, {23, 7083}, {58, 3622}, {89, 3246}, {100, 14997}, {109, 9095}, {145, 595}, {390, 2361}, {580, 20070}, {601, 15717}, {602, 3522}, {614, 23958}, {643, 4779}, {902, 3240}, {1001, 14996}, {1191, 16948}, {1397, 9544}, {1399, 5265}, {1460, 13595} et al

X(30653) = anticomplement of X(25959)
X(30653) = {X(2),X(31)}-harmonic conjugate of X(30652)


X(30654) = X(659)X(4435)∩X(663)X(788)

Barycentrics    a^2 (b^3 - c^3) (a^4 - b^2 c^2) : :

X(30654) is the intersection of perspectrices of every pair of {ABC, Gemini triangle 33, Gemini triangle 34}.

X(30654) lies on these lines: {659, 4435}, {663, 788}, {824, 4560}, {3287, 3805}, {5029, 30671} et al

X(30654) = barycentric product X(i)*X(j) for these {i,j}: {31, 30639}, {171, 30665}, {238, 3805}, {788, 1966}, {824, 1691}, {869, 14296}, {1491, 1580}
X(30654) = barycentric quotient X(i)/X(j) for these (i,j): (788, 1581), (1491, 1934), (1580, 789), (1691, 4586), (3805, 334), (14296, 871), (30639, 561), (30665, 7018)


X(30655) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 33

Barycentrics    a (b^3 - c^3) (a^2 - b c) (2 a^3 - b^3 - c^3) : :

The side-triangle of ABC and Gemini triangle 33 is degenerate, lying on the perspectrix, line X(659)X(4435).

X(30655) lies on these lines: {659, 4435}, {824, 6546}, {4809, 14402} et al

X(30655) = reflection of X(30656) in X(14402)
X(30655) = tripolar centroid of X(238)
X(30655) = barycentric product X(i)*X(j) for these {i,j}: {752, 30665}, {2243, 4486}, {3783, 4809}, {17493, 30656}
X(30655) = barycentric quotient X(30656)/X(30669)


X(30656) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 34

Barycentrics    a (b^3 - c^3) (a^2 + b c) (2 a^3 - b^3 - c^3) : :

The side-triangle of ABC and Gemini triangle 34 is degenerate, lying on the perspectrix, line X(3287)X(3805).

X(30656) lies on these lines: {3287, 3805}, {4809, 14402}

X(30656) = reflection of X(30655) in X(14402)
X(30656) = tripolar centroid of X(171)
X(30656) = barycentric product X(i)*X(j) for these {i,j}: {752, 3805}, {30655, 30669}
X(30656) = barycentric quotient X(30655)/X(17493)


X(30657) = X(238)X(292)∩X(1966)X(30642)

Barycentrics    a^2 (a^2 + b c)/(a^2 - b c)^2 : :

Let A33B33C33 be Gemini triangle 33. Let A' be the center of conic {{A,B,C,B33,C33}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30657).

X(30657) lies on these lines: {238, 292}, {1966, 30642}, {4447, 18787}

X(30657) = barycentric product X(i)*X(j) for these {i,j}: {31, 30642}, {171, 30663}, {291, 18787}, {292, 30669}
X(30657) = barycentric quotient X(i)/X(j) for these (i,j): (172, 4366), (1911, 18786), (1581, 30643), (9468, 30658), (18787, 350), (30642, 561), (30663, 7018), (30669, 1921)


X(30658) = X(238)X(893)∩X(17493)X(30643)

Barycentrics    a^2 (a^2 - b c)/(a^2 + b c)^2 : :

Let A34B34C34 be Gemini triangle 34. Let A' be the center of conic {{A,B,C,B34,C34}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30658).

X(30658) lies on these lines: {238, 893}, {17493, 30643}

X(30658) = barycentric product X(i)*X(j) for these {i,j}: {31, 30643}, {256, 18786}, {893, 17493}
X(30658) = barycentric quotient X(i)/X(j) for these (i,j): (893, 30669), (1581, 30642), (1914, 6645), (9468, 30657), (17493, 1920), (18786, 1909), (30643, 561)


X(30659) = CENTROID OF CROSS-TRIANGLE OF GEMINI TRIANGLES 31 AND 33

Barycentrics    (a^2 - b c) (a^4 (b^3 + c^3) - 3 a^3 b^2 c^2 + a^2 b c (b^3 + c^3) - 3 a b^3 c^3 + b^2 c^2 (b^3 + c^3)) : :

X(30659) lies on these lines: {2, 30666}, {31, 561}, {740, 3873}, {874, 4418} et al

X(30659) = reflection of X(30666) in X(2)


X(30660) = PERSPECTOR OF GEMINI TRIANGLE 31 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 31 AND 33

Barycentrics    b c (a^4 - a^2 b c - a b^3 - a c^3 - 2 b^2 c^2) : :

X(30660) lies on these lines: {2, 893}, {7, 310}, {8, 3978}, {69, 4485}, {312, 18037}, {321, 1909}, {329, 26735}, {561, 4388}, {1920, 4645} et al

X(30660) = anticomplement of X(893)
X(30660) = {X(2),X(30662)}-harmonic conjugate of X(30661)
X(30660) = barycentric product X(75)*X(17739)
X(30660) = barycentric quotient X(17739)/X(2)


X(30661) = PERSPECTOR OF GEMINI TRIANGLE 33 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 31 AND 33

Barycentrics    2 a^4 b c + a^3 b^3 + a^3 c^3 + a^2 b^2 c^2 - b^3 c^3 : :

Let A'B'C' be the medial triangle. Let A1 and A2 be the 1st and 2nd bicentrics of A', resp., and define B1, B2, C1, C2 cyclically. The lines AB1, BC1, CA1 concur in P(8) (the 1st bicentric of X(2)). The lines AC2, BA2, CB2 concur in U(8) (the 2nd bicentric of X(2)). Let A" = A1C2∩B1A2, B" = B1A2∩C1B2, C" = C1B2∩A1C2. Triangle A"B"C" is homothetic to ABC at X(893), to the medial triangle at X(1966), and to the anticomplementary triangle at X(30661).

X(30661) lies on these lines: {2, 893}, {31, 19580}, {42, 894}, {55, 192}, {63, 24579}, {81, 330}, {846, 1655}, {1908, 1920} et al

X(30661) = anticomplement of X(7018)
X(30661) = {X(2),X(30662)}-harmonic conjugate of X(30660)


X(30662) = {X(30660),X(30661)}-HARMONIC CONJUGATE OF X(2)

Barycentrics    (a^2 - b c) (3 a^2 b c + a b^3 + a c^3 + 3 b^2 c^2) : :

X(30662) lies on these lines: {2, 893}, {145, 740}, {722, 20064}, {3619, 12264}, {3924, 4043} et al

X(30662) = anticomplement of X(17493)
X(30662) = {X(30660),X(30661)}-harmonic conjugate of X(2)


X(30663) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 31 AND 33

Barycentrics    a/(a^2 - b c)^2 : :

X(30663) lies on these lines: {1, 3252}, {37, 9505}, {238, 292}, {241, 1463}, {291, 518}, {334, 1921}, {335, 726}, {660, 1757}, {716, 7245}, {740, 4562}, {984, 22116}, {1581, 4645}, {1920, 30633} et al

X(30663) = isogonal conjugate of X(8300)
X(30663) = isotomic conjugate of complement of X(30669)
X(30663) = trilinear square of X(291)
X(30663) = barycentric product X(i)*X(j) for these {i,j}: {291, 335}, {292, 334}, {660, 4444}, {876, 4562}, {893, 30642}, {1581, 30669}, {7018, 30657}
X(30663) = barycentric quotient X(i)/X(j) for these (i,j): (1, 4366), (6, 8300), (291, 239), (292, 238), (334, 1921), (335, 350), (660, 3570), (876, 812), (4444, 3766), (4562, 874), (30642, 1920), (30657, 171), (30669, 1966)


X(30664) = TRILINEAR POLE OF LINE X(6)X(291)

Barycentrics    a/((b^3 - c^3) (a^2 - b c)) : :

X(30664) is the intersection, other than X(789), of the circumcircle and the tangent at X(789) to hyperbola {A,B,C,X(789),PU(6)}.

X(30664) is the intersection, other than X(825), of the circumcircle and the tangent at X(825) to hyperbola {A,B,C,X(825),PU(12)}.

X(30664) lies on the circumcircle and these lines: {99, 4613}, {100, 4562}, {101, 660}, {110, 4584}, {291, 753}, {292, 743}, {334, 9075}, {335, 761}, {701, 14598}, {717, 1922}, {731, 1911}, {813, 3573}, {870, 9073}, {985, 2382}, {14621, 14665} et al

X(30664) = isogonal conjugate of X(30665)
X(30664) = trilinear pole of line X(6)X(291)
X(30664) = Ψ(X(6), X(291))
X(30664) = Λ(X(659), X(4435))
X(30664) = Λ(X(876), X(2254))
X(30664) = Λ(X(3766), X(4010))
X(30664) = barycentric product X(i)*X(j) for these {i,j}: {291, 4586}, {292, 789}, {335, 1492}, {660, 14621}, {813, 870}, {985, 4562}, {30670, 30669}
X(30664) = barycentric quotient X(i)/X(j) for these (i,j): (6, 30665), (291, 824), (292, 1491), (660, 3661), (789, 1921), (813, 984), (894, 30639, (985, 812), (1492, 239), (4586, 350), (14621, 3766), (30670, 17493)


X(30665) = ISOGONAL CONJUGATE OF X(30664)

Barycentrics    a (b^3 - c^3) (a^2 - b c) : :

X(30665) is the infinite point of the perspectrices of every pair of {ABC, Gemini triangle 31, Gemini triangle 33}.

X(30665) lies on these lines: {30, 511}, {659, 4435}, {876, 2254}, {3766, 4010}

X(30665) = isogonal conjugate of X(30664)
X(30665) = crossdifference of every pair of points on line X(6)X(291)
X(30665) = barycentric product X(i)*X(j) for these {i,j}: {238, 824}, {239, 1491}, {350, 3250}, {893, 30639}, {2276, 3766}, {3805, 17493}
X(30665) = barycentric quotient X(i)/X(j) for these (i,j): (6, 30664), (238, 4586), (239, 789), (869, 813), (1491, 335), (1914, 1492), (2276, 660), (3250, 291), (3805, 30669), (8632, 985), (30639, 1920), (30655, 752)


X(30666) = CENTROID OF CROSS-TRIANGLE OF GEMINI TRIANGLES 32 AND 34

Barycentrics    (a^2 - b c) (a^4 (b^3 + c^3) - a^3 b^2 c^2 - a^2 b c (b^3 + c^3) - a b^3 c^3 + b^2 c^2 (b^3 + c^3)) : :

X(30666) lies on these lines: {2, 30659}, {740, 3681}, {748, 1966}, {2022, 19551} et al

X(30666) = reflection of X(30659) in X(2)


X(30667) = PERSPECTOR OF GEMINI TRIANGLE 34 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 32 AND 34

Barycentrics    2 a^4 b c - a^3 b^3 - a^3 c^3 - a^2 b^2 c^2 + b^3 c^3 : :

X(30667) lies on these lines: {1, 1655}, {2, 292}, {6, 190}, {31, 19580}, {105, 330}, {239, 672}, {894, 9359}, {1931 ,2109}, {2112, 2145}, {2113, 17493} et al

X(30667) = anticomplement of X(334)
X(30667) = {X(2),X(30668)}-harmonic conjugate of X(20345)


X(30668) = {X(20345),X(30667)}-HARMONIC CONJUGATE OF X(2)

Barycentrics    a^3 (b + c) (b^2 + b c + c^2) + a b c (a + 2 b + 2 c) (2 b^2 - b c + 2 c^2) - 3 b^3 c^3 : :

X(30668) lies on these lines: {2, 292}, {141, 1278}, {145, 17794}

X(30668) = anticomplement of X(30669)
X(30668) = {X(20345),X(30667)}-harmonic conjugate of X(2)


X(30669) = ISOTOMIC CONJUGATE OF X(17493)

Barycentrics    (a^2 + b c)/(a^2 - b c) : :

X(30669) lies on these lines: {2, 292}, {7, 192}, {8, 291}, {69, 3862}, {171, 7369}, {193, 7077}, {194, 3864}, {295, 20096}, {385, 4447}, {660, 20072}, {894, 7184}, {1581, 4645}, {1655, 6625}, {1909, 7187}, {1911, 17379}, {1966, 30642}, {2295, 6645} et al

X(30669) = isotomic conjugate of X(17493)
X(30669) = complement of X(30668)
X(30669) = anticomplement of isotomic conjugate of X(30663)
X(30669) = barycentric product X(i)*X(j) for these {i,j}: {335, 894}, {238, 30642}, {1921, 30657}
X(30669) = barycentric quotient X(i)/X(j) for these (i,j): (2, 17493), (893, 30658), (894, 239), (7018, 30643), (30641, 30640), (30642, 334), (30656, 30655), (30657, 292), (30664, 30670)


X(30670) = TRILINEAR POLE OF LINE X(6)X(256)

Barycentrics    a/((b^3 - c^3) (a^2 + b c)) : :

X(30670) is the intersection, other than A, B, and C, of the circumcircle and conic {{A,B,C,PU(36)}}. X(30670) is also the isogonal conjugate of X(3805), which is the infinite point of perspectrices of every pair of {ABC, Gemini triangle 32, Gemini triangle 34}.

X(30670) lies on the circumcircle and these lines: {82, 733}, {99, 7260}, {100, 27805}, {101, 3903}, {109, 1492}, {110, 4603}, {256, 753}, {257, 761}, {662, 805}, {717, 7104}, {731, 904}, {741, 985}, {743, 893}, {813, 4579}, {932, 4586}, {1916, 8301}, {1967, 8300} et al

X(30670) = isogonal conjugate of X(3805)
X(30670) = trilinear pole of line X(6)X(256)
X(30670) = Ψ(X(6), X(256))
X(30670) = Λ(X(38), X(661))
X(30670) = barycentric product X(i)*X(j) for these {i,j}: {256, 4586}, {257, 1492}, {789, 893}, {985, 27805}, {3903, 14621}, {17493, 30664}
X(30670) = barycentric quotient X(i)/X(j) for these (i,j): (6, 3805), (239, 30639), (256, 824), (789, 1920), (893, 1491), (985, 4369), (1492, 894), (3903, 3661), (4586, 1909), (14621, 4374), (30664, 30669)


X(30671) = X(38)X(661)∩X(42)X(649)

Barycentrics    a^2 (b^3 - c^3)/(a^2 - b c) : :

X(30671) is the intersection of perspectrices of every pair of {Gemini triangles 31, 32, 33, 34}.

X(30671) lies on these lines: {38, 661}, {42, 649}, {321, 693}, {876, 2254}, {882, 3569}, {5029, 30654} et al

X(30671) = barycentric product X(i)*X(j) for these {i,j}: {6, 23596}, {291, 1491}, {292, 824}, {334, 788}, {335, 3250}, {876, 984}, {2276, 4444}, {3572, 3661}
X(30671) = barycentric quotient X(i)/X(j) for these (i,j): (291, 789), (292, 4586), (788, 238), (824, 1921), (875, 985), (876, 870), (984, 874), (1491, 350), (1911, 1492), (2276, 3570), (3250, 239), (3572, 14621), (3661, 27853), (23596, 76)


X(30672) = CENTROID OF GEMINI TRIANGLE 35

Barycentrics    a^6 + 6 a^5 (b + c) - a^4 (b^2 - 32 b c + c^2) - 4 a^3 (b + c) (3 b^2 - b c + 3 c^2) - a^2 (b + c)^2 (b^2 + 30 b c + c^2) + 2 a (b - c)^2 (b + c) (3 b^2 + 4 b c + 3 c^2) + (b^2 - c^2)^2 (b^2 + c^2) : :

X(30672) lies on the line {2, 30679}


X(30673) = CENTROID OF GEMINI TRIANGLE 36

Barycentrics    a^6 - 6 a^5 (b + c) - a^4 (b^2 - 32 b c + c^2) + 4 a^3 (b + c) (3 b^2 - 5 b c + 3 c^2) - a^2 (b - c)^2 (b^2 + 34 b c + c^2) - 2 a (b - c)^2 (b + c) (3 b^2 - 4 b c + 3 c^2) + (b^2 - c^2)^2 (b^2 + c^2) : :

X(30673) lies on these lines: {2, 30680}, {100, 999}, {6349, 26740}


X(30674) = PERSPECTOR OF GEMINI TRIANLGE 35 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 35 AND 36

Barycentrics    a (a^2 - b^2 - c^2) (a^6 - a^4 (b + c)^2 - 4 a^3 b c (b + c) - a^2 (b^2 - c^2)^2 + 4 a b c (b - c)^2 (b + c) + (b - c)^2 (b + c)^4) : :

X(30674) lies on these lines: {2, 19}, {3, 4512}, {9, 223}, {40, 18641}, {57, 17073}, {63, 348}, {77, 6508}, {221, 960}, {1001, 1040}, {1528, 6908} et al

X(30674) = {X(2),X(30675)}-harmonic conjugate of X(10319)


X(30675) = {X(10319),X(30674)}-HARMONIC CONJUGATE OF X(2)

Barycentrics    a (a^2 - b^2 - c^2) (a^6 - a^4 (b^2 + c^2) - 4 a^3 b c (b + c) - a^2 (b^2 - c^2)^2 + 4 a b c (b - c)^2 (b + c) + (b^2 - c^2)^2 (b^2 + c^2)) : :
Barycentrics    cos A + sec B + sec C : :

X(30675) lies on these lines: {2, 19}, {3, 392}, {9, 6350}, {57, 6349}, {63, 77}, {343, 8897}, {1038, 3869}, {1040, 1621}, {1158, 2360} et al

X(30675) = {X(2),X(4329)}-harmonic conjugate of X(30687)
X(30675) = {X(10319),X(30674)}-harmonic conjugate of X(2)


X(30676) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 35 AND 36

Barycentrics    a/((a^2 + b^2 + c^2 - 2 b c) (a^3 - a^2 (b + c) + a (b^2 + 4 b c + c^2) - (b + c) (b^2 + c^2))) : :

The perspectrix of Gemini triangles 35 and 36 passes through X(663).

X(30676) lies on these lines: {6, 7131}, {9, 30701}, {57, 30705}

X(30676) = isogonal conjugate of X(30677)


X(30677) = PERSPECTOR OF UNARY COFACTOR TRIANGLES OF GEMINI TRIANGLES 35 AND 36

Barycentrics    a (a^2 + b^2 + c^2 - 2 b c) (a^3 - a^2 (b + c) + a (b^2 + 4 b c + c^2) - (b + c) (b^2 + c^2)) : :

X(30677) lies on these lines: {2, 7}, {19, 1611}, {169, 5272}, {444, 3291}, {614, 1184}, {2128, 21216} et al

X(30677) = isogonal conjugate of X(30676)


X(30678) = EIGENCENTER OF GEMINI TRIANGLE 36

Barycentrics    a^8 - a^7 (b + c) + a^6 (b^2 - b c + c^2) - a^5 (b + c) (3 b^2 + 2 b c + 3 c^2) + a^4 (b + c)^2 (3 b^2 - b c + 3 c^2) - a^3 (3 b^5 - b^4 c - 6 b^3 c^2 - 6 b^2 c^3 - b c^4 + 3 c^5) + a^2 (3 b^6 - 3 b^5 c - 7 b^4 c^2 + 2 b^3 c^3 - 7 b^2 c^4 - 3 b c^5 + 3 c^6) - a (b - c)^2 (b + c) (b^2 + c^2) (b^2 - 4 b c + c^2) - b c (b - c)^2 (b^2 + c^2)^2 : :

X(30678) lies on these lines: {2, 294}, {57, 7075}, {100, 11329}, {3912, 9441} et al


X(30679) = X(21)X(999)∩X(78)X(4001)

Barycentrics    (a^2 - b^2 - c^2)/(a^2 - b^2 - c^2 - 4 b c) : :
Barycentrics    1/(2 sec A + 1) : :

Let A35B35C35 be Gemini triangle 35. Let LA be the line through A35 parallel to BC, and define LB, LC cyclically. Let A'35 = LB∩LC, and define B'35, C'35 cyclically. Triangle A'35B'35C'35 is homothetic to ABC at X(30679).

X(30679) lies on these lines: {2, 30672}, {21, 999}, {78, 4001}, {280, 6360}, {347, 7361}, {1214, 30680}, {1812, 22129}, {2339, 3218}, {3219, 7131} et al

X(30679) = isotomic conjugate of polar conjugate of X(3296)


X(30680) = X(21)X(145)∩X(78)X(3977)

Barycentrics    (a^2 - b^2 - c^2)/(a^2 - b^2 - c^2 + 4 b c) : :
Barycentrics    1/(2 sec A - 1) : :

Let A36B36C36 be Gemini triangle 36. Let LA be the line through A36 parallel to BC, and define LB, LC cyclically. Let A'36 = LB∩LC, and define B'36, C'36 cyclically. Triangle A'36B'36C'36 is homothetic to ABC at X(30680).

X(30680) lies on these lines: {2, 30673}, {21, 145}, {78, 3977}, {1214, 30679}, {2339, 3219}, {3218, 7131} et al

X(30680) = isotomic conjugate of polar conjugate of X(1000)


X(30681) = X(341)X(346)∩X(345)X(3694)

Barycentrics    (a^2 - b^2 - c^2) (a - b - c)^3 : :

Let A35B35C35 be Gemini triangle 35. Let A' be the center of conic {{A,B,C,B35,C35}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30681).

X(30681) lies on these lines: {341, 346}, {345, 3694}, {480, 5423}, {1265, 3692}, {3161, 3871} et al

X(30681) = isotomic conjugate of polar conjugate of X(5423)
X(30681) = barycentric product X(i)*X(j) for these {i,j}: {8, 1265}, {69, 5423}, {78, 341}, {305, 480}, {312, 3692}, {345, 346}
X(30681) = barycentric quotient X(i)/X(j) for these (i,j): (8, 1119), (69, 479), (78, 269), (312, 1847), (341, 273), (345, 279), (346, 278), (480, 25), (1265, 7), (3692, 57), (3926, 30682), (5423, 4)


X(30682) = X(77)X(1040)∩X(348)X(17073)

Barycentrics    (a^2 - b^2 - c^2)/(a - b - c)^3 : :

Let A36B36C36 be Gemini triangle 36. Let A' be the center of conic {{A,B,C,B36,C36}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30682).

X(30682) lies on these lines: {7, 4626}, {77, 1040}, {279, 1418}, {348, 17073}, {479, 1014}, {910, 9533}, {934, 1486}, {1088, 1440} et al

X(30682) = isotomic conjugate of polar conjugate of X(479)
X(30682) = barycentric product X(i)*X(j) for these {i,j}: {7, 7056}, {69, 479}, {77, 1088}, {279, 348}, {4025, 4626}
X(30682) = barycentric quotient X(i)/X(j) for these (i,j): (7, 7046), (69, 5423), (77, 200), (279, 281), (348, 346), (479, 4), (1088, 318), (3926, 30681), (4025, 4163), (4626, 1897), (7056, 8)


X(30683) = CENTROID OF GEMINI TRIANGLE 37

Barycentrics    2 a^6 + a^5 (b + c) - 2 a^4 (b + c)^2 - a^3 (b + c) (2 b^2 + b c + 2 c^2) - 2 a^2 (b^4 - b^2 c^2 + c^4) + a (b - c)^2 (b + c) (b^2 + 3 b c + c^2) + 2 (b - c)^2 (b + c)^4 : :

X(30683) lies on these lines: {2, 7110}, {191, 6175}, {30684, 30685}

X(30683) = reflection of X(30684) in X(30685)


X(30684) = CENTROID OF GEMINI TRIANGLE 38

Barycentrics    2 a^6 - a^5 (b + c) - 2 a^4 (b - c)^2 + a^3 (b - 2 c) (2 b - c) (b + c) - 2 a^2 (b^4 - 3 b^2 c^2 + c^4) - a (b - c)^2 (b + c) (b^2 - 3 b c + c^2) + 2 (b - c)^4 (b + c)^2 : :

X(30684) lies on these lines: {1, 10031}, {2, 2006}, {693, 3960}, {908, 5723}, {2990, 28609}, {30683, 30685} et al

X(30684) = reflection of X(30683) in X(30685)


X(30685) = CENTROID OF MID-TRIANGLE OF GEMINI TRIANGLES 37 AND 38

Barycentrics    2 a^12 - 5 a^10 (b^2 + c^2) + 2 a^8 (b^4 + 4 b^2 c^2 + c^4) + a^6 (2 b^6 - 3 b^4 c^2 - 3 b^2 c^4 + 2 c^6) + 2 a^4 (b^8 - 3 b^6 c^2 + 3 b^4 c^4 - 3 b^2 c^6 + c^8) - a^2 (b^2 - c^2)^2 (b^2 + c^2) (5 b^4 - 9 b^2 c^2 + 5 c^4) + 2 (b^2 - c^2)^4 (b^4 + c^4) : :
Barycentrics    SA^4 (SB + SC)^2 + 4 SA^3 (SB + SC) (SB^2 + SB SC + SC^2) - SA^2 (5 SB^4 + 4 SB^3 SC + 30 SB^2 SC^2 + 4 SB SC^3 + 5 SC^4) - 2 SA SB SC (SB + SC) (5 SB^2 - 12 SB SC + 5 SC^2) - SB^2 SC^2 (5 SB^2 - 22 SB SC + 5 SC^2) : :

X(30685) lies on these lines: {2, 94}, {110, 381}, {115, 14389}, {136, 7576}, {542, 13448}, {2641, 15073}, {30683, 30684} et al

X(30685) = midpoint of X(30683) and X(30684)


X(30686) = PERSPECTOR OF GEMINI TRIANGLE 37 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 37 AND 38

Barycentrics    (a^4 (b + c) + 2 a^3 b c - 2 a^2 (b^3 + c^3) - 2 a b c (b + c)^2 + (b - c)^2 (b + c) (b^2 + c^2))/(a^2 - b^2 - c^2) : :

X(30686) lies on these lines: {2, 19}, {4, 12565}, {33, 2550}, {34, 28629}, {40, 406}, {65, 13567}, {85, 92}, {142, 278}, {196, 226}, {204, 4307}, {207, 388}, {329, 7079}, {451, 6197}, {516, 4183}, {946, 7498}, {1435, 9776}, {1519, 7551}, {1842, 11109}, {1859, 1861}, {1871, 8728} et al

X(30686) = polar conjugate of isogonal conjugate of X(4300)
X(30686) = {X(2),X(30687)}-harmonic conjugate of X(1848)


X(30687) = {X(1848),X(30686)}-HARMONIC CONJUGATE OF X(2)

Barycentrics    (a^4 (b + c) - 2 a^2 (b^3 + c^3) - 4 a b^2 c^2 + (b - c)^2 (b + c) (b^2 + c^2))/(a^2 - b^2 - c^2) : :
Barycentrics    sec A + cos B + cos C : :

X(30687) lies on these lines: {2, 19}, {29, 102}, {33, 3434}, {77, 278}, {92, 226}, {142, 17923}, {281, 908}, {406, 5250}, {442, 1871}, {516, 1013}, {962, 4194}, {1432, 16082}, {1748, 5745}, {1838, 12609}, {1844, 10916}, {1859, 2886} et al

X(30687) = polar conjugate of X(1065)
X(30687) = {X(2),X(4329)}-harmonic conjugate of X(30675)
X(30687) = {X(1848),X(30686)}-harmonic conjugate of X(2)


X(30688) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 37 AND 38

Barycentrics    1/((a^2 - b^2 - c^2) (a^2 + b^2 + c^2 - 2 b c) (a^2 b - a b^2 + a^2 c - a c^2 - a b c + b^2 c + b c^2)) : :

The perspectrix of Gemini triangles 37 and 38 passes through X(18344).

X(30688) lies on these lines: {19, 4209}, {278, 30705}, {281, 17786}

X(30688) = isogonal conjugate of X(30689)


X(30689) = PERSPECTOR OF UNARY COFACTOR TRIANGLES OF GEMINI TRIANGLES 37 AND 38

Barycentrics    a^2 (a^2 - b^2 - c^2) (a^2 + b^2 + c^2 - 2 b c) (a^2 b - a b^2 + a^2 c - a c^2 - a b c + b^2 c + b c^2) : :

X(30689) lies on these lines: {48, 1613}, {63, 77}, {614, 1184}

X(30689) = isogonal conjugate of X(30688)


X(30690) = ISOGONAL CONJUGATE OF X(2174)

Barycentrics    b c/(a^2 - b^2 - c^2 - b c) : :
Barycentrics    1/(1 + 2 cos A) : :
Barycentrics    1 + 2 cos B + 2 cos C + 4 cos B cos C : :

Let A37B37C37 be Gemini triangle 37. Let LA be the line through A37 parallel to BC, and define LB, LC cyclically. Let A'37 = LB∩LC, and define B'37, C'37 cyclically. Triangle A'37B'37C'37 is homothetic to ABC at X(30690).

X(30690) lies on these lines: {2, 7110}, {7, 2994}, {8, 79}, {29, 1870}, {75, 3578}, {92, 445}, {94, 226}, {312, 1230}, {321, 4102}, {333, 2160}, {554, 7043}, {1081, 7026} et al

X(30690) = isogonal conjugate of X(2174)
X(30690) = isotomic conjugate of X(3219)
X(30690) = anticomplement of X(16585)
X(30690) = polar conjugate of X(6198)
X(30690) = trilinear pole of line X(522)X(4823) (the polar wrt polar circle of X(6198))
X(30690) = barycentric product X(i)*X(j) for these {i,j}: {1, 20565}, {75, 79}, {76, 2160}, {85, 7110}, {94, 3218}, {328, 1870}, {20932, 30602}
X(30690) = barycentric quotient X(i)/X(j) for these (i,j): (1, 35), (2, 3219), (4, 6198), (6, 2174), (75, 319), (79, 1), (85, 17095), (94, 18359), (1870, 186), (2160, 6), (3218, 323), (7110, 9), (20565, 75), (30602, 267)


X(30691) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 37

Barycentrics    a (b - c) (a^4 (b + c) - a^3 (b^2 + c^2) - a^2 (b - c)^2 (b + c) + a (b^2 - c^2)^2 - 2 b c (b - c)^2 (b + c)) : :

The side-triangle of ABC and Gemini triangle 37 is degenerate, lying on the perspectrix, line X(513)X(1835).

X(30691) lies on these lines: {65, 676}, {244, 665}, {354, 6366}, {513, 1835}, {928, 5902}, {942, 10015}, {1637, 9391} et al

X(30691) = tripolar centroid of X(278)


X(30692) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 38

Barycentrics    a (b - c) (a - b - c)^2 (a^4 (b + c) - a^3 (b^2 + c^2) - a^2 (b - c)^2 (b + c) + a (b^2 - c^2)^2 - 2 b c (b - c)^2 (b + c)) : :

The side-triangle of ABC and Gemini triangle 38 is degenerate, lying on the perspectrix, line X(3064)X(3700).

X(30692) lies on these lines: {926, 4120}, {2310, 3119}, {3064, 3700} et al

X(30692) = tripolar centroid of X(281)


X(30693) = ISOTOMIC CONJUGATE OF X(738)

Barycentrics    b c (b + c - a)^3 : :

Let A37B37C37 be Gemini triangle 37. Let A' be the center of conic {{A,B,C,B37,C37}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30693).

X(30693) lies on these lines: {85, 17786}, {312, 2321}, {322, 4033}, {341, 346}, {594, 3959}, {646, 3718}, {2324, 3699} et al

X(30693) = isotomic conjugate of X(738)
X(30693) = barycentric product X(i)*X(j) for these {i,j}: {8, 341}, {75, 5423}, {76, 728}, {312, 346}, {646, 3239}, {3699, 4397}, {3718, 7046}
X(30693) = barycentric quotient X(i)/X(j) for these (i,j): (2, 738), (8, 269), (75, 479), (76, 23062), (312, 279), (341, 7), (346, 57), (646, 658), (728, 6), (3239, 3669), (3699, 934), (3718, 7056), (4397, 3676), (5423, 1), (7046, 34)


X(30694) = PERSPECTOR OF GEMINI TRIANGLE 37 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 35 AND 37

Barycentrics    a^4 - 2 a^3 (b + c) + 2 a^2 (b^2 - b c + c^2) - 2 a (b - c)^2 (b + c) + (b - c)^2 (b^2 + 4 b c + c^2) : :
Barycentrics    1/(1 + sec A) - 1/(1 + sec B) - 1/(1 + sec C) : :

X(30694) lies on these lines: {2, 85}, {4, 3732}, {8, 10025}, {10, 30625}, {63, 3691}, {92, 6392}, {100, 9305}, {120, 11681}, {144, 1654}, {145, 10405}, {193, 5942}, {329, 3661}, {346, 17786}, {1146, 6604} et al

X(30694) = anticomplement of X(348)
X(30694) = polar conjugate of X(7)-cross conjugate of X(4)
X(30694) = {X(2),X(30695)}-harmonic conjugate of X(3177)
X(30694) = {X(6392),X(21216)}-harmonic conjugate of X(30699)


X(30695) = {X(3177),X(30694)}-HARMONIC CONJUGATE OF X(2)

Barycentrics    a^4 - 4 a^3 (b + c) + a^2 (6 b^2 - 4 b c + 6 c^2) - 4 a (b - c)^2 (b + c) + (b - c)^2 (b^2 + 6 b c + c^2) : :
Barycentrics    tan^2(A/2) - tan^2(B/2) - tan^2(C/2) : :

X(30695) lies on these lines: {2, 85}, {8, 144}, {20, 3732}, {63, 28638}, {145, 10025}, {193, 20008}, {346, 16284}, {3621, 20111} et al

X(30695) = anticomplement of X(279)
X(30695) = {X(3177),X(30694)}-harmonic conjugate of X(2)


X(30696) = PERSPECTOR OF GEMINI TRIANGLE 38 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 35 AND 38

Barycentrics    a^9 - a^8 (b + c) - 2 a^7 (b^2 + b c + c^2) + 2 a^6 (b + c) (b^2 - 5 b c + c^2) + 2 a^5 b c (b^2 - 10 b c + c^2) + 18 a^4 b c (b - c)^2 (b + c) + 2 a^3 (b + c)^2 (b^4 - b^3 c + 12 b^2 c^2 - b c^3 + c^4) - 2 a^2 (b + c) (b^6 + 3 b^5 c - 13 b^4 c^2 + 2 b^3 c^3 - 13 b^2 c^4 + 3 b c^5 + c^6) - a (b - c)^2 (b + c)^4 (b^2 + c^2) + (b - c)^2 (b + c)^3 (b^4 - 2 b^3 c + 10 b^2 c^2 - 2 b c^3 + c^4) : :

X(30696) lies on these lines: {2, 800}, {278, 318}, {5809, 10453}

X(30696) = {X(2),X(30698)}-harmonic conjugate of X(30697)


X(30697) = PERSPECTOR OF GEMINI TRIANGLE 36 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 36 AND 37

Barycentrics    a^8 (b + c) - a^7 (b^2 - b c + c^2) - a^6 (b + c) (3 b^2 - 5 b c + 3 c^2) + a^5 (3 b^4 - b^3 c - 12 b^2 c^2 - b c^3 + 3 c^4) + a^4 (b + c) (3 b^4 - 9 b^3 c - 4 b^2 c^2 - 9 b c^3 + 3 c^4) - a^3 (3 b^6 + b^5 c - 7 b^4 c^2 + 22 b^3 c^3 - 7 b^2 c^4 + b c^5 + 3 c^6) - a^2 (b + c)^3 (b^4 - 5 b^3 c + 4 b^2 c^2 - 5 b c^3 + c^4) + a (b^2 - c^2)^2 (b^4 + b^3 c + 8 b^2 c^2 + b c^3 + c^4) + b c (b - c)^2 (b + c)^5 : :

X(30697) lies on these lines: {2, 800}, {8, 57}, {345, 17786}

X(30697) = {X(2),X(30698)}-harmonic conjugate of X(30696)


X(30698) = {X(30696),X(30697)}-HARMONIC CONJUGATE OF X(2)

Barycentrics    a^8 - 4 a^6 (b^2 + c^2) + a^4 (6 b^4 - 44 b^2 c^2 + 6 c^4) - 4 a^2 (b^6 - 9 b^4 c^2 - 9 b^2 c^4 + c^6) + (b^2 - c^2)^2 (b^4 + 14 b^2 c^2 + c^4) : :
Barycentrics    4 SA^3 (SB + SC) + SA^2 (5 SB^2 + 6 SB SC + 5 SC^2) + 2 SA SB SC (SB + SC) - 3 SB^2 SC^2 : :

X(30698) lies on these lines: {2, 800}, {69, 3146}, {75, 279}, {253, 1370}, {346, 18750}, {394, 17037}, {14360, 23974} et al

X(30698) = anticomplement of polar conjugate of isotomic conjugate of X(15740)
X(30698) = {X(30696),X(30697)}-harmonic conjugate of X(2)


X(30699) = PERSPECTOR OF GEMINI TRIANGLE 38 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 36 AND 38

Barycentrics    a^3 + a^2 (b + c) + a (b^2 + c^2) + (b + c) (b^2 - 4 b c + c^2) : :
Barycentrics    1/(1 - sec A) - 1/(1 - sec B) - 1/(1 - sec C) : :

X(30699) lies on these lines: {2, 37}, {7, 1999}, {8, 3914}, {31, 24280}, {57, 1266}, {69, 3782}, {92, 6392}, {145, 388}, {149, 7391}, {193, 1839}, {225, 11851}, {226, 3875}, {239, 329}, {333, 4419}, {377, 20009}, {1722, 2899} et al

X(30699) = anticomplement of X(345)
X(30699) = polar conjugate of X(8)-cross conjugate of X(4)
X(30699) = {X(2),X(4452)}-harmonic conjugate of X(3210)
X(30699) = {X(6392),X(21216)}-harmonic conjugate of X(30694)


X(30700) = CENTROID OF SIDE-TRIANGLE OF GEMINI TRIANGLES 35 AND 37

Barycentrics    a (b - c) (a^3 (b + c) - 2 a^2 (b + c)^2 + a (b + c) (b^2 + 4 b c + c^2) - 2 b c (b^2 + b c + c^2)) : :

The side-triangle of Gemini triangles 35 and 37 is degenerate, lying on the perspectrix, line X(513)X(4468).

X(30700) lies on these lines: {2, 30704}, {210, 918}, {513, 4468}, {518, 1639}, {654, 5220}, {668, 891}, {926, 3681}, {984, 3310}, {1635, 4712}, {1638, 3740}, {3219, 6139}, {3695, 18289}, {3887, 14740} et al

X(30700) = reflection of X(30704) in X(2)


X(30701) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 35 AND 37

Barycentrics    1/(a^2 + (b - c)^2) : :
Barycentrics    1/(b c - SW) : :
Barycentrics    1/(cot^2(B/2) + cot^2(C/2)) : :

X(30701) lies on these lines: {1, 344}, {8, 105}, {28, 1043}, {57, 345}, {69, 17742}, {75, 277}, {81, 7123}, {88, 17740}, {220, 4437}, {274, 2345}, {278, 312}, {279, 304}, {281, 17786}, {291, 3976}, {306, 2333}, {321, 15474}, {668, 6554}, {959, 7672}, {961, 1037}, {985, 5255}, {1002, 3889}, {1390, 3616} et al

X(30701) = isogonal conjugate of X(16502)
X(30701) = isotomic conjugate of X(4000)
X(30701) = polar conjugate of X(1851)
X(30701) = trilinear pole of line X(513)X(4468) (the polar of X(1851) wrt polar circle)
X(30701) = barycentric product X(i)*X(j) for these {i,j}: {8, 8817}, {76, 7123}, {561, 7084}
X(30701) = barycentric quotient X(i)/X(j) for these (i,j): (1, 614), (2, 4000), (4, 1851), (6, 16502), (8, 497), (9, 2082), (55, 7083), (220, 30706), (7084, 31), (7123, 6), (8817, 7)


X(30702) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 36 AND 37

Barycentrics    1/((a^2 + b^2 + c^2 - 2 b c) (a^5 - 3 a^4 (b + c) + 2 a^3 (b + c)^2 + 2 a^2 (b - c)^2 (b + c) - a (3 b^4 + 4 b^3 c + 10 b^2 c^2 + 4 b c^3 + 3 c^4) + (b + c) (b^4 + 4 b^3 c - 2 b^2 c^2 + 4 b c^3 + c^4))) : :

X(30702) lies on these lines: {2257, 7131}, {3824, 8237}

X(30702) = isogonal conjugate of X(30703)


X(30703) = PERSPECTOR OF UNARY COFACTOR TRIANGLES OF GEMINI TRIANGLES 36 AND 37

Barycentrics    a^2 (a^2 + b^2 + c^2 - 2 b c) (a^5 - 3 a^4 (b + c) + 2 a^3 (b + c)^2 + 2 a^2 (b - c)^2 (b + c) - a (3 b^4 + 4 b^3 c + 10 b^2 c^2 + 4 b c^3 + 3 c^4) + (b + c) (b^4 + 4 b^3 c - 2 b^2 c^2 + 4 b c^3 + c^4)) : :

X(30703) lies on these lines: {57, 219}, {614, 1184}

X(30703) = isogonal conjugate of X(30702)


X(30704) = CENTROID OF SIDE-TRIANGLE OF GEMINI TRIANGLES 36 AND 38

Barycentrics    a (b - c) (a^3 (b + c) - 2 a^2 (b^2 + c^2) + a (b + c) (b^2 + c^2) - 2 b c (b^2 - b c + c^2)) : :

The side-triangle of Gemini triangles 36 and 38 is degenerate, lying on the perspectrix, line X(3900)X(4025).

X(30704) lies on these lines: {2, 30700}, {354, 918}, {518, 1638}, {891, 3227}, {926, 3873}, {982, 3310}, {1639, 3742}, {3218, 6139}, {3738, 4458}, {3900, 4025}, {4083, 4786} et al

X(30704) = reflection of X(30700) in X(2)


X(30705) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 36 AND 38

Barycentrics    1/((a - b - c)^2 (a^2 + (b - c)^2)) : :
Barycentrics    1/(tan^2(B/2) + tan^2(C/2)) : :

X(30705) lies on these lines: {7, 1037}, {9, 348}, {69, 200}, {77, 1041}, {85, 281}, {86, 4183}, {279, 304}, {1323, 21629} et al

X(30705) = isogonal conjugate of X(30706)
X(30705) = isotomic conjugate of X(6554)
X(30705) = polar conjugate of X(1863)
X(30705) = trilinear pole of line X(3900)X(4025) (the polar of X(1863) wrt polar circle)


X(30706) = PERSPECTOR OF UNARY COFACTOR TRIANGLES OF GEMINI TRIANGLES 36 AND 38

Barycentrics    a^2 (a - b - c)^2 (a^2 + (b - c)^2) : :
Barycentrics    a^2 (tan^2(B/2) + tan^2(C/2)) : :

X(30706) lies on these lines: {2, 294}, {6, 57}, {25, 41}, {31, 1200}, {43, 170}, {55, 2195}, {100, 7123}, {198, 800}, {200, 220}, {213, 3198}, {219, 3169}, {440, 2238}, {613, 8300}, {614, 1184}, {650, 11502}, {851, 21753}, {1185, 9449}, {1202, 1471}, {1253, 8012}, {1436, 5065} et al

X(30706) = isogonal conjugate of X(30705)
X(30706) = crossdifference of every pair of points on line X(3900)X(4025)
X(30706) = barycentric product X(i)*X(j) for these {i,j}: {1, 4319}, {3, 1863}, {6, 6554}, {8, 7083}, {9, 2082}, {55, 497}, {56, 4012}, {57, 28070}, {100, 17115}, {200, 614}, {220, 4000}
X(30706) = barycentric quotient X(i)/X(j) for these (i,j): (6, 30705), (55, 8817), (220, 30701), (497, 6063), (614, 1088), (1863, 264), (2082, 85), (4012, 3596), (4319, 75), (6554, 76), (7083, 7), (17115, 693), (28070, 312)


X(30707) = CENTROID OF GEMINI TRIANGLE 39

Barycentrics    3 a^3 - 25 a^2 (b + c) - a (43 b^2 + 82 b c + 43 c^2) - (b + c) (5 b + 3 c) (3 b + 5 c) : :

X(30707) lies on these lines: {2, 1449}, {1770, 18231}

X(30707) = anticomplement of X(28617)


X(30708) = CENTROID OF GEMINI TRIANGLE 40

Barycentrics    3 a^3 + 31 a^2 (b + c) + 13 a (b + c)^2 - (3 b - 5 c) (5 b - 3 c) (b + c) : :

X(30708) lies on these lines: {1, 6556}, {2, 1743}, {950, 18220}, {3731, 3982}, {3752, 4052}, {3817, 6686}, {3950, 4358}, {3986, 5316} et al


X(30709) = CENTROID OF SIDE-TRIANGLE OF GEMINI TRIANGLES 39 AND 40

Barycentrics    (b - c) (a^3 + a b^2 + a c^2 + 3 a b c - 3 b^2 c - 3 b c^2) : :

The side-triangle of Gemini triangles 39 and 40 is degenerate, lying on the perspectrix, line X(513)X(2517).

X(30709) lies on these lines: {2, 2787}, {4, 2775}, {8, 4010}, {80, 885}, {405, 16158}, {513, 2517}, {668, 891}, {671, 690}, {812, 14430}, {1022, 4013} et al

X(30709) = anticomplement of X(14419)


X(30710) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 39 AND 40

Barycentrics b c/(b^2 + c^2 + a b + a c) : :

X(30710) lies on these lines: {1, 312}, {2, 1240}, {8, 181}, {28, 1791}, {57, 75}, {81, 314}, {88, 4359}, {89, 28605}, {105, 3757}, {226, 1432}, {239, 1258}, {264, 278}, {274, 1920}, {279, 6063}, {291, 3741}, {309, 1422}, {341, 7322}, {668, 1211}, {957, 3421}, {961, 4968}, {985, 4362}, {1002, 10453}, {1255, 4358}, {1402, 7081} et al

X(30710) = isogonal conjugate of X(2300)
X(30710) = isotomic conjugate of X(3666)
X(30710) = polar conjugate of X(1829)
X(30710) = trilinear pole of line X(513)X(2517) (the polar of X(1829) wrt polar circle)


X(30711) = X(2)X(1449)∩X(92)X(144)

Barycentrics    (a - b - c)/(a + 3 b + 3 c) : :

Let A39B39C39 be Gemini triangle 39. Let LA be the line through A39 parallel to BC, and define LB, LC cyclically. Let A'39 = LB∩LC, and define B'39, C'39 cyclically. Triangle A'39B'39C'39 is homothetic to ABC at X(30711).

X(30711) lies on these lines: {2, 1449}, {8, 4314}, {63, 10405}, {85, 4359}, {92, 144}, {145, 4981}, {312, 391}, {346, 4102}, {3663, 3943} et al

X(30711) = isotomic conjugate of anticomplement of X(5273)


X(30712) = ISOTOMIC CONJUGATE OF X(3617)

Barycentrics    1/(a - 3 b - 3 c) : :

Let A40B40C40 be Gemini triangle 40. Let LA be the line through A40 parallel to BC, and define LB, LC cyclically. Let A'40 = LB∩LC, and define B'40, C'40 cyclically. Triangle A'40B'40C40' is homothetic to ABC at X(30712).

X(30712) lies on these lines: {1, 4373}, {2, 1743}, {7, 1420}, {27, 8025}, {69, 1268}, {75, 145}, {86, 16948}, {273, 7518}, {335, 4704}, {673, 17379}, {903, 3672}, {1215, 3633}, {1292, 3625}, {1440, 10586} et al

X(30712) = isotomic conjugate of X(3617)


X(30713) = ISOTOMIC CONJUGATE OF X(1412)

Barycentrics    b^2 c^2 (b + c) (a - b - c) : :

Let A39B39C39 be Gemini triangle 39. Let A' be the center of conic {{A,B,C,B39,C39}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(30713).

X(30713) lies on these lines: {2, 3264}, {306, 4033}, {312, 2321}, {313, 321}, {318, 341}, {333, 17787}, {349, 6358}, {561, 1233}, {1215, 4710}, {1269, 28605}, {1334, 3975} et al

X(30713) = isogonal conjugate of X(16947)
X(30713) = isotomic conjugate of X(1412)
X(30713) = barycentric product X(i)*X(j) for these {i,j}: {8, 313}, {10, 3596}, {75, 3701}, {76, 2321}, {210, 561}, {306, 7017}, {312, 321}, {318, 20336}, {333, 28654}, {341, 1441}, {346, 349}, {1334, 1502}, {4033, 4391}
X(30713) = barycentric quotient X(i)/X(j) for these (i,j): (2, 1412), (6, 16947), (8, 58), (10, 56), (75, 1014), (76, 1434), (210, 31), (306, 222), (312, 81), (313, 7), (318, 28), (321, 57), (333, 593), (341, 21), (346, 284), (349, 279), (1334, 32), (1441, 269), (2321, 6), (3701, 1), (4033, 651), (4391, 1019), (7017, 27), (20336, 77), (28654, 226)


X(30714) = MIDPOINT OF X(3) AND X(23236)

Barycentrics    -4 a^10+10 a^8 (b^2+c^2)+(b^2-c^2)^4 (b^2+c^2)-7 a^6 (b^2+c^2)^2-a^2 (b^2-c^2)^2 (b^4+c^4)+a^4 (b^6+5 b^4 c^2+5 b^2 c^4+c^6) : :
X(30714) = X2*X[2]-3*X[11693], X[4]-3*X[110], 2*X[5]-3*X[5642], 3*X[74]-5*X[3522], 3*X[146]+X[5059], 3*X[265]-5*X[1656], 3*X[376]-X[15054], 3*X[381]-7*X[15039], X[382]-3*X[5655], 3*X[549]-2*X[20379], 2*X[576]-3*X[15303], 5*X[631]-3*X[9140], 5*X[632]-4*X[20396], 3*X[1495]-2*X[16619], 3*X[1568]-2*X[18572], X[3146]-3*X[10706], 2*X[3233]-X[25641], 3*X[3448]-7*X[3523], 9*X[3524]-7*X[15057], 7*X[3526]-5*X[15027], 7*X[3528]-5*X[15021], 17*X[3533]-15*X[15059], 9*X[3545]-7*X[15044], 2*X[3628]-3*X[11694], 4*X[3850]-3*X[10113], 7*X[3851]-6*X[7687], 11*X[3855]-13*X[15029], 11*X[5056]-12*X[12900], 13*X[5067]-11*X[15025], X[5073]-6*X[6053], 2*X[5446]-3*X[12824], 3*X[5891]-2*X[15738], X[7722]+X[12273], 3*X[10264]-5*X[15712], 13*X[10299]-15*X[15051], 3*X[10540]-X[18325], 5*X[11522]-6*X[11723], 3*X[11562]-2*X[13148], 3*X[11597]-2*X[12242], 3*X[11702]-2*X[11803], 3*X[11720]-2*X[13464], 2*X[11800]-3*X[16222], X[11801]-2*X[13392], X[12219]+X[15102], 2*X[12236]-3*X[16223], X[12308]+X[20127], X[12317]-3*X[15055], 2*X[14156]-X[25739], 2*X[14708]-X[21649], 9*X[14845]-8*X[15465], 5*X[15040]-3*X[15061], 3*X[16164]-2*X[16617], 3*X[16165]-2*X[16618]

Let A'B'C' be the 2nd Euler triangle. Let L, M, N be lines through A', B' and C', respectively, parallel to the Euler line. Let L' be the reflection of L in sideline BC, and define M' and N' cyclically. The lines L', M', N' concur in X(30714); cf. X(i) for i = 74, 113, 399, 1147, 1511, 5504, 5609, 5655, 10692, 12383, 14094.

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28804.

X(30714) lies on these lines: {2,11693}, {3,67}, {4,110}, {5,5642}, {20,541}, {24,12828}, {30,3292}, {52,5095}, {74,3522}, {125,128}, {146,5059}, {186,539}, {265,1656}, {376,15054}, {381,15039}, {382,5655}, {399,1498}, {549,20379}, {550,5562}, {567,25555}, {569,15118}, {576,15303}, {631,9140}, {632,20396}, {690,10992}, {1092,11750}, {1154,14448}, {1495,16619}, {1503,10564}, {1568,18572}, {1986,13431}, {2771,12757}, {2781,10625}, {2836,12675}, {2854,8550}, {2929,5898}, {2931,3515}, {3146,10706}, {3233,25641}, {3448,3523}, {3516,12168}, {3517,12310}, {3519,21394}, {3524,15057}, {3526,15027}, {3528,15021}, {3533,15059}, {3545,15044}, {3581,5965}, {3628,11694}, {3850,10113}, {3851,7687}, {3855,15029}, {4857,12896}, {5056,12900}, {5067,15025}, {5073,6053}, {5094,15115}, {5270,18968}, {5446,12824}, {5449,11449}, {5622,13336}, {5891,15738}, {7495,18475}, {7533,15033}, {7574,15139}, {7722,12273}, {8542,11179}, {8674,10993}, {8907,12893}, {9033,12790}, {9517,14900}, {9703,18388}, {9977,15037}, {10116,22467}, {10264,15712}, {10295,13754}, {10299,15051}, {10540,18325}, {10620,11850}, {11411,25712}, {11430,18553}, {11522,11723}, {11561,14049}, {11562,13148},{11597,12242}, {11702,11803}, {11720,13464}, {11800,16222}, {11801,13392}, {12038,12827}, {12134,18488}, {12219,15102}, {12236,16223}, {12308,20127}, {12317,15055}, {13403,18350}, {14156,25739}, {14708,21649}, {14805,24206}, {14845,15465}, {15040,15061}, {15473,19504}, {16164,16617}, {16165,16618}, {16176,17834}, {18555,20771}

X(30714) = midpoint of X(i) and X(j) for these {i,j}: {3,23236}, {74,14683}, {110,12383}, {12219,15102}
X(30714) = reflection of X(i) in X(j) for these {i,j}: {4,16534}, {52,25711}, {113,110}, {125,1511}, {265,5972}, {3448,6699}, {5181,12584}, {7728,6053}, {10113,10272}, {11801,13392}, {12295,113}, {12902,7687}, {15063,5609}, {15133,15115}, {16003,3}, {21649,14708}, {25641,3233},{25739,14156}
X(30714) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4,110,16534}, {4,16534,113}, {20,9143,14094}, {265,5972,23515}, {550,10990,16111}, {631,9140,20397}, {3448,15035,6699}, {9140,15020,631}, {10990,16163,550}, {12902,14643,7687}






leftri  Centers associated with line-reflected triangles: X(30715) -X(30721)  rightri

In the plane of a triangle ABC, suppose that A'B'C' is a triangle and L is a line. Let A'' be the reflection A' in L, and define B'' and C'' cyclically. The triangle A''B''C'' is here named the L-reflection of A'B'C'. If A'B'C' is a central triangle and L a central line, then A''B''C'' is a central triangle. (Clark Kimberling), January 11, 2019)

Let T denote the Euler-line-reflection of ABC. Peter Moses (January 12, 2019) found that T is perspective to the following triangles, with perspectors as indicated:

ABC: X(523)
Schroeter (anticevian triangle of X(523); see X(8286), X(10276)): X(523)
tangential: X(30715)
Macbeath: X(30716)
orthic-of-medial (anti-6th-mixtilinear; see X(11363)): X(30717)
5th Euler (see X(3758): X(30718)
circum-medial: X(23)
Gemini 44: X(23)
Gossard: X(30)
reflection of ABC in X(3): X(30)
infinite altitude: X(74)
circum-orthic: X(186)
Carnot (Johnson, the reflection of ABC in X(5)): X(30)
Kosnita: X(186)

The triangle T is also perspective to these triangles: Euler, Trinh, 2nd Euler, 5th Euler, Artzt, anti-Artzt, tangential of tangential, anti-1st-Euler, anti-Hutson intouch, anti-incircle-circles (see X(11363), orthic-of-medial, Ehrmann side-triangle.

The locus of a point P such that the cevian triangle of P is perspective to T is the cubic pK(14618,264). The locus of P such that the anticevian triangle of P is perspective to T is the cubic pK(112,648). (Peter Moses, January 13, 2019)

For the Nagel-line-reflection of ABC, see X(30719)-X(30721).

underbar




X(30715) = PERSPECTOR OF THESE TRIANGLES: EULER-LINE-REFLECTION OF ABC AND TANGENTIAL

Barycentrics    a^2*(a^12 - 2*a^10*b^2 + a^8*b^4 - a^4*b^8 + 2*a^2*b^10 - b^12 - 2*a^10*c^2 + 4*a^8*b^2*c^2 - 2*a^6*b^4*c^2 + 2*a^4*b^6*c^2 - 4*a^2*b^8*c^2 + 2*b^10*c^2 + a^8*c^4 - 2*a^6*b^2*c^4 - a^4*b^4*c^4 + 2*a^2*b^6*c^4 - b^8*c^4 + 2*a^4*b^2*c^6 + 2*a^2*b^4*c^6 - a^4*c^8 - 4*a^2*b^2*c^8 - b^4*c^8 + 2*a^2*c^10 + 2*b^2*c^10 - c^12) : :

X(30715) lies on these lines: {6, 250}, {23, 230}, {157, 2453}, {186, 1503}, {523, 3447}, {5099, 11641}, {14729, 21006}

X(30715) = midpoint of X(3447) and X(7669)
X(30715) = reflection of X(3447) in the Euler line
X(30715) = tangential-isogonal conjugate of X(110)
X(30715) = X(338)-Ceva conjugate of X(6)
X(30715) = X(59)-of-tangential-triangle if ABC is acute


X(30716) = PERSPECTOR OF THESE TRIANGLES: EULER-LINE-REFLECTION OF ABC AND MACBEATH

Barycentrics    (a^2 - b^2)*(a^2 - c^2)*(a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*(a^6 - a^4*b^2 + a^2*b^4 - b^6 - a^4*c^2 - a^2*b^2*c^2 + b^4*c^2 + a^2*c^4 + b^2*c^4 - c^6) : :
X(30716) = X[648] - 4 X[7473]

X(30716) lies on the cubic K556 and these lines: {4, 14884}, {20, 13573}, {99, 935}, {107, 476}, {110, 6368}, {112, 1287}, {242, 2074}, {250, 523}, {264, 2453}, {691, 1289}, {925, 1304}, {1286, 10423}, {1288, 10420}, {1316, 23635}, {1325, 10538}, {2409, 16237}, {4226, 14590}, {10421, 17702}, {13619, 29012}

X(30716) = reflection of X(i) and X(j) for these {i,j}: {250, 7473}, {648, 250}
X(30716) = reflection of X(250) in the Euler line
X(30716) = X(264)-Ceva conjugate of X(648)
X(30716) = X(i)-isoconjugate of X(j) for these (i,j): {656, 3447}, {810, 13485}, {4575, 6328}
X(30716) = trilinear pole of line {3448, 22146}
X(30716) = barycentric product X(i)*X(j) for these {i,j}: {162, 20941}, {648, 3448}, {811, 16562}, {6331, 7669}, {6528, 22146}, {14366, 14618}
X(30716) = barycentric quotient X(i) / X(j) for these {i,j}: {112, 3447}, {648, 13485}, {2501, 6328}, {3448, 525}, {7669, 647}, {8574, 20975}, {14366, 4558}, {16562, 656}, {20941, 14208}, {21092, 4064}, {21203, 4466}, {22146, 520}
X(30716) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {476, 7480, 107}, {935, 7482, 99}


X(30717) = PERSPECTOR OF THESE TRIANGLES: EULER-LINE-REFLECTION OF ABC AND ORTHIC-OF-MEDIAL

Barycentrics    a^2*(a^12 - 3*a^10*b^2 + 4*a^8*b^4 - 2*a^6*b^6 - 3*a^4*b^8 + 5*a^2*b^10 - 2*b^12 - 3*a^10*c^2 + 6*a^8*b^2*c^2 - 6*a^6*b^4*c^2 + 7*a^4*b^6*c^2 - 7*a^2*b^8*c^2 + 3*b^10*c^2 + 4*a^8*c^4 - 6*a^6*b^2*c^4 + a^4*b^4*c^4 + a^2*b^6*c^4 - 2*b^8*c^4 - 2*a^6*c^6 + 7*a^4*b^2*c^6 + a^2*b^4*c^6 + 2*b^6*c^6 - 3*a^4*c^8 - 7*a^2*b^2*c^8 - 2*b^4*c^8 + 5*a^2*c^10 + 3*b^2*c^10 - 2*c^12) : :

X(30717) lies on these lines: {186, 249}, {523, 14060}, {858, 16320}, {3563, 16760}, {5968, 7485}


X(30718) = PERSPECTOR OF THESE TRIANGLES: EULER-LINE-REFLECTION OF ABC AND 5TH EULER

Barycentrics    a^12 - 2*a^10*b^2 + 2*a^8*b^4 + 6*a^6*b^6 - 5*a^4*b^8 - 4*a^2*b^10 + 2*b^12 - 2*a^10*c^2 + 2*a^8*b^2*c^2 - 8*a^6*b^4*c^2 + a^4*b^6*c^2 + 11*a^2*b^8*c^2 - 2*b^10*c^2 + 2*a^8*c^4 - 8*a^6*b^2*c^4 + 9*a^4*b^4*c^4 - 7*a^2*b^6*c^4 - 2*b^8*c^4 + 6*a^6*c^6 + a^4*b^2*c^6 - 7*a^2*b^4*c^6 + 4*b^6*c^6 - 5*a^4*c^8 + 11*a^2*b^2*c^8 - 2*b^4*c^8 - 4*a^2*c^10 - 2*b^2*c^10 + 2*c^12 : :

X(30718) lies on these lines: {2, 8877}, {23, 3849}, {99, 5189}, {111, 5099}, {112, 468}, {523, 10415}, {858, 10717}, {2453, 6032}

X(30718) = reflection of X(10415) in the Euler line


X(30719) = X(190)-CEVA CONJUGATE OF X(57)

Barycentrics    (3 a-b-c) (b-c) (a+b-c) (a-b+c) : :
X(30719) = 3 X[3669] - X[7178],3 X[3676] - 2 X[7178],3 X[8643] + X[23764]

Let T be the Nagel-line-reflection of ABC. The locus of a ponit P such that T is perspective to the cevian triangle of P is the cubic pK(30719,7). (Peter Moses, January 13, 2019)

X(30719) lies on these lines: {56, 4401}, {57, 4498}, {109, 2737}, {190, 5382}, {241, 514}, {278, 1022}, {522, 4318}, {651, 25737}, {664, 1016}, {1420, 8643}, {1422, 2401}, {2403, 5435}, {3476, 28591}, {3667, 4162}, {4017, 4778}, {4077, 4801}, {4462, 4521}, {4546, 4925}, {4560, 7203}, {4905, 28292}, {6332, 21222}, {10106, 28470}, {24099, 29324}, {25576, 28846}

X(30719) = midpoint of X(6332) and X(21222)
X(30719) = reflection of X(i) and X(j) for these {i,j}: {3676, 3669}, {4462, 4521}, {4546, 4925}, {14837, 3960}, {21120, 7658}
X(30719) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {2137, 150}, {8051, 21293}
X(30719) = X(i)-complementary conjugate of X(j) for these (i,j): {1415, 15347}, {30236, 141}
X(30719) = X(i)-Ceva conjugate of X(j) for these (i,j): {190, 57}, {664, 145}
X(30719) = X(i)-cross conjugate of X(j) for these (i,j): {3667, 3676}, {4394, 3667}
X(30719) = X(i)-isoconjugate of X(j) for these (i,j): {9, 1293}, {55, 27834}, {101, 3680}, {644, 3445}, {663, 5382}, {692, 6557}, {1320, 2429}, {1415, 6556}, {3939, 8056}, {6558, 16945}
X(30719) = crosspoint of X(i) and X(j) for these (i,j): {190, 18743}, {279, 664}
X(30719) = crosssum of X(220) and X(663)
X(30719) = trilinear pole of line {2976, 3756}
X(30719) = crossdifference of every pair of points on line {55, 2347}
X(30719) = barycentric product X(i)*X(j) for these {i,j}: {7, 3667}, {57, 4462}, {85, 4394}, {145, 3676}, {279, 4521}, {479, 4546}, {514, 5435}, {658, 4534}, {664, 3756}, {693, 1420}, {934, 4939}, {1014, 4404}, {1088, 4162}, {1434, 14321}, {1743, 24002}, {2403, 3911}, {3669, 18743}, {3950, 17096}, {4077, 16948}, {4248, 17094}, {4504, 7249}, {4573, 21950}, {4626, 4953}, {4848, 7192}, {4998, 23764}, {6063, 8643}
X(30719) = barycentric quotient X(i) / X(j) for these {i,j}: {56, 1293}, {57, 27834}, {145, 3699}, {513, 3680}, {514, 6557}, {522, 6556}, {651, 5382}, {1404, 2429}, {1420, 100}, {1743, 644}, {2403, 4997}, {2441, 2316}, {2976, 5853}, {3052, 3939}, {3158, 4578}, {3161, 6558}, {3667, 8}, {3669, 8056}, {3676, 4373}, {3756, 522}, {3911, 2415}, {4162, 200}, {4394, 9}, {4404, 3701}, {4462, 312}, {4504, 7081}, {4521, 346}, {4534, 3239}, {4546, 5423}, {4729, 210}, {4848, 3952}, {4849, 4069}, {4855, 4571}, {4925, 3717}, {4939, 4397}, {4949, 4007}, {4953, 4163}, {5435, 190}, {7178, 4052}, {7200, 27831}, {8643, 55}, {14284, 6736}, {14321, 2321}, {14425, 2325}, {16948, 643}, {18211, 3737}, {18743, 646}, {20818, 4587}, {21950, 3700}, {23764, 11}


X(30720) = X(190)-CEVA CONJUGATE OF X(3699)

Barycentrics    (3 a-b-c) (a-b) (a-c) (a-b-c) : :

Let T be the Nagel-line-reflection of ABC. The locus of a ponit P such that T is perspective to the anticevian triangle of P is the cubic pK(30720,190). (Peter Moses, January 13, 2019)

The triangle T is perspective to the following triangles, with perspectors as indicated: ABC: X(3667)
intouch: X(30721)
Caelum (5th mixtilinear; see X(5603)): X(519)
outer Garcia: X(519)
Yff contact: X(4076)

The triangle T is also perspective to the following triangles: hexyl, 6th mixtilinear, Hutson intouch, Artzt, reflection of X(1) in sides of ABC, 3rd Conway (see X(10434), incircle-circles (see X(10434), anti-Artzt (see X(11147), and Jenkins (vertices are the centers of the Jenkings circles).

X(30720) lies on these lines: {8, 3021}, {101, 6079}, {190, 2415}, {312, 4986}, {346, 4370}, {644, 1639}, {664, 1016}, {728, 21384}, {1018, 28521}, {3161, 4534}, {4115, 4752}, {4513, 16969}

X(30720) = X(644),X(6558)}-harmonic conjugate of X(3699)
X(30720) = X(i)-Ceva conjugate of X(j) for these (i,j): {190, 3699}, {1016, 145}, {4076, 15519}, {8706, 4578}
X(30720) = X(i)-cross conjugate of X(j) for these (i,j): {4521, 3161}, {15519, 4076}
X(30720) = X(i)-isoconjugate of X(j) for these (i,j): {514, 16945}, {649, 19604}, {667, 27818}, {1357, 27834}, {3445, 3669}, {4394, 16079}
X(30720) = cevapoint of X(i) and X(j) for these (i,j): {3161, 4521}, {4546, 4936}
X(30720) = trilinear pole of line {3158, 3161}
X(30720) = crossdifference of every pair of points on line {1357, 17071}
X(30720) = barycentric product X(i)*X(j) for these {i,j}: {145, 3699}, {190, 3161}, {644, 18743}, {645, 3950}, {646, 1743}, {664, 6555}, {668, 3158}, {1016, 4521}, {3667, 4076}, {4162, 7035}, {4534, 6632}, {4546, 4998}, {4554, 4936}, {4848, 7256}, {4849, 7257}, {5435, 6558}, {8706, 12640}
X(30720) = barycentric quotient X(i) / X(j) for these {i,j}: {100, 19604}, {145, 3676}, {190, 27818}, {644, 8056}, {692, 16945}, {1293, 16079}, {1332, 27832}, {1743, 3669}, {3158, 513}, {3161, 514}, {3667, 1358}, {3699, 4373}, {3939, 3445}, {3950, 7178}, {4162, 244}, {4513, 27837}, {4521, 1086}, {4534, 6545}, {4546, 11}, {4578, 3680}, {4849, 4017}, {4936, 650}, {4953, 21132}, {6065, 1293}, {6555, 522}, {6558, 6557}, {8643, 1357}, {15519, 3667}, {16948, 7203}, {18743, 24002}


X(30721) = PERSPECTOR OF THESE TRIANGLES: NAGEL-LINE-REFLECTION OF ABC AND INTOUCH

Barycentrics    (a - b)*(a - c)*(a^3 - 2*a^2*b + 4*a*b^2 - b^3 - 2*a^2*c - 3*a*b*c + 4*a*c^2 - c^3) : :

X(30721) lies on these lines: {109, 765}, {190, 3667}, {513, 3699}, {519, 1738}, {2234, 5524}, {2748, 6012}, {4582, 4962}, {5205, 9362}, {5730, 6790}, {6163, 17780}

X(30721) = reflection of X(4076) in the Nagel line


X(30722) = X(241)X(514)∩X(1434)X(17906)

Barycentrics    (b - c)*(-a + b - c)*(a + b - c)*(4*a + b + c) : :
X(30722) = 3 X[3669] + 2 X[3676],4 X[3669] + X[7178],7 X[3669] - 2 X[30719],8 X[3676] - 3 X[7178],7 X[3676] + 3 X[30719],4 X[3960] + X[21104],7 X[7178] + 8 X[30719]

Centers X(30722)-X(30726) are given by first barycentrics (b-c)(a-b+c)(a+b-c)(na+b+c), for n = 4,-3, 2, -2, -4 respectively; these points lie on the line X(241)X(514). See also X(30727).

X(30722) lies on these lines: {241, 514}, {1434, 17096}, {3649, 4017}

X(30722) = X(9)-isoconjugate of X(28210)
X(30722) = barycentric product X(i)*X(j) for these {i,j}: {7, 28209}, {514, 4031}, {551, 3676}, {1358, 4781}, {3669, 24589}, {4714, 7203}, {7178, 26860}, {16666, 24002}
X(30722) = barycentric quotient X(i)/X(j) for these {i,j}: {56, 28210}, {551, 3699}, {3707, 6558}, {4031, 190}, {4781, 4076}, {7178, 27797}, {14435, 2325}, {16666, 644}, {21747, 3939}, {21806, 4069}, {22357, 4587}, {24589, 646}, {26860, 645}, {28209, 8}


X(30723) = X(241)X(514)∩X(1019)X(17096)

Barycentrics    (b - c)*(-a + b - c)*(a + b - c)*(3*a + b + c) : :
X(30723) = 3 X[3669] + X[7178],3 X[3669] - X[30719],3 X[3676] - X[7178],3 X[3676] + X[30719],X[17496] + 3 X[21183]

See X(30722).

X(30723) lies on these lines: {241, 514}, {1019, 17096}, {3667, 4017}, {4040, 28225}, {4504, 28296}, {4765, 4801}, {4778, 16533}, {4978, 24002}, {7265, 22042}, {17496, 21183}, {28161, 30572}

X(30723) = midpoint of X(i) and X(j) for these {i,j}: {3669, 3676}, {4765, 4801}, {7178, 30719}
X(30723) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3669, 7178, 30719}, {3676, 30719, 7178}
X(30723) = X(i)-Ceva conjugate of X(j) for these (i,j): {664, 5586}, {4624, 7}
X(30723) = X(4790)-cross conjugate of X(4778)
X(30723) = X(i)-isoconjugate of X(j) for these (i,j): {9, 8694}, {55, 4606}, {101, 4866}, {210, 4627}, {644, 2334}, {1253, 4624}, {1334, 4614}, {3939, 25430}, {4515, 5545}
X(30723) = crosspoint of X(7) and X(4624)
X(30723) = crossdifference of every pair of points on line {55, 3217}
X(30723) = barycentric product X(i)*X(j) for these {i,j}: {7, 4778}, {57, 4801}, {85, 4790}, {269, 4811}, {279, 4765}, {514, 21454}, {693, 3361}, {1014, 4815}, {1434, 4841}, {1449, 24002}, {3616, 3676}, {3669, 19804}, {3671, 7192}, {4827, 23062}, {4830, 7233}, {5257, 17096}
X(30723) = barycentric quotient X(i)/X(j) for these {i,j}: {56, 8694}, {57, 4606}, {279, 4624}, {391, 6558}, {513, 4866}, {1014, 4614}, {1412, 4627}, {1434, 4633}, {1449, 644}, {3361, 100}, {3616, 3699}, {3669, 25430}, {3671, 3952}, {3676, 5936}, {4512, 4578}, {4652, 4571}, {4765, 346}, {4773, 2325}, {4778, 8}, {4790, 9}, {4801, 312}, {4811, 341}, {4815, 3701}, {4818, 3790}, {4822, 210}, {4827, 728}, {4830, 3685}, {4832, 1334}, {4839, 3985}, {4841, 2321}, {4843, 4082}, {5586, 4756}, {19804, 646}, {21454, 190}


X(30724) = X(241)X(514)∩X(693)X(26732)

Barycentrics    (b - c)*(-a + b - c)*(a + b - c)*(2*a + b + c) : :
X(30724) = 2 X[905] + X[21104],2 X[1019] + X[23729],4 X[2487] - X[4498],X[3669] + 2 X[3676],2 X[3669] + X[7178],5 X[3669] - 2 X[30719],4 X[3676] - X[7178],5 X[3676] + X[30719],X[4801] + 2 X[17069],X[4976] + 2 X[4978],5 X[7178] + 4 X[30719],X[21120] - 4 X[21188]

See X(30722).

X(30724) lies on these lines: {241, 514}, {693, 26732}, {900, 4017}, {918, 28779}, {1019, 23729}, {2487, 4498}, {3649, 4992}, {3910, 4453}, {4773, 29302}, {4801, 17069}, {4897, 28493}, {4927, 6002}, {4976, 4978}, {4977, 5298}, {6545, 29162}, {21183, 23880}

X(30724) = {X(3669),X(3676)}-harmonic conjugate of X(7178)
X(30724) = X(1434)-Ceva conjugate of X(1358)
X(30724) = X(4979)-cross conjugate of X(4977)
X(30724) = X(i)-isoconjugate of X(j) for these (i,j): {9, 8701}, {41, 6540}, {210, 4629}, {644, 1126}, {692, 4102}, {1171, 4069}, {1255, 3939}, {1334, 4596}, {3699, 28615}
X(30724) = crosspoint of X(3676) and X(17096)
X(30724) = crossdifference of every pair of points on line {55, 7064}
X(30724) = barycentric product X(i)*X(j) for these {i,j}: {7, 4977}, {57, 4978}, {85, 4979}, {269, 4985}, {279, 4976}, {479, 4990}, {514, 553}, {552, 6367}, {1014, 30591}, {1100, 24002}, {1125, 3676}, {1213, 17096}, {1358, 4427}, {1434, 4988}, {3649, 7192}, {3669, 4359}, {4017, 16709}, {4647, 7203}, {5298, 6548}, {7178, 8025}
X(30724) = barycentric quotient X(i)/X(j) for these {i,j}: {7, 6540}, {56, 8701}, {514, 4102}, {553, 190}, {1014, 4596}, {1100, 644}, {1125, 3699}, {1358, 4608}, {1412, 4629}, {1434, 4632}, {1962, 4069}, {2308, 3939}, {3649, 3952}, {3669, 1255}, {3676, 1268}, {3683, 4578}, {3686, 6558}, {3916, 4571}, {4359, 646}, {4427, 4076}, {4856, 30720}, {4870, 4767}, {4976, 346}, {4977, 8}, {4978, 312}, {4979, 9}, {4983, 210}, {4984, 2325}, {4985, 341}, {4988, 2321}, {4990, 5423}, {4992, 27538}, {5298, 17780}, {6367, 6057}, {7178, 6539}, {7341, 6578}, {8025, 645}, {8663, 7064}, {16709, 7257}, {22054, 4587}, {30581, 4612}, {30591, 3701}, {30592, 4009}


X(30725) = X(1)X(2826)∩X(241)X(514)

Barycentrics    (2*a - b - c)*(b - c)*(a + b - c)*(a - b + c) : :
X(30725) = 2 X[676] - 3 X[14413],3 X[1638] - 4 X[3960],3 X[1638] - 2 X[10015],3 X[1639] - 2 X[3762],3 X[3669] - 2 X[3676],4 X[3676] - 3 X[7178],X[3676] - 3 X[30719],X[4467] - 3 X[17496],X[4895] - 3 X[30573],X[7178] - 4 X[30719],3 X[14413] - X[21132]

See X(30722).

X(30725) lies on these lines: {1, 2826}, {7, 6009}, {8, 4925}, {12, 3837}, {56, 659}, {57, 21385}, {65, 891}, {88, 2403}, {190, 644}, {241, 514}, {523, 7286}, {676, 14413}, {764, 29240}, {812, 14759}, {900, 1317}, {1022, 2006}, {1086, 1358}, {1319, 1960}, {1388, 25569}, {1407, 24115}, {1639, 3762}, {2099, 21343}, {2254, 6366}, {2401, 21786}, {2821, 3057}, {2827, 15558}, {3910, 4467}, {4017, 4977}, {4391, 29005}, {4449, 6362}, {4453, 30577}, {4462, 26695}, {4504, 28487}, {4560, 18199}, {4762, 30181}, {4773, 23888}, {4897, 28468}, {4905, 28473}, {5433, 24093}, {6180, 24098}, {7288, 24128}, {9001, 14307}, {10074, 19916}, {14284, 30198}, {14425, 21129}, {23729, 29126}, {28585, 28591}

X(30725) = midpoint of X(i) and X(j) for these {i,j}: {659, 24097}, {2254, 21105}, {3904, 21222}
X(30725) = reflection of X(i) and X(j) for these {i,j}: {8, 4925}, {3669, 30719}, {3837, 24099}, {7178, 3669}, {10015, 3960}, {21120, 905}, {21129, 14425}, {21132, 676}
X(30725) = isogonal conjugate of X(5548)
X(30725) = isotomic conjugate of X(4582)
X(30725) = X(i)-anticomplementary conjugate of X(j) for these (i,j): {6079, 21286}, {8686, 150}
X(30725) = X(2743)-complementary conjugate of X(141)
X(30725) = X(i)-Ceva conjugate of X(j) for these (i,j): {7, 14027}, {655, 57}, {664, 1317}, {2006, 1086}
X(30725) = X(i)-cross conjugate of X(j) for these (i,j): {1635, 900}, {2087, 1319}, {4530, 1877}, {14027, 7}
X(30725) = X(i)-isoconjugate of X(j) for these (i,j): {1, 5548}, {9, 901}, {31, 4582}, {41, 4555}, {55, 3257}, {88, 3939}, {100, 2316}, {101, 1320}, {106, 644}, {210, 4591}, {650, 9268}, {663, 5376}, {692, 4997}, {1022, 6065}, {1023, 1318}, {1252, 23838}, {1334, 4622}, {1417, 6558}, {2170, 6551}, {3689, 4638}, {3699, 9456}, {4571, 8752}, {4674, 5546}, {4792, 5549}
X(30725) = cevapoint of X(i) and X(j) for these (i,j): {900, 14425}, {2087, 6550}, {3310, 6085}
X(30725) = crosspoint of X(i) and X(j) for these (i,j): {514, 2401}, {655, 14628}, {2415, 4358}
X(30725) = crosssum of X(i) and X(j) for these (i,j): {101, 2427}, {2441, 9456}
X(30725) = trilinear pole of line {1647, 3259}
X(30725) = crossdifference of every pair of points on line {55, 2316}
X(30725) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {3960, 10015, 1638}, {14413, 21132, 676}
X(30725) = barycentric product X(i)*X(j) for these {i,j}: {7, 900}, {44, 24002}, {57, 3762}, {85, 1635}, {86, 30572}, {269, 4768}, {279, 1639}, {331, 22086}, {479, 4528}, {514, 3911}, {519, 3676}, {658, 4530}, {664, 1647}, {693, 1319}, {1088, 4895}, {1111, 23703}, {1317, 6548}, {1358, 17780}, {1404, 3261}, {1434, 4120}, {1847, 14418}, {1877, 4025}, {1960, 6063}, {2087, 4554}, {3669, 4358}, {3943, 17096}, {3960, 14628}, {3992, 7203}, {4448, 7233}, {4453, 14584}, {4555, 14027}, {4608, 5298}, {4922, 7249}, {4998, 6550}, {7178, 16704}, {7209, 14408}, {14425, 27818}, {14427, 23062}
X(30725) = barycentric quotient X(i)/X(j) for these {i,j}: {2, 4582}, {6, 5548}, {7, 4555}, {44, 644}, {56, 901}, {57, 3257}, {59, 6551}, {109, 9268}, {244, 23838}, {513, 1320}, {514, 4997}, {519, 3699}, {649, 2316}, {651, 5376}, {900, 8}, {902, 3939}, {1014, 4622}, {1317, 17780}, {1319, 100}, {1357, 23345}, {1358, 6548}, {1404, 101}, {1412, 4591}, {1434, 4615}, {1635, 9}, {1639, 346}, {1647, 522}, {1877, 1897}, {1960, 55}, {2087, 650}, {2325, 6558}, {3251, 3689}, {3259, 2804}, {3285, 5546}, {3669, 88}, {3676, 903}, {3689, 4578}, {3762, 312}, {3911, 190}, {4017, 4674}, {4120, 2321}, {4358, 646}, {4448, 3685}, {4528, 5423}, {4530, 3239}, {4542, 4528}, {4730, 210}, {4768, 341}, {4773, 391}, {4895, 200}, {4922, 7081}, {4958, 4007}, {4984, 3686}, {4998, 6635}, {5298, 4427}, {5440, 4571}, {6544, 2325}, {6550, 11}, {7178, 4080}, {8661, 3271}, {14027, 900}, {14122, 6163}, {14407, 1334}, {14408, 3208}, {14418, 3692}, {14425, 3161}, {14427, 728}, {14429, 3710}, {14435, 3707}, {14442, 4530}, {16704, 645}, {17460, 23705}, {17780, 4076}, {21805, 4069}, {22086, 219}, {22356, 4587}, {23344, 6065}, {23345, 1318}, {23703, 765}, {23757, 6735}, {23888, 5233}, {24002, 20568}, {24188, 21132}, {24816, 23354}, {30572, 10}, {30573, 6745}, {30576, 4612}, {30583, 4009}


X(30726) = X(241)X(514)∩X(918)X(29002)

Barycentrics    (4*a - b - c)*(b - c)*(a + b - c)*(a - b + c) : :
X(30726) = 5 X[3669] - 2 X[3676],4 X[3669] - X[7178],X[3669] + 2 X[30719],8 X[3676] - 5 X[7178],X[3676] + 5 X[30719],4 X[3960] - X[21120],X[7178] + 8 X[30719]

See X(30722).

X(30726) lies on these lines: {241, 514}, {918, 29002}, {4017, 28209}, {4897, 28501}, {4927, 28490}, {17496, 26732}

X(30726) = X(9)-isoconjugate of X(28218)
X(30726) = barycentric product X(i)*X(j) for these {i,j}: {7, 28217}, {3244, 3676}, {16669, 24002}
X(30726) = barycentric quotient X(i)/X(j) for these {i,j}: {56, 28218}, {3244, 3699}, {16669, 644}, {28217, 8}


X(30727) = X(100)X(4427)∩X(644)X(1639)

Barycentrics    (-a + b)*(a - c)*(a - b - c)*(4*a + b + c) : :

Centers X(30727)-X(30732) are given by first barycentrics (a-b)(a-c)(a-b-c)(na+b+c), for n = -4, 3, 2, 0, -2, -4, respectively; these points lie on the line X(644)X(1639). See also X(30722).

X(30727) lies on these lines: {101, 4427}, {644, 1639}, {645, 4560}, {663, 4069}, {1023, 3952}, {4752, 17780}, {4781, 14435}

X(30727) = barycentric product X(i)*X(j) for these {i,j}: {8, 4781}, {100, 3902}, {190, 3707}, {551, 3699}, {643, 4714}, {644, 24589}, {646, 16666}, {4031, 6558}, {4076, 28209}, {7257, 21806}
X(30727) = barycentric quotient X(i)/X(j) for these {i,j}: {551, 3676}, {3707, 514}, {3902, 693}, {4714, 4077}, {4781, 7}, {6065, 28210}, {16666, 3669}, {21806, 4017}, {24589, 24002}, {26860, 17096}, {28209, 1358}


X(30728) = X(99)X(101)∩X(644)X(1639)

Barycentrics    (-a + b)*(a - c)*(a - b - c)*(3*a + b + c) : :

See X(30727).

X(30728) lies on these lines: {99, 101}, {644, 1639}, {1043, 27415}, {4997, 26074}, {9057, 29163}

X(30728) = X(4765)-cross conjugate of X(391)
X(30728) = X(i)-isoconjugate of X(j) for these (i,j): {1357, 4606}, {2334, 3669}, {3125, 5545}, {3248, 4624}
X(30728) = cevapoint of X(391) and X(4765),trilinear pole of line {391, 4061}
X(30728) = crossdifference of every pair of points on line {1357, 3122}
X(30728) = {X(644),X(3699)}-harmonic conjugate of X(6558)
X(30728) = barycentric product X(i)*X(j) for these {i,j}: {99, 4061}, {100, 4673}, {190, 391}, {461, 4561}, {644, 19804}, {645, 5257}, {646, 1449}, {668, 4512}, {765, 4811}, {1016, 4765}, {1978, 4258}, {3616, 3699}, {3671, 7256}, {4076, 4778}, {4571, 5342}, {4582, 4700}, {4600, 4843}, {6558, 21454}
X(30728) = barycentric quotient X(i)/X(j) for these {i,j}: {391, 514}, {461, 7649}, {644, 25430}, {1016, 4624}, {1449, 3669}, {3616, 3676}, {3699, 5936}, {3939, 2334}, {4061, 523}, {4101, 17094}, {4258, 649}, {4512, 513}, {4570, 5545}, {4578, 4866}, {4673, 693}, {4765, 1086}, {4771, 7212}, {4778, 1358}, {4811, 1111}, {4819, 30572}, {4827, 2170}, {4843, 3120}, {5257, 7178}, {6065, 8694}, {8653, 3122}, {19804, 24002}


X(30729) = X(101)X(835)∩X(644)X(1639)

Barycentrics    (-a + b)*(a - c)*(a - b - c)*(2*a + b + c) : :

See X(30727). X(30729) lies on these lines: {8, 4919}, {101, 835}, {644, 1639}, {645, 4612}, {1018, 17780}, {1023, 4103}, {4115, 4427}, {4723, 6603}

X(30729) = X(i)-cross conjugate of X(j) for these (i,j): {4976, 3686}, {4990, 3702}
X(30729) = X(i)-isoconjugate of X(j) for these (i,j): {604, 4608}, {1126, 3669}, {1171, 4017}, {3676, 28615}
X(30729) = cevapoint of X(i) and X(j) for these (i,j): {3686, 4976}, {3700, 4060}
X(30729) = crosspoint of X(645) and X(3699)
X(30729) = trilinear pole of line {3683, 3686}
X(30729) = barycentric product X(i)*X(j) for these {i,j}: {8, 4427}, {99, 4046}, {100, 3702}, {190, 3686}, {333, 4115}, {553, 6558}, {643, 4647}, {644, 4359}, {645, 1213}, {646, 1100}, {668, 3683}, {765, 4985}, {1016, 4976}, {1125, 3699}, {1230, 5546}, {1269, 3939}, {1962, 7257}, {3649, 7256}, {4069, 16709}, {4076, 4977}, {4582, 4969}, {4631, 21816}, {4990, 4998}, {6064, 6367}
X(30729) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 4608}, {644, 1255}, {1100, 3669}, {1125, 3676}, {1213, 7178}, {1962, 4017}, {3683, 513}, {3686, 514}, {3699, 1268}, {3702, 693}, {3939, 1126}, {4046, 523}, {4076, 6540}, {4115, 226}, {4359, 24002}, {4427, 7}, {4587, 1796}, {4647, 4077}, {4856, 30719}, {4976, 1086}, {4977, 1358}, {4985, 1111}, {4990, 11}, {5546, 1171}, {6065, 8701}, {6367, 1365}, {6558, 4102}, {8025, 17096}, {20970, 7180}


X(30730) = X(2)X(4986)∩X(644)X(1639)

Barycentrics    (-a + b)*(a - c)*(a - b - c)*(b + c) : :

See X(30727). X(30730) lies on these lines: {2, 4986}, {8, 2170}, {10, 21950}, {101, 9059}, {190, 4606}, {346, 27546}, {514, 25272}, {644, 1639}, {646, 4526}, {1018, 3952}, {1930, 26757}, {2321, 21030}, {3212, 29715}, {3693, 4723}, {3701, 4515}, {3807, 4595}, {3950, 21041}, {4006, 17751}, {4033, 4552}, {4482, 17136}, {4560, 7257}, {4568, 21272}, {4674, 22035}, {4738, 24036}, {5546, 7256}, {17164, 21067}, {17314, 17465}, {20247, 29400}, {24403, 27076}, {24786, 27043}, {26752, 28598}

X(30730) = isotomic conjugate of X(17096)
X(30730) = X(i)-Ceva conjugate of X(j) for these (i,j): {3699, 4069}, {4033, 3952}, {4076, 6057}
X(30730) = X(i)-cross conjugate of X(j) for these (i,j): {3700, 2321}, {3709, 210}, {4041, 8}, {4069, 3952}, {6057, 4076}
X(30730) = X(i)-isoconjugate of X(j) for these (i,j): {6, 7203}, {31, 17096}, {34, 7254}, {56, 1019}, {57, 3733}, {58, 3669}, {59, 8042}, {60, 7216}, {109, 16726}, {163, 1358}, {244, 4565}, {269, 7252}, {513, 1412}, {514, 1408}, {552, 798}, {593, 4017}, {603, 17925}, {604, 7192}, {649, 1014}, {661, 7341}, {662, 1357}, {667, 1434}, {693, 16947}, {738, 21789}, {757, 7180}, {849, 7178}, {1015, 1414}, {1021, 7023}, {1106, 4560}, {1333, 3676}, {1395, 15419}, {1396, 1459}, {1397, 7199}, {1407, 3737}, {1415, 17205}, {1431, 18200}, {1435, 23189}, {1461, 18191}, {1577, 7342}, {1977, 4625}, {2185, 7250}, {2206, 24002}, {3248, 4573}, {3271, 4637}, {4620, 8027}, {7153, 16695}, {7253, 7366}
X(30730) = cevapoint of X(i) and X(j) for these (i,j): {37, 14321}, {210, 3709}, {522, 3706}, {523, 21949}, {650, 3686}, {2321, 3700}, {3239, 3965}
X(30730) = crosspoint of X(646) and X(3699)
X(30730) = trilinear pole of line {210, 2321}
X(30730) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1018, 4103, 3952}, {3699, 6558, 644}, {4103, 4169, 1018}, {4568, 23891, 21272}
X(30730) = crossdifference of every pair of points on line {1357, 8650}
X(30730) = barycentric product X(i)*X(j) for these {i,j}: {8, 3952}, {9, 4033}, {10, 3699}, {12, 7256}, {37, 646}, {55, 27808}, {75, 4069}, {99, 6057}, {100, 3701}, {101, 30713}, {190, 2321}, {210, 668}, {226, 6558}, {312, 1018}, {313, 3939}, {321, 644}, {333, 4103}, {341, 4551}, {346, 4552}, {523, 4076}, {594, 645}, {643, 1089}, {664, 4082}, {670, 7064}, {756, 7257}, {762, 4631}, {765, 4086}, {850, 6065}, {1016, 3700}, {1020, 30693}, {1334, 1978}, {1441, 4578}, {1897, 3710}, {2171, 7258}, {3596, 4557}, {3694, 6335}, {3704, 8707}, {3790, 4613}, {3943, 4582}, {3985, 4562}, {4041, 7035}, {4046, 6540}, {4052, 30720}, {4095, 27805}, {4102, 4115}, {4136, 4621}, {4169, 4997}, {4433, 4583}, {4515, 4554}, {4566, 5423}, {4574, 7017}, {5546, 28654}, {6358, 7259}, {6632, 21044}, {8706, 21031}
X(30730) = barycentric quotient X(i)/X(j) for these {i,j}: {1, 7203}, {2, 17096}, {8, 7192}, {9, 1019}, {10, 3676}, {37, 3669}, {55, 3733}, {99, 552}, {100, 1014}, {101, 1412}, {110, 7341}, {181, 7250}, {190, 1434}, {200, 3737}, {210, 513}, {219, 7254}, {220, 7252}, {281, 17925}, {312, 7199}, {321, 24002}, {341, 18155}, {345, 15419}, {346, 4560}, {480, 21789}, {512, 1357}, {522, 17205}, {523, 1358}, {594, 7178}, {643, 757}, {644, 81}, {645, 1509}, {646, 274}, {650, 16726}, {692, 1408}, {728, 1021}, {756, 4017}, {765, 1414}, {1016, 4573}, {1018, 57}, {1020, 738}, {1089, 4077}, {1252, 4565}, {1260, 23189}, {1334, 649}, {1500, 7180}, {1576, 7342}, {1783, 1396}, {2170, 8042}, {2171, 7216}, {2318, 1459}, {2321, 514}, {2329, 18200}, {3208, 18197}, {3239, 17197}, {3694, 905}, {3695, 17094}, {3699, 86}, {3700, 1086}, {3701, 693}, {3704, 3004}, {3709, 1015}, {3710, 4025}, {3711, 4833}, {3715, 4840}, {3717, 23829}, {3900, 18191}, {3939, 58}, {3950, 30719}, {3952, 7}, {3985, 812}, {4007, 4960}, {4033, 85}, {4037, 7212}, {4041, 244}, {4046, 4977}, {4061, 4778}, {4069, 1}, {4076, 99}, {4082, 522}, {4086, 1111}, {4095, 4369}, {4097, 4401}, {4103, 226}, {4111, 6372}, {4115, 553}, {4136, 3776}, {4140, 7200}, {4147, 23824}, {4162, 18211}, {4169, 3911}, {4171, 2170}, {4391, 16727}, {4433, 659}, {4513, 18199}, {4515, 650}, {4516, 764}, {4524, 3271}, {4538, 830}, {4551, 269}, {4552, 279}, {4557, 56}, {4559, 1407}, {4564, 4637}, {4566, 479}, {4571, 1444}, {4574, 222}, {4578, 21}, {4587, 1790}, {4612, 763}, {4998, 4616}, {5423, 7253}, {5546, 593}, {6057, 523}, {6065, 110}, {6066, 1576}, {6558, 333}, {6632, 4620}, {6735, 23788}, {7035, 4625}, {7064, 512}, {7081, 17212}, {7256, 261}, {7257, 873}, {7259, 2185}, {8611, 3942}, {17787, 16737}, {21044, 6545}, {21859, 1427}, {23067, 7053}, {24394, 2832}, {25268, 18600}, {27538, 17217}, {27808, 6063}, {30681, 15411}, {30713, 3261}


X(30731) = X(8)X(9)∩X(644)X(1639)

Barycentrics    (a - b)*(a - c)*(a - b - c)*(2*a - b - c) : :

See X(30727). X(30731) lies on these lines: {8, 9}, {190, 6009}, {644, 1639}, {1018, 4427}, {1023, 4169}, {2415, 21129}, {2429, 14425}, {3952, 4752}

X(30731) = X(i)-Ceva conjugate of X(j) for these (i,j): {4076, 4152}, {4582, 3699}, {24004, 17780}
X(30731) = X(i)-cross conjugate of X(j) for these (i,j): {1639, 2325}, {4152, 4076}, {4528, 4723}, {4543, 8}
X(30731) = X(i)-isoconjugate of X(j) for these (i,j): {56, 1022}, {57, 23345}, {106, 3669}, {514, 1417}, {604, 6548}, {1357, 3257}, {1407, 23838}, {1408, 4049}, {1415, 6549}, {2403, 16945}, {2441, 19604}, {3676, 9456}
X(30731) = cevapoint of X(i) and X(j) for these (i,j): {44, 14425}, {1639, 2325}
X(30731) = crosspoint of X(i) and X(j) for these (i,j): {190, 6079}, {2415, 17780}, {3699, 4582}
X(30731) = crosssum of X(i) and X(j) for these (i,j): {649, 6085}, {2441, 23345}
X(30731) = trilinear pole of line {2325, 3689}
X(30731) = crossdifference of every pair of points on line {1357, 8054}
X(30731) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {1023, 4169, 17780}, {6558, 30720, 644}
X(30731) = barycentric product X(i)*X(j) for these {i,j}: {8, 17780}, {9, 24004}, {44, 646}, {100, 4723}, {190, 2325}, {312, 1023}, {333, 4169}, {341, 23703}, {519, 3699}, {643, 3992}, {644, 4358}, {645, 3943}, {668, 3689}, {765, 4768}, {900, 4076}, {1016, 1639}, {2415, 3161}, {3264, 3939}, {3596, 23344}, {3911, 6558}, {4103, 30606}, {4152, 4555}, {4370, 4582}, {4528, 4998}, {4530, 6632}, {4542, 6635}, {4895, 7035}, {7257, 21805}
X(30731) = barycentric quotient X(i)/X(j) for these {i,j}: {8, 6548}, {9, 1022}, {44, 3669}, {55, 23345}, {200, 23838}, {519, 3676}, {522, 6549}, {644, 88}, {646, 20568}, {692, 1417}, {900, 1358}, {1023, 57}, {1639, 1086}, {1960, 1357}, {2321, 4049}, {2325, 514}, {2415, 27818}, {3161, 2403}, {3689, 513}, {3699, 903}, {3711, 23352}, {3939, 106}, {3943, 7178}, {3992, 4077}, {4069, 4674}, {4076, 4555}, {4152, 900}, {4169, 226}, {4358, 24002}, {4528, 11}, {4530, 6545}, {4542, 6550}, {4543, 1647}, {4578, 1320}, {4587, 1797}, {4723, 693}, {4768, 1111}, {4873, 23598}, {4895, 244}, {5548, 2226}, {6065, 901}, {6558, 4997}, {14418, 3942}, {14427, 2170}, {16704, 17096}, {17780, 7}, {21805, 4017}, {23344, 56}, {23703, 269}, {24004, 85}


X(30732) = X(644)X(1639)∩X(649)X(1018)

Barycentrics    (a - b)*(a - c)*(a - b - c)*(4*a - b - c) : :

See X(30727). X(30732) lies on these lines: {644, 1639}, {649, 1018}

X(30732) = barycentric product X(i)*X(j) for these {i,j}: {646, 16669}, {3244, 3699}, {4076, 28217}
X(30732) = barycentric quotient X(i)/X(j) for these {i,j}: {3244, 3676}, {6065, 28218}, {16669, 3669}, {28217, 1358}


X(30733) = ISOGONAL CONJUGATE OF X(28787)

Barycentrics    a (a+b) (a+c) (a^2+b^2-c^2) (a^2-b^2+c^2) (a^3-a^2 b-a b^2+b^3-a^2 c-2 a b c+b^2 c-a c^2+b c^2+c^3) : :

As a point on the Euler line, X(30733) has Shinagawa coefficients {(2R + r - p)(p + r + 2 R)(2R + r), p^2 (r + R) - r (r + 2 R) (r + 3 R)}.

See Kadir Altintas and Ercole Suppa, Hyacinthos 28809.

X(30733) lies on these lines: {1,2299}, {2,3}, {9,1474}, {19,5248}, {34,18593}, {37,943}, {72,2203}, {104,1301}, {105,1289}, {107,915}, {112,15344}, {158,2218}, {241,1396}, {1068,8747}, {1104,1870}, {1612,5317}, {1708,1780}, {1848,5259}, {1891,5251}, {1974,10477}, {2164,7040}, {2189,2303}, {2204,5089}, {2332,7719}, {2360,18446}, {2687,22239}, {2752,10423}, {3487,27802}, {5436,7713}, {11517,17776}, {14344,21789}, {20626,26707}

X(30733) = isogonal conjugate of X(28787)
X(30733) = barycentric product of X(i) and X(j) for these {i,j}: {27,3811}, {28,17776}, {29,1708}, {92,1780}, {286,2911}, {648,15313}, {1289,26217}, {1896,3173}, {2322,4341}, {4567,5521}
X(30733) = trilinear product of X(i) and X(j) for these {i,j}: {4,1780}, {27,2911}, {28,3811}, {162,15313}, {1172,1708}, {1474,17776}, {1896,3215}, {3173,8748}, {4183,4341}, {4570,5521}, {8747,11517}
X(30733) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {4,186,3651}, {4,3147,6889}, {4,6878,3541}, {4,7505,6829}, {21,28,4227}, {21,4233,28}, {25,405,4}, {25,468,4239}, {25,17520,28}, {28,2074,21}, {28,4183,4}, {29,13739,28}, {405,2915,440}, {440,2915,3651}, {2074,4233,4227}, {3089,6987,4}, {3575,8226,4}, {4248,17515,28}, {5047,7466,5142}, {6846,7487,4}


X(30734) = EULER LINE INTERCEPT OF X(32)X(21448)

Barycentrics    a^2 (5 a^4-5 b^4+26 b^2 c^2-5 c^4) : :
Barycentrics    (36 R^2-5 SW)S^2 + 5 SW SB SC : :

As a point on the Euler line, X(30734) has Shinagawa coefficients {4 E - 5 F, 5 E + 5 F)}.

See Kadir Altintas and Ercole Suppa, Hyacinthos 28811.

X(30734) lies on these lines: {2,3}, {32,21448}, {111,9605}, {373,26864}, {576,3066}, {1351,10545}, {1495,10541}, {3053,8585}, {3167,15019}, {3292,9777}, {5050,5643}, {5544,6800}, {5640,11482}, {5644,9544}, {5651,11477}, {5943,22234}, {8780,11451}, {9306,22330}, {10314,15860}, {11465,14530}, {14924,22112}, {15034,15465}

X(30734) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,5020,16042}, {3,12105,9715}, {3,16042,11284}, {1995,5020,11284}, {1995,11284,25}, {1995,16042,3}


X(30735) = (name pending)

Barycentrics    b^2 c^2 (b^2 - c^2) (3a^4 + b^4 + c^4 - 2 b^2 c^2) : :

X(30735) lies on these lines:


X(30736) = (name pending)

Barycentrics    b^2 c^2 (2 b^2 c^2 - a^2 b^2 - a^2 c^2) : :

X(30736) lies on these lines:


X(30737) = (name pending)

Barycentrics    b^2 c^2 (-2a^6 + b^6 + c^6 + a^4 b^2 + a^4 c^2 - b^4 c^2 - b^2 c^4) : :

X(30737) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 101: X(30738) - X(30803)  rightri

Extending the preambles just before X(24537), X(26153), and X(27378), Gemini triangles A'B'C', indexed as 101 to 111, are introduced here, given by barycentrics for A', followed by the range of associated triangle centers. Each range of centers is preceded by a preamble.

Gemini triangle 101: A' = a^2 - b^2 - c^2 : 2 a^2 : 2 a ^2; range X(30738)-X(30803)
Gemini triangle 102: A' = a - b - c: 2 a : 2 a; range X(30808)-X(30869)
Gemini triangle 103: A' = - a^3 : b^3 + c^3 : b^3 + c^3; range X(30878)-X(30939)
Gemini triangle 104: A' = - b c : a(b + c) : a(b + c); range X(30942)-X(31007)
Gemini triangle 105: A' = - b - c : 2a + b + c : 2a + b + c; range X(31014)-X(31063)
Gemini triangle 106: A' = - b^2 - c^2 : 2a^2 + b^2 + c^2 : 2a^2 + b^2 + c^2; range X(31071)-X(31132)
Gemini triangle 107: A' = -1 : 2 : 2; range X(31133)-X(31181)
Gemini triangle 108: A' = 3a - b - c : 2(a - b - c) : 2(a - b - c); range X(31183)-X(31234)
Gemini triangle 109: A' = 1 : 2 : 2; range X(31235)-X(31283)
Gemini triangle 110: A' = 2 : 1 : 1; range X(31284)-X(31289)
Gemini triangle 111: A' = -3 : 1 : 1; range X(31290)-X(31205)

Let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 101, as in centers X(30738)-X(30803). Then

m(X) = a^3 x - (a + c) (a^2 - a c + c^2) y - (a + b) (a^2 - a b + b^2) z : :

and m(X) is a self-inverse mapping such that m(X) is collinear with X(2) and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. Also, m(nine-point circle) = nine-point circle. The fixed points of m are X(2) and every point on the line X(325)X(523), which is the isotomic conjugate of the circumcircle. Among the fixed points are X(i) for these i: 2, 325, 523, 684, 850 ,3260, 3265, 3266, 3267, 20735, 30736, 30737. (Clark Kimberling, January 19, 2019)

underbar




X(30738) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^6 + a^5 b - 2 a^4 b^2 - 4 a^3 b^3 - a^2 b^4 + 3 a b^5 + 2 b^6 + a^5 c - 2 a^4 b c + 4 a^2 b^3 c - 5 a b^4 c + 2 b^5 c - 2 a^4 c^2 + 2 a^2 b^2 c^2 + 2 a b^3 c^2 - 2 b^4 c^2 - 4 a^3 c^3 + 4 a^2 b c^3 + 2 a b^2 c^3 - 4 b^3 c^3 - a^2 c^4 - 5 a b c^4 - 2 b^2 c^4 + 3 a c^5 + 2 b c^5 + 2 c^6 : :

X(30738) lies on these lines:


X(30739) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    -a^4 b^2 + b^6 - a^4 c^2 + 8 a^2 b^2 c^2 - b^4 c^2 - b^2 c^4 + c^6 : :

X(30739) lies on these lines:


X(30740) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^4 - 4 a^2 b^2 + 3 b^4 + 2 a^2 b c + 4 a b^2 c - 2 b^3 c - 4 a^2 c^2 + 4 a b c^2 - 2 b^2 c^2 - 2 b c^3 + 3 c^4 : :

X(30740) lies on these lines:


X(30741) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^3 - a^2 b - 3 a b^2 + 3 b^3 - a^2 c - b^2 c - 3 a c^2 - b c^2 + 3 c^3 : :

X(30741) lies on these lines:


X(30742) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^4 - a^3 b - a^2 b^2 - a b^3 + 2 b^4 - a^3 c + a b^2 c - 2 b^3 c - a^2 c^2 + a b c^2 - a c^3 - 2 b c^3 + 2 c^4 : :

X(30742) lies on these lines:


X(30743) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^7 - 2 a^5 b^2 + a^3 b^4 - 2 a^2 b^5 + 2 b^7 - 2 a^5 c^2 + 2 a^3 b^2 c^2 - 2 b^5 c^2 + a^3 c^4 - 2 a^2 c^5 - 2 b^2 c^5 + 2 c^7 : :

X(30743) lies on these lines:


X(30744) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^6 - 2 a^4 b^2 - a^2 b^4 + 2 b^6 - 2 a^4 c^2 + 4 a^2 b^2 c^2 - 2 b^4 c^2 - a^2 c^4 - 2 b^2 c^4 + 2 c^6 : :

X(30744) lies on these lines:


X(30745) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^6 - 2 a^4 b^2 - a^2 b^4 + 2 b^6 - 2 a^4 c^2 + 5 a^2 b^2 c^2 - 2 b^4 c^2 - a^2 c^4 - 2 b^2 c^4 + 2 c^6 : :

X(30745) lies on these lines:


X(30746) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^5 - a^3 b^2 + 2 b^5 - a^3 c^2 + 2 c^5 : :

X(30746) lies on these lines:


X(30747) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^6 - a^4 b^2 + 2 b^6 - a^4 c^2 + 2 c^6 : :

X(30747) lies on these lines:


X(30748) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^3 b + a b^3 + a^3 c - a b^2 c + 2 b^3 c - a b c^2 + a c^3 + 2 b c^3 : :

X(30748) lies on these lines:


X(30749) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^4 b^2 + a^2 b^4 + a^4 c^2 - 2 a^2 b^2 c^2 + 2 b^4 c^2 + a^2 c^4 + 2 b^2 c^4 : :

X(30749) lies on these lines:


X(30750) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^6 - a^5 b - a^4 b^2 + a^3 b^3 - 2 a b^5 + 2 b^6 - a^5 c + a^3 b^2 c - 2 b^5 c - a^4 c^2 + a^3 b c^2 + a^3 c^3 - 2 a c^5 - 2 b c^5 + 2 c^6 : :

X(30750) lies on these lines:


X(30751) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^4 b - a^2 b^3 + 2 a b^4 + a^4 c - a^2 b^2 c + 2 b^4 c - a^2 b c^2 - a^2 c^3 + 2 a c^4 + 2 b c^4 : :

X(30751) lies on these lines:


X(30752) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^4 b - a^2 b^3 + 2 a b^4 + a^4 c - a^3 b c - a^2 b^2 c - a b^3 c + 2 b^4 c - a^2 b c^2 - a^2 c^3 - a b c^3 + 2 a c^4 + 2 b c^4 : :

X(30752) lies on these lines:


X(30753) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    2 a^4 - a^3 b - 2 a^2 b^2 - a b^3 + 4 b^4 - a^3 c + a b^2 c - 2 b^3 c - 2 a^2 c^2 + a b c^2 - a c^3 - 2 b c^3 + 4 c^4 : :

X(30753) lies on these lines:


X(30754) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^4 - 2 a^3 b - a^2 b^2 - 2 a b^3 + 2 b^4 - 2 a^3 c + 2 a b^2 c - 4 b^3 c - a^2 c^2 + 2 a b c^2 - 2 a c^3 - 4 b c^3 + 2 c^4 : :

X(30754) lies on these lines:


X(30755) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^5 - a^4 b - a^3 b^2 + a^2 b^3 - 2 a b^4 + 2 b^5 - a^4 c + a^2 b^2 c - 2 b^4 c - a^3 c^2 + a^2 b c^2 + a^2 c^3 - 2 a c^4 - 2 b c^4 + 2 c^5 : :

X(30755) lies on these lines:


X(30756) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^6 - 2 a^4 b^2 - a^2 b^4 + 2 b^6 + 2 a^4 b c - 2 a^2 b^3 c + 4 a b^4 c - 2 a^4 c^2 + 2 a^2 b^2 c^2 - 2 b^4 c^2 - 2 a^2 b c^3 - a^2 c^4 + 4 a b c^4 - 2 b^2 c^4 + 2 c^6 : :

X(30756) lies on these lines:


X(30757) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^5 - 2 a^3 b^2 - 2 a^2 b^3 + a b^4 + 2 b^5 + 2 a^3 b c + 2 a b^3 c - 2 a^3 c^2 + 2 a b^2 c^2 - 2 b^3 c^2 - 2 a^2 c^3 + 2 a b c^3 - 2 b^2 c^3 + a c^4 + 2 c^5 : :

X(30757) lies on these lines:


X(30758) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    b c (-a^2 - 2 a b + b^2 - 2 a c + c^2) : :

X(30758) lies on these lines:


X(30759) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^6 + a^5 b - a^4 b^2 - a^3 b^3 + 2 a b^5 + 2 b^6 + a^5 c + a^4 b c - a^3 b^2 c - a^2 b^3 c + 2 a b^4 c + 2 b^5 c - a^4 c^2 - a^3 b c^2 - a^3 c^3 - a^2 b c^3 + 2 a b c^4 + 2 a c^5 + 2 b c^5 + 2 c^6 : :

X(30759) lies on these lines:


X(30760) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^5 + a^4 b - a^3 b^2 - a^2 b^3 + 2 a b^4 + 2 b^5 + a^4 c + a^3 b c - a^2 b^2 c + a b^3 c + 2 b^4 c - a^3 c^2 - a^2 b c^2 - a^2 c^3 + a b c^3 + 2 a c^4 + 2 b c^4 + 2 c^5 : :

X(30760) lies on these lines:


X(30761) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^4 + a^3 b - a^2 b^2 + a b^3 + 2 b^4 + a^3 c + a^2 b c + a b^2 c + b^3 c - a^2 c^2 + a b c^2 + a c^3 + b c^3 + 2 c^4 : :

X(30761) lies on these lines:


X(30762) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    3 a^3 - a^2 b - 5 a b^2 + 7 b^3 - a^2 c - b^2 c - 5 a c^2 - b c^2 + 7 c^3 : :

X(30762) lies on these lines:


X(30763) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^4 - a^2 b^2 + 2 b^4 - a^2 b c - 2 a b^2 c + b^3 c - a^2 c^2 - 2 a b c^2 + b c^3 + 2 c^4 : :

X(30763) lies on these lines:


X(30764) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    (b - c) (a^4 - a^2 b^2 - 2 a b^3 - 2 a b^2 c + 2 b^3 c - a^2 c^2 - 2 a b c^2 + 2 b^2 c^2 - 2 a c^3 + 2 b c^3) : :

X(30764) lies on these lines:


X(30765) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    (-b + c) (a^3 b - 2 a^2 b^2 - a b^3 + a^3 c - 2 a^2 b c - a b^2 c - 2 a^2 c^2 - a b c^2 + 2 b^2 c^2 - a c^3) : :

X(30765) lies on these lines:


X(30766) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    (b - c) (a^5 - a^3 b^2 - 2 a b^4 - 2 a b^3 c + 2 b^4 c - a^3 c^2 - 2 a b^2 c^2 + 2 b^3 c^2 - 2 a b c^3 + 2 b^2 c^3 - 2 a c^4 + 2 b c^4) : :

X(30766) lies on these lines:


X(30767) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^5 - a^3 b^2 + 2 b^5 - a^3 b c - a b^3 c - a^3 c^2 - a b c^3 + 2 c^5 : :

X(30767) lies on these lines:


X(30768) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    2 a^3 + a^2 b + 3 b^3 + a^2 c + b^2 c + b c^2 + 3 c^3 : :

X(30768) lies on these lines:


X(30769) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    3 a^6 - 7 a^4 b^2 - 3 a^2 b^4 + 7 b^6 - 7 a^4 c^2 + 14 a^2 b^2 c^2 - 7 b^4 c^2 - 3 a^2 c^4 - 7 b^2 c^4 + 7 c^6 : :

X(30769) lies on these lines:


X(30770) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^6 - 2 a^4 b^2 - a^2 b^4 + 2 b^6 - a^4 b c - a^3 b^2 c - a^2 b^3 c - a b^4 c - 2 a^4 c^2 - a^3 b c^2 + 2 a^2 b^2 c^2 - a b^3 c^2 - 2 b^4 c^2 - a^2 b c^3 - a b^2 c^3 - a^2 c^4 - a b c^4 - 2 b^2 c^4 + 2 c^6 : :

X(30770) lies on these lines:


X(30771) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    (a^2 - b^2 - c^2) (a^4 - a^2 b^2 - 2 b^4 - a^2 c^2 + 4 b^2 c^2 - 2 c^4) : :

X(30771) lies on these lines:


X(30772) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    (a^2 - b^2 - c^2) (a^6 + a^5 b - 2 a^3 b^3 - 3 a^2 b^4 - 3 a b^5 - 2 b^6 + a^5 c + a^4 b c - 2 a^3 b^2 c - 2 a^2 b^3 c - 3 a b^4 c - 3 b^5 c - 2 a^3 b c^2 + 2 a^2 b^2 c^2 + 6 a b^3 c^2 + 2 b^4 c^2 - 2 a^3 c^3 - 2 a^2 b c^3 + 6 a b^2 c^3 + 6 b^3 c^3 - 3 a^2 c^4 - 3 a b c^4 + 2 b^2 c^4 - 3 a c^5 - 3 b c^5 - 2 c^6) : :

X(30772) lies on these lines:


X(30773) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    (a^2 - b^2 - c^2) (a^7 + a^6 b - 3 a^3 b^4 - 3 a^2 b^5 - 2 a b^6 - 2 b^7 + a^6 c + a^5 b c - 2 a^3 b^3 c - 3 a^2 b^4 c - 3 a b^5 c - 2 b^6 c + 2 a^3 b^2 c^2 + 2 a^2 b^3 c^2 + 2 a b^4 c^2 + 2 b^5 c^2 - 2 a^3 b c^3 + 2 a^2 b^2 c^3 + 6 a b^3 c^3 + 2 b^4 c^3 - 3 a^3 c^4 - 3 a^2 b c^4 + 2 a b^2 c^4 + 2 b^3 c^4 - 3 a^2 c^5 - 3 a b c^5 + 2 b^2 c^5 - 2 a c^6 - 2 b c^6 - 2 c^7) : :

X(30773) lies on these lines:


X(30774) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    (a^2 - b^2 - c^2) (a^7 - a^5 b^2 + 2 a^4 b^3 - a^3 b^4 + a b^6 - 2 b^7 - a^5 b c + a^4 b^2 c + 2 a^3 b^3 c + 2 a^2 b^4 c + 3 a b^5 c + b^6 c - a^5 c^2 + a^4 b c^2 + 6 a^3 b^2 c^2 - 2 a^2 b^3 c^2 - a b^4 c^2 + 5 b^5 c^2 + 2 a^4 c^3 + 2 a^3 b c^3 - 2 a^2 b^2 c^3 - 6 a b^3 c^3 - 4 b^4 c^3 - a^3 c^4 + 2 a^2 b c^4 - a b^2 c^4 - 4 b^3 c^4 + 3 a b c^5 + 5 b^2 c^5 + a c^6 + b c^6 - 2 c^7) : :

X(30774) lies on these lines:


X(30775) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    5 a^6 - 11 a^4 b^2 - 5 a^2 b^4 + 11 b^6 - 11 a^4 c^2 + 18 a^2 b^2 c^2 - 11 b^4 c^2 - 5 a^2 c^4 - 11 b^2 c^4 + 11 c^6 : :

X(30775) lies on these lines:


X(30776) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^6 - 3 a^4 b^2 - a^2 b^4 + 3 b^6 + 2 a^4 b c + 2 a^3 b^2 c + 2 a^2 b^3 c + 2 a b^4 c - 3 a^4 c^2 + 2 a^3 b c^2 + 10 a^2 b^2 c^2 + 2 a b^3 c^2 - 3 b^4 c^2 + 2 a^2 b c^3 + 2 a b^2 c^3 - a^2 c^4 + 2 a b c^4 - 3 b^2 c^4 + 3 c^6 : :

X(30776) lies on these lines:


X(30777) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^6 - a^4 b^2 + 2 b^6 - a^4 c^2 + 5 a^2 b^2 c^2 - b^4 c^2 - b^2 c^4 + 2 c^6 : :

X(30777) lies on these lines:


X(30778) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    -a^4 b^2 + b^6 + 2 a^4 b c - 2 a^2 b^3 c + 4 a b^4 c - a^4 c^2 + 8 a^2 b^2 c^2 - b^4 c^2 - 2 a^2 b c^3 + 4 a b c^4 - b^2 c^4 + c^6 : :

X(30778) lies on these lines:


X(30779) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    (a^2 - b^2 - c^2) (a^5 - 2 a^2 b^3 - a b^4 - 2 b^5 + 2 a b^2 c^2 + 2 b^3 c^2 - 2 a^2 c^3 + 2 b^2 c^3 - a c^4 - 2 c^5) : :

X(30779) lies on these lines:


X(30780) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^6 - 2 a^4 b^2 - a^2 b^4 + 2 b^6 - a^4 b c + a^2 b^3 c - 2 a b^4 c - 2 a^4 c^2 + 2 a^2 b^2 c^2 - 2 b^4 c^2 + a^2 b c^3 - a^2 c^4 - 2 a b c^4 - 2 b^2 c^4 + 2 c^6 : :

X(30780) lies on these lines:


X(30781) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^6 - 2 a^4 b^2 - a^2 b^4 + 2 b^6 + a^4 b c - a^2 b^3 c + 2 a b^4 c - 2 a^4 c^2 + 2 a^2 b^2 c^2 - 2 b^4 c^2 - a^2 b c^3 - a^2 c^4 + 2 a b c^4 - 2 b^2 c^4 + 2 c^6 : :

X(30781) lies on these lines:


X(30782) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^5 - 2 a^3 b^2 - 2 a^2 b^3 + a b^4 + 2 b^5 - 2 a^3 c^2 + 2 a b^2 c^2 - 2 b^3 c^2 - 2 a^2 c^3 - 2 b^2 c^3 + a c^4 + 2 c^5 : :

X(30782) lies on these lines:


X(30783) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^5 b - 4 a^3 b^3 + 3 a b^5 + a^5 c + 2 a^2 b^3 c - a b^4 c + 2 b^5 c + 2 a b^3 c^2 - 4 a^3 c^3 + 2 a^2 b c^3 + 2 a b^2 c^3 - 4 b^3 c^3 - a b c^4 + 3 a c^5 + 2 b c^5 : :

X(30783) lies on these lines:


X(30784) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^7 - a^3 b^4 + 2 a^2 b^5 + 2 b^7 - a^3 b^2 c^2 + 2 a^2 b^3 c^2 - a b^4 c^2 + 2 b^5 c^2 + 2 a^2 b^2 c^3 - a^3 c^4 - a b^2 c^4 + 2 a^2 c^5 + 2 b^2 c^5 + 2 c^7 : :

X(30784) lies on these lines:


X(30785) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^6 + a^2 b^4 + 2 b^6 + 3 a^2 b^2 c^2 + b^4 c^2 + a^2 c^4 + b^2 c^4 + 2 c^6 : :

X(30785) lies on these lines:


X(30786) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    (a^2 + b^2 - 2 c^2) (a^2 - b^2 - c^2) (a^2 - 2 b^2 + c^2) : :

X(30786) lies on these lines:


X(30787) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^5 - a^4 b - a^3 b^2 + a^2 b^3 - 2 a b^4 + 2 b^5 - a^4 c + a^3 b c + a^2 b^2 c + a b^3 c - 2 b^4 c - a^3 c^2 + a^2 b c^2 + a^2 c^3 + a b c^3 - 2 a c^4 - 2 b c^4 + 2 c^5 : :

X(30787) lies on these lines:


X(30788) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^6 - a^5 b - a^4 b^2 + a^3 b^3 - 2 a b^5 + 2 b^6 - a^5 c + a^4 b c + a^3 b^2 c - a^2 b^3 c + 2 a b^4 c - 2 b^5 c - a^4 c^2 + a^3 b c^2 + a^3 c^3 - a^2 b c^3 + 2 a b c^4 - 2 a c^5 - 2 b c^5 + 2 c^6 : :

X(30788) lies on these lines:


X(30789) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^8 - 2 a^6 b^2 + a^4 b^4 - 2 a^2 b^6 + 2 b^8 - 2 a^6 c^2 + 3 a^4 b^2 c^2 + a^2 b^4 c^2 - 2 b^6 c^2 + a^4 c^4 + a^2 b^2 c^4 - 2 a^2 c^6 - 2 b^2 c^6 + 2 c^8 : :

X(30789) lies on these lines:


X(30790) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^4 - a^3 b - a^2 b^2 - a b^3 + 2 b^4 - a^3 c + a^2 b c + 3 a b^2 c - 3 b^3 c - a^2 c^2 + 3 a b c^2 - a c^3 - 3 b c^3 + 2 c^4 : :

X(30790) lies on these lines:


X(30791) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^3 b + a b^3 + a^3 c - a^2 b c - 3 a b^2 c + 3 b^3 c - 3 a b c^2 + a c^3 + 3 b c^3 : :

X(30791) lies on these lines:


X(30792) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    (b - c) (2 a^3 + a^2 b - 6 a b^2 + b^3 + a^2 c - 4 a b c + 5 b^2 c - 6 a c^2 + 5 b c^2 + c^3) : :

X(30792) lies on these lines:


X(30793) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^4 b^2 + a^2 b^4 + a^4 c^2 - 7 a^2 b^2 c^2 + 3 b^4 c^2 + a^2 c^4 + 3 b^2 c^4 : :

X(30793) lies on these lines:


X(30794) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^10 - 3 a^8 b^2 - a^2 b^8 + 3 b^10 - 3 a^8 c^2 + 4 a^4 b^4 c^2 - b^8 c^2 + 4 a^4 b^2 c^4 + 2 a^2 b^4 c^4 - 2 b^6 c^4 - 2 b^4 c^6 - a^2 c^8 - b^2 c^8 + 3 c^10 : :

X(30794) lies on these lines:


X(30795) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    (b - c) (a^3 - 2 a b^2 - a b c + 2 b^2 c - 2 a c^2 + 2 b c^2) : :

X(30795) lies on these lines:


X(30796) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    b c (a^4 - 2 a^3 b - 2 a^2 b^2 + 2 a b^3 + b^4 - 2 a^3 c + 10 a^2 b c - 2 a b^2 c - 2 b^3 c - 2 a^2 c^2 - 2 a b c^2 + 2 b^2 c^2 + 2 a c^3 - 2 b c^3 + c^4) : :

X(30796) lies on these lines:


X(30797) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^5 b^2 - a^3 b^4 + 2 a^2 b^5 - 2 a^5 b c + 2 a^3 b^3 c - 4 a b^5 c + a^5 c^2 - 3 a^3 b^2 c^2 - 2 a^2 b^3 c^2 + a b^4 c^2 + 2 b^5 c^2 + 2 a^3 b c^3 - 2 a^2 b^2 c^3 - a^3 c^4 + a b^2 c^4 + 2 a^2 c^5 - 4 a b c^5 + 2 b^2 c^5 : :

X(30797) lies on these lines:


X(30798) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^4 - a^2 b^2 + 2 b^4 + a^2 b c + 2 a b^2 c - b^3 c - a^2 c^2 + 2 a b c^2 - b c^3 + 2 c^4 : :

X(30798) lies on these lines:


X(30799) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    2 a^5 - 3 a^3 b^2 - 2 a^2 b^3 + a b^4 + 4 b^5 - 3 a^3 c^2 + 2 a b^2 c^2 - 2 b^3 c^2 - 2 a^2 c^3 - 2 b^2 c^3 + a c^4 + 4 c^5 : :

X(30799) lies on these lines:


X(30800) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^3 b^2 + 2 a^2 b^3 - a b^4 + a^3 b c + a b^3 c + a^3 c^2 - 2 a b^2 c^2 + 2 b^3 c^2 + 2 a^2 c^3 + a b c^3 + 2 b^2 c^3 - a c^4 : :

X(30800) lies on these lines:


X(30801) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^3 b^2 + 2 a^2 b^3 - a b^4 + a^3 c^2 - 2 a b^2 c^2 + 2 b^3 c^2 + 2 a^2 c^3 + 2 b^2 c^3 - a c^4 : :

X(30801) lies on these lines:


X(30802) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(24), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    (a^2 - b^2 - c^2) (a^10 - 2 a^8 b^2 - 2 a^6 b^4 + 4 a^4 b^6 + a^2 b^8 - 2 b^10 - 2 a^8 c^2 + 2 a^6 b^2 c^2 + 2 a^2 b^6 c^2 + 6 b^8 c^2 - 2 a^6 c^4 - 6 a^2 b^4 c^4 - 4 b^6 c^4 + 4 a^4 c^6 + 2 a^2 b^2 c^6 - 4 b^4 c^6 + a^2 c^8 + 6 b^2 c^8 - 2 c^10) : :

X(30802) lies on these lines:


X(30803) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(26), WHERE A'B'C' = GEMINI TRIANGLE 101

Barycentrics    a^12 - 3 a^10 b^2 + 6 a^6 b^6 - 3 a^4 b^8 - 3 a^2 b^10 + 2 b^12 - 3 a^10 c^2 + 4 a^8 b^2 c^2 + 4 a^6 b^4 c^2 - a^2 b^8 c^2 - 4 b^10 c^2 + 4 a^6 b^2 c^4 - 10 a^4 b^4 c^4 + 4 a^2 b^6 c^4 - 2 b^8 c^4 + 6 a^6 c^6 + 4 a^2 b^4 c^6 + 8 b^6 c^6 - 3 a^4 c^8 - a^2 b^2 c^8 - 2 b^4 c^8 - 3 a^2 c^10 - 4 b^2 c^10 + 2 c^12 : :

X(30803) lies on these lines:


X(30804) = (name pending)

Barycentrics    b c (b - c) (3a^2 + b^2 + c^2 - 2 b c) : :

X(30804) lies on these lines:


X(30805) = (name pending)

Barycentrics    (b - c) (a^2 - b^2 - c^2)^2 : :

X(30805) lies on these lines:


X(30806) = (name pending)

Barycentrics    b c (-2 a^2 + b^2 + c^2 - 2 b c + a b + a c) : :

X(30806) lies on these lines:


X(30807) = (name pending)

Barycentrics    b c (-2 a^3 + b^3 + c^3 + a^ b + a^2 c - b^2 c - b c^2) : :

X(30807) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 102: X(30808) - X(30869)  rightri

Extending the preambles just before X(24537), X(26153), X(27378), X(30738), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 102, as in centers X(30808)-X(30869). Then

m(X) = (a - b - c) x + 2 b y + 2 c z : :

and m(X) is a self-inverse mapping such that m(X) is collinear with X(2) and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. The fixed points of m are X(2) and every point on the line X(514)X(661), which is the isotomic conjugate of the circumellipse {{A,B,C,X(88),X(100),X(162),X(190)}}. Among the fixed points are X(i) for these i: 2, 514, 693, 3912, 6381, 4358, 15413, 30804, 30805, 30806, 30807. (Clark Kimberling, January 19, 2019)

underbar




X(30808) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^5 - a^4 b - a^3 b^2 - a^2 b^3 + 2 b^5 - a^4 c + a^2 b^2 c - a^3 c^2 + a^2 b c^2 - 2 b^3 c^2 - a^2 c^3 - 2 b^2 c^3 + 2 c^5 : :

X(30808) lies on these lines:


X(30809) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^5 - 3 a^4 b - a b^4 + 3 b^5 - 3 a^4 c + 4 a^2 b^2 c - b^4 c + 4 a^2 b c^2 + 2 a b^2 c^2 - 2 b^3 c^2 - 2 b^2 c^3 - a c^4 - b c^4 + 3 c^5 : :

X(30809) lies on these lines:


X(30810) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    2 a^4 b - a^3 b^2 - a^2 b^3 + a b^4 - b^5 + 2 a^4 c - 3 a^2 b^2 c + b^4 c - a^3 c^2 - 3 a^2 b c^2 - 2 a b^2 c^2 - a^2 c^3 + a c^4 + b c^4 - c^5 : :

X(30810) lies on these lines:


X(30811) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^3 - a^2 b + 2 b^3 - a^2 c + 2 c^3 : :

X(30811) lies on these lines:


X(30812) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    (a - b - c) (a^3 b - 2 a^2 b^2 - 3 a b^3 + a^3 c + a b^2 c - 2 b^3 c - 2 a^2 c^2 + a b c^2 + 4 b^2 c^2 - 3 a c^3 - 2 b c^3) : :

X(30812) lies on these lines:


X(30813) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^4 - 3 a^3 b + 5 a^2 b^2 - 5 a b^3 + 2 b^4 - 3 a^3 c + 6 a^2 b c + a b^2 c - 4 b^3 c + 5 a^2 c^2 + a b c^2 + 4 b^2 c^2 - 5 a c^3 - 4 b c^3 + 2 c^4 : :

X(30813) lies on these lines:


X(30814) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^7 - a^6 b - 2 a^4 b^3 - a^3 b^4 + a^2 b^5 + 2 b^7 - a^6 c + a^2 b^4 c - 2 a^4 c^3 - 2 b^4 c^3 - a^3 c^4 + a^2 b c^4 - 2 b^3 c^4 + a^2 c^5 + 2 c^7 : :

X(30814) lies on these lines:


X(30815) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^7 - a^6 b - 2 a^4 b^3 - a^3 b^4 + a^2 b^5 + 2 b^7 - a^6 c + a^2 b^4 c + a^3 b^2 c^2 + a^2 b^3 c^2 - 2 a^4 c^3 + a^2 b^2 c^3 - 2 b^4 c^3 - a^3 c^4 + a^2 b c^4 - 2 b^3 c^4 + a^2 c^5 + 2 c^7 : :

X(30815) lies on these lines:


X(30816) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^4 - a^3 b + 2 b^4 - a^3 c + 2 c^4 : :

X(30816) lies on these lines:


X(30817) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^5 - a^4 b + 2 b^5 - a^4 c + 2 c^5 : :

X(30817) lies on these lines:


X(30818) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^2 b + a b^2 + a^2 c - 2 a b c + 2 b^2 c + a c^2 + 2 b c^2 : :

X(30818) lies on these lines:


X(30819) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^3 b^2 + a^2 b^3 - a^2 b^2 c + a^3 c^2 - a^2 b c^2 + 2 b^3 c^2 + a^2 c^3 + 2 b^2 c^3 : :

X(30819) lies on these lines:


X(30820) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^5 - 2 a^4 b + a^3 b^2 - 2 a b^4 + 2 b^5 - 2 a^4 c + 2 a^3 b c - 2 b^4 c + a^3 c^2 - 2 a c^4 - 2 b c^4 + 2 c^5 : :

X(30820) lies on these lines:


X(30821) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^3 b - a^2 b^2 + 2 a b^3 + a^3 c - 2 a^2 b c + 2 b^3 c - a^2 c^2 + 2 a c^3 + 2 b c^3 : :

X(30821) lies on these lines:


X(30822) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^3 b - a^2 b^2 + 2 a b^3 + a^3 c - 3 a^2 b c - a b^2 c + 2 b^3 c - a^2 c^2 - a b c^2 + 2 a c^3 + 2 b c^3 : :

X(30822) lies on these lines:


X(30823) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    2 a^3 - 3 a^2 b - a b^2 + 4 b^3 - 3 a^2 c + 2 a b c - 2 b^2 c - a c^2 - 2 b c^2 + 4 c^3 : :

X(30823) lies on these lines:


X(30824) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^3 - 3 a^2 b - 2 a b^2 + 2 b^3 - 3 a^2 c + 4 a b c - 4 b^2 c - 2 a c^2 - 4 b c^2 + 2 c^3 : :

X(30824) lies on these lines:


X(30825) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^4 - 2 a^3 b + a^2 b^2 - 2 a b^3 + 2 b^4 - 2 a^3 c + 2 a^2 b c - 2 b^3 c + a^2 c^2 - 2 a c^3 - 2 b c^3 + 2 c^4 : :

X(30825) lies on these lines:


X(30826) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    (a - b - c) (a^4 - a^2 b^2 - 2 a b^3 - 2 b^4 + 2 a^2 b c + 2 b^3 c - a^2 c^2 - 2 a c^3 + 2 b c^3 - 2 c^4) : :

X(30826) lies on these lines:


X(30827) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    (a - b - c) (a^2 - a b - 2 b^2 - a c + 4 b c - 2 c^2) : :

X(30827) lies on these lines:


X(30828) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^3 - 3 a^2 b - a b^2 + 3 b^3 - 3 a^2 c - b^2 c - a c^2 - b c^2 + 3 c^3 : :

X(30828) lies on these lines:


X(30829) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    b c (-5 a + b + c) : :

X(30829) lies on these lines:


X(30830) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    b c (-2 a^2 b - 2 a^2 c - a b c + b^2 c + b c^2) : :

X(30830) lies on these lines:


X(30831) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^3-a^2 b+2 b^3-a^2 c+a b c+2 c^3 : :

X(30831) lies on these lines:


X(30832) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^3 + a b^2 + 2 b^3 + 3 a b c + b^2 c + a c^2 + b c^2 + 2 c^3 : :

X(30832) lies on these lines:


X(30833) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    3 a^2 - 6 a b + 7 b^2 - 6 a c - 2 b c + 7 c^2 : :

X(30833) lies on these lines:


X(30834) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^3 - 2 a^2 b - a b^2 + 2 b^3 - 2 a^2 c - b^2 c - a c^2 - b c^2 + 2 c^3 : :

X(30834) lies on these lines:


X(30835) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    (b - c) (a^2 - 2 a b - 2 a c + 2 b c) : :

X(30835) lies on these lines:


X(30836) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    (b - c) (a^4 - a^3 b - 2 a b^3 - a^3 c - 2 a b^2 c + 2 b^3 c - 2 a b c^2 + 2 b^2 c^2 - 2 a c^3 + 2 b c^3) : :

X(30836) lies on these lines:


X(30837) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^4 - a^3 b + 2 b^4 - a^3 c - a^2 b c - a b^2 c - a b c^2 + 2 c^4 : :

X(30837) lies on these lines:


X(30838) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    3 a^5 - 5 a^4 b - 2 a^3 b^2 - 2 a^2 b^3 - a b^4 + 7 b^5 - 5 a^4 c + 6 a^2 b^2 c - b^4 c - 2 a^3 c^2 + 6 a^2 b c^2 + 2 a b^2 c^2 - 6 b^3 c^2 - 2 a^2 c^3 - 6 b^2 c^3 - a c^4 - b c^4 + 7 c^5 : :

X(30838) lies on these lines:


X(30839) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^5 - a^4 b - a^3 b^2 - a^2 b^3 + 2 b^5 - a^4 c - a^3 b c - a^2 b^2 c - a b^3 c - a^3 c^2 - a^2 b c^2 - 2 a b^2 c^2 - 2 b^3 c^2 - a^2 c^3 - a b c^3 - 2 b^2 c^3 + 2 c^5 : :

X(30839) lies on these lines:


X(30840) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^7 - a^6 b - 2 a^4 b^3 - a^3 b^4 + a^2 b^5 + 2 b^7 - a^6 c + a^2 b^4 c + 2 a^3 b^2 c^2 + 2 a^2 b^3 c^2 - 2 a^4 c^3 + 2 a^2 b^2 c^3 - 2 b^4 c^3 - a^3 c^4 + a^2 b c^4 - 2 b^3 c^4 + a^2 c^5 + 2 c^7 : :

X(30840) lies on these lines:


X(30841) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^6 - a^5 b - 2 a^4 b^2 - a^2 b^4 + a b^5 + 2 b^6 - a^5 c - 3 a^4 b c + 2 a^3 b^2 c + 2 a^2 b^3 c - a b^4 c + b^5 c - 2 a^4 c^2 + 2 a^3 b c^2 + 6 a^2 b^2 c^2 - 2 b^4 c^2 + 2 a^2 b c^3 - 2 b^3 c^3 - a^2 c^4 - a b c^4 - 2 b^2 c^4 + a c^5 + b c^5 + 2 c^6 : :

X(30841) lies on these lines:


X(30842) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^8 - a^6 b^2 - 2 a^5 b^3 - 3 a^4 b^4 + a^2 b^6 + 2 a b^7 + 2 b^8 - a^6 b c - 3 a^5 b^2 c - 2 a^4 b^3 c + a^2 b^5 c + 3 a b^6 c + 2 b^7 c - a^6 c^2 - 3 a^5 b c^2 + 2 a^4 b^2 c^2 + 8 a^3 b^3 c^2 + 3 a^2 b^4 c^2 - a b^5 c^2 - 2 a^5 c^3 - 2 a^4 b c^3 + 8 a^3 b^2 c^3 + 6 a^2 b^3 c^3 - 4 a b^4 c^3 - 2 b^5 c^3 - 3 a^4 c^4 + 3 a^2 b^2 c^4 - 4 a b^3 c^4 - 4 b^4 c^4 + a^2 b c^5 - a b^2 c^5 - 2 b^3 c^5 + a^2 c^6 + 3 a b c^6 + 2 a c^7 + 2 b c^7 + 2 c^8 : :

X(30842) lies on these lines:


X(30843) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^8 - a^7 b + a^6 b^2 + a^5 b^3 - 3 a^4 b^4 + a^3 b^5 - a^2 b^6 - a b^7 + 2 b^8 - a^7 c + 3 a^6 b c + 5 a^5 b^2 c - 3 a^4 b^3 c - 3 a^3 b^4 c + a^2 b^5 c - a b^6 c - b^7 c + a^6 c^2 + 5 a^5 b c^2 - 6 a^3 b^3 c^2 + a^2 b^4 c^2 + a b^5 c^2 - 2 b^6 c^2 + a^5 c^3 - 3 a^4 b c^3 - 6 a^3 b^2 c^3 - 2 a^2 b^3 c^3 + a b^4 c^3 + b^5 c^3 - 3 a^4 c^4 - 3 a^3 b c^4 + a^2 b^2 c^4 + a b^3 c^4 + a^3 c^5 + a^2 b c^5 + a b^2 c^5 + b^3 c^5 - a^2 c^6 - a b c^6 - 2 b^2 c^6 - a c^7 - b c^7 + 2 c^8 : :

X(30843) lies on these lines:


X(30844) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    5 a^5 - 7 a^4 b - 4 a^3 b^2 - 4 a^2 b^3 - a b^4 + 11 b^5 - 7 a^4 c + 8 a^2 b^2 c - b^4 c - 4 a^3 c^2 + 8 a^2 b c^2 + 2 a b^2 c^2 - 10 b^3 c^2 - 4 a^2 c^3 - 10 b^2 c^3 - a c^4 - b c^4 + 11 c^5 : :

X(30844) lies on these lines:


X(30845) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^5 - 3 a^4 b - a b^4 + 3 b^5 - 3 a^4 c + 2 a^3 b c + 8 a^2 b^2 c + 2 a b^3 c - b^4 c + 8 a^2 b c^2 + 6 a b^2 c^2 - 2 b^3 c^2 + 2 a b c^3 - 2 b^2 c^3 - a c^4 - b c^4 + 3 c^5 : :

X(30845) lies on these lines:


X(30846) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^5 - a^4 b + 2 b^5 - a^4 c + 2 a^2 b^2 c + 2 a^2 b c^2 + a b^2 c^2 - b^3 c^2 - b^2 c^3 + 2 c^5 : :

X(30846) lies on these lines:


X(30847) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    (a - b - c) (2 a^3 b + a^2 b^2 + b^4 + 2 a^3 c + 2 a^2 b c + 2 a b^2 c - 2 b^3 c + a^2 c^2 + 2 a b c^2 + 2 b^2 c^2 - 2 b c^3 + c^4) : :

X(30847) lies on these lines:


X(30848) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^5 - a^4 b - a^3 b^2 - a^2 b^3 + 2 b^5 - a^4 c - a^3 b c + 2 a^2 b^2 c - 2 a b^3 c - a^3 c^2 + 2 a^2 b c^2 - 2 b^3 c^2 - a^2 c^3 - 2 a b c^3 - 2 b^2 c^3 + 2 c^5 : :

X(30848) lies on these lines:


X(30849) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^5 - a^4 b - a^3 b^2 - a^2 b^3 + 2 b^5 - a^4 c + a^3 b c + 2 a b^3 c - a^3 c^2 - 2 b^3 c^2 - a^2 c^3 + 2 a b c^3 - 2 b^2 c^3 + 2 c^5 : :

X(30849) lies on these lines:


X(30850) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    3 a^2 b^2 - a b^3 - a b^2 c + 3 a^2 c^2 - a b c^2 + 4 b^2 c^2 - a c^3 : :

X(30850) lies on these lines:


X(30851) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^5 - a^3 b^2 + 2 a b^4 + 2 b^5 - a^3 b c - a^2 b^2 c + 2 a b^3 c + 2 b^4 c - a^3 c^2 - a^2 b c^2 + 2 a b c^3 + 2 a c^4 + 2 b c^4 + 2 c^5 : :

X(30851) lies on these lines:


X(30852) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^3 - 2 a^2 b - a b^2 + 2 b^3 - 2 a^2 c + 4 a b c - 2 b^2 c - a c^2 - 2 b c^2 + 2 c^3 : :

X(30852) lies on these lines:


X(30853) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^5 - a^4 b + a^3 b^2 + a^2 b^3 + 2 b^5 - a^4 c + a^2 b^2 c + a^3 c^2 + a^2 b c^2 + a b^2 c^2 + b^3 c^2 + a^2 c^3 + b^2 c^3 + 2 c^5 : :

X(30853) lies on these lines:


X(30854) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    b c (-a + b + c) (3 a^2 + b^2 - 2 b c + c^2) : :

X(30854) lies on these lines:


X(30855) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^3 - 3 a^2 b - 2 a b^2 + 2 b^3 - 3 a^2 c + 13 a b c - 4 b^2 c - 2 a c^2 - 4 b c^2 + 2 c^3 : :

X(30855) lies on these lines:


X(30856) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^5 - a^4 b - a^3 b^2 - a^2 b^3 + 2 b^5 - a^4 c + 3 a^2 b^2 c - a^3 c^2 + 3 a^2 b c^2 + a b^2 c^2 - 3 b^3 c^2 - a^2 c^3 - 3 b^2 c^3 + 2 c^5 : :

X(30856) lies on these lines:


X(30857) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^4 - 2 a^3 b + a^2 b^2 - 2 a b^3 + 2 b^4 - 2 a^3 c + 3 a^2 b c + a b^2 c - 2 b^3 c + a^2 c^2 + a b c^2 - 2 a c^3 - 2 b c^3 + 2 c^4 : :

X(30857) lies on these lines:


X(30858) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^5 - 2 a^4 b + a^3 b^2 - 2 a b^4 + 2 b^5 - 2 a^4 c + 3 a^3 b c - a^2 b^2 c + 2 a b^3 c - 2 b^4 c + a^3 c^2 - a^2 b c^2 + 2 a b c^3 - 2 a c^4 - 2 b c^4 + 2 c^5 : :

X(30858) lies on these lines:


X(30859) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^7 - a^6 b - a^5 b^2 + a^4 b^3 - 2 a^2 b^5 + 2 b^7 - a^6 c + a^4 b^2 c - a^5 c^2 + a^4 b c^2 + a^3 b^2 c^2 + a^2 b^3 c^2 - 2 b^5 c^2 + a^4 c^3 + a^2 b^2 c^3 - 2 a^2 c^5 - 2 b^2 c^5 + 2 c^7 : :

X(30859) lies on these lines:


X(30860) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    (b + c) (-2 a^4 - a b^3 + b^4 + 4 a^2 b c + a b^2 c - 2 b^3 c + a b c^2 - a c^3 - 2 b c^3 + c^4) : :

X(30860) lies on these lines:


X(30861) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^2 b + a b^2 + a^2 c - 7 a b c + 3 b^2 c + a c^2 + 3 b c^2 : :

X(30861) lies on these lines:


X(30862) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    2 a^5 - 2 a^4 b - a^3 b^2 - a^2 b^3 + 4 b^5 - 2 a^4 c + a^2 b^2 c - a^3 c^2 + a^2 b c^2 - 2 b^3 c^2 - a^2 c^3 - 2 b^2 c^3 + 4 c^5 : :

X(30862) lies on these lines:


X(30863) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^3 b^2 + a^2 b^3 - 3 a^2 b^2 c + a^3 c^2 - 3 a^2 b c^2 - a b^2 c^2 + 3 b^3 c^2 + a^2 c^3 + 3 b^2 c^3 : :

X(30863) lies on these lines:


X(30864) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    (b - c) (a^4 b - a^2 b^3 + a^4 c - 2 a^3 b c + a^2 b^2 c + a^2 b c^2 - 2 a b^2 c^2 + 2 b^3 c^2 - a^2 c^3 + 2 b^2 c^3) : :

X(30864) lies on these lines:


X(30865) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    (b - c) (a^4 - a^3 b - 2 a b^3 - a^3 c + a^2 b c - a b^2 c + 2 b^3 c - a b c^2 + 2 b^2 c^2 - 2 a c^3 + 2 b c^3) : :

X(30865) lies on these lines:


X(30866) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    b c (-a^3 + 2 a^2 b - 3 a b^2 + 2 a^2 c + a b c + b^2 c - 3 a c^2 + b c^2) : :

X(30866) lies on these lines:


X(30867) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    a^3 - a^2 b + 2 b^3 - a^2 c + 5 a b c - b^2 c - b c^2 + 2 c^3 : :

X(30867) lies on these lines:


X(30868) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    2 a^4 - 2 a^3 b - 3 a^2 b^2 + a b^3 + 4 b^4 - 2 a^3 c + a b^2 c - 3 a^2 c^2 + a b c^2 - 4 b^2 c^2 + a c^3 + 4 c^4 : :

X(30868) lies on these lines:


X(30869) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 102

Barycentrics    3 a^2 b^2 - a b^3 + a^2 b c + 3 a^2 c^2 + 4 b^2 c^2 - a c^3 : :

X(30869) lies on these lines:


X(30870) = (name pending)

Barycentrics    b^3 c^3 (b^3 - c^3) : :

X(30870) lies on these lines:


X(30871) = (name pending)

Barycentrics    b^3 c^3 (b^3 - c^3) (3a^6 + b^6 + c^6 - 2b^3 c^3) : :

X(30871) lies on these lines:


X(30872) = (name pending)

Barycentrics    b^3 c^3 (b^3 - c^3) (-a^3 + b^3 + c^3) : :

X(30872) lies on these lines:


X(30873) = (name pending)

Barycentrics    (b^3 - c^3) (-a^3 + b^3 + c^3)^2 : :

X(30873) lies on these lines:


X(30874) = (name pending)

Barycentrics    b^3 c^3 (2 a^3 - b^3 - c^3) : :

X(30874) lies on these lines:


X(30875) = (name pending)

Barycentrics    b^3 c^3 (2 b^3 c^3 - a^3 b^3 - a^3 c^3) : :

X(30875) lies on these lines:


X(30876) = (name pending)

Barycentrics    b^3 c^3 (-2 a^6 + b^6 + c^6 + a^3 b^3 + a^3 c^3 - 2 b^3 c^3) : :

X(30876) lies on these lines:


X(30877) = (name pending)

Barycentrics    b^6 + c^6 - a^3 b^3 - a^3 c^3 : :

X(30877) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 103: X(30878) - X(30937)  rightri

Extending the preambles just before X(24537), X(26153), X(27378), X(30738), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 103, as in centers X(30878)-X(30937). Then

m(X) = a^3 x - (a + c)(a^2 - a c + c^2) y - (a + b) (a^2 - a b + b^2) z : :

and m(X) is a self-inverse mapping such that m(X) is collinear with X(2) and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. The fixed points of m are X(2) and every point on the line X(824)X(30870). Among the fixed points are X(i) for these i: 2, 824, 30870, 30871, 30872, 30873, 30874, 30875, 30876, 30877. (Clark Kimberling, January 20, 2019)

underbar




X(30878) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a^2 (a^6 - a^4 b^2 + a^3 b^3 - a b^5 - a^4 c^2 + a b^3 c^2 + a^3 c^3 + a b^2 c^3 + 2 b^3 c^3 - a c^5) : :

X(30878) lies on these lines:


X(30879) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a^7 - a^3 b^4 + 2 a^3 b^2 c^2 + a^2 b^3 c^2 + b^5 c^2 + a^2 b^2 c^3 - b^4 c^3 - a^3 c^4 - b^3 c^4 + b^2 c^5 : :

X(30879) lies on these lines:


X(30880) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    3 a^7 - 2 a^5 b^2 + a^4 b^3 - a^3 b^4 - 2 a^2 b^5 + b^7 - 2 a^5 c^2 + 2 a^3 b^2 c^2 + a^4 c^3 - b^4 c^3 - a^3 c^4 - b^3 c^4 - 2 a^2 c^5 + c^7 : :

X(30880) lies on these lines:


X(30881) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    2 a^7 - 2 a^5 b^2 + a^4 b^3 - 2 a^2 b^5 + b^7 - 2 a^5 c^2 - a^2 b^3 c^2 - b^5 c^2 + a^4 c^3 - a^2 b^2 c^3 - 2 a^2 c^5 - b^2 c^5 + c^7 : :

X(30881) lies on these lines:


X(30882) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a^5 - a^3 b^2 - a^3 c^2 - b^3 c^2 - b^2 c^3 : :

X(30882) lies on these lines:


X(30883) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    3 a^5-2 a^4 b-a^3 b^2+a^2 b^3-2 a b^4+b^5-2 a^4 c+2 a^3 b c-a^3 c^2-b^3 c^2+a^2 c^3-b^2 c^3-2 a c^4+c^5 : :

X(30883) lies on these lines:


X(30884) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    3 a^4 - a^3 b + a b^3 + b^4 - a^3 c - b^3 c + a c^3 - b c^3 + c^4 : :

X(30884) lies on these lines:


X(30885) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a^5 - a^3 b^2 + 2 a^3 b c + a b^3 c + b^4 c - a^3 c^2 - b^3 c^2 + a b c^3 - b^2 c^3 + b c^4 : :

X(30885) lies on these lines:


X(30886) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    2 a^4 + a b^3 + b^4 + a c^3 + c^4 : :

X(30886) lies on these lines:


X(30887) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    2 a^6 - 2 a^5 b + a^3 b^3 - a^2 b^4 - a b^5 + b^6 - 2 a^5 c + 2 a^4 b c - a^2 b^3 c + 2 a b^4 c - b^5 c + a^3 c^3 - a^2 b c^3 - a^2 c^4 + 2 a b c^4 - a c^5 - b c^5 + c^6 : :

X(30887) lies on these lines:


X(30888) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a^9 + a^7 b^2 - a^5 b^4 - a^3 b^6 + a^7 c^2 + a^4 b^3 c^2 + a^3 b^4 c^2 + b^7 c^2 + a^4 b^2 c^3 - b^6 c^3 - a^5 c^4 + a^3 b^2 c^4 - a^3 c^6 - b^3 c^6 + b^2 c^7 : :

X(30888) lies on these lines:


X(30889) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a^9 + a^7 b^2 - a^5 b^4 - a^3 b^6 + a^7 c^2 - a^5 b^2 c^2 + a^4 b^3 c^2 + a^3 b^4 c^2 - a^2 b^5 c^2 + b^7 c^2 + a^4 b^2 c^3 - b^6 c^3 - a^5 c^4 + a^3 b^2 c^4 - a^2 b^2 c^5 - a^3 c^6 - b^3 c^6 + b^2 c^7 : :

X(30889) lies on these lines:


X(30890) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a^6 - a^3 b^3 - a^3 c^3 - 2 b^3 c^3 : :

X(30890) lies on these lines:


X(30891) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a^7 - a^3 b^4 - b^4 c^3 - a^3 c^4 - b^3 c^4 : :

X(30891) lies on these lines:


X(30892) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    b c (2 a^3 + a b^2 + b^3 + a c^2 + c^3) : :

X(30892) lies on these lines:


X(30893) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    b^2 c^2 (2 a^3 + a^2 b + b^3 + a^2 c + c^3) : :

X(30893) lies on these lines:


X(30894) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a (a^6 - a^5 b + a^3 b^3 - a^2 b^4 - a^5 c + a^2 b^3 c + a^3 c^3 + a^2 b c^3 + 2 b^3 c^3 - a^2 c^4) : :

X(30894) lies on these lines:


X(30895) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a^5 b - a^4 b^2 + a^5 c - a^3 b^2 c - a^4 c^2 - a^3 b c^2 - a b^3 c^2 - b^4 c^2 - a b^2 c^3 - b^2 c^4 : :

X(30895) lies on these lines:


X(30896) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a^5 b - a^4 b^2 + a^5 c + a^4 b c - a^3 b^2 c + a b^4 c - a^4 c^2 - a^3 b c^2 - a b^3 c^2 - b^4 c^2 - a b^2 c^3 + a b c^4 - b^2 c^4 : :

X(30896) lies on these lines:


X(30897) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    2 a^5 - 2 a^3 b^2 + 2 a^3 b c + a b^3 c + b^4 c - 2 a^3 c^2 - 2 b^3 c^2 + a b c^3 - 2 b^2 c^3 + b c^4 : :

X(30897) lies on these lines:


X(30898) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a^5 - a^3 b^2 + 4 a^3 b c + 2 a b^3 c + 2 b^4 c - a^3 c^2 - b^3 c^2 + 2 a b c^3 - b^2 c^3 + 2 b c^4 : :

X(30898) lies on these lines:


X(30899) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a^6 - a^5 b + a^4 b^2 - a^3 b^3 - a^5 c + a^3 b^2 c + a^4 c^2 + a^3 b c^2 + a b^3 c^2 + b^4 c^2 - a^3 c^3 + a b^2 c^3 - 2 b^3 c^3 + b^2 c^4 : :

X(30899) lies on these lines:


X(30900) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a^7 - a^3 b^4 + 2 a^5 b c - 2 a^4 b^2 c - 2 a^4 b c^2 + 2 a^3 b^2 c^2 + a^2 b^3 c^2 - 2 a b^4 c^2 + b^5 c^2 + a^2 b^2 c^3 - b^4 c^3 - a^3 c^4 - 2 a b^2 c^4 - b^3 c^4 + b^2 c^5 : :

X(30900) lies on these lines:


X(30901) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a^6 + a^5 b - a^4 b^2 - a^3 b^3 + a^5 c - 2 a^4 b c + a^3 b^2 c + a^2 b^3 c - 2 a b^4 c + b^5 c - a^4 c^2 + a^3 b c^2 - a^3 c^3 + a^2 b c^3 - 2 b^3 c^3 - 2 a b c^4 + b c^5 : :

X(30901) lies on these lines:


X(30902) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    3 a^5 - a^3 b^2 + a^2 b^3 + b^5 - a^3 c^2 - b^3 c^2 + a^2 c^3 - b^2 c^3 + c^5 : :

X(30902) lies on these lines:


X(30903) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a (a^3 b + b^4 + a^3 c - a^2 b c + c^4) : :

X(30903) lies on these lines:


X(30904) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a^2 (a^3 b^2 + b^5 + a^3 c^2 - a b^2 c^2 + c^5) : :

X(30904) lies on these lines:


X(30905) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    (a + b) (a + c) (a^4 - a^2 b^2 - 2 a^2 b c + 2 a b^2 c - b^3 c - a^2 c^2 + 2 a b c^2 - 2 b^2 c^2 - b c^3) : :

X(30905) lies on these lines:


X(30906) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    (a + b) (a + c) (a^3 - 2 a^2 b + a b^2 - b^3 - 2 a^2 c + 2 a b c - b^2 c + a c^2 - b c^2 - c^3) : :

X(30906) lies on these lines:


X(30907) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    5 a^4 - 3 a^3 b + a b^3 + b^4 - 3 a^3 c - 3 b^3 c + a c^3 - 3 b c^3 + c^4 : :

X(30907) lies on these lines:


X(30908) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a^5 + a^4 b - a^3 b^2 + a b^4 + a^4 c - a^3 b c - a^3 c^2 - b^3 c^2 - b^2 c^3 + a c^4 : :

X(30908) lies on these lines:


X(30909) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    (b - c) (a^5 + a^4 b + a^4 c - a^3 b c - a b^2 c^2 + b^3 c^2 + b^2 c^3) : :

X(30909) lies on these lines:


X(30910) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    b c (b - c) (2 a^3 - a^2 b + b^3 - a^2 c + a b c + c^3) : :

X(30910) lies on these lines:


X(30911) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    (b - c) (a^5 + a^4 b + a^4 c + a^3 b c - a^2 b^2 c + b^4 c - a^2 b c^2 + b^3 c^2 + b^2 c^3 + b c^4) : :

X(30911) lies on these lines:


X(30912) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    (b - c) (a^6 + a^4 b^2 + a^4 b c - a^3 b^2 c + a^4 c^2 - a^3 b c^2 + b^3 c^3) : :

X(30912) lies on these lines:


X(30913) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a (b - c) (a^4 - a^3 b + a b^3 - b^4 - a^3 c - a^2 b c + a b^2 c - b^3 c + a b c^2 - b^2 c^2 + a c^3 - b c^3 - c^4) : :

X(30913) lies on these lines:


X(30914) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    (a^2 - b c) (a^4 - a b^3 + 2 a^2 b c + 2 b^2 c^2 - a c^3) : :

X(30914) lies on these lines:


X(30915) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    2 a^3 b + a b^3 + b^4 + 2 a^3 c + 2 b^3 c + a c^3 + 2 b c^3 + c^4 : :

X(30915) lies on these lines:


X(30916) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    5 a^7 - 2 a^5 b^2 + a^4 b^3 - 3 a^3 b^4 - 2 a^2 b^5 + b^7 - 2 a^5 c^2 + 6 a^3 b^2 c^2 + 2 a^2 b^3 c^2 + 2 b^5 c^2 + a^4 c^3 + 2 a^2 b^2 c^3 - 3 b^4 c^3 - 3 a^3 c^4 - 3 b^3 c^4 - 2 a^2 c^5 + 2 b^2 c^5 + c^7 : :

X(30916) lies on these lines:


X(30917) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    (a + b) (a + c) (a^5 - a^4 b + a^3 b^2 - a^2 b^3 - a^4 c + 2 a^3 b c - a^2 b^2 c + a b^3 c + b^4 c + a^3 c^2 - a^2 b c^2 + 2 a b^2 c^2 - b^3 c^2 - a^2 c^3 + a b c^3 - b^2 c^3 + b c^4) : :

X(30917) lies on these lines:


X(30918) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a^9 + a^7 b^2 - a^5 b^4 - a^3 b^6 + a^7 c^2 - 2 a^5 b^2 c^2 + a^4 b^3 c^2 + a^3 b^4 c^2 - 2 a^2 b^5 c^2 + b^7 c^2 + a^4 b^2 c^3 - b^6 c^3 - a^5 c^4 + a^3 b^2 c^4 - 2 a^2 b^2 c^5 - a^3 c^6 - b^3 c^6 + b^2 c^7 : :

X(30918) lies on these lines:


X(30919) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    (a + b) (a + c) (a^7 + 2 a^6 b - a^5 b^2 - a^4 b^3 + a^3 b^4 - 2 a^2 b^5 - a b^6 + b^7 + 2 a^6 c - 2 a^5 b c - a^4 b^2 c + 2 a^3 b^3 c - 2 a^2 b^4 c + b^6 c - a^5 c^2 - a^4 b c^2 + 2 a^3 b^2 c^2 + a b^4 c^2 - b^5 c^2 - a^4 c^3 + 2 a^3 b c^3 - b^4 c^3 + a^3 c^4 - 2 a^2 b c^4 + a b^2 c^4 - b^3 c^4 - 2 a^2 c^5 - b^2 c^5 - a c^6 + b c^6 + c^7) : :

X(30919) lies on these lines:


X(30920) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    (a + b) (a + c) (a^8 + a^6 b^2 - a^4 b^4 - a^2 b^6 + 2 a^6 b c - 2 a^5 b^2 c - a^4 b^3 c + 2 a^3 b^4 c - 2 a^2 b^5 c + b^7 c + a^6 c^2 - 2 a^5 b c^2 + 2 a^3 b^3 c^2 - a^2 b^4 c^2 - a^4 b c^3 + 2 a^3 b^2 c^3 - b^5 c^3 - a^4 c^4 + 2 a^3 b c^4 - a^2 b^2 c^4 - 2 a^2 b c^5 - b^3 c^5 - a^2 c^6 + b c^7) : :

X(30920) lies on these lines:


X(30921) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    (a + b) (a + c) (a^8 - 3 a^7 b + a^6 b^2 + 2 a^5 b^3 - 2 a^4 b^4 + a^3 b^5 + a^2 b^6 - b^8 - 3 a^7 c + 2 a^6 b c - 2 a^4 b^3 c + 3 a^3 b^4 c + a^6 c^2 - a^2 b^4 c^2 + 2 a^5 c^3 - 2 a^4 b c^3 - 2 a^4 c^4 + 3 a^3 b c^4 - a^2 b^2 c^4 + 2 b^4 c^4 + a^3 c^5 + a^2 c^6 - c^8) : :

X(30921) lies on these lines:


X(30922) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    7 a^7 - 2 a^5 b^2 + a^4 b^3 - 5 a^3 b^4 - 2 a^2 b^5 + b^7 - 2 a^5 c^2 + 10 a^3 b^2 c^2 + 4 a^2 b^3 c^2 + 4 b^5 c^2 + a^4 c^3 + 4 a^2 b^2 c^3 - 5 b^4 c^3 - 5 a^3 c^4 - 5 b^3 c^4 - 2 a^2 c^5 + 4 b^2 c^5 + c^7 : :

X(30922) lies on these lines:


X(30923) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    3 a^7 - 2 a^5 b^2 + a^4 b^3 - a^3 b^4 - 2 a^2 b^5 + b^7 - 2 a^5 b c - 2 a^4 b^2 c - 2 a^2 b^4 c - 2 a b^5 c - 2 a^5 c^2 - 2 a^4 b c^2 + 2 a^3 b^2 c^2 - 2 a b^4 c^2 + a^4 c^3 - b^4 c^3 - a^3 c^4 - 2 a^2 b c^4 - 2 a b^2 c^4 - b^3 c^4 - 2 a^2 c^5 - 2 a b c^5 + c^7 : :

X(30923) lies on these lines:


X(30924) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a^7 - a^5 b^2 - a^3 b^4 - a^2 b^5 - a^5 c^2 + a^3 b^2 c^2 - b^4 c^3 - a^3 c^4 - b^3 c^4 - a^2 c^5 : :

X(30924) lies on these lines:


X(30925) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a^5 b - a^4 b^2 + a^5 c + a^3 b^2 c + a^2 b^3 c + b^5 c - a^4 c^2 + a^3 b c^2 + a^2 b c^3 + b c^5 : :

X(30925) lies on these lines:


X(30926) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a^6 + a^5 b - a^4 b^2 - a^3 b^3 + a^5 c + a^3 b^2 c + a^2 b^3 c + b^5 c - a^4 c^2 + a^3 b c^2 - a^3 c^3 + a^2 b c^3 - 2 b^3 c^3 + b c^5 : :

X(30926) lies on these lines:


X(30927) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    (a + b) (a + c) (a^5 - a^4 b - a b^4 - a^4 c + a^3 b c + b^4 c + 2 a b^2 c^2 - b^3 c^2 - b^2 c^3 - a c^4 + b c^4) : :

X(30927) lies on these lines:


X(30928) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a^6 - a^5 b + a^4 b^2 - a^3 b^3 - a^5 c - a^4 b c + a^3 b^2 c - a b^4 c + a^4 c^2 + a^3 b c^2 + a b^3 c^2 + b^4 c^2 - a^3 c^3 + a b^2 c^3 - 2 b^3 c^3 - a b c^4 + b^2 c^4 : :

X(30928) lies on these lines:


X(30929) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    2 a^7 - 2 a^5 b^2 + a^4 b^3 - 2 a^2 b^5 + b^7 - 2 a^5 c^2 + 2 a^3 b^2 c^2 + a^4 c^3 - 2 a^2 c^5 + c^7 : :

X(30929) lies on these lines:


X(30930) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a^5 - a^4 b - a^3 b^2 - a b^4 - a^4 c + 3 a^3 b c + a b^3 c + b^4 c - a^3 c^2 - b^3 c^2 + a b c^3 - b^2 c^3 - a c^4 + b c^4 : :

X(30930) lies on these lines:


X(30931) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a^4 b + a b^4 + a^4 c - 3 a^3 b c - a b^3 c - b^4 c - a b c^3 + a c^4 - b c^4 : :

X(30931) lies on these lines:


X(30932) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    2 a^7 - 2 a^3 b^4 + 2 a^3 b^2 c^2 + a^2 b^3 c^2 + b^5 c^2 + a^2 b^2 c^3 - 2 b^4 c^3 - 2 a^3 c^4 - 2 b^3 c^4 + b^2 c^5 : :

X(30932) lies on these lines:


X(30933) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a^5 b^2 + a^2 b^5 + a^5 c^2 - 3 a^3 b^2 c^2 - a^2 b^3 c^2 - b^5 c^2 - a^2 b^2 c^3 + a^2 c^5 - b^2 c^5 : :

X(30933) lies on these lines:


X(30934) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    (b - c) (a^7 b + a^5 b^3 + a^7 c + a^5 b^2 c + a^5 b c^2 - a^3 b^3 c^2 + a^5 c^3 - a^3 b^2 c^3 + a^2 b^3 c^3 + b^4 c^4) : :

X(30934) lies on these lines:


X(30935) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a^5 - a^4 b - a^3 b^2 - a b^4 - a^4 c + a^3 b c - a^3 c^2 - b^3 c^2 - b^2 c^3 - a c^4 : :

X(30935) lies on these lines:


X(30936) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a^5 b - a^4 b^2 + a^5 c + a^4 b c + a^3 b^2 c + a^2 b^3 c + a b^4 c + b^5 c - a^4 c^2 + a^3 b c^2 + a^2 b c^3 + a b c^4 + b c^5 : :

X(30936) lies on these lines:


X(30937) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = GEMINI TRIANGLE 103

Barycentrics    a^6 + 2 a^5 b - a^3 b^3 + a^2 b^4 + a b^5 + 2 a^5 c + a^2 b^3 c + b^5 c - a^3 c^3 + a^2 b c^3 - 2 b^3 c^3 + a^2 c^4 + a c^5 + b c^5 : :

X(30937) lies on these lines:


X(30938) = (name pending)

Barycentrics    b c (a + b)(a + c)(a^2 b^2 + a^2 c^2 - 2 b^2 c^2) : :

X(30938) lies on these lines:


X(30939) = (name pending)

Barycentrics    b c (a + b)(a + c)(2 a - b - c) : :

X(30939) lies on these lines:


X(30940) = (name pending)

Barycentrics    b c (a + b)(a + c)(a^2 - b c) : :

X(30940) lies on these lines:


X(30941) = (name pending)

Barycentrics    (a + b)(a + c)(b^2 + c^2 - a b - a c) : :

X(30941) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 104: X(30942) - X(31007)  rightri

Extending the preambles just before X(24537), X(26153), X(27378), X(30738), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 104, as in centers X(30942)-X(31007). Then

m(X) = - b c x + b (a + c) y + c (a + b) z : :

and m(X) is a self-inverse mapping such that m(X) is collinear with X(2) and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. The fixed points of m are X(2) and every point on the line X(513)X(693). Among the fixed points are X(i) for these i: 2, 513, 693, 3250, 7912, 30938, 30939, 30940, 30941, 31008. (Clark Kimberling, January 20, 2019)

underbar




X(30942) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a b^2 - a b c + b^2 c + a c^2 + b c^2 : :

X(30942) lies on these lines:


X(30943) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a^5 b - a b^5 + a^5 c + 3 a^4 b c - 2 a^3 b^2 c - 2 a^2 b^3 c + a b^4 c - b^5 c - 2 a^3 b c^2 - 2 a^2 b c^3 + 2 b^3 c^3 + a b c^4 - a c^5 - b c^5 : :

X(30943) lies on these lines:


X(30944) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a (a^4 b - a^2 b^3 + a^4 c + 2 a^3 b c - 2 a^2 b^2 c - 2 a b^3 c + b^4 c - 2 a^2 b c^2 - b^3 c^2 - a^2 c^3 - 2 a b c^3 - b^2 c^3 + b c^4) : :

X(30944) lies on these lines:


X(30945) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a b^3 - a^2 b c + b^3 c + a c^3 + b c^3 : :

X(30945) lies on these lines:


X(30946) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a^3 b - a b^3 + a^3 c - a^2 b c - a b^2 c - b^3 c - a b c^2 + 2 b^2 c^2 - a c^3 - b c^3 : :

X(30946) lies on these lines:


X(30947) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a^2 b - a b^2 + a^2 c + 5 a b c - b^2 c - a c^2 - b c^2 : :

X(30947) lies on these lines:


X(30948) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a^2 b - 3 a b^2 + a^2 c + 7 a b c - 3 b^2 c - 3 a c^2 - 3 b c^2 : :

X(30948) lies on these lines:


X(30949) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    -a^2 b^2 + a b^3 - a^2 b c - a b^2 c + b^3 c - a^2 c^2 - a b c^2 - 2 b^2 c^2 + a c^3 + b c^3 : :

X(30949) lies on these lines:


X(30950) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a (a b + a c + 4 b c) : :

X(30950) lies on these lines:


X(30951) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    -a^5 b^3 + a b^7 - a^6 b c - a^4 b^3 c + a^2 b^5 c + b^7 c - a^5 c^3 - a^4 b c^3 - a b^4 c^3 - b^5 c^3 - a b^3 c^4 + a^2 b c^5 - b^3 c^5 + a c^7 + b c^7 : :

X(30951) lies on these lines:


X(30952) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    -a^5 b^3 + a b^7 - a^6 b c - a^4 b^3 c + a^2 b^5 c + b^7 c + a^3 b^3 c^2 - a^5 c^3 - a^4 b c^3 + a^3 b^2 c^3 + a^2 b^3 c^3 - a b^4 c^3 - b^5 c^3 - a b^3 c^4 + a^2 b c^5 - b^3 c^5 + a c^7 + b c^7 : :

X(30952) lies on these lines:


X(30953) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a b^4 - a^3 b c + b^4 c + a c^4 + b c^4 : :

X(30953) lies on these lines:


X(30954) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a b^5 - a^4 b c + b^5 c + a c^5 + b c^5 : :

X(30954) lies on these lines:


X(30955) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a^3 b^3 + a b^3 c^2 + a^3 c^3 + a b^2 c^3 + 2 b^3 c^3 : :

X(30955) lies on these lines:


X(30956) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    -a^2 b^4 + a b^5 - a^4 b c + a^3 b^2 c - 2 a b^4 c + b^5 c + a^3 b c^2 - b^4 c^2 - a^2 c^4 - 2 a b c^4 - b^2 c^4 + a c^5 + b c^5 : :

X(30956) lies on these lines:


X(30957) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a b^2 - 3 a b c + b^2 c + a c^2 + b c^2 : :

X(30957) lies on these lines:


X(30958) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    2 a^2 b^2 - a b^3 + a^2 b c + 2 a b^2 c - b^3 c + 2 a^2 c^2 + 2 a b c^2 + 4 b^2 c^2 - a c^3 - b c^3 : :

X(30958) lies on these lines:


X(30959) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    -a^2 b^3 + a b^4 - a^3 b c + a^2 b^2 c - 2 a b^3 c + b^4 c + a^2 b c^2 - b^3 c^2 - a^2 c^3 - 2 a b c^3 - b^2 c^3 + a c^4 + b c^4 : :

X(30959) lies on these lines:


X(30960) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    -a^3 b^3 + a b^5 - a^4 b c + 2 a^2 b^3 c + b^5 c - 2 a^2 b^2 c^2 + a b^3 c^2 - a^3 c^3 + 2 a^2 b c^3 + a b^2 c^3 - 2 b^3 c^3 + a c^5 + b c^5 : :

X(30960) lies on these lines:


X(30961) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    -a^3 b^2 + a b^4 - a^3 b c + a^2 b^2 c + a b^3 c + b^4 c - a^3 c^2 + a^2 b c^2 - b^3 c^2 + a b c^3 - b^2 c^3 + a c^4 + b c^4 : :

X(30961) lies on these lines:


X(30962) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a^3 b - a b^3 + a^3 c + 3 a^2 b c + a b^2 c - b^3 c + a b c^2 - a c^3 - b c^3 : :

X(30962) lies on these lines:


X(30963) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    b c (-2 a^2 - a b - a c + b c) : :

X(30963) lies on these lines:


X(30964) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    b c (-a^3 b - a^2 b^2 - a^3 c - a^2 c^2 + b^2 c^2): :

X(30964) lies on these lines:


X(30965) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    (a + b) (a + c) (-b^3 + a b c - b^2 c - b c^2 - c^3) : :

X(30965) lies on these lines:


X(30966) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    (a + b) (a + c) (b^2 + b c + c^2) : :

X(30966) lies on these lines:


X(30967) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a b^3 - 3 a^2 b c - a b^2 c + b^3 c - a b c^2 + b^2 c^2 + a c^3 + b c^3 : :

X(30967) lies on these lines:


X(30968) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    (b - c) (a^2 b^3 + a^3 b c + a^2 b^2 c + a^2 b c^2 - b^3 c^2 + a^2 c^3 - b^2 c^3) : :

X(30968) lies on these lines:


X(30969) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a b^4 - a^3 b c - a^2 b^2 c + b^4 c - a^2 b c^2 - a b^2 c^2 + a c^4 + b c^4 : :

X(30969) lies on these lines:


X(30970) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a^2 b + 2 a b^2 + a^2 c + 2 a b c + 2 b^2 c + 2 a c^2 + 2 b c^2 : :

X(30970) lies on these lines:


X(30971) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a^5 b + 2 a^3 b^3 - 3 a b^5 + a^5 c + 5 a^4 b c - 2 a^3 b^2 c - 2 a^2 b^3 c + a b^4 c - 3 b^5 c - 2 a^3 b c^2 + 2 a b^3 c^2 + 2 a^3 c^3 - 2 a^2 b c^3 + 2 a b^2 c^3 + 6 b^3 c^3 + a b c^4 - 3 a c^5 - 3 b c^5 : :

X(30971) lies on these lines:


X(30972) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    -a^5 b^3 + a b^7 - a^6 b c - a^4 b^3 c + a^2 b^5 c + b^7 c + 2 a^3 b^3 c^2 - a^5 c^3 - a^4 b c^3 + 2 a^3 b^2 c^3 + 2 a^2 b^3 c^3 - a b^4 c^3 - b^5 c^3 - a b^3 c^4 + a^2 b c^5 - b^3 c^5 + a c^7 + b c^7 : :

X(30972) lies on these lines:


X(30973) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    (a + b) (a + c) (a^4 b^2 - b^6 + 3 a^4 b c - 2 a^2 b^3 c - b^5 c + a^4 c^2 - 4 a^2 b^2 c^2 + b^4 c^2 - 2 a^2 b c^3 + 2 b^3 c^3 + b^2 c^4 - b c^5 - c^6) : :

X(30973) lies on these lines:


X(30974) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    (a + b) (a + c) (a^4 b^3 - b^7 + a^5 b c + a^4 b^2 c - a b^5 c - b^6 c + a^4 b c^2 - 4 a^2 b^3 c^2 + b^5 c^2 + a^4 c^3 - 4 a^2 b^2 c^3 + 2 a b^3 c^3 + b^4 c^3 + b^3 c^4 - a b c^5 + b^2 c^5 - b c^6 - c^7) : :

X(30974) lies on these lines:


X(30975) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    (a + b) (a + c) (a^5 b^2 - a^4 b^3 - a b^6 + b^7 + a^5 b c + 2 a^4 b^2 c - 2 a^3 b^3 c - 2 a^2 b^4 c + a b^5 c + a^5 c^2 + 2 a^4 b c^2 - 4 a^3 b^2 c^2 + 2 a^2 b^3 c^2 + a b^4 c^2 - 2 b^5 c^2 - a^4 c^3 - 2 a^3 b c^3 + 2 a^2 b^2 c^3 - 2 a b^3 c^3 + b^4 c^3 - 2 a^2 b c^4 + a b^2 c^4 + b^3 c^4 + a b c^5 - 2 b^2 c^5 - a c^6 + c^7) : :

X(30975) lies on these lines:


X(30976) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a^5 b + 4 a^3 b^3 - 5 a b^5 + a^5 c + 7 a^4 b c - 2 a^3 b^2 c - 2 a^2 b^3 c + a b^4 c - 5 b^5 c - 2 a^3 b c^2 + 4 a b^3 c^2 + 4 a^3 c^3 - 2 a^2 b c^3 + 4 a b^2 c^3 + 10 b^3 c^3 + a b c^4 - 5 a c^5 - 5 b c^5 : :

X(30976) lies on these lines:


X(30977) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a^5 b - a b^5 + a^5 c + 3 a^4 b c - 4 a^3 b^2 c - 4 a^2 b^3 c + a b^4 c - b^5 c - 4 a^3 b c^2 - 6 a^2 b^2 c^2 - 2 a b^3 c^2 - 4 a^2 b c^3 - 2 a b^2 c^3 + 2 b^3 c^3 + a b c^4 - a c^5 - b c^5 : :

X(30977) lies on these lines:


X(30978) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a b^5 - a^4 b c + a^3 b^2 c + a^2 b^3 c + b^5 c + a^3 b c^2 + a^2 b c^3 - b^3 c^3 + a c^5 + b c^5 : :

X(30978) lies on these lines:


X(30979) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a (a^4 b - a^2 b^3 + a^4 c + 2 a^3 b c - 2 a^2 b^2 c - 4 a b^3 c + b^4 c - 2 a^2 b c^2 + 2 a b^2 c^2 - 3 b^3 c^2 - a^2 c^3 - 4 a b c^3 - 3 b^2 c^3 + b c^4) : :

X(30979) lies on these lines:


X(30980) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    -a^3 b^3 + a b^5 - a^4 b c - a^2 b^3 c + b^5 c + a^2 b^2 c^2 - 2 a b^3 c^2 - a^3 c^3 - a^2 b c^3 - 2 a b^2 c^3 - 2 b^3 c^3 + a c^5 + b c^5 : :

X(30980) lies on these lines:


X(30981) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    -a^3 b^3 + a b^5 - a^4 b c + a^2 b^3 c + b^5 c - a^2 b^2 c^2 - a^3 c^3 + a^2 b c^3 - 2 b^3 c^3 + a c^5 + b c^5 : :

X(30981) lies on these lines:


X(30982) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a^3 b^2 + a^2 b^2 c - a b^3 c + a^3 c^2 + a^2 b c^2 + 2 a b^2 c^2 + b^3 c^2 - a b c^3 + b^2 c^3 : :

X(30982) lies on these lines:


X(30983) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    -a^3 b^4 + a b^6 - a^5 b c + a^3 b^3 c - a^2 b^4 c + b^6 c - a b^4 c^2 + a^3 b c^3 - b^4 c^3 - a^3 c^4 - a^2 b c^4 - a b^2 c^4 - b^3 c^4 + a c^6 + b c^6 : :

X(30983) lies on these lines:


X(30984) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    (a + b) (a + c) (-b^4 + a^2 b c - b^3 c - b c^3 - c^4) : :

X(30984) lies on these lines:


X(30985) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    -a^3 b^2 + a b^4 - a^3 b c - a^2 b^2 c + a b^3 c + b^4 c - a^3 c^2 - a^2 b c^2 - 2 a b^2 c^2 - b^3 c^2 + a b c^3 - b^2 c^3 + a c^4 + b c^4 : :

X(30985) lies on these lines:


X(30986) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a (a^3 b^2 - a b^4 + a^2 b^2 c - 3 b^4 c + a^3 c^2 + a^2 b c^2 - 4 a b^2 c^2 + b^3 c^2 + b^2 c^3 - a c^4 - 3 b c^4) : :

X(30986) lies on these lines:


X(30987) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a^3 b^3 + a b^5 - a^4 b c + a^3 b^2 c + a^2 b^3 c + b^5 c + a^3 b c^2 + a b^3 c^2 + a^3 c^3 + a^2 b c^3 + a b^2 c^3 + b^3 c^3 + a c^5 + b c^5 : :

X(30987) lies on these lines:


X(30988) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    b c (-2 a^4 + a^3 b + 2 a^2 b^2 - a b^3 + a^3 c - a^2 b c + a b^2 c + b^3 c + 2 a^2 c^2 + a b c^2 - 2 b^2 c^2 - a c^3 + b c^3) : :

X(30988) lies on these lines:


X(30989) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a^3 b^4 - a^3 b^3 c - a^2 b^4 c - a^2 b^3 c^2 - a b^4 c^2 - a^3 b c^3 - a^2 b^2 c^3 + a b^3 c^3 + b^4 c^3 + a^3 c^4 - a^2 b c^4 - a b^2 c^4 + b^3 c^4 : :

X(30989) lies on these lines:


X(30990) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    2 a^3 b^2 + a^2 b^3 - a b^4 + a^3 b c - 4 a^2 b^2 c - b^4 c + 2 a^3 c^2 - 4 a^2 b c^2 - a b^2 c^2 + 3 b^3 c^2 + a^2 c^3 + 3 b^2 c^3 - a c^4 - b c^4 : :

X(30990) lies on these lines:


X(30991) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    -2 a^2 b + 2 a b^2 + 4 b^3 - 2 a^2 c + a b c + 2 a c^2 + 4 c^3 : :

X(30991) lies on these lines:


X(30992) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    (a + b) (a + c) (b^2 + a c - b c - c^2) (a b - b^2 - b c + c^2) : :

X(30992) lies on these lines:


X(30993) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    -a^2 b^3 + a b^4 - a^3 b c + 2 a^2 b^2 c - 2 a b^3 c + b^4 c + 2 a^2 b c^2 + a b^2 c^2 - b^3 c^2 - a^2 c^3 - 2 a b c^3 - b^2 c^3 + a c^4 + b c^4 : :

X(30993) lies on these lines:


X(30994) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    -a^2 b^4 + a b^5 - a^4 b c + a^3 b^2 c + a^2 b^3 c - 2 a b^4 c + b^5 c + a^3 b c^2 - a^2 b^2 c^2 + a b^3 c^2 - b^4 c^2 + a^2 b c^3 + a b^2 c^3 - a^2 c^4 - 2 a b c^4 - b^2 c^4 + a c^5 + b c^5 : :

X(30994) lies on these lines:


X(30995) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    (a + b) (a + c) (a b^5 - b^6 + a^4 b c - a^3 b^2 c - a^3 b c^2 + a^2 b^2 c^2 - a b^3 c^2 + b^4 c^2 - a b^2 c^3 + b^2 c^4 + a c^5 - c^6) : :

X(30995) lies on these lines:


X(30996) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a^5 b + a^5 c + 2 a^4 b c - 2 a^3 b^2 c - 2 a^2 b^3 c + a b^4 c - 2 a^3 b c^2 - 2 a^2 b c^3 + 2 b^3 c^3 + a b c^4 : :

X(30996) lies on these lines:


X(30997) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    -a^2 b^2 + a b^3 + a^2 b c + b^3 c - a^2 c^2 - 3 b^2 c^2 + a c^3 + b c^3 : :

X(30997) lies on these lines:


X(30998) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a^2 b^2 - 2 a^2 b c + a^2 c^2 + 3 b^2 c^2 : :

X(30998) lies on these lines:


X(30999) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    -a^3 b^3 + 2 a b^5 - 2 a^4 b c + 2 b^5 c - a b^3 c^2 - a^3 c^3 - a b^2 c^3 - 2 b^3 c^3 + 2 a c^5 + 2 b c^5 : :

X(30999) lies on these lines:


X(31000) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a^3 b^3 - a^3 b^2 c - a^2 b^3 c - a^3 b c^2 + a b^3 c^2 + a^3 c^3 - a^2 b c^3 + a b^2 c^3 + 3 b^3 c^3 : :

X(31000) lies on these lines:


X(31001) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    (a + b) (a + c) (a b^4 - b^5 + a^3 b c - a^2 b^2 c - a^2 b c^2 - a b^2 c^2 + b^3 c^2 + b^2 c^3 + a c^4 - c^5) : :

X(31001) lies on these lines:


X(31002) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    b c (a b - 2 a c + b c) (-2 a b + a c + b c) : :

X(31002) lies on these lines:


X(31003) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    (b - c) (a^2 b^3 + a^3 b c + a b^2 c^2 - b^3 c^2 + a^2 c^3 - b^2 c^3) : :

X(31003) lies on these lines:


X(31004) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a b^3 + a^2 b c + a b^2 c + b^3 c + a b c^2 - b^2 c^2 + a c^3 + b c^3 : :

X(31004) lies on these lines:


X(31005) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a^3 b^2 + 2 a^2 b^2 c - a b^3 c + a^3 c^2 + 2 a^2 b c^2 + 3 a b^2 c^2 + b^3 c^2 - a b c^3 + b^2 c^3 : :

X(31005) lies on these lines:


X(31006) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    a^3 b^2 - a b^4 + 3 a^3 b c + 3 a^2 b^2 c - b^4 c + a^3 c^2 + 3 a^2 b c^2 + 2 a b^2 c^2 - a c^4 - b c^4 : :

X(31006) lies on these lines:


X(31007) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(26), WHERE A'B'C' = GEMINI TRIANGLE 104

Barycentrics    -a^9 b^3 + 2 a^7 b^5 - 2 a^3 b^9 + a b^11 - a^10 b c + a^8 b^3 c + 2 a^6 b^5 c - 2 a^4 b^7 c - a^2 b^9 c + b^11 c + 2 a^7 b^3 c^2 - 2 a b^9 c^2 - a^9 c^3 + a^8 b c^3 + 2 a^7 b^2 c^3 + 4 a^6 b^3 c^3 - 2 a^5 b^4 c^3 - 2 a^4 b^5 c^3 + 2 a^3 b^6 c^3 - a b^8 c^3 - 3 b^9 c^3 - 2 a^5 b^3 c^4 + 2 a^7 c^5 + 2 a^6 b c^5 - 2 a^4 b^3 c^5 + 2 a^2 b^5 c^5 + 2 a b^6 c^5 + 2 b^7 c^5 + 2 a^3 b^3 c^6 + 2 a b^5 c^6 - 2 a^4 b c^7 + 2 b^5 c^7 - a b^3 c^8 - 2 a^3 c^9 - a^2 b c^9 - 2 a b^2 c^9 - 3 b^3 c^9 + a c^11 + b c^11 : :

X(31007) lies on these lines:


X(31008) = (name pending)

Barycentrics    b c (a + b) (a + c) (a b + a c - b c) : :

X(31008) lies on these lines:


X(31009) = (name pending)

Barycentrics    (b - c) (a + 2 b + c)(a + b + 2c)(3a^2 + b^2 + c^2 + 3a b + 3 a c + b c) : :

X(31009) lies on these lines:


X(31010) = (name pending)

Barycentrics    (b^2 - c^2) (a + 2 b + c)(a + b + 2c) : :

X(31010) lies on these lines:


X(31011) = (name pending)

Barycentrics    (2a - b - c) (a + 2 b + c)(a + b + 2c) : :

X(31011) lies on these lines:


X(31012) = (name pending)

Barycentrics    (a + 2 b + c)(a + b + 2c)(2 a^2 - b^2 - c^2 + 2 a b + 2 a c - 4 b c) : :

X(31012) lies on these lines:


X(31013) = (name pending)

Barycentrics    (a + 2 b + c)(a + b + 2c)(2 a^2 - b^2 - c^2) : :

X(31013) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 105: X(31014) - X(31063)  rightri

Extending the preambles just before X(24537), X(26153), X(27378), X(30738), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 105, as in centers X(31014)-X(31063). Then

m(X) = - (b + c) x + (a + 2b + c) y + (a + b + 2c) z : :

and m(X) is a self-inverse mapping such that m(X) is collinear with X(2) and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. The fixed points of m are X(2) and every point on the line X(514)X(4024). Among the fixed points are X(i) for these i: 2, 514, 4024, 4608, 6542, 31009, 31010, 31011, 31012, 31013, 31064. (Clark Kimberling, January 21, 2019)

underbar




X(31014) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    -a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + 2 b^5 - a^4 c + b^4 c - a^3 c^2 - 2 a b^2 c^2 - 3 b^3 c^2 - a^2 c^3 - 3 b^2 c^3 + a c^4 + b c^4 + 2 c^5 : :

X(31014) lies on these lines:


X(31015) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    a^5 + 2 a^4 b - a^3 b^2 - a^2 b^3 - b^5 + 2 a^4 c - 2 a^2 b^2 c - a^3 c^2 - 2 a^2 b c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3 - c^5 : :

X(31015) lies on these lines:


X(31016) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    2 a^5 + 3 a^4 b - 3 a^3 b^2 - 3 a^2 b^3 + a b^4 + 3 a^4 c - 4 a^2 b^2 c + b^4 c - 3 a^3 c^2 - 4 a^2 b c^2 - 2 a b^2 c^2 - b^3 c^2 - 3 a^2 c^3 - b^2 c^3 + a c^4 + b c^4 : :

X(31016) lies on these lines:


X(31017) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    -a^2 b + a b^2 + 2 b^3 - a^2 c + b^2 c + a c^2 + b c^2 + 2 c^3 : :

X(31017) lies on these lines:


X(31018) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    a^3 + a^2 b - a b^2 - b^3 + a^2 c - 4 a b c + b^2 c - a c^2 + b c^2 - c^3 : :

X(31018) lies on these lines:


X(31019) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    a^2 b - b^3 + a^2 c + a b c + b^2 c + b c^2 - c^3 : :

X(31019) lies on these lines:


X(31020) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    2 a^4 + a^3 b - 4 a^2 b^2 + a b^3 + a^3 c - 2 a^2 b c + 2 a b^2 c + b^3 c - 4 a^2 c^2 + 2 a b c^2 - 2 b^2 c^2 + a c^3 + b c^3 : :

X(31020) lies on these lines:


X(31021) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    -a^6 b - a^5 b^2 - 2 a^4 b^3 + a^2 b^5 + a b^6 + 2 b^7 - a^6 c - a^4 b^2 c + a^2 b^4 c + b^6 c - a^5 c^2 - a^4 b c^2 - a b^4 c^2 - b^5 c^2 - 2 a^4 c^3 - 2 b^4 c^3 + a^2 b c^4 - a b^2 c^4 - 2 b^3 c^4 + a^2 c^5 - b^2 c^5 + a c^6 + b c^6 + 2 c^7 : :

X(31021) lies on these lines:


X(31022) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    -a^6 b - a^5 b^2 - 2 a^4 b^3 + a^2 b^5 + a b^6 + 2 b^7 - a^6 c - a^4 b^2 c + a^2 b^4 c + b^6 c - a^5 c^2 - a^4 b c^2 + 2 a^3 b^2 c^2 + 2 a^2 b^3 c^2 - a b^4 c^2 - b^5 c^2 - 2 a^4 c^3 + 2 a^2 b^2 c^3 - 2 b^4 c^3 + a^2 b c^4 - a b^2 c^4 - 2 b^3 c^4 + a^2 c^5 - b^2 c^5 + a c^6 + b c^6 + 2 c^7 : :

X(31022) lies on these lines:


X(31023) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    -a^3 b + a b^3 + 2 b^4 - a^3 c + b^3 c + a c^3 + b c^3 + 2 c^4 : :

X(31023) lies on these lines:


X(31024) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    -a^4 b + a b^4 + 2 b^5 - a^4 c + b^4 c + a c^4 + b c^4 + 2 c^5 : :

X(31024) lies on these lines:


X(31025) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    (b + c) (a^2 + a b + a c + 3 b c) : :

X(31025) lies on these lines:


X(31026) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    a^3 b^2 + a^2 b^3 + a^3 c^2 + 2 a b^2 c^2 + 3 b^3 c^2 + a^2 c^3 + 3 b^2 c^3 : :

X(31026) lies on these lines:


X(31027) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    a b^3 - a^2 b c + a b^2 c + b^3 c + a b c^2 + b^2 c^2 + a c^3 + b c^3 : :

X(31027) lies on these lines:


X(31028) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    a b^3 - 2 a^2 b c + b^3 c + b^2 c^2 + a c^3 + b c^3 : :

X(31028) lies on these lines:


X(31029) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    -3 a^2 b + a b^2 + 4 b^3 - 3 a^2 c - 2 a b c - b^2 c + a c^2 - b c^2 + 4 c^3 : :

X(31029) lies on these lines:


X(31030) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    3 a^2 b + a b^2 - 2 b^3 + 3 a^2 c + 4 a b c + 5 b^2 c + a c^2 + 5 b c^2 - 2 c^3 : :

X(31030) lies on these lines:


X(31031) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    -a^3 b - a b^3 + 2 b^4 - a^3 c + 2 a^2 b c - 2 a b^2 c - b^3 c - 2 a b c^2 - 2 b^2 c^2 - a c^3 - b c^3 + 2 c^4 : :

X(31031) lies on these lines:


X(31032) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    -a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + 2 b^5 - a^4 c + 4 a b^3 c + b^4 c - a^3 c^2 + 2 a b^2 c^2 - 3 b^3 c^2 - a^2 c^3 + 4 a b c^3 - 3 b^2 c^3 + a c^4 + b c^4 + 2 c^5 : :

X(31032) lies on these lines:


X(31033) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    -a^4 b + a^3 b^2 - a^2 b^3 - a b^4 + 2 b^5 - a^4 c + 2 a^3 b c - 2 a b^3 c - b^4 c + a^3 c^2 - b^3 c^2 - a^2 c^3 - 2 a b c^3 - b^2 c^3 - a c^4 - b c^4 + 2 c^5 : :

X(31033) lies on these lines:


X(31034) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    a^3 + 2 a^2 b - b^3 + 2 a^2 c - c^3 : :

X(31034) lies on these lines:


X(31035) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    a^2 b + a b^2 + a^2 c + 4 a b c - b^2 c + a c^2 - b c^2 : :

X(31035) lies on these lines:


X(31036) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    a^3 b^2 + a^2 b^3 + 2 a^2 b^2 c + a^3 c^2 + 2 a^2 b c^2 - b^3 c^2 + a^2 c^3 - b^2 c^3 : :

X(31036) lies on these lines:


X(31037) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    (b + c) (-a^2 + a b + 2 b^2 + a c - b c + 2 c^2) : :

X(31037) lies on these lines:


X(31038) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    a^2 b^2 - 2 a b^3 + b^4 + 5 a^2 b c - b^3 c + a^2 c^2 - 2 a c^3 - b c^3 + c^4 : :

X(31038) lies on these lines:


X(31039) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    2 a^5 + 3 a^4 b - 3 a^3 b^2 - 3 a^2 b^3 + a b^4 + 3 a^4 c - 4 a^2 b^2 c - 4 a b^3 c + b^4 c - 3 a^3 c^2 - 4 a^2 b c^2 - 6 a b^2 c^2 - b^3 c^2 - 3 a^2 c^3 - 4 a b c^3 - b^2 c^3 + a c^4 + b c^4 : :

X(31039) lies on these lines:


X(31040) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    (b - c) (a^3 b + a^2 b^2 + 2 a b^3 + a^3 c + a^2 b c + 2 a b^2 c - 2 b^3 c + a^2 c^2 + 2 a b c^2 - 3 b^2 c^2 + 2 a c^3 - 2 b c^3) : :

X(31040) lies on these lines:


X(31041) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    -a^3 b + a b^3 + 2 b^4 - a^3 c - 2 a^2 b c - 2 a b^2 c + b^3 c - 2 a b c^2 + a c^3 + b c^3 + 2 c^4 : :

X(31041) lies on these lines:


X(31042) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    a^5 + 3 a^4 b - a b^4 - 3 b^5 + 3 a^4 c - 2 a^2 b^2 c - b^4 c - 2 a^2 b c^2 + 2 a b^2 c^2 + 4 b^3 c^2 + 4 b^2 c^3 - a c^4 - b c^4 - 3 c^5 : :

X(31042) lies on these lines:


X(31043) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    (b + c) (-a^4 - a^3 b - a^2 b^2 + a b^3 + 2 b^4 - a^3 c - 3 a^2 b c - 3 a b^2 c - b^3 c - a^2 c^2 - 3 a b c^2 - 2 b^2 c^2 + a c^3 - b c^3 + 2 c^4) : :

X(31043) lies on these lines:


X(31044) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    -a^6 b - a^5 b^2 - 2 a^4 b^3 + a^2 b^5 + a b^6 + 2 b^7 - a^6 c - a^4 b^2 c + a^2 b^4 c + b^6 c - a^5 c^2 - a^4 b c^2 + 4 a^3 b^2 c^2 + 4 a^2 b^3 c^2 - a b^4 c^2 - b^5 c^2 - 2 a^4 c^3 + 4 a^2 b^2 c^3 - 2 b^4 c^3 + a^2 b c^4 - a b^2 c^4 - 2 b^3 c^4 + a^2 c^5 - b^2 c^5 + a c^6 + b c^6 + 2 c^7 : :

X(31044) lies on these lines:


X(31045) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    (b + c) (3 a^5 + 2 a^4 b - 2 a^3 b^2 - a b^4 - 2 b^5 + 2 a^4 c - a^3 b c - 3 a^2 b^2 c + a b^3 c + b^4 c - 2 a^3 c^2 - 3 a^2 b c^2 + b^3 c^2 + a b c^3 + b^2 c^3 - a c^4 + b c^4 - 2 c^5) : :

X(31045) lies on these lines:


X(31046) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    (b + c) (-a^7 - 2 a^6 b - 3 a^5 b^2 - 2 a^4 b^3 + a^3 b^4 + 2 a^2 b^5 + 3 a b^6 + 2 b^7 - 2 a^6 c - 3 a^5 b c + a^4 b^2 c + 2 a^3 b^3 c + a b^5 c + b^6 c - 3 a^5 c^2 + a^4 b c^2 + 10 a^3 b^2 c^2 + 4 a^2 b^3 c^2 - 3 a b^4 c^2 - b^5 c^2 - 2 a^4 c^3 + 2 a^3 b c^3 + 4 a^2 b^2 c^3 - 2 a b^3 c^3 - 2 b^4 c^3 + a^3 c^4 - 3 a b^2 c^4 - 2 b^3 c^4 + 2 a^2 c^5 + a b c^5 - b^2 c^5 + 3 a c^6 + b c^6 + 2 c^7) : :

X(31046) lies on these lines:


X(31047) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    (b + c) (a^7 + 4 a^6 b - a^5 b^2 - 6 a^4 b^3 - a^3 b^4 + a b^6 + 2 b^7 + 4 a^6 c + 7 a^5 b c - a^4 b^2 c - 6 a^3 b^3 c - 2 a^2 b^4 c - a b^5 c - b^6 c - a^5 c^2 - a^4 b c^2 + 2 a^3 b^2 c^2 + 2 a^2 b^3 c^2 - a b^4 c^2 - b^5 c^2 - 6 a^4 c^3 - 6 a^3 b c^3 + 2 a^2 b^2 c^3 + 2 a b^3 c^3 - a^3 c^4 - 2 a^2 b c^4 - a b^2 c^4 - a b c^5 - b^2 c^5 + a c^6 - b c^6 + 2 c^7) : :

X(31047) lies on these lines:


X(31048) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    a^5 + 4 a^4 b + a^3 b^2 + a^2 b^3 - 2 a b^4 - 5 b^5 + 4 a^4 c - 2 a^2 b^2 c - 2 b^4 c + a^3 c^2 - 2 a^2 b c^2 + 4 a b^2 c^2 + 7 b^3 c^2 + a^2 c^3 + 7 b^2 c^3 - 2 a c^4 - 2 b c^4 - 5 c^5 : :

X(31048) lies on these lines:


X(31049) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    a^5 + 2 a^4 b - a^3 b^2 - a^2 b^3 - b^5 + 2 a^4 c - 2 a^3 b c - 6 a^2 b^2 c - 2 a b^3 c - a^3 c^2 - 6 a^2 b c^2 - 4 a b^2 c^2 + b^3 c^2 - a^2 c^3 - 2 a b c^3 + b^2 c^3 - c^5 : :

X(31049) lies on these lines:


X(31050) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    -a^4 b + a^3 b^2 + a^2 b^3 + a b^4 + 2 b^5 - a^4 c + 2 a^2 b^2 c + b^4 c + a^3 c^2 + 2 a^2 b c^2 - b^3 c^2 + a^2 c^3 - b^2 c^3 + a c^4 + b c^4 + 2 c^5 : :

X(31050) lies on these lines:


X(31051) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    -a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + 2 b^5 - a^4 c + 2 a b^3 c + b^4 c - a^3 c^2 - 3 b^3 c^2 - a^2 c^3 + 2 a b c^3 - 3 b^2 c^3 + a c^4 + b c^4 + 2 c^5 : :

X(31051) lies on these lines:


X(31052) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    a^3 b + 2 a^2 b^2 - a b^3 + a^3 c + 2 a^2 b c + b^3 c + 2 a^2 c^2 + 4 b^2 c^2 - a c^3 + b c^3 : :

X(31052) lies on these lines:


X(31053) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    a^2 b - b^3 + a^2 c - a b c + b^2 c + b c^2 - c^3 : :

X(31053) lies on these lines:


X(31054) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    a^9 + 2 a^8 b - a^5 b^4 - a^4 b^5 - b^9 + 2 a^8 c - 2 a^4 b^4 c - a^5 c^4 - 2 a^4 b c^4 + b^5 c^4 - a^4 c^5 + b^4 c^5 - c^9 : :

X(31054) lies on these lines:


X(31055) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    a^2 b^4 - a^2 b^3 c - 2 a b^4 c - 2 a^2 b^2 c^2 + b^4 c^2 - a^2 b c^3 + b^3 c^3 + a^2 c^4 - 2 a b c^4 + b^2 c^4 : :

X(31055) lies on these lines:


X(31056) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    -3 a^2 b + a b^2 + 4 b^3 - 3 a^2 c + 7 a b c - b^2 c + a c^2 - b c^2 + 4 c^3 : :

X(31056) lies on these lines:


X(31057) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    (b + c) (-a^4 + a b^3 + 2 b^4 + 2 a^2 b c - a b^2 c - b^3 c - a b c^2 - 3 b^2 c^2 + a c^3 - b c^3 + 2 c^4) : :

X(31057) lies on these lines:


X(31058) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    -a^3 b - a b^3 + 2 b^4 - a^3 c + 4 a^2 b c - b^3 c - 2 b^2 c^2 - a c^3 - b c^3 + 2 c^4 : :

X(31058) lies on these lines:


X(31059) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    (a + b) (a + c) (2 a - b - c) (a^2 + a b - b^2 + a c - b c - c^2) : :

X(31059) lies on these lines:


X(31060) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    b c (-a^2 b - a^2 c + a b c + 2 b^2 c + 2 b c^2) : :

X(31060) lies on these lines:


X(31061) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    a^3 b^2 + a^2 b^3 - 2 a^3 b c - 2 a^2 b^2 c + 2 a b^3 c + a^3 c^2 - 2 a^2 b c^2 - b^3 c^2 + a^2 c^3 + 2 a b c^3 - b^2 c^3 : :

X(31061) lies on these lines:


X(31062) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    -3 a^3 b - 2 a^2 b^2 + 3 a b^3 + 4 b^4 - 3 a^3 c - 2 a^2 b c + b^3 c - 2 a^2 c^2 - 4 b^2 c^2 + 3 a c^3 + b c^3 + 4 c^4 : :

X(31062) lies on these lines:


X(31063) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 105

Barycentrics    a^3 b + 2 a^2 b^2 - a b^3 + a^3 c + 4 a^2 b c + 2 a b^2 c + b^3 c + 2 a^2 c^2 + 2 a b c^2 + 4 b^2 c^2 - a c^3 + b c^3 : :

X(31063) lies on these lines:


X(31064) = (name pending)

Barycentrics    (a + 2 b + c)(a + b + 2c)(2a^3 - b^3 - c^3 + a^2 b + a^2 c - a b^2 - a c^2) : :

X(31064) lies on these lines:


X(31065) = (name pending)

Barycentrics    (b^2 - c^2)(a^2 + 2 b^2 + c^2)(a^2 + b^2 + 2 c^2) : :

X(31065) lies on these lines:


X(31066) = (name pending)

Barycentrics    (b^2 - c^2)(a^2 + 2 b^2 + c^2)(a^2 + b^2 + 2 c^2)(3 a^4 + b^4 + c^4 + 3 a^2 b^2 + 3 a^2 c^2 + b^2 c^2) : :

X(31066) lies on these lines:


X(31067) = (name pending)

Barycentrics    (b^4 - c^4)(a^2 + 2 b^2 + c^2)(a^2 + b^2 + 2 c^2) : :

X(31067) lies on these lines:


X(31068) = (name pending)

Barycentrics    (2a^2 - b^2 - c^2)(a^2 + 2 b^2 + c^2)(a^2 + b^2 + 2 c^2) : :

X(31068) lies on these lines:


X(31069) = (name pending)

Barycentrics    (2a^2 - b^2 - c^2)(a^2 + 2 b^2 + c^2)(2 a^4 - b^4 - c^4 + 2 a^2 b^2 + 2 a^2 c^2 - 4 b^2 c^2) : :

X(31069) lies on these lines:


X(31070) = (name pending)

Barycentrics    (2 a^4 - b^4 - c^4)(a^2 + 2b^2 - c^2)(a^2 + b^2 + 2c^2) : :

X(31070) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 106: X(31071) - X(31132)  rightri

Extending the preambles just before X(24537), X(26153), X(27378), X(30738), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 106, as in centers X(31071)-X(31132). Then

m(X) = - (b^2 + c^2) x + (a^2 + 2 b^2 + c^2) y + (a^2 + b^2 + 2 c^2) z : :

and m(X) is a self-inverse mapping such that m(X) is collinear with X(2) and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. The fixed points of m are X(2) and every point on the line X(523)X(2528). Among the fixed points are X(i) for these i: 2, 523, 2528, 31065, 31066, 31067, 31068, 31069, 31070. (Clark Kimberling, January 21, 2019)

underbar




X(31071) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    -a^3 b - a b^3 + 2 b^4 - a^3 c - 2 a^2 b c - 3 b^3 c + 2 b^2 c^2 - a c^3 - 3 b c^3 + 2 c^4 : :

X(31071) lies on these lines:


X(31072) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    (b^2 - c^2) (a^4 - a^2 b^2 - a^2 c^2 + 3 b^2 c^2) : :

X(31072) lies on these lines:


X(31073) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    a (a^4 - a^3 b + a^2 b^2 - a b^3 - a^3 c + 2 a^2 b c - 2 a b^2 c + 2 b^3 c + a^2 c^2 - 2 a b c^2 - b^2 c^2 - a c^3 + 2 b c^3) : :

X(31073) lies on these lines:


X(31074) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    -a^4 b^2 + b^6 - a^4 c^2 + a^2 b^2 c^2 - b^4 c^2 - b^2 c^4 + c^6 : :

X(31074) lies on these lines:


X(31075) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    -a^3 b^2 + a^2 b^3 + 2 b^5 - a^3 c^2 + b^3 c^2 + a^2 c^3 + b^2 c^3 + 2 c^5 : :

X(31075) lies on these lines:


X(31076) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    -a^4 b^2 + a^2 b^4 + 2 b^6 - a^4 c^2 + b^4 c^2 + a^2 c^4 + b^2 c^4 + 2 c^6 : :

X(31076) lies on these lines:


X(31077) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    a^3 b + a b^3 + a^3 c + 2 a^2 b c + 3 b^3 c + a c^3 + 3 b c^3 : :

X(31077) lies on these lines:


X(31078) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    (b^2 + c^2) (a^4 + a^2 b^2 + a^2 c^2 + 3 b^2 c^2) : :

X(31078) lies on these lines:


X(31079) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    a^2 b - a b^2 + 2 b^3 + a^2 c + b^2 c - a c^2 + b c^2 + 2 c^3 : :

X(31079) lies on these lines:


X(31080) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    a^4 - a^3 b + 2 a^2 b^2 - a b^3 - b^4 - a^3 c - 2 a b^2 c + b^3 c + 2 a^2 c^2 - 2 a b c^2 - a c^3 + b c^3 - c^4 : :

X(31080) lies on these lines:


X(31081) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    a^3 b^2 - a^2 b^3 + 2 a b^4 - 2 a^3 b c - 2 a b^3 c + 2 b^4 c + a^3 c^2 + 2 a b^2 c^2 + b^3 c^2 - a^2 c^3 - 2 a b c^3 + b^2 c^3 + 2 a c^4 + 2 b c^4 : :

X(31081) lies on these lines:


X(31082) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    -a^3 b - a b^3 + 4 b^4 - a^3 c - 2 a^2 b c - 3 b^3 c + 4 b^2 c^2 - a c^3 - 3 b c^3 + 4 c^4 : :

X(31082) lies on these lines:


X(31083) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    a^3 b + a b^3 - b^4 + a^3 c + 2 a^2 b c + 3 b^3 c - b^2 c^2 + a c^3 + 3 b c^3 - c^4 : :

X(31083) lies on these lines:


X(31084) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    -a^3 b^2 + a^2 b^3 - a b^4 + b^5 - b^4 c - a^3 c^2 - a b^2 c^2 + a^2 c^3 - a c^4 - b c^4 + c^5 : :

X(31084) lies on these lines:


X(31085) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    -a^4 b^2 + b^6 + a^3 b^2 c - a^2 b^3 c + 2 a b^4 c - a^4 c^2 + a^3 b c^2 - a^2 b^2 c^2 + a b^3 c^2 - b^4 c^2 - a^2 b c^3 + a b^2 c^3 + 2 a b c^4 - b^2 c^4 + c^6 : :

X(31085) lies on these lines:


X(31086) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    -a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + 2 b^5 - a^4 c + 4 a^3 b c - 2 a^2 b^2 c + 4 a b^3 c - b^4 c - a^3 c^2 - 2 a^2 b c^2 + 2 a b^2 c^2 - b^3 c^2 - a^2 c^3 + 4 a b c^3 - b^2 c^3 + a c^4 - b c^4 + 2 c^5 : :

X(31086) lies on these lines:


X(31087) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    a^3 b + a b^3 + a^3 c + 2 a b^2 c - b^3 c + 2 a b c^2 + a c^3 - b c^3 : :

X(31087) lies on these lines:


X(31088) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    a^4 b^2 + a^2 b^4 + a^4 c^2 + 4 a^2 b^2 c^2 - b^4 c^2 + a^2 c^4 - b^2 c^4 : :

X(31088) lies on these lines:


X(31089) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    a b^4 + b^5 + a^3 b c + a b^3 c + b^4 c + a b^2 c^2 + b^3 c^2 + a b c^3 + b^2 c^3 + a c^4 + b c^4 + c^5 : :

X(31089) lies on these lines:


X(31090) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    a^3 b + a b^3 + b^4 + a^3 c + a^2 b c + a b^2 c + b^3 c + a b c^2 + b^2 c^2 + a c^3 + b c^3 + c^4 : :

X(31090) lies on these lines:


X(31091) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    a^3 - a^2 b + 3 a b^2 - 3 b^3 - a^2 c - b^2 c + 3 a c^2 - b c^2 - 3 c^3 : :

X(31091) lies on these lines:


X(31092) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    a^4 b - 3 a^3 b^2 + 5 a^2 b^3 - 5 a b^4 + 2 b^5 + a^4 c + 4 a^3 b c + 2 a^2 b^2 c + 4 a b^3 c - 3 b^4 c - 3 a^3 c^2 + 2 a^2 b c^2 - 6 a b^2 c^2 + b^3 c^2 + 5 a^2 c^3 + 4 a b c^3 + b^2 c^3 - 5 a c^4 - 3 b c^4 + 2 c^5 : :

X(31092) lies on these lines:


X(31093) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    -a^3 b - a b^3 + 2 b^4 - a^3 c - 2 a b^2 c + b^3 c - 2 a b c^2 + 2 b^2 c^2 - a c^3 + b c^3 + 2 c^4 : :

X(31093) lies on these lines:


X(31094) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    (b - c) (a^3 b + a^2 b^2 + 2 a b^3 + a^3 c - a^2 b c + 2 a b^2 c - 2 b^3 c + a^2 c^2 + 2 a b c^2 - b^2 c^2 + 2 a c^3 - 2 b c^3) : :

X(31094) lies on these lines:


X(31095) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    (b - c) (a^4 + 2 a^2 b^2 + a b^3 + 2 a^2 b c + a b^2 c + b^3 c + 2 a^2 c^2 + a b c^2 - b^2 c^2 + a c^3 + b c^3) : :

X(31095) lies on these lines:


X(31096) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    (b - c) (2 a^3 b^2 + 2 a b^4 + a^3 b c - a^2 b^2 c + 2 a b^3 c - 2 b^4 c + 2 a^3 c^2 - a^2 b c^2 + 3 a b^2 c^2 - 2 b^3 c^2 + 2 a b c^3 - 2 b^2 c^3 + 2 a c^4 - 2 b c^4) : :

X(31096) lies on these lines:


X(31097) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    -a^3 b^2 + a^2 b^3 + 2 b^5 - 2 a^3 b c - 2 a b^3 c - a^3 c^2 + b^3 c^2 + a^2 c^3 - 2 a b c^3 + b^2 c^3 + 2 c^5 : :

X(31097) lies on these lines:


X(31098) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    2 a^3 + 3 a^2 b + a b^2 + 4 b^3 + 3 a^2 c + 3 b^2 c + a c^2 + 3 b c^2 + 4 c^3 : :

X(31098) lies on these lines:


X(31099) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    a^6 + 3 a^4 b^2 - a^2 b^4 - 3 b^6 + 3 a^4 c^2 - 2 a^2 b^2 c^2 + 3 b^4 c^2 - a^2 c^4 + 3 b^2 c^4 - 3 c^6 : :

X(31099) lies on these lines:


X(31100) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    -a^4 b^2 + b^6 - a^4 b c - a^3 b^2 c - a^2 b^3 c - a b^4 c - a^4 c^2 - a^3 b c^2 - a^2 b^2 c^2 - a b^3 c^2 - b^4 c^2 - a^2 b c^3 - a b^2 c^3 - a b c^4 - b^2 c^4 + c^6 : :

X(31100) lies on these lines:


X(31101) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    -a^4 b^2 + b^6 - a^4 c^2 + 3 a^2 b^2 c^2 - b^4 c^2 - b^2 c^4 + c^6 : :

X(31101) lies on these lines:


X(31102) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    -a^7 b - a^6 b^2 - a^5 b^3 - a^4 b^4 + a^3 b^5 + a^2 b^6 + a b^7 + b^8 - a^7 c - a^6 b c - a^5 b^2 c - a^4 b^3 c + a^3 b^4 c + a^2 b^5 c + a b^6 c + b^7 c - a^6 c^2 - a^5 b c^2 + a^4 b^2 c^2 + 4 a^3 b^3 c^2 + 2 a^2 b^4 c^2 - a b^5 c^2 - a^5 c^3 - a^4 b c^3 + 4 a^3 b^2 c^3 + 4 a^2 b^3 c^3 - a b^4 c^3 - b^5 c^3 - a^4 c^4 + a^3 b c^4 + 2 a^2 b^2 c^4 - a b^3 c^4 - 2 b^4 c^4 + a^3 c^5 + a^2 b c^5 - a b^2 c^5 - b^3 c^5 + a^2 c^6 + a b c^6 + a c^7 + b c^7 + c^8 : :

X(31102) lies on these lines:


X(31103) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    -a^7 b^2 - a^6 b^3 - a^5 b^4 - a^4 b^5 + a^3 b^6 + a^2 b^7 + a b^8 + b^9 - a^7 b c - a^6 b^2 c - a^5 b^3 c - a^4 b^4 c + a^3 b^5 c + a^2 b^6 c + a b^7 c + b^8 c - a^7 c^2 - a^6 b c^2 + a^5 b^2 c^2 + a^4 b^3 c^2 + 2 a^3 b^4 c^2 + 2 a^2 b^5 c^2 - a^6 c^3 - a^5 b c^3 + a^4 b^2 c^3 + 4 a^3 b^3 c^3 + 2 a^2 b^4 c^3 - a b^5 c^3 - a^5 c^4 - a^4 b c^4 + 2 a^3 b^2 c^4 + 2 a^2 b^3 c^4 - 2 a b^4 c^4 - 2 b^5 c^4 - a^4 c^5 + a^3 b c^5 + 2 a^2 b^2 c^5 - a b^3 c^5 - 2 b^4 c^5 + a^3 c^6 + a^2 b c^6 + a^2 c^7 + a b c^7 + a c^8 + b c^8 + c^9 : :

X(31103) lies on these lines:


X(31104) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    a^8 b - 2 a^4 b^5 + b^9 + a^8 c + a^7 b c + a^6 b^2 c + a^5 b^3 c - a^4 b^4 c - a^3 b^5 c - a^2 b^6 c - a b^7 c + a^6 b c^2 + 3 a^5 b^2 c^2 - 2 a^4 b^3 c^2 - 3 a^3 b^4 c^2 + 2 a^2 b^5 c^2 - b^7 c^2 + a^5 b c^3 - 2 a^4 b^2 c^3 - 4 a^3 b^3 c^3 - a^2 b^4 c^3 + a b^5 c^3 + b^6 c^3 - a^4 b c^4 - 3 a^3 b^2 c^4 - a^2 b^3 c^4 - b^5 c^4 - 2 a^4 c^5 - a^3 b c^5 + 2 a^2 b^2 c^5 + a b^3 c^5 - b^4 c^5 - a^2 b c^6 + b^3 c^6 - a b c^7 - b^2 c^7 + c^9 : :

X(31104) lies on these lines:


X(31105) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    a^6 + 5 a^4 b^2 - a^2 b^4 - 5 b^6 + 5 a^4 c^2 + 5 b^4 c^2 - a^2 c^4 + 5 b^2 c^4 - 5 c^6 : :

X(31105) lies on these lines:


X(31106) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    a^6 + a^4 b^2 - a^2 b^4 - b^6 - 2 a^4 b c - 2 a^3 b^2 c - 2 a^2 b^3 c - 2 a b^4 c + a^4 c^2 - 2 a^3 b c^2 - 4 a^2 b^2 c^2 - 2 a b^3 c^2 + b^4 c^2 - 2 a^2 b c^3 - 2 a b^2 c^3 - a^2 c^4 - 2 a b c^4 + b^2 c^4 - c^6 : :

X(31106) lies on these lines:


X(31107) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    (b^2 + c^2) (b^4 + c^4 + a^2 b^2 + a^2 c^2 - b^2 c^2) : :

X(31107) lies on these lines:


X(31108) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    a (a^5 - a b^4 - a^2 b^2 c + a b^3 c - 2 b^4 c - a^2 b c^2 - 5 a b^2 c^2 - b^3 c^2 + a b c^3 - b^2 c^3 - a c^4 - 2 b c^4) : :

X(31108) lies on these lines:


X(31109) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    -2 a^4 b^2 + 2 b^6 - a^3 b^2 c + a^2 b^3 c - 2 a b^4 c - 2 a^4 c^2 - a^3 b c^2 - 2 a^2 b^2 c^2 - a b^3 c^2 - 2 b^4 c^2 + a^2 b c^3 - a b^2 c^3 - 2 a b c^4 - 2 b^2 c^4 + 2 c^6 : :

X(31109) lies on these lines:


X(31110) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    -2 a^4 b^2 + 2 b^6 + a^3 b^2 c - a^2 b^3 c + 2 a b^4 c - 2 a^4 c^2 + a^3 b c^2 - 2 a^2 b^2 c^2 + a b^3 c^2 - 2 b^4 c^2 - a^2 b c^3 + a b^2 c^3 + 2 a b c^4 - 2 b^2 c^4 + 2 c^6 : :

X(31110) lies on these lines:


X(31111) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    a^4 b + 2 a^2 b^3 - a b^4 + a^4 c + 2 a^2 b^2 c + b^4 c + 2 a^2 b c^2 - 2 a b^2 c^2 + 2 b^3 c^2 + 2 a^2 c^3 + 2 b^2 c^3 - a c^4 + b c^4 : :

X(31111) lies on these lines:


X(31112) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    -a^5 b^2 - a^4 b^3 + a^3 b^4 - a^2 b^5 + 2 b^7 - a^5 c^2 + 2 a^3 b^2 c^2 - 2 a^2 b^3 c^2 - b^5 c^2 - a^4 c^3 - 2 a^2 b^2 c^3 - b^4 c^3 + a^3 c^4 - b^3 c^4 - a^2 c^5 - b^2 c^5 + 2 c^7 : :

X(31112) lies on these lines:


X(31113) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    -a^4 b^2 + a^2 b^4 + 2 a b^5 + 2 b^6 + 2 a b^4 c + 2 b^5 c - a^4 c^2 + 2 a b^3 c^2 + b^4 c^2 + 2 a b^2 c^3 + 2 b^3 c^3 + a^2 c^4 + 2 a b c^4 + b^2 c^4 + 2 a c^5 + 2 b c^5 + 2 c^6 : :

X(31113) lies on these lines:


X(31114) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(26), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    -a^10 b^2 + a^8 b^4 + 2 a^6 b^6 - 2 a^4 b^8 - a^2 b^10 + b^12 - a^10 c^2 + 3 a^8 b^2 c^2 + 3 a^6 b^4 c^2 - a^4 b^6 c^2 - 2 a^2 b^8 c^2 - 2 b^10 c^2 + a^8 c^4 + 3 a^6 b^2 c^4 - 4 a^4 b^4 c^4 + 3 a^2 b^6 c^4 - b^8 c^4 + 2 a^6 c^6 - a^4 b^2 c^6 + 3 a^2 b^4 c^6 + 4 b^6 c^6 - 2 a^4 c^8 - 2 a^2 b^2 c^8 - b^4 c^8 - a^2 c^10 - 2 b^2 c^10 + c^12 : :

X(31114) lies on these lines:


X(31115) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    a^4 b + 2 a^2 b^3 - a b^4 + a^4 c + 2 a^3 b c + 2 a^2 b^2 c + 2 a b^3 c + b^4 c + 2 a^2 b c^2 - 2 a b^2 c^2 + 2 b^3 c^2 + 2 a^2 c^3 + 2 a b c^3 + 2 b^2 c^3 - a c^4 + b c^4 : :

X(31115) lies on these lines:


X(31116) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    -a^4 b - 2 a^3 b^2 + a b^4 + 4 b^5 - a^4 c - 2 a^2 b^2 c - b^4 c - 2 a^3 c^2 - 2 a^2 b c^2 + 2 a b^2 c^2 + a c^4 - b c^4 + 4 c^5 : :

X(31116) lies on these lines:


X(31117) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    a^3 b + a b^3 + 2 b^4 + a^3 c + 2 a b^2 c - b^3 c + 2 a b c^2 + 2 b^2 c^2 + a c^3 - b c^3 + 2 c^4 : :

X(31117) lies on these lines:


X(31118) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    a^4 b^2 + a^2 b^4 - 2 a^4 b c - 4 a^2 b^3 c + 2 a b^4 c + a^4 c^2 + 4 a^2 b^2 c^2 - b^4 c^2 - 4 a^2 b c^3 + a^2 c^4 + 2 a b c^4 - b^2 c^4 : :

X(31118) lies on these lines:


X(31119) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    a^3 b^2 - a^2 b^3 + 2 a b^4 + 2 b^4 c + a^3 c^2 + 2 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3 + 2 a c^4 + 2 b c^4 : :

X(31119) lies on these lines:


X(31120) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    a^4 b + a^3 b^2 + a^2 b^3 - b^5 + a^4 c + 2 a^2 b^2 c + a^3 c^2 + 2 a^2 b c^2 + a b^2 c^2 + a^2 c^3 - c^5 : :

X(31120) lies on these lines:


X(31121) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    -a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + 2 b^5 - a^4 c - 2 a^2 b^2 c - b^4 c - a^3 c^2 - 2 a^2 b c^2 + 2 a b^2 c^2 - b^3 c^2 - a^2 c^3 - b^2 c^3 + a c^4 - b c^4 + 2 c^5 : :

X(31121) lies on these lines:


X(31122) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    a^5 b + 2 a^3 b^3 - 3 a b^5 + a^5 c - 2 a^4 b c + a^3 b^2 c - 3 a^2 b^3 c - b^5 c + a^3 b c^2 - 3 a b^3 c^2 + 2 a^3 c^3 - 3 a^2 b c^3 - 3 a b^2 c^3 + 2 b^3 c^3 - 3 a c^5 - b c^5 : :

X(31122) lies on these lines:


X(31123) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    a^10 + 2 a^8 b^2 - a^6 b^4 - a^4 b^6 - b^10 + 2 a^8 c^2 - 2 a^4 b^4 c^2 - a^6 c^4 - 2 a^4 b^2 c^4 + b^6 c^4 - a^4 c^6 + b^4 c^6 - c^10 : :

X(31123) lies on these lines:


X(31124) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    (b^2 + c^2) (a^4 + 3 a^2 b^2 + 2 b^4 + 3 a^2 c^2 + b^2 c^2 + 2 c^4) : :

X(31124) lies on these lines:


X(31125) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    (a^2 + b^2 - 2 c^2) (a^2 - 2 b^2 + c^2) (b^2 + c^2) : :

X(31125) lies on these lines:


X(31126) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    -a^3 b^2 + a^2 b^3 - a b^4 + b^5 + a^3 b c + a b^3 c - b^4 c - a^3 c^2 - a b^2 c^2 + a^2 c^3 + a b c^3 - a c^4 - b c^4 + c^5 : :

X(31126) lies on these lines:


X(31127) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    -a^6 b^2 - a^2 b^6 + 2 b^8 - a^6 c^2 + 4 a^4 b^2 c^2 - b^6 c^2 - 2 b^4 c^4 - a^2 c^6 - b^2 c^6 + 2 c^8 : :

X(31127) lies on these lines:


X(31128) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    (2 a^2 - b^2 - c^2) (a^4 + a^2 b^2 + a^2 c^2 - 3 b^2 c^2) : :

X(31128) lies on these lines:


X(31129) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    b^4 - a^2 b c + a b^2 c - 2 b^3 c + a b c^2 + b^2 c^2 - 2 b c^3 + c^4 : :

X(31129) lies on these lines:


X(31130) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    b c (a^2 - a b + 2 b^2 - a c + 2 c^2) : :

X(31130) lies on these lines:


X(31131) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    (b - c) (a^3 - a^2 b + 3 a b^2 - b^3 - a^2 c + a b c - 2 b^2 c + 3 a c^2 - 2 b c^2 - c^3) : :

X(31131) lies on these lines:


X(31132) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = GEMINI TRIANGLE 106

Barycentrics    -3 a^4 b^2 + a^2 b^4 + 4 b^6 - 3 a^4 c^2 - 2 a^2 b^2 c^2 - b^4 c^2 + a^2 c^4 - b^2 c^4 + 4 c^6 : :

X(31132) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 107: X(31133) - X(31181)  rightri

Extending the preambles just before X(24537), X(26153), X(27378), X(30738), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 107, as in centers X(31133)-X(31181). Then

m(X) = x - 2 y - 2 z : -2 x + y - 2 z : -2 x - 2 y + z : :

and m(X) is a self-inverse mapping; indeed, m(X) = reflection of X in X(2); the fixed points are X(2) and every point on the line at infinity. (Clark Kimberling, January 22, 2019)

underbar




X(31133) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^6 + 2 a^4 b^2 - a^2 b^4 - 2 b^6 + 2 a^4 c^2 + 2 b^4 c^2 - a^2 c^4 + 2 b^2 c^4 - 2 c^6 : :

X(31133) lies on these lines:


X(31134) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^3 - 2 b^3 - 2 c^3 : :

X(31134) lies on these lines:


X(31135) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^4 - a^3 b + 2 a b^3 - 2 b^4 - a^3 c + 2 b^3 c + 2 a c^3 + 2 b c^3 - 2 c^4 : :

X(31135) lies on these lines:


X(31136) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^2 b - 2 a b^2 + a^2 c - 2 b^2 c - 2 a c^2 - 2 b c^2 : :

X(31136) lies on these lines:


X(31137) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^2 b - 2 a b^2 + a^2 c + 3 a b c - 2 b^2 c - 2 a c^2 - 2 b c^2 : :

X(31137) lies on these lines:


X(31138) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    2 a^2 + a b - 4 b^2 + a c + 4 b c - 4 c^2 : :

X(31138) lies on these lines:


X(31139) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^2 + 2 a b - 2 b^2 + 2 a c + 8 b c - 2 c^2 : :

X(31139) lies on these lines:


X(31140) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^3 - a^2 b + 2 a b^2 - 2 b^3 - a^2 c + 2 b^2 c + 2 a c^2 + 2 b c^2 - 2 c^3 : :

X(31140) lies on these lines:


X(31141) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^4 + a^2 b^2 - 2 b^4 + 2 a^2 b c - 4 a b^2 c + a^2 c^2 - 4 a b c^2 + 4 b^2 c^2 - 2 c^4 : :

X(31141) lies on these lines:


X(31142) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^3 + 2 a^2 b - a b^2 - 2 b^3 + 2 a^2 c - 6 a b c + 2 b^2 c - a c^2 + 2 b c^2 - 2 c^3 : :

X(31142) lies on these lines:


X(31143) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^3 + a^2 b - 2 a b^2 - 2 b^3 + a^2 c - 3 a b c - 2 b^2 c - 2 a c^2 - 2 b c^2 - 2 c^3 : :

X(31143) lies on these lines:


X(31144) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^2 - 3 a b - 2 b^2 - 3 a c - 3 b c - 2 c^2 : :

X(31144) lies on these lines:


X(31145) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    7 a - 5 b - 5 c : :

X(31145) lies on these lines:


X(31146) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^3 - 4 a^2 b + 5 a b^2 - 2 b^3 - 4 a^2 c - 6 a b c + 2 b^2 c + 5 a c^2 + 2 b c^2 - 2 c^3 : :

X(31146) lies on these lines:


X(31147) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    (b - c) (-a^2 - 2 a b - 2 a c + 2 b c) : :

X(31147) lies on these lines:


X(31148) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    (b - c) (2 a^2 + a b + a c + 2 b c) : :

X(31148) lies on these lines:


X(31149) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    (b - c) (-a^3 - 2 a b^2 - 2 a b c + 2 b^2 c - 2 a c^2 + 2 b c^2) : :

X(31149) lies on these lines:


X(31150) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    (b - c) (-2 a^2 + 2 a b + 2 a c + b c) : :

X(31150) lies on these lines:


X(31151) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^3 - 2 b^3 + 3 a b c - 2 c^3 : :

X(31151) lies on these lines:


X(31152) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^6 + 2 a^4 b^2 - a^2 b^4 - 2 b^6 + 2 a^4 c^2 - 6 a^2 b^2 c^2 + 2 b^4 c^2 - a^2 c^4 + 2 b^2 c^4 - 2 c^6 : :

X(31152) lies on these lines:


X(31153) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^6 + 5 a^5 b + 2 a^4 b^2 - 4 a^3 b^3 - a^2 b^4 - a b^5 - 2 b^6 + 5 a^5 c + 5 a^4 b c - 4 a^3 b^2 c - 4 a^2 b^3 c - a b^4 c - b^5 c + 2 a^4 c^2 - 4 a^3 b c^2 - 6 a^2 b^2 c^2 + 2 a b^3 c^2 + 2 b^4 c^2 - 4 a^3 c^3 - 4 a^2 b c^3 + 2 a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 - a b c^4 + 2 b^2 c^4 - a c^5 - b c^5 - 2 c^6 : :

X(31153) lies on these lines:


X(31154) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^7 + a^6 b + 2 a^5 b^2 + 2 a^4 b^3 - a^3 b^4 - a^2 b^5 - 2 a b^6 - 2 b^7 + a^6 c + 5 a^5 b c + 2 a^4 b^2 c - 4 a^3 b^3 c - a^2 b^4 c - a b^5 c - 2 b^6 c + 2 a^5 c^2 + 2 a^4 b c^2 - 6 a^3 b^2 c^2 - 6 a^2 b^3 c^2 + 2 a b^4 c^2 + 2 b^5 c^2 + 2 a^4 c^3 - 4 a^3 b c^3 - 6 a^2 b^2 c^3 + 2 a b^3 c^3 + 2 b^4 c^3 - a^3 c^4 - a^2 b c^4 + 2 a b^2 c^4 + 2 b^3 c^4 - a^2 c^5 - a b c^5 + 2 b^2 c^5 - 2 a c^6 - 2 b c^6 - 2 c^7 : :

X(31154) lies on these lines:


X(31155) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^7 - 4 a^6 b - 3 a^5 b^2 + 6 a^4 b^3 + 3 a^3 b^4 - a b^6 - 2 b^7 - 4 a^6 c - 5 a^5 b c + a^4 b^2 c + 4 a^3 b^3 c + 4 a^2 b^4 c + a b^5 c - b^6 c - 3 a^5 c^2 + a^4 b c^2 + 2 a^3 b^2 c^2 - 4 a^2 b^3 c^2 + a b^4 c^2 + 3 b^5 c^2 + 6 a^4 c^3 + 4 a^3 b c^3 - 4 a^2 b^2 c^3 - 2 a b^3 c^3 + 3 a^3 c^4 + 4 a^2 b c^4 + a b^2 c^4 + a b c^5 + 3 b^2 c^5 - a c^6 - b c^6 - 2 c^7 : :

X(31155) lies on these lines:


X(31156) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    (a - b - c) (5 a^3 + 5 a^2 b + a b^2 + b^3 + 5 a^2 c + 4 a b c - b^2 c + a c^2 - b c^2 + c^3) : :

X(31156) lies on these lines:


X(31157) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    4 a^4 - 5 a^2 b^2 + b^4 + 2 a^2 b c - 4 a b^2 c - 5 a^2 c^2 - 4 a b c^2 - 2 b^2 c^2 + c^4 : :

X(31157) lies on these lines:


X(31158) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^5 + 2 a^4 b - a b^4 - 2 b^5 + 2 a^4 c - 4 a^2 b^2 c + 2 b^4 c - 4 a^2 b c^2 + 2 a b^2 c^2 - a c^4 + 2 b c^4 - 2 c^5 : :

X(31158) lies on these lines:


X(31159) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^4 + a^2 b^2 - 2 b^4 - a^2 b c + 2 a b^2 c + a^2 c^2 + 2 a b c^2 + 4 b^2 c^2 - 2 c^4 : :

X(31159) lies on these lines:


X(31160) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^4 + a^2 b^2 - 2 b^4 + a^2 b c - 2 a b^2 c + a^2 c^2 - 2 a b c^2 + 4 b^2 c^2 - 2 c^4 : :

X(31160) lies on these lines:


X(31161) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    2 a^2 b - a b^2 + 2 a^2 c + 2 b^2 c - a c^2 + 2 b c^2 : :

X(31161) lies on these lines:


X(31162) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^4 + 3 a^3 b + a^2 b^2 - 3 a b^3 - 2 b^4 + 3 a^3 c - 6 a^2 b c + 3 a b^2 c + a^2 c^2 + 3 a b c^2 + 4 b^2 c^2 - 3 a c^3 - 2 c^4 : :

X(31162) lies on these lines:


X(31163) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^5 - a^3 b^2 + 2 a^2 b^3 - 2 b^5 - a^3 c^2 + 2 b^3 c^2 + 2 a^2 c^3 + 2 b^2 c^3 - 2 c^5 : :

X(31163) lies on these lines:


X(31164) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^3 + 2 a^2 b - a b^2 - 2 b^3 + 2 a^2 c + 2 b^2 c - a c^2 + 2 b c^2 - 2 c^3 : :

X(31164) lies on these lines:


X(31165) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a (3 a^2 b - 3 b^3 + 3 a^2 c - 4 a b c - b^2 c - b c^2 - 3 c^3) : :

X(31165) lies on these lines:


X(31166) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    5 a^8 - 4 a^4 b^4 - b^8 - 4 a^4 c^4 + 2 b^4 c^4 - c^8 : :

X(31166) lies on these lines:


X(31167) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^5 + a^3 b^2 - 2 a^2 b^3 - 2 b^5 - 2 a^2 b^2 c + a^3 c^2 - 2 a^2 b c^2 + a b^2 c^2 - 2 b^3 c^2 - 2 a^2 c^3 - 2 b^2 c^3 - 2 c^5 : :

X(31167) lies on these lines:


X(31168) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^4 - 3 a^2 b^2 - 2 b^4 - 3 a^2 c^2 - 3 b^2 c^2 - 2 c^4 : :

X(31168) lies on these lines:


X(31169) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    2 a^3 b - 4 a^2 b^2 + 2 a b^3 + 2 a^3 c + a^2 b c - 2 a b^2 c - b^3 c - 4 a^2 c^2 - 2 a b c^2 + 2 b^2 c^2 + 2 a c^3 - b c^3 : :

X(31169) lies on these lines:


X(31170) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^3 b^2 - 2 a^2 b^3 - 2 a^3 b c + 2 a^2 b^2 c + 4 a b^3 c + a^3 c^2 + 2 a^2 b c^2 - a b^2 c^2 - 2 b^3 c^2 - 2 a^2 c^3 + 4 a b c^3 - 2 b^2 c^3 : :

X(31170) lies on these lines:


X(31171) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^3 + 3 a^2 b - 2 b^3 + 3 a^2 c - 15 a b c + 6 b^2 c + 6 b c^2 - 2 c^3 : :

X(31171) lies on these lines:


X(31172) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    4 a^3 + 6 a^2 b - 6 a b^2 - 8 b^3 + 6 a^2 c - 15 a b c - 6 a c^2 - 8 c^3 : :

X(31172) lies on these lines:


X(31173) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    2 a^4 + a^2 b^2 - 4 b^4 + a^2 c^2 + 4 b^2 c^2 - 4 c^4 : :

X(31173) lies on these lines:


X(31174) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    (b^2 - c^2) (a^4 - a^2 b^2 - a^2 c^2 + 4 b^2 c^2) : :

X(31174) lies on these lines:


X(31175) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^5 - a^3 b^2 + 2 a^2 b^3 - 2 b^5 - 2 a^2 b^2 c - a^3 c^2 - 2 a^2 b c^2 + a b^2 c^2 + 2 b^3 c^2 + 2 a^2 c^3 + 2 b^2 c^3 - 2 c^5 : :

X(31175) lies on these lines:


X(31176) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    (b^2 - c^2) (-a^4 - 2 a^2 b^2 - 2 a^2 c^2 + 2 b^2 c^2) : :

X(31176) lies on these lines:


X(31177) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    2 a^3 + 2 a^2 b - a b^2 - 4 b^3 + 2 a^2 c + 2 b^2 c - a c^2 + 2 b c^2 - 4 c^3 : :

X(31177) lies on these lines:


X(31178) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    2 a^2 b - a b^2 + 2 a^2 c + 3 a b c + 2 b^2 c - a c^2 + 2 b c^2 : :

X(31178) lies on these lines:


X(31179) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^3 + 4 a^2 b + a b^2 - 2 b^3 + 4 a^2 c + b^2 c + a c^2 + b c^2 - 2 c^3 : :

X(31179) lies on these lines:


X(31180) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(24), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^10 - 4 a^6 b^4 + 2 a^4 b^6 + 3 a^2 b^8 - 2 b^10 - 6 a^6 b^2 c^2 + 6 a^4 b^4 c^2 - 6 a^2 b^6 c^2 + 6 b^8 c^2 - 4 a^6 c^4 + 6 a^4 b^2 c^4 + 6 a^2 b^4 c^4 - 4 b^6 c^4 + 2 a^4 c^6 - 6 a^2 b^2 c^6 - 4 b^4 c^6 + 3 a^2 c^8 + 6 b^2 c^8 - 2 c^10 : :

X(31180) lies on these lines:


X(31181) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(26), WHERE A'B'C' = GEMINI TRIANGLE 107

Barycentrics    a^10 - 4 a^6 b^4 + 2 a^4 b^6 + 3 a^2 b^8 - 2 b^10 - 8 a^6 b^2 c^2 + 4 a^4 b^4 c^2 - 2 a^2 b^6 c^2 + 6 b^8 c^2 - 4 a^6 c^4 + 4 a^4 b^2 c^4 - 2 a^2 b^4 c^4 - 4 b^6 c^4 + 2 a^4 c^6 - 2 a^2 b^2 c^6 - 4 b^4 c^6 + 3 a^2 c^8 + 6 b^2 c^8 - 2 c^10 : :

X(31181) lies on these lines:


X(31182) = (name pending)

Barycentrics    (b - c) (3a - b - c)^2 : :

X(31182) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 108: X(31183) - X(31234)  rightri

Extending the preambles just before X(24537), X(26153), X(27378), X(30738), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 108, as in centers X(31183)-X(31234). Then

m(X) = (3a - b - c) x - 2(a - b + c) y - 2(a + b - c) z : :

and m(X) is a self-inverse mapping such that m(X) is collinear with X(2) and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. The fixed points of m are X(2) and every point on the line X(241)X(514). Among the fixed points are X(i) for these i: 2, 241, 514, 650, 1323, 3008, 3676,3911,30719, 31182, 31182. (Clark Kimberling, January 23, 2019)

underbar




X(31183) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    3 a^2 - 3 a b + 2 b^2 - 3 a c - 4 b c + 2 c^2 : :

X(31133) lies on these lines:


X(31184) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    3 a^5 - a^4 b - a^3 b^2 - a^2 b^3 - 2 a b^4 + 2 b^5 - a^4 c + 3 a^2 b^2 c - 2 b^4 c - a^3 c^2 + 3 a^2 b c^2 + 4 a b^2 c^2 - a^2 c^3 - 2 a c^4 - 2 b c^4 + 2 c^5 : :

X(31184) lies on these lines:


X(31185) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    7 a^5 - a^4 b - 4 a^3 b^2 - 4 a^2 b^3 - 3 a b^4 + 5 b^5 - a^4 c + 4 a^2 b^2 c - 3 b^4 c - 4 a^3 c^2 + 4 a^2 b c^2 + 6 a b^2 c^2 - 2 b^3 c^2 - 4 a^2 c^3 - 2 b^2 c^3 - 3 a c^4 - 3 b c^4 + 5 c^5 : :

X(31185) lies on these lines:


X(31186) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    4 a^5 - 3 a^3 b^2 - 3 a^2 b^3 - a b^4 + 3 b^5 + a^2 b^2 c - b^4 c - 3 a^3 c^2 + a^2 b c^2 + 2 a b^2 c^2 - 2 b^3 c^2 - 3 a^2 c^3 - 2 b^2 c^3 - a c^4 - b c^4 + 3 c^5 : :

X(31186) lies on these lines:


X(31187) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    3 a^3 - a^2 b - 2 a b^2 + 2 b^3 - a^2 c - 2 b^2 c - 2 a c^2 - 2 b c^2 + 2 c^3 : :

X(31187) lies on these lines:


X(31188) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    (7 a - 5 b - 5 c) (a + b - c) (a - b + c) : :

X(31188) lies on these lines:


X(31189) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    7 a^2 - 4 a b + 5 b^2 - 4 a c - 6 b c + 5 c^2 : :

X(31189) lies on these lines:


X(31190) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    3 a^3 - 2 a^2 b - 3 a b^2 + 2 b^3 - 2 a^2 c + 10 a b c - 2 b^2 c - 3 a c^2 - 2 b c^2 + 2 c^3 : :

X(31190) lies on these lines:


X(31191) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    4 a^2 - a b + 3 b^2 - a c - 2 b c + 3 c^2 : :

X(31191) lies on these lines:


X(31192) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    4 a^4 - 4 a^3 b + a^2 b^2 - 4 a b^3 + 3 b^4 - 4 a^3 c + 2 a^2 b c + 4 a b^2 c - 4 b^3 c + a^2 c^2 + 4 a b c^2 + 2 b^2 c^2 - 4 a c^3 - 4 b c^3 + 3 c^4 : :

X(31192) lies on these lines:


X(31193) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    3 a^7 - a^6 b + 2 a^5 b^2 - 2 a^4 b^3 - 3 a^3 b^4 + a^2 b^5 - 2 a b^6 + 2 b^7 - a^6 c + 2 a^4 b^2 c + a^2 b^4 c - 2 b^6 c + 2 a^5 c^2 + 2 a^4 b c^2 + 2 a b^4 c^2 + 2 b^5 c^2 - 2 a^4 c^3 - 2 b^4 c^3 - 3 a^3 c^4 + a^2 b c^4 + 2 a b^2 c^4 - 2 b^3 c^4 + a^2 c^5 + 2 b^2 c^5 - 2 a c^6 - 2 b c^6 + 2 c^7 : :

X(31193) lies on these lines:


X(31194) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    3 a^7 - a^6 b + 2 a^5 b^2 - 2 a^4 b^3 - 3 a^3 b^4 + a^2 b^5 - 2 a b^6 + 2 b^7 - a^6 c + 2 a^4 b^2 c + a^2 b^4 c - 2 b^6 c + 2 a^5 c^2 + 2 a^4 b c^2 - a^3 b^2 c^2 - a^2 b^3 c^2 + 2 a b^4 c^2 + 2 b^5 c^2 - 2 a^4 c^3 - a^2 b^2 c^3 - 2 b^4 c^3 - 3 a^3 c^4 + a^2 b c^4 + 2 a b^2 c^4 - 2 b^3 c^4 + a^2 c^5 + 2 b^2 c^5 - 2 a c^6 - 2 b c^6 + 2 c^7 : :

X(31194) lies on these lines:


X(31195) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    3 a^4 - a^3 b - 2 a b^3 + 2 b^4 - a^3 c - 2 b^3 c - 2 a c^3 - 2 b c^3 + 2 c^4 : :

X(31195) lies on these lines:


X(31196) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    3 a^5 - a^4 b - 2 a b^4 + 2 b^5 - a^4 c - 2 b^4 c - 2 a c^4 - 2 b c^4 + 2 c^5 : :

X(31196) lies on these lines:


X(31197) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    a (a b + b^2 + a c - 10 b c + c^2) : :

X(31197) lies on these lines:


X(31198) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    a (a^2 b^2 + a b^3 - 3 a b^2 c + a^2 c^2 - 3 a b c^2 - 4 b^2 c^2 + a c^3) : :

X(31198) lies on these lines:


X(31199) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    3 a^3 b - 3 a^2 b^2 + 2 a b^3 + 3 a^3 c - 2 a^2 b c - 4 a b^2 c + 2 b^3 c - 3 a^2 c^2 - 4 a b c^2 - 4 b^2 c^2 + 2 a c^3 + 2 b c^3 : :

X(31199) lies on these lines:


X(31200) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    3 a^3 b - 3 a^2 b^2 + 2 a b^3 + 3 a^3 c - a^2 b c - 3 a b^2 c + 2 b^3 c - 3 a^2 c^2 - 3 a b c^2 - 4 b^2 c^2 + 2 a c^3 + 2 b c^3 : :

X(31200) lies on these lines:


X(31201) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    6 a^3 - 3 a^2 b - 5 a b^2 + 4 b^3 - 3 a^2 c + 10 a b c - 4 b^2 c - 5 a c^2 - 4 b c^2 + 4 c^3 : :

X(31201) lies on these lines:


X(31202) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    3 a^3 - 3 a^2 b - 4 a b^2 + 2 b^3 - 3 a^2 c + 20 a b c - 2 b^2 c - 4 a c^2 - 2 b c^2 + 2 c^3 : :

X(31202) lies on these lines:


X(31203) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    3 a^4 - 4 a^3 b + 3 a^2 b^2 - 4 a b^3 + 2 b^4 - 4 a^3 c + 2 a^2 b c + 4 a b^2 c - 4 b^3 c + 3 a^2 c^2 + 4 a b c^2 + 4 b^2 c^2 - 4 a c^3 - 4 b c^3 + 2 c^4 : :

X(31203) lies on these lines:


X(31204) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    3 a^3 - a^2 b - 2 a b^2 + 2 b^3 - a^2 c - a b c - 2 b^2 c - 2 a c^2 - 2 b c^2 + 2 c^3 : :

X(31204) lies on these lines:


X(31205) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    3 a^3 - 2 a^2 b - 3 a b^2 + 2 b^3 - 2 a^2 c - 3 a b c - 3 b^2 c - 3 a c^2 - 3 b c^2 + 2 c^3 : :

X(31205) lies on these lines:


X(31206) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(24), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    3 a^11 - a^10 b - 4 a^9 b^2 - 4 a^7 b^4 + 4 a^6 b^5 + 6 a^5 b^6 - 2 a^4 b^7 + a^3 b^8 - 3 a^2 b^9 - 2 a b^10 + 2 b^11 - a^10 c + 4 a^8 b^2 c - 4 a^6 b^4 c - 2 a^4 b^6 c + 5 a^2 b^8 c - 2 b^10 c - 4 a^9 c^2 + 4 a^8 b c^2 - 2 a^7 b^2 c^2 - 2 a^6 b^3 c^2 + 2 a^5 b^4 c^2 + 2 a^4 b^5 c^2 - 2 a^3 b^6 c^2 - 2 a^2 b^7 c^2 + 6 a b^8 c^2 - 2 b^9 c^2 - 2 a^6 b^2 c^3 + 2 a^4 b^4 c^3 - 2 a^2 b^6 c^3 + 2 b^8 c^3 - 4 a^7 c^4 - 4 a^6 b c^4 + 2 a^5 b^2 c^4 + 2 a^4 b^3 c^4 + 2 a^3 b^4 c^4 + 2 a^2 b^5 c^4 - 4 a b^6 c^4 - 4 b^7 c^4 + 4 a^6 c^5 + 2 a^4 b^2 c^5 + 2 a^2 b^4 c^5 + 4 b^6 c^5 + 6 a^5 c^6 - 2 a^4 b c^6 - 2 a^3 b^2 c^6 - 2 a^2 b^3 c^6 - 4 a b^4 c^6 + 4 b^5 c^6 - 2 a^4 c^7 - 2 a^2 b^2 c^7 - 4 b^4 c^7 + a^3 c^8 + 5 a^2 b c^8 + 6 a b^2 c^8 + 2 b^3 c^8 - 3 a^2 c^9 - 2 b^2 c^9 - 2 a c^10 - 2 b c^10 + 2 c^11 : :

X(31206) lies on these lines:


X(31207) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    (b - c) (3 a^2 - 2 a b - 2 a c + 2 b c) : :

X(31207) lies on these lines:


X(31208) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    (b - c) (3 a^4 - a^3 b + 2 a^2 b^2 - 2 a b^3 - a^3 c + 2 a^2 b c - 2 a b^2 c + 2 b^3 c + 2 a^2 c^2 - 2 a b c^2 - 2 a c^3 + 2 b c^3) : :

X(31208) lies on these lines:


X(31209) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    (b - c) (2 a^2 - 2 a b - 2 a c + b c) : :

X(31209) lies on these lines:


X(31210) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    3 a^4 - a^3 b - 2 a b^3 + 2 b^4 - a^3 c + a^2 b c + a b^2 c - 2 b^3 c + a b c^2 - 2 a c^3 - 2 b c^3 + 2 c^4 : :

X(31210) lies on these lines:


X(31211) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    2 a^2 - 5 a b + b^2 - 5 a c - 6 b c + c^2 : :

X(31211) lies on these lines:


X(31212) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    13 a^5 - 3 a^4 b - 6 a^3 b^2 - 6 a^2 b^3 - 7 a b^4 + 9 b^5 - 3 a^4 c + 10 a^2 b^2 c - 7 b^4 c - 6 a^3 c^2 + 10 a^2 b c^2 + 14 a b^2 c^2 - 2 b^3 c^2 - 6 a^2 c^3 - 2 b^2 c^3 - 7 a c^4 - 7 b c^4 + 9 c^5 : :

X(31212) lies on these lines:


X(31213) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    3 a^5 - a^4 b - a^3 b^2 - a^2 b^3 - 2 a b^4 + 2 b^5 - a^4 c + a^3 b c + 5 a^2 b^2 c + a b^3 c - 2 b^4 c - a^3 c^2 + 5 a^2 b c^2 + 6 a b^2 c^2 - a^2 c^3 + a b c^3 - 2 a c^4 - 2 b c^4 + 2 c^5 : :

X(31213) lies on these lines:


X(31214) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    3 a^7 - a^6 b + 2 a^5 b^2 - 2 a^4 b^3 - 3 a^3 b^4 + a^2 b^5 - 2 a b^6 + 2 b^7 - a^6 c + 2 a^4 b^2 c + a^2 b^4 c - 2 b^6 c + 2 a^5 c^2 + 2 a^4 b c^2 - 2 a^3 b^2 c^2 - 2 a^2 b^3 c^2 + 2 a b^4 c^2 + 2 b^5 c^2 - 2 a^4 c^3 - 2 a^2 b^2 c^3 - 2 b^4 c^3 - 3 a^3 c^4 + a^2 b c^4 + 2 a b^2 c^4 - 2 b^3 c^4 + a^2 c^5 + 2 b^2 c^5 - 2 a c^6 - 2 b c^6 + 2 c^7 : :

X(31214) lies on these lines:


X(31215) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    3 a^6 + 3 a^5 b - 2 a^4 b^2 - 4 a^3 b^3 - 3 a^2 b^4 + a b^5 + 2 b^6 + 3 a^5 c - a^4 b c - 3 a b^4 c + b^5 c - 2 a^4 c^2 + 6 a^2 b^2 c^2 + 2 a b^3 c^2 - 2 b^4 c^2 - 4 a^3 c^3 + 2 a b^2 c^3 - 2 b^3 c^3 - 3 a^2 c^4 - 3 a b c^4 - 2 b^2 c^4 + a c^5 + b c^5 + 2 c^6 : :

X(31215) lies on these lines:


X(31216) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    3 a^8 + 2 a^7 b + a^6 b^2 - 5 a^4 b^4 - 2 a^3 b^5 - a^2 b^6 + 2 b^8 + 2 a^7 c + 5 a^6 b c + 3 a^5 b^2 c - 4 a^4 b^3 c - 6 a^3 b^4 c - a^2 b^5 c + a b^6 c + a^6 c^2 + 3 a^5 b c^2 + 2 a^4 b^2 c^2 + a^2 b^4 c^2 + a b^5 c^2 - 4 a^4 b c^3 + 2 a^2 b^3 c^3 - 2 a b^4 c^3 - 5 a^4 c^4 - 6 a^3 b c^4 + a^2 b^2 c^4 - 2 a b^3 c^4 - 4 b^4 c^4 - 2 a^3 c^5 - a^2 b c^5 + a b^2 c^5 - a^2 c^6 + a b c^6 + 2 c^8 : :

X(31216) lies on these lines:


X(31217) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    3 a^8 - 5 a^7 b - 5 a^6 b^2 + 5 a^5 b^3 + 3 a^4 b^4 + 5 a^3 b^5 - 3 a^2 b^6 - 5 a b^7 + 2 b^8 - 5 a^7 c - 7 a^6 b c + 3 a^5 b^2 c + 5 a^4 b^3 c + 5 a^3 b^4 c + 7 a^2 b^5 c - 3 a b^6 c - 5 b^7 c - 5 a^6 c^2 + 3 a^5 b c^2 + 4 a^4 b^2 c^2 - 10 a^3 b^3 c^2 + 3 a^2 b^4 c^2 + 7 a b^5 c^2 - 2 b^6 c^2 + 5 a^5 c^3 + 5 a^4 b c^3 - 10 a^3 b^2 c^3 - 14 a^2 b^3 c^3 + a b^4 c^3 + 5 b^5 c^3 + 3 a^4 c^4 + 5 a^3 b c^4 + 3 a^2 b^2 c^4 + a b^3 c^4 + 5 a^3 c^5 + 7 a^2 b c^5 + 7 a b^2 c^5 + 5 b^3 c^5 - 3 a^2 c^6 - 3 a b c^6 - 2 b^2 c^6 - 5 a c^7 - 5 b c^7 + 2 c^8 : :

X(31217) lies on these lines:


X(31218) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    19 a^5 - 5 a^4 b - 8 a^3 b^2 - 8 a^2 b^3 - 11 a b^4 + 13 b^5 - 5 a^4 c + 16 a^2 b^2 c - 11 b^4 c - 8 a^3 c^2 + 16 a^2 b c^2 + 22 a b^2 c^2 - 2 b^3 c^2 - 8 a^2 c^3 - 2 b^2 c^3 - 11 a c^4 - 11 b c^4 + 13 c^5 : :

X(31218) lies on these lines:


X(31219) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    7 a^5 - a^4 b - 4 a^3 b^2 - 4 a^2 b^3 - 3 a b^4 + 5 b^5 - a^4 c - 2 a^3 b c - 2 a b^3 c - 3 b^4 c - 4 a^3 c^2 + 2 a b^2 c^2 - 2 b^3 c^2 - 4 a^2 c^3 - 2 a b c^3 - 2 b^2 c^3 - 3 a c^4 - 3 b c^4 + 5 c^5 : :

X(31219) lies on these lines:


X(31220) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    3 a^5 - a^4 b - 2 a^3 b^2 - 2 a^2 b^3 - 2 a b^4 + 2 b^5 - a^4 c + 2 a^2 b^2 c - 2 b^4 c - 2 a^3 c^2 + 2 a^2 b c^2 + 3 a b^2 c^2 - b^3 c^2 - 2 a^2 c^3 - b^2 c^3 - 2 a c^4 - 2 b c^4 + 2 c^5 : :

X(31220) lies on these lines:


X(31221) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    (a + b - c) (a - b + c) (4 a^3 + a b^2 - 3 b^3 - 2 a b c - 5 b^2 c + a c^2 - 5 b c^2 - 3 c^3) : :

X(31221) lies on these lines:


X(31222) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    3 a^5 - a^4 b - a^3 b^2 - a^2 b^3 - 2 a b^4 + 2 b^5 - a^4 c + 3 a^3 b c + 2 a b^3 c - 2 b^4 c - a^3 c^2 - a^2 c^3 + 2 a b c^3 - 2 a c^4 - 2 b c^4 + 2 c^5 : :

X(31222) lies on these lines:


X(31223) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    2 a^3 b - 5 a^2 b^2 + a b^3 + 2 a^3 c + 4 a^2 b c + 3 a b^2 c + 2 b^3 c - 5 a^2 c^2 + 3 a b c^2 - 4 b^2 c^2 + a c^3 + 2 b c^3 : :

X(31223) lies on these lines:


X(31224) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    3 a^3 - 2 a^2 b - 3 a b^2 + 2 b^3 - 2 a^2 c + 8 a b c - 2 b^2 c - 3 a c^2 - 2 b c^2 + 2 c^3 : :

X(31224) lies on these lines:


X(31225) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    (a + b - c) (a - b + c) (2 a^2 b - 2 a b^2 + 2 a^2 c - a b c - b^2 c - 2 a c^2 - b c^2) : :

X(31225) lies on these lines:


X(31226) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    3 a^4 - 4 a^3 b + 3 a^2 b^2 - 4 a b^3 + 2 b^4 - 4 a^3 c + a^2 b c + 3 a b^2 c - 4 b^3 c + 3 a^2 c^2 + 3 a b c^2 + 4 b^2 c^2 - 4 a c^3 - 4 b c^3 + 2 c^4 : :

X(31226) lies on these lines:


X(31227) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    (a - 2 b + c) (a + b - 2 c) (3 a - b - c) : :

X(31227) lies on these lines:


X(31228) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    3 a^2 b + 3 a b^2 + 3 a^2 c - 17 a b c + b^2 c + 3 a c^2 + b c^2 : :

X(31228) lies on these lines:


X(31229) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    3 a^3 - a b^2 + 2 b^3 - b^2 c - a c^2 - b c^2 + 2 c^3 : :

X(31229) lies on these lines:


X(31230) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    (a + b - c) (a - b + c) (3 a^3 - a^2 b + 2 a b^2 - 2 b^3 - a^2 c - 2 b^2 c + 2 a c^2 - 2 b c^2 - 2 c^3) : :

X(31230) lies on these lines:


X(31231) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    (3 a - 2 b - 2 c) (a + b - c) (a - b + c) : :

X(31231) lies on these lines:


X(31232) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    7 a^3 - a^2 b - 3 a b^2 + 5 b^3 - a^2 c - 3 b^2 c - 3 a c^2 - 3 b c^2 + 5 c^3 : :

X(31232) lies on these lines:


X(31233) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    2 a^2 b + 2 a b^2 + 2 a^2 c - 7 a b c + b^2 c + 2 a c^2 + b c^2 : :

X(31233) lies on these lines:


X(31234) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 108

Barycentrics    2 a^3 b^2 + 2 a^2 b^3 - 2 a^2 b^2 c + 2 a^3 c^2 - 2 a^2 b c^2 - 3 a b^2 c^2 + b^3 c^2 + 2 a^2 c^3 + b^2 c^3 : :

X(31234) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 109: X(31235) - X(31283)  rightri

Extending the preambles just before X(24537), X(26153), X(27378), X(30738), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 109, as in centers X(31235)-X(31283). Then

m(X) = x + 2 y + 2 z : :

and m(X) is collinear with X(2) and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. (Clark Kimberling, January 24, 2019)

underbar




X(31235) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    4 a^3 - 4 a^2 b - 3 a b^2 + 3 b^3 - 4 a^2 c + 10 a b c - 3 b^2 c - 3 a c^2 - 3 b c^2 + 3 c^3 : :

X(31235) lies on these lines:


X(31236) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^6 - 2 a^4 b^2 - a^2 b^4 + 2 b^6 - 2 a^4 c^2 - 2 b^4 c^2 - a^2 c^4 - 2 b^2 c^4 + 2 c^6 : :

X(31236) lies on these lines:


X(31237) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^3 + 2 b^3 + 2 c^3 : :

X(31237) lies on these lines:


X(31238) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    3 a b + 3 a c + 4 b c : :

X(31238) lies on these lines:


X(31239) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    3 a^2 b^2 + 3 a^2 c^2 + 4 b^2 c^2 : :

X(31239) lies on these lines:


X(31240) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^4 - a^3 b - 2 a b^3 + 2 b^4 - a^3 c - 2 b^3 c - 2 a c^3 - 2 b c^3 + 2 c^4 : :

X(31240) lies on these lines:


X(31241) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^2 b + 2 a b^2 + a^2 c + 2 b^2 c + 2 a c^2 + 2 b c^2 : :

X(31241) lies on these lines:


X(31242) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^2 b + 2 a b^2 + a^2 c - 5 a b c + 2 b^2 c + 2 a c^2 + 2 b c^2 : :

X(31242) lies on these lines:


X(31243) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    2 a^2 - 3 a b + 4 b^2 - 3 a c - 4 b c + 4 c^2 : :

X(31243) lies on these lines:


X(31244) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^2 - 6 a b + 2 b^2 - 6 a c - 8 b c + 2 c^2 : :

X(31244) lies on these lines:


X(31245) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^3 - a^2 b - 2 a b^2 + 2 b^3 - a^2 c - 2 b^2 c - 2 a c^2 - 2 b c^2 + 2 c^3 : :

X(31245) lies on these lines:


X(31246) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^4 - 3 a^2 b^2 + 2 b^4 + 2 a^2 b c + 4 a b^2 c - 3 a^2 c^2 + 4 a b c^2 - 4 b^2 c^2 + 2 c^4 : :

X(31246) lies on these lines:


X(31247) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^3 + a^2 b + 2 a b^2 + 2 b^3 + a^2 c + 5 a b c + 2 b^2 c + 2 a c^2 + 2 b c^2 + 2 c^3 : :

X(31247) lies on these lines:


X(31248) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^2 + 5 a b + 2 b^2 + 5 a c + 5 b c + 2 c^2 : :

X(31248) lies on these lines:


X(31249) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^3 - 3 a b^2 + 2 b^3 + 10 a b c - 2 b^2 c - 3 a c^2 - 2 b c^2 + 2 c^3 : :

X(31249) lies on these lines:


X(31250) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    (b - c) (3 a^2 - 3 a b - 3 a c + 4 b c) : :

X(31250) lies on these lines:


X(31251) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    (b - c) (a^3 - 2 a b^2 - 2 a b c + 2 b^2 c - 2 a c^2 + 2 b c^2) : :

X(31251) lies on these lines:


X(31252) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^3 + 2 b^3 - 5 a b c + 2 c^3 : :

X(31252) lies on these lines:


X(31253) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    6 a + 7 b + 7 c : :

X(31253) lies on these lines:


X(31254) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^4 - 3 a^2 b^2 + 2 b^4 - 5 a^2 b c - 5 a b^2 c - 3 a^2 c^2 - 5 a b c^2 - 4 b^2 c^2 + 2 c^4 : :

X(31254) lies on these lines:


X(31255) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^6 - 2 a^4 b^2 - a^2 b^4 + 2 b^6 - 2 a^4 c^2 + 10 a^2 b^2 c^2 - 2 b^4 c^2 - a^2 c^4 - 2 b^2 c^4 + 2 c^6 : :

X(31255) lies on these lines:


X(31256) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^6 - 3 a^5 b - 2 a^4 b^2 + 4 a^3 b^3 - a^2 b^4 - a b^5 + 2 b^6 - 3 a^5 c - 3 a^4 b c + 4 a^3 b^2 c + 4 a^2 b^3 c - a b^4 c - b^5 c - 2 a^4 c^2 + 4 a^3 b c^2 + 10 a^2 b^2 c^2 + 2 a b^3 c^2 - 2 b^4 c^2 + 4 a^3 c^3 + 4 a^2 b c^3 + 2 a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 - a b c^4 - 2 b^2 c^4 - a c^5 - b c^5 + 2 c^6 : :

X(31256) lies on these lines:


X(31257) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^7 + a^6 b - 2 a^5 b^2 - 2 a^4 b^3 - a^3 b^4 - a^2 b^5 + 2 a b^6 + 2 b^7 + a^6 c - 3 a^5 b c - 2 a^4 b^2 c + 4 a^3 b^3 c - a^2 b^4 c - a b^5 c + 2 b^6 c - 2 a^5 c^2 - 2 a^4 b c^2 + 10 a^3 b^2 c^2 + 10 a^2 b^3 c^2 - 2 a b^4 c^2 - 2 b^5 c^2 - 2 a^4 c^3 + 4 a^3 b c^3 + 10 a^2 b^2 c^3 + 2 a b^3 c^3 - 2 b^4 c^3 - a^3 c^4 - a^2 b c^4 - 2 a b^2 c^4 - 2 b^3 c^4 - a^2 c^5 - a b c^5 - 2 b^2 c^5 + 2 a c^6 + 2 b c^6 + 2 c^7 : :

X(31257) lies on these lines:


X(31258) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^7 + 4 a^6 b + a^5 b^2 - 6 a^4 b^3 - 5 a^3 b^4 + 3 a b^6 + 2 b^7 + 4 a^6 c + 3 a^5 b c - 3 a^4 b^2 c - 4 a^3 b^3 c - 4 a^2 b^4 c + a b^5 c + 3 b^6 c + a^5 c^2 - 3 a^4 b c^2 + 2 a^3 b^2 c^2 + 4 a^2 b^3 c^2 - 3 a b^4 c^2 - b^5 c^2 - 6 a^4 c^3 - 4 a^3 b c^3 + 4 a^2 b^2 c^3 - 2 a b^3 c^3 - 4 b^4 c^3 - 5 a^3 c^4 - 4 a^2 b c^4 - 3 a b^2 c^4 - 4 b^3 c^4 + a b c^5 - b^2 c^5 + 3 a c^6 + 3 b c^6 + 2 c^7 : :

X(31258) lies on these lines:


X(31259) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    3 a^4 - 4 a^2 b^2 + b^4 - 10 a^2 b c - 10 a b^2 c - 4 a^2 c^2 - 10 a b c^2 - 2 b^2 c^2 + c^4 : :

X(31259) lies on these lines:


X(31260) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    4 a^4 - 7 a^2 b^2 + 3 b^4 - 2 a^2 b c - 4 a b^2 c - 7 a^2 c^2 - 4 a b c^2 - 6 b^2 c^2 + 3 c^4 : :

X(31260) lies on these lines:


X(31261) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^5 - 2 a^4 b - a b^4 + 2 b^5 - 2 a^4 c + 4 a^2 b^2 c - 2 b^4 c + 4 a^2 b c^2 + 2 a b^2 c^2 - a c^4 - 2 b c^4 + 2 c^5 : :

X(31261) lies on these lines:


X(31262) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^4 - 3 a^2 b^2 + 2 b^4 - a^2 b c - 2 a b^2 c - 3 a^2 c^2 - 2 a b c^2 - 4 b^2 c^2 + 2 c^4 : :

X(31262) lies on these lines:


X(31263) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^4 - 3 a^2 b^2 + 2 b^4 + a^2 b c + 2 a b^2 c - 3 a^2 c^2 + 2 a b c^2 - 4 b^2 c^2 + 2 c^4 : :

X(31263) lies on these lines:


X(31264) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    2 a^2 b + a b^2 + 2 a^2 c + 2 b^2 c + a c^2 + 2 b c^2 : :

X(31264) lies on these lines:


X(31265) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(48), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^5 - a^3 b^2 - 2 a^2 b^3 + 2 b^5 - a^3 c^2 - 2 b^3 c^2 - 2 a^2 c^3 - 2 b^2 c^3 + 2 c^5 : :

X(31265) lies on these lines:


X(31266) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^3 - 2 a^2 b - a b^2 + 2 b^3 - 2 a^2 c - 2 b^2 c - a c^2 - 2 b c^2 + 2 c^3 : :

X(31266) lies on these lines:


X(31267) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    3 a^8 - 4 a^4 b^4 + b^8 - 4 a^4 c^4 - 2 b^4 c^4 + c^8 : :

X(31267) lies on these lines:


X(31268) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^4 + 5 a^2 b^2 + 2 b^4 + 5 a^2 c^2 + 5 b^2 c^2 + 2 c^4 : :

X(31268) lies on these lines:


X(31269) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    2 a^3 b - 4 a^2 b^2 + 2 a b^3 + 2 a^3 c - a^2 b c - 2 a b^2 c + b^3 c - 4 a^2 c^2 - 2 a b c^2 - 2 b^2 c^2 + 2 a c^3 + b c^3 : :

X(31269) lies on these lines:


X(31270) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^3 b^2 + 2 a^2 b^3 - 2 a^3 b c - 2 a^2 b^2 c - 4 a b^3 c + a^3 c^2 - 2 a^2 b c^2 - a b^2 c^2 + 2 b^3 c^2 + 2 a^2 c^3 - 4 a b c^3 + 2 b^2 c^3 : :

X(31270) lies on these lines:


X(31271) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^3 - 5 a^2 b - 4 a b^2 + 2 b^3 - 5 a^2 c + 25 a b c - 6 b^2 c - 4 a c^2 - 6 b c^2 + 2 c^3 : :

X(31271) lies on these lines:


X(31272) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^3 - a^2 b - 2 a b^2 + 2 b^3 - a^2 c + 5 a b c - 2 b^2 c - 2 a c^2 - 2 b c^2 + 2 c^3 : :

X(31272) lies on these lines:


X(31273) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^4 - a^3 b - 2 a b^3 + 2 b^4 - a^3 c + a^2 b c + 2 a b^2 c - 2 b^3 c + 2 a b c^2 - 2 a c^3 - 2 b c^3 + 2 c^4 : :

X(31273) lies on these lines:


X(31274) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    4 a^4 - 4 a^2 b^2 + 3 b^4 - 4 a^2 c^2 - 2 b^2 c^2 + 3 c^4 : :

X(31274) lies on these lines:


X(31275) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    2 a^4 - 3 a^2 b^2 + 4 b^4 - 3 a^2 c^2 - 4 b^2 c^2 + 4 c^4 : :

X(31275) lies on these lines:


X(31276) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^2 b^2 + a^2 c^2 + 3 b^2 c^2 : :

X(31276) lies on these lines:


X(31277) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    (b^2 - c^2) (3 a^4 - 3 a^2 b^2 - 3 a^2 c^2 + 4 b^2 c^2) : :

X(31277) lies on these lines:


X(31278) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^5 - a^3 b^2 - 2 a^2 b^3 + 2 b^5 + 2 a^2 b^2 c - a^3 c^2 + 2 a^2 b c^2 + a b^2 c^2 - 2 b^3 c^2 - 2 a^2 c^3 - 2 b^2 c^3 + 2 c^5 : :

X(31278) lies on these lines:


X(31279) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    (b^2 - c^2) (a^4 - 2 a^2 b^2 - 2 a^2 c^2 + 2 b^2 c^2) : :

X(31279) lies on these lines:


X(31280) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    2 a^3 - 2 a^2 b - a b^2 + 4 b^3 - 2 a^2 c - 2 b^2 c - a c^2 - 2 b c^2 + 4 c^3 : :

X(31280) lies on these lines:


X(31281) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^3 - 4 a^2 b - 3 a b^2 + 2 b^3 - 4 a^2 c - 3 b^2 c - 3 a c^2 - 3 b c^2 + 2 c^3 : :

X(31281) lies on these lines:


X(31282) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(24), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^10 - 4 a^8 b^2 + 4 a^6 b^4 + 2 a^4 b^6 - 5 a^2 b^8 + 2 b^10 - 4 a^8 c^2 + 10 a^6 b^2 c^2 - 10 a^4 b^4 c^2 + 10 a^2 b^6 c^2 - 6 b^8 c^2 + 4 a^6 c^4 - 10 a^4 b^2 c^4 - 10 a^2 b^4 c^4 + 4 b^6 c^4 + 2 a^4 c^6 + 10 a^2 b^2 c^6 + 4 b^4 c^6 - 5 a^2 c^8 - 6 b^2 c^8 + 2 c^10 : :

X(31282) lies on these lines:


X(31283) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(26), WHERE A'B'C' = GEMINI TRIANGLE 109

Barycentrics    a^10 - 4 a^8 b^2 + 4 a^6 b^4 + 2 a^4 b^6 - 5 a^2 b^8 + 2 b^10 - 4 a^8 c^2 + 8 a^6 b^2 c^2 - 4 a^4 b^4 c^2 + 6 a^2 b^6 c^2 - 6 b^8 c^2 + 4 a^6 c^4 - 4 a^4 b^2 c^4 - 2 a^2 b^4 c^4 + 4 b^6 c^4 + 2 a^4 c^6 + 6 a^2 b^2 c^6 + 4 b^4 c^6 - 5 a^2 c^8 - 6 b^2 c^8 + 2 c^10 : :

X(31283) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 110: X(31284) - X(31289)  rightri

Extending the preambles just before X(24537), X(26153), X(27378), X(30738), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 110, as in centers X(31284)-X(31289). Then

m(X) = 2 x + y + z : : = complement(complement(X))

and m(X) is collinear with X(2) and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. (Clark Kimberling, January 25, 2019)

See the preamble just before X(6666).

underbar




X(31284) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 110

Barycentrics    2 a^4 - 2 a^3 b - a b^3 + b^4 - 2 a^3 c - b^3 c - a c^3 - b c^3 + c^4 : :

X(31284) lies on these lines:


X(31285) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 110

Barycentrics    2 a^2 - 6 a b + b^2 - 6 a c - 4 b c + c^2 : :

X(31285) lies on these lines:


X(31286) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 110

Barycentrics    (b - c) (2 a^2 - a b - a c + b c) : :

X(31286) lies on these lines:


X(31287) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 110

Barycentrics    (b - c) (3 a^2 - 3 a b - 3 a c + 2 b c) : :

X(31287) lies on these lines:


X(31288) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 110

Barycentrics    (b - c) (2 a^3 - a b^2 - a b c + b^2 c - a c^2 + b c^2) : :

X(31288) lies on these lines:


X(31289) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 110

Barycentrics    2 a^3 + b^3 - 4 a b c + c^3 : :

X(31289) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 111: X(31290) - X(31305)  rightri

Extending the preambles just before X(24537), X(26153), X(27378), X(30738), let m(X) denote the (A,B,C,X(2); A',B',C',X(2)) collineation image of X = x : y : z, where A'B'C' = Gemini triangle 111, as in centers X(31290)-X(31305). Then

m(X) = 3 x - y - z : : = anticomplement(anticomplement(X))

and m(X) is collinear with X(2) and X; e.g., m(Euler line) = Euler line and m(Nagel line) = Nagel line. (Clark Kimberling, January 25, 2019)

See the preambles just before X(6666) and X(31281).

underbar




X(31290) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics    (b - c) (a^2 + 3 a b + 3 a c + b c) : :

X(31290) lies on these lines:


X(31291) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics    (b - c) (-3 a^3 - a b^2 - a b c + b^2 c - a c^2 + b c^2) : :

X(31291) lies on these lines:


X(31292) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics    3 a^6 + 5 a^5 b + a^4 b^2 - 2 a^3 b^3 - 3 a^2 b^4 - 3 a b^5 - b^6 + 5 a^5 c + 5 a^4 b c - 2 a^3 b^2 c - 2 a^2 b^3 c - 3 a b^4 c - 3 b^5 c + a^4 c^2 - 2 a^3 b c^2 + 2 a^2 b^2 c^2 + 6 a b^3 c^2 + b^4 c^2 - 2 a^3 c^3 - 2 a^2 b c^3 + 6 a b^2 c^3 + 6 b^3 c^3 - 3 a^2 c^4 - 3 a b c^4 + b^2 c^4 - 3 a c^5 - 3 b c^5 - c^6 : :

X(31292) lies on these lines:


X(31293) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics    3 a^7 + 3 a^6 b + a^5 b^2 + a^4 b^3 - 3 a^3 b^4 - 3 a^2 b^5 - a b^6 - b^7 + 3 a^6 c + 5 a^5 b c + a^4 b^2 c - 2 a^3 b^3 c - 3 a^2 b^4 c - 3 a b^5 c - b^6 c + a^5 c^2 + a^4 b c^2 + 2 a^3 b^2 c^2 + 2 a^2 b^3 c^2 + a b^4 c^2 + b^5 c^2 + a^4 c^3 - 2 a^3 b c^3 + 2 a^2 b^2 c^3 + 6 a b^3 c^3 + b^4 c^3 - 3 a^3 c^4 - 3 a^2 b c^4 + a b^2 c^4 + b^3 c^4 - 3 a^2 c^5 - 3 a b c^5 + b^2 c^5 - a c^6 - b c^6 - c^7 : :

X(31293) lies on these lines:


X(31294) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics    3 a^7 - 2 a^6 b - 4 a^5 b^2 + 3 a^4 b^3 - a^3 b^4 + 2 a b^6 - b^7 - 2 a^6 c - 5 a^5 b c - 2 a^4 b^2 c + 2 a^3 b^3 c + 2 a^2 b^4 c + 3 a b^5 c + 2 b^6 c - 4 a^5 c^2 - 2 a^4 b c^2 + 6 a^3 b^2 c^2 - 2 a^2 b^3 c^2 - 2 a b^4 c^2 + 4 b^5 c^2 + 3 a^4 c^3 + 2 a^3 b c^3 - 2 a^2 b^2 c^3 - 6 a b^3 c^3 - 5 b^4 c^3 - a^3 c^4 + 2 a^2 b c^4 - 2 a b^2 c^4 - 5 b^3 c^4 + 3 a b c^5 + 4 b^2 c^5 + 2 a c^6 + 2 b c^6 - c^7 : :

X(31294) lies on these lines:


X(31295) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics    5 a^4 - 2 a^2 b^2 - 3 b^4 + 2 a^2 b c + 2 a b^2 c - 2 a^2 c^2 + 2 a b c^2 + 6 b^2 c^2 - 3 c^4 : :

X(31295) lies on these lines:


X(31296) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics    (b^2 - c^2) (-a^4 + a^2 b^2 + a^2 c^2 + b^2 c^2) : :

X(31296) lies on these lines:


X(31297) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics    3 a^5 - 3 a^3 b^2 + a^2 b^3 - b^5 - a^2 b^2 c - 3 a^3 c^2 - a^2 b c^2 + 3 a b^2 c^2 + b^3 c^2 + a^2 c^3 + b^2 c^3 - c^5 : :

X(31297) lies on these lines:


X(31298) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics    a^2 b^2 - 5 a^2 b c + 3 a b^2 c + a^2 c^2 + 3 a b c^2 - 3 b^2 c^2 : :

X(31298) lies on these lines:


X(31299) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics    (b^2 - c^2) (-3 a^4 - a^2 b^2 - a^2 c^2 + b^2 c^2) : :

X(31299) lies on these lines:


X(31300) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics    3 a^2 - a b - b^2 - a c + 3 b c - c^2 : :

X(31300) lies on these lines:


X(31301) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics    6 a^3 + a^2 b - 3 a b^2 - 2 b^3 + a^2 c + b^2 c - 3 a c^2 + b c^2 - 2 c^3 : :

X(31301) lies on these lines:


X(31302) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics    a^2 b - 3 a b^2 + a^2 c - a b c + b^2 c - 3 a c^2 + b c^2 : :

X(31302) lies on these lines:


X(31303) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics    3 a^3 + 2 a^2 b - 2 a b^2 - b^3 + 2 a^2 c - 2 b^2 c - 2 a c^2 - 2 b c^2 - c^3 : :

X(31303) lies on these lines:


X(31304) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(24), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics    3 a^10 - 5 a^8 b^2 - 2 a^6 b^4 + 6 a^4 b^6 - a^2 b^8 - b^10 - 5 a^8 c^2 + 2 a^6 b^2 c^2 - 2 a^4 b^4 c^2 + 2 a^2 b^6 c^2 + 3 b^8 c^2 - 2 a^6 c^4 - 2 a^4 b^2 c^4 - 2 a^2 b^4 c^4 - 2 b^6 c^4 + 6 a^4 c^6 + 2 a^2 b^2 c^6 - 2 b^4 c^6 - a^2 c^8 + 3 b^2 c^8 - c^10 : :

X(31304) lies on these lines:


X(31305) = (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(26), WHERE A'B'C' = GEMINI TRIANGLE 111

Barycentrics    3 a^10 - 5 a^8 b^2 - 2 a^6 b^4 + 6 a^4 b^6 - a^2 b^8 - b^10 - 5 a^8 c^2 - 4 a^6 b^2 c^2 + 2 a^4 b^4 c^2 + 4 a^2 b^6 c^2 + 3 b^8 c^2 - 2 a^6 c^4 + 2 a^4 b^2 c^4 - 6 a^2 b^4 c^4 - 2 b^6 c^4 + 6 a^4 c^6 + 4 a^2 b^2 c^6 - 2 b^4 c^6 - a^2 c^8 + 3 b^2 c^8 - c^10 : :

X(31305) lies on these lines:






leftri  Perspectors involving Gemini triangles 1 to 111: X(31306) - X(31352)  rightri

This preamble and centers X(31306)-X(31352) were contributed by César Eliud Lozada, January 26, 2019.

The perspector of Gemini triangles i and j is X(2) for i, j ∈ {1, 2, 9, 10, 11, 12, 13, 14, 20, 21, 22, 23, 24, 27, 28, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111}

Also, the appearance of (i, j, k) in the following list means that the perspector of Gemini triangles i and j is X(k):
(1, 8, 21), (1, 25, 1125), (1, 29, 1), (1, 63, 31306), (2, 7, 81), (2, 15, 28606), (2, 30, 63), (3, 4, 86), (3, 5, 2), (3, 6, 1), (3, 7, 1), (3, 16, 2), (3, 18, 2), (3, 19, 86), (3, 23, 1125), (3, 25, 86), (3, 26, 2), (3, 40, 3616), (3, 62, 31307), (3, 63, 31308), (3, 73, 31309), (3, 104, 3720), (4, 5, 9), (4, 6, 192), (4, 8, 9), (4, 17, 192), (4, 19, 86), (4, 25, 86), (4, 63, 31310), (4, 107, 190), (4, 109, 31311), (4, 110, 31312), (4, 111, 31313), (5, 8, 9), (5, 16, 2), (5, 18, 2), (5, 26, 2), (5, 110, 1), (6, 7, 1), (6, 16, 8), (6, 17, 192), (6, 23, 10), (6, 40, 8), (6, 63, 31314), (6, 64, 31315), (6, 104, 42), (7, 15, 2), (7, 16, 24174), (7, 17, 2), (7, 21, 24174), (7, 23, 24161), (7, 25, 2), (7, 30, 88), (7, 60, 330), (7, 110, 4859), (8, 18, 8), (8, 29, 1320), (8, 71, 31316), (9, 63, 31317), (11, 15, 31318), (11, 63, 31319), (12, 15, 31320), (12, 26, 31321), (13, 25, 1), (13, 29, 8), (13, 63, 31322), (14, 63, 31323), (15, 17, 2), (15, 19, 37), (15, 25, 2), (15, 60, 192), (15, 61, 31324), (15, 65, 31325), (15, 108, 31326), (16, 18, 2), (16, 21, 24174), (16, 22, 31327), (16, 23, 1), (16, 26, 2), (16, 40, 8), (16, 62, 31328), (16, 63, 31329), (16, 104, 31330), (17, 25, 2), (17, 105, 3995), (17, 111, 145), (18, 26, 2), (18, 111, 1278), (19, 25, 86), (19, 63, 31331), (19, 107, 31332), (19, 109, 31333), (19, 110, 142), (19, 111, 31334), (20, 63, 29576), (23, 63, 31335), (25, 29, 10), (25, 63, 31336), (26, 32, 5224), (26, 39, 9780), (26, 57, 31337), (26, 68, 31338), (26, 80, 31339), (26, 81, 31340), (26, 104, 31341), (28, 63, 239), (29, 63, 31342), (29, 71, 31343), (29, 111, 20059), (30, 100, 20348), (30, 111, 3621), (31, 63, 31344), (32, 62, 31345), (38, 63, 31346), (40, 63, 31347), (62, 64, 2), (62, 100, 27424), (63, 104, 31348), (63, 107, 31349), (63, 109, 31350), (63, 110, 31351), (63, 111, 31352)

underbar

X(31306) = PERSPECTOR OF THESE TRIANGLES: GEMINI 1 AND GEMINI 63

Barycentrics    3*(b+c)*a^3+2*(b^2+4*b*c+c^2)*a^2+(b+c)*(b^2+4*b*c+c^2)*a+2*(b^2+b*c+c^2)*b*c : :
X(31306) = 3*X(2)+X(31314) = X(319)-7*X(4751) = X(1100)+2*X(3739) = 2*X(17239)-5*X(31238)

X(31306) lies on these lines: {1,27474}, {2,210}, {37,17339}, {75,4470}, {86,239}, {319,4648}, {335,29614}, {872,28254}, {984,29603}, {1125,17755}, {3666,31348}, {3696,4393}, {3775,24603}, {3797,15569}, {4359,18157}, {4384,4649}, {4670,14621}, {4698,29609}, {4699,17014}, {4725,31351}, {4755,31349}, {4883,31027}, {5045,27274}, {10436,20172}, {17023,24325}, {17234,17239}, {17348,20159}, {17769,29574}, {19684,27476}, {28581,29584}

X(31306) = midpoint of X(27495) and X(31314)
X(31306) = complement of X(27495)
X(31306) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 27474, 31342), (2, 27484, 31322), (2, 31314, 27495)


X(31307) = PERSPECTOR OF THESE TRIANGLES: GEMINI 3 AND GEMINI 62

Barycentrics
((b+c)^3*a^7+(2*b^2-b*c+2*c^2)*(b+c)^2*a^6+3*(b^3+c^3)*(b^2+b*c+c^2)*a^5+(2*b^6+2*c^6+(b^4+c^4+(4*b^2+9*b*c+4*c^2)*b*c)*b*c)*a^4+(b+c)*(b^6+c^6+(5*b^2-6*b*c+5*c^2)*b^2*c^2)*a^3+(b^6+c^6+(b+c)^2*b^2*c^2)*b*c*a^2-(b^3+c^3)*(b^2+b*c+c^2)*b^2*c^2*a-(b^2+b*c+c^2)^2*b^3*c^3)*((b-c)*a+b*c)*((b-c)*a-b*c) : :

X(31307) lies on these lines: {24661,27444}, {24669,27429}


X(31308) = PERSPECTOR OF THESE TRIANGLES: GEMINI 3 AND GEMINI 63

Barycentrics    3*(b+c)*a^3+(5*b^2+11*b*c+5*c^2)*a^2+(b+c)*(b^2+4*b*c+c^2)*a-(b^2+b*c+c^2)*b*c : :
X(31308) = 3*X(2)-4*X(31336) = 4*X(37)-X(1654) = 2*X(86)+X(192) = 4*X(1213)-7*X(27268) = 2*X(3993)+X(24342) = 5*X(4699)-8*X(6707) = 5*X(4704)+X(20090) = 4*X(5625)-X(24349)

X(31308) lies on these lines: {1,6651}, {2,740}, {37,319}, {75,28640}, {86,192}, {144,1959}, {524,17488}, {726,29580}, {984,29588}, {1001,4393}, {1125,31335}, {1213,17233}, {3616,31347}, {3661,25354}, {3720,31348}, {3797,15569}, {3993,16826}, {3995,19565}, {4664,4795}, {4699,6707}, {5625,24349}, {5880,6650}, {9791,17316}, {17293,31248}, {17302,27487}, {17499,24051}, {20016,31323}

X(31308) = reflection of X(i) in X(j) for these (i,j): (27483, 31336), (31310, 6651)
X(31308) = anticomplement of X(27483)
X(31308) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 27481, 31314), (27474, 31319, 2), (27483, 31336, 2)


X(31309) = PERSPECTOR OF THESE TRIANGLES: GEMINI 3 AND GEMINI 73

Barycentrics
a*((b^6+c^6-(b^2-c^2)^2*b*c)*a^6+(b+c)*(2*b^2-3*b*c+2*c^2)*b^2*c^2*a^5-(b^2-c^2)^2*b^2*c^2*a^4+(b+c)*(3*b^2-5*b*c+3*c^2)*b^3*c^3*a^3-(b^2-4*b*c+c^2)*b^4*c^4*a^2+(b^2-c^2)*(b-c)*b^4*c^4*a-(b-c)^2*b^5*c^5) : :

X(31309) lies on these lines: {86,24662}, {24656,27880}


X(31310) = PERSPECTOR OF THESE TRIANGLES: GEMINI 4 AND GEMINI 63

Barycentrics
3*(b+c)*a^5+(4*b^2+13*b*c+4*c^2)*a^4-(b+c)*(5*b^2-4*b*c+5*c^2)*a^3-(4*b^4+4*c^4+(11*b^2+12*b*c+11*c^2)*b*c)*a^2-(b+c)*(b^2+b*c+c^2)*(b^2-3*b*c+c^2)*a+(b^2+b*c+c^2)*(b^2+5*b*c+c^2)*b*c : :
X(31310) = X(6650)-4*X(17755)

X(31310) lies on these lines: {1,6651}, {86,27949}, {190,20142}, {335,31336}, {740,31349}, {5698,27484}, {6650,17755}, {20158,27480}

X(31310) = reflection of X(i) in X(j) for these (i,j): (335, 31336), (6650, 27483), (27483, 17755), (31308, 6651)


X(31311) = PERSPECTOR OF THESE TRIANGLES: GEMINI 4 AND GEMINI 109

Barycentrics    3*a^2-7*(b+c)*a-5*b*c : :

X(31311) lies on these lines: {1,872}, {2,7232}, {10,31333}, {45,4772}, {86,16669}, {190,3739}, {192,16675}, {239,31350}, {1268,4422}, {1654,31285}, {2476,3826}, {3731,4764}, {3758,31312}, {4360,16674}, {4678,17233}, {4686,16815}


X(31312) = PERSPECTOR OF THESE TRIANGLES: GEMINI 4 AND GEMINI 110

Barycentrics    3*a^2+7*(b+c)*a+8*b*c : :
X(31312) = 3*X(2)+X(30712) = 5*X(1698)-2*X(15593)

X(31312) lies on these lines: {1,3696}, {2,1743}, {86,16667}, {142,3624}, {165,24220}, {190,3731}, {192,16673}, {239,31313}, {1125,4779}, {1418,25086}, {1449,31238}, {1698,4648}, {2999,5333}, {3247,4686}, {3634,4869}, {3636,4402}, {3729,29578}, {3758,31311}, {3946,28641}, {3973,4670}, {4000,25055}, {4364,4902}, {4677,17390}, {4699,29597}, {4751,16833}, {4772,16826}, {4816,4916}, {4821,29595}, {4888,5257}, {4898,29624}, {4967,29602}, {5234,21246}, {6707,17306}, {8056,10455}, {17175,18186}, {17296,19875}, {17304,29612}, {17313,19876}, {19701,23511}

X(31312) = complement of the complement of X(30712)
X(31312) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (86, 16832, 16667), (16831, 25590, 16673)


X(31313) = PERSPECTOR OF THESE TRIANGLES: GEMINI 4 AND GEMINI 111

Barycentrics    8*a^2+7*(b+c)*a+3*b*c : :
X(31313) = 3*X(2)-4*X(30598)

X(31313) lies on these lines: {1,1278}, {2,319}, {9,29570}, {86,4772}, {190,4704}, {239,31312}, {1386,20080}, {1449,29595}, {3616,17343}, {3622,5625}, {3636,17364}, {4670,4788}, {4821,17393}, {4909,17397}, {5698,11038}, {14996,21769}, {16667,16826}, {16669,27268}, {16673,17350}, {17018,24661}, {17300,26104}, {17349,31311}, {17375,29586}

X(31313) = anticomplement of the anticomplement of X(30598)
X(31313) = {X(25417), X(30562)}-harmonic conjugate of X(2)


X(31314) = PERSPECTOR OF THESE TRIANGLES: GEMINI 6 AND GEMINI 63

Barycentrics    3*(b+c)*a^3+(b^2+7*b*c+c^2)*a^2-(b^2-c^2)*(b-c)*a+(b^2+b*c+c^2)*b*c : :
X(31314) = 3*X(2)-4*X(31306) = X(192)-4*X(1100) = 2*X(319)-5*X(4699) = 2*X(4649)+X(24349)

X(31314) lies on these lines: {1,6651}, {2,210}, {8,31329}, {10,31335}, {42,31348}, {75,20016}, {192,1100}, {239,27478}, {319,4675}, {726,29584}, {984,29586}, {3797,29588}, {3807,24512}, {4393,4649}, {4740,4795}, {6542,27474}, {17029,31063}, {17300,27487}, {17755,29569}, {17778,27476}, {20072,24357}, {24325,27483}, {29592,31323}

X(31314) = reflection of X(27495) in X(31306)
X(31314) = anticomplement of X(27495)
X(31314) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 27481, 31308), (27495, 31306, 2)


X(31315) = PERSPECTOR OF THESE TRIANGLES: GEMINI 6 AND GEMINI 64

Barycentrics
((b-c)*a-b*c)*((b-c)*a+b*c)*((b+c)^3*a^7-9*(b+c)^2*b*c*a^6-(b+c)*(b^4+c^4-(6*b^2+23*b*c+6*c^2)*b*c)*a^5+(2*b^6+2*c^6-(9*b^4+9*c^4-(10*b^2-67*b*c+10*c^2)*b*c)*b*c)*a^4-(b+c)*(b^6+c^6-(10*b^4+10*c^4-(35*b^2-64*b*c+35*c^2)*b*c)*b*c)*a^3-(b^6+c^6+(2*b^4+2*c^4-3*(11*b^2-18*b*c+11*c^2)*b*c)*b*c)*b*c*a^2+(b+c)*(b^4+c^4-(10*b^2-13*b*c+10*c^2)*b*c)*b^2*c^2*a+(b^4+b^2*c^2+c^4)*b^3*c^3) : :

X(31315) lies on these lines: {}


X(31316) = PERSPECTOR OF THESE TRIANGLES: GEMINI 8 AND GEMINI 71

Barycentrics    a*(a^2-3*(2*b-c)*a+b^2+3*b*c-2*c^2)*(a+b-3*c)*(a^2+3*(b-2*c)*a-2*b^2+3*b*c+c^2)*(a-3*b+c)*(-a+b+c) : :

X(31316) lies on the Feuerbach hyperbola and these lines: {1,27834}, {9,31343}, {2505,23836}

X(31316) = trilinear pole of the line {650, 3680}


X(31317) = PERSPECTOR OF THESE TRIANGLES: GEMINI 9 AND GEMINI 63

Barycentrics    (b+c)*a^3+2*b*c*a^2+(b+c)*b*c*a+(b^2+b*c+c^2)*b*c : :
X(31317) = 5*X(4699)-X(4741)

X(31317) lies on these lines: {1,3797}, {2,38}, {6,75}, {7,1654}, {37,17339}, {56,27954}, {190,24357}, {192,5749}, {321,17027}, {354,31028}, {518,3661}, {726,17023}, {740,4393}, {871,3978}, {1001,6651}, {1278,17014}, {1757,4384}, {1921,3765}, {3008,27478}, {3252,19584}, {3662,3739}, {3666,16606}, {3696,28538}, {3873,31027}, {3923,4366}, {4372,17103}, {4645,24693}, {4675,27487}, {4687,29609}, {4688,4715}, {4692,30114}, {4732,31329}, {4751,17291}, {4850,31348}, {4968,17033}, {4974,20158}, {5220,16815}, {5695,20162}, {6542,27474}, {6645,16822}, {9055,17369}, {9318,27931}, {16584,28606}, {16825,20142}, {16975,31344}, {17048,31276}, {17141,26035}, {17251,31139}, {17266,27475}, {17308,27495}, {20678,26241}, {24046,27324}, {24231,24603}, {24327,27913}, {24331,27949}, {24346,27941}, {24514,26234}, {25253,26807}, {26240,27912}, {26561,30177}, {31041,31079}

X(31317) = midpoint of X(75) and X(3758)
X(31317) = reflection of X(17237) in X(3739)
X(31317) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 24349, 335), (1215, 24631, 2), (17755, 24325, 2)


X(31318) = PERSPECTOR OF THESE TRIANGLES: GEMINI 11 AND GEMINI 15

Barycentrics    a*((b+c)*a^2+(2*b^2+7*b*c+2*c^2)*a+(b+c)*(b^2+4*b*c+c^2)) : :

X(31318) lies on these lines: {1,210}, {2,3743}, {6,5506}, {35,1486}, {36,27802}, {37,3624}, {45,6763}, {192,6533}, {312,25512}, {474,4436}, {764,14349}, {975,5259}, {1125,3971}, {1203,5287}, {1698,4646}, {1962,17749}, {2650,5692}, {3338,3731}, {3616,3952}, {3666,31320}, {3670,25502}, {3678,29814}, {3720,5904}, {3746,5268}, {4011,25526}, {4673,19870}, {4687,19863}, {4850,19878}, {4868,19877}, {4975,19853}, {5054,8143}, {5284,30142}, {5312,15569}, {5563,25579}, {5697,22300}, {5747,24933}, {7294,26742}, {7611,19514}, {12047,29571}, {16296,20326}, {17263,19846}, {18398,26102}, {19862,28606}, {19883,27782}, {24945,25079}

X(31318) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 25430, 25431), (3646, 25430, 1)


X(31319) = PERSPECTOR OF THESE TRIANGLES: GEMINI 11 AND GEMINI 63

Barycentrics    3*(b+c)*a^3+2*(b+2*c)*(2*b+c)*a^2+(b+2*c)*(2*b+c)*(b+c)*a+(b^2+b*c+c^2)*b*c : :
X(31319) = 5*X(4687)+X(17393)

X(31319) lies on these lines: {1,27495}, {2,740}, {37,17339}, {75,29609}, {192,31347}, {239,4687}, {335,3616}, {984,29586}, {1001,14621}, {1125,27478}, {3661,15569}, {3797,29603}, {3842,4393}, {4698,17233}, {5625,20145}, {16929,17733}, {18230,26626}, {24325,27494}, {28566,29622}, {28606,31348}, {31238,31351}

X(31319) = complement of X(31329)
X(31319) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 27480, 27483), (2, 31308, 27474)


X(31320) = PERSPECTOR OF THESE TRIANGLES: GEMINI 12 AND GEMINI 15

Barycentrics    a*(3*(b+c)*a^2+(6*b^2+11*b*c+6*c^2)*a+(b+c)*(3*b^2+4*b*c+3*c^2)) : :

X(31320) lies on these lines: {1,3683}, {8,3743}, {10,27783}, {35,27802}, {37,1698}, {46,16673}, {79,2335}, {191,16777}, {594,31321}, {1089,4704}, {1125,24165}, {1486,3746}, {1962,5904}, {3666,31318}, {27268,28611}

X(31320) = {X(46), X(16673)}-harmonic conjugate of X(25431)


X(31321) = PERSPECTOR OF THESE TRIANGLES: GEMINI 12 AND GEMINI 26

Barycentrics    3*(b+c)*a^3+(8*b^2+15*b*c+8*c^2)*a^2+(b+c)*(7*b^2+12*b*c+7*c^2)*a+2*(b+c)^4 : :

X(31321) lies on these lines: {6,1698}, {10,4970}, {46,3929}, {594,31320}, {1330,9780}, {3828,27783}


X(31322) = PERSPECTOR OF THESE TRIANGLES: GEMINI 13 AND GEMINI 63

Barycentrics    (4*b^2+7*b*c+4*c^2)*a^2+2*(b+c)*(b^2+4*b*c+c^2)*a+(b^2+4*b*c+c^2)*b*c : :
X(31322) = 5*X(4687)-2*X(16777)

X(31322) lies on these lines: {1,31336}, {2,210}, {8,31342}, {10,27474}, {37,27480}, {75,1213}, {239,4687}, {984,24603}, {1698,17755}, {3739,31347}, {3797,31329}, {3842,4384}, {3876,27156}, {4751,17291}, {4971,31350}, {5224,27487}, {14621,17335}, {16826,20156}, {16830,20154}, {16972,17277}, {17337,17397}, {19804,31348}, {31349,31351}

X(31322) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 27484, 31306), (2, 27495, 27475)


X(31323) = PERSPECTOR OF THESE TRIANGLES: GEMINI 14 AND GEMINI 63

Barycentrics    (2*b^2+3*b*c+2*c^2)*a^2+(b+c)*(b^2+3*b*c+c^2)*a+b^2*c^2 : :

X(31323) lies on these lines: {1,16912}, {2,38}, {9,14621}, {10,3797}, {37,239}, {45,20172}, {75,1213}, {190,25384}, {192,5296}, {518,16826}, {726,24603}, {742,17256}, {1757,20132}, {3509,16993}, {3589,4687}, {3661,3932}, {3681,17032}, {3739,17305}, {3912,27495}, {4359,18152}, {4389,25357}, {4393,4974}, {4451,17038}, {4664,28309}, {4698,29609}, {4704,27480}, {4981,31027}, {5220,20131}, {5263,6651}, {7384,29054}, {15569,29584}, {17143,21816}, {17237,27487}, {18230,26626}, {20016,31308}, {20343,25145}, {24589,31348}, {27474,29593}, {27475,29581}, {29592,31314}, {31238,31335}

X(31323) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 984, 335), (3842, 17755, 2)


X(31324) = PERSPECTOR OF THESE TRIANGLES: GEMINI 15 AND GEMINI 61

Barycentrics
a*((b+c)*a^6-(2*b^2+7*b*c+2*c^2)*a^5-(b^3+c^3)*a^4+2*(b+2*c)*(2*b+c)*(b^2+c^2)*a^3-(b+c)*(b^4+c^4-2*(b^2+5*b*c+c^2)*b*c)*a^2-(b^2-c^2)^2*(2*b^2+3*b*c+2*c^2)*a+(b^2-c^2)^2*(b+c)*(b^2-3*b*c+c^2)) : :

X(31324) lies on these lines: {1,2287}, {2,5831}, {37,5703}, {281,1895}, {938,966}, {2285,6986}, {3998,7229}, {10445,21617}, {13411,17355}, {20223,27404}


X(31325) = PERSPECTOR OF THESE TRIANGLES: GEMINI 15 AND GEMINI 65

Barycentrics    a^5-3*(b+c)*a^4+2*(b^2+c^2)*a^3+2*(b+c)*(b^2-8*b*c+c^2)*a^2-3*(b^2-c^2)^2*a+(b^2-c^2)^2*(b+c) : :

X(31325) lies on these lines: {1,346}, {2,20270}, {37,6554}, {938,2321}, {1766,3600}, {2171,11036}, {2345,14986}, {3085,25081}, {3161,9369}, {3247,5703}, {3672,4552}, {3945,28968}, {4072,6744}, {8165,21074}, {9785,10445}, {9819,10443}, {13405,16673}


X(31326) = PERSPECTOR OF THESE TRIANGLES: GEMINI 15 AND GEMINI 108

Barycentrics    3*a^3-8*(b+c)*a^2-3*(3*b^2-2*b*c+3*c^2)*a+2*(b^2-c^2)*(b-c) : :

X(31326) lies on these lines: {1,3523}, {226,4862}, {1018,5437}, {1699,17593}, {3586,13442}, {3633,14829}, {3666,26742}, {3752,16601}, {4384,14949}, {4850,25080}, {4888,21454}, {5256,5483}, {6692,16673}, {8056,24181}, {17760,29598}


X(31327) = PERSPECTOR OF THESE TRIANGLES: GEMINI 16 AND GEMINI 22

Barycentrics    (b^2+b*c+c^2)*a^2+(b+c)*(b^2+5*b*c+c^2)*a+2*(b+c)^2*b*c : :

X(31327) lies on these lines: {1,3696}, {8,2650}, {10,312}, {40,5788}, {72,3679}, {75,24214}, {304,4967}, {740,19853}, {764,4802}, {958,4436}, {982,24176}, {984,4647}, {986,31330}, {1046,4042}, {1698,4646}, {3617,3952}, {3702,26037}, {3741,24174}, {3976,4359}, {4197,21027}, {4714,10479}, {4732,9534}, {5082,20539}, {5255,5271}, {5263,16478}, {16819,27474}, {16828,22316}, {17063,28611}, {17592,19858}, {25079,26038}

X(31327) = {X(4714), X(10479)}-harmonic conjugate of X(24440)


X(31328) = PERSPECTOR OF THESE TRIANGLES: GEMINI 16 AND GEMINI 62

Barycentrics    ((b^2+c^2)*(b+c)^3*a^5+(b^4+6*b^2*c^2+c^4)*b*c*a^4+2*(b+c)*(3*b^2-b*c+3*c^2)*b^2*c^2*a^3+4*b^4*c^4*a^2+(b+c)*b^4*c^4*a-b^5*c^5)*((b-c)*a+b*c)*((b-c)*a-b*c) : :

X(31328) lies on these lines: {3741,27436}, {17792,27429}, {25121,27444}, {25124,27447}


X(31329) = PERSPECTOR OF THESE TRIANGLES: GEMINI 16 AND GEMINI 63

Barycentrics    (b^2+4*b*c+c^2)*a^2+2*(b^2+4*b*c+c^2)*(b+c)*a+(4*b^2+7*b*c+4*c^2)*b*c : :
X(31329) = 4*X(3739)-X(17393)

X(31329) lies on these lines: {1,31335}, {2,740}, {8,31314}, {10,27481}, {75,4377}, {1921,28605}, {2550,27484}, {3661,27478}, {3696,4393}, {3739,17393}, {3797,31322}, {4648,4699}, {4709,17397}, {4732,31317}, {17230,27475}, {20055,24325}, {20145,24342}, {29612,31351}, {31330,31348}

X(31329) = anticomplement of X(31319)
X(31329) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (8, 31347, 31314), (27474, 27483, 2)


X(31330) = PERSPECTOR OF THESE TRIANGLES: GEMINI 16 AND GEMINI 104

Barycentrics    (b^2+b*c+c^2)*a+b*c*(b+c) : :

X(31330) lies on these lines: {1,2}, {6,4042}, {9,3588}, {11,5743}, {31,333}, {37,3706}, {38,75}, {45,4387}, {55,5737}, {63,4418}, {141,3779}, {149,4368}, {171,1150}, {192,3989}, {238,5278}, {244,19804}, {274,17208}, {312,756}, {321,984}, {350,5224}, {354,3739}, {355,4192}, {497,966}, {517,30981}, {518,30969}, {594,2276}, {649,25128}, {672,2345}, {726,7226}, {740,28606}, {748,2209}, {750,14829}, {851,5794}, {908,4104}, {941,5257}, {956,11358}, {958,1011}, {960,1985}, {964,5247}, {968,3886}, {982,4359}, {986,31327}, {993,4184}, {1001,19732}, {1008,5015}, {1010,1468}, {1043,10448}, {1107,21877}, {1211,2886}, {1215,3681}, {1376,4191}, {1458,27339}, {1573,21838}, {1621,5235}, {1654,4388}, {1740,16738}, {1836,4643}, {1861,4196}, {2049,19714}, {2051,10886}, {2177,3996}, {2238,3966}, {2550,6817}, {2551,6818}, {2887,3775}, {2975,13588}, {3056,15985}, {3097,6539}, {3120,17794}, {3169,24392}, {3218,3980}, {3219,3923}, {3295,16345}, {3303,16355}, {3419,4199}, {3662,21027}, {3666,3696}, {3670,28612}, {3740,30818}, {3742,3846}, {3758,4722}, {3790,6535}, {3816,5241}, {3826,25961}, {3844,31005}, {3847,11680}, {3868,30984}, {3869,14009}, {3873,24325}, {3878,14008}, {3896,17592}, {3911,16878}, {3914,4357}, {3944,26580}, {3971,4671}, {4008,11031}, {4011,27065}, {4041,21259}, {4083,30968}, {4113,4849}, {4147,4893}, {4210,25440}, {4279,27631}, {4307,14552}, {4361,17599}, {4363,24690}, {4364,4854}, {4379,17072}, {4392,24165}, {4423,17259}, {4479,17250}, {4524,30864}, {4649,19684}, {4655,20292}, {4660,24259}, {4665,4884}, {4688,21342}, {4698,4891}, {4703,5057}, {4709,4970}, {4713,17251}, {4716,17600}, {4732,4850}, {4733,24643}, {4751,17450}, {4804,24718}, {5044,25591}, {5232,17220}, {5258,11322}, {5361,17126}, {5739,26098}, {5741,17717}, {5790,19540}, {5791,8731}, {5793,16405}, {5835,21677}, {5836,30960}, {6376,18152}, {6382,19562}, {6384,7148}, {6536,17248}, {7396,18659}, {8692,19751}, {9552,10475}, {9708,16058}, {9709,16059}, {10436,30941}, {10439,24220}, {10456,20245}, {10458,27164}, {10472,10473}, {10478,29311}, {12514,14956}, {16062,19787}, {16468,19742}, {16684,17293}, {16748,16887}, {16975,23632}, {17063,24589}, {17065,23444}, {17140,25294}, {17142,17238}, {17143,31008}, {17147,17163}, {17165,31025}, {17178,25528}, {17184,17889}, {17228,21026}, {17237,21949}, {17239,21264}, {17271,31134}, {17272,20347}, {17289,26061}, {17303,24512}, {17495,17591}, {18792,27163}, {20220,30975}, {20594,22230}, {21231,30983}, {21232,30994}, {21238,25624}, {21241,25958}, {21254,31001}, {21384,26035}, {21727,25667}, {21805,31264}, {23791,26824}, {24003,30993}, {24046,28611}, {24686,24694}, {25121,30998}, {25301,27345}, {25385,31053}, {25627,30835}, {26066,30944}, {28595,30982}, {31329,31348}

X(31330) = midpoint of X(7226) and X(28605)
X(31330) = isotomic conjugate of X(2296)
X(31330) = complement of X(17018)
X(31330) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 29839, 29661), (899, 31241, 2), (3661, 29641, 15523)


X(31331) = PERSPECTOR OF THESE TRIANGLES: GEMINI 19 AND GEMINI 63

Barycentrics
(5*b^2+8*b*c+5*c^2)*a^4+2*(b+c)*(7*b^2+13*b*c+7*c^2)*a^3+(7*b^4+7*c^4+(38*b^2+63*b*c+38*c^2)*b*c)*a^2+(b+c)*(b^4+c^4+(b+4*c)*(4*b+c)*b*c)*a-(b^4+c^4+(3*b^2+b*c+3*c^2)*b*c)*b*c : :
X(31331) = X(27481)+2*X(31336)

X(31331) lies on these lines: {86,27949}, {1001,4393}, {3797,24603}, {4078,27495}, {9055,31332}, {24325,27481}, {27478,31350}


X(31332) = PERSPECTOR OF THESE TRIANGLES: GEMINI 19 AND GEMINI 107

Barycentrics    5*a^2+13*(b+c)*a+2*b^2+b*c+2*c^2 : :
X(31332) = 5*X(17319)+4*X(28633) = 4*X(17319)+5*X(31248)

These triangles are triple-perspective

X(31332) lies on these lines: {2,3943}, {37,24625}, {86,545}, {190,551}, {519,25354}, {523,27811}, {903,16826}, {1001,6172}, {1022,28840}, {3247,17240}, {4370,29586}, {4664,24325}, {4715,29580}, {9055,31331}, {10022,29592}, {16590,29584}, {16673,17342}, {16777,17271}, {17045,31333}, {17319,28633}, {24441,29570}, {25361,31153}


X(31333) = PERSPECTOR OF THESE TRIANGLES: GEMINI 19 AND GEMINI 109

Barycentrics    3*a^2-5*(b+c)*a+2*b^2-b*c+2*c^2 : :

X(31333) lies on these lines: {2,4398}, {8,344}, {9,17241}, {10,31311}, {37,29630}, {45,17236}, {86,4422}, {142,190}, {144,17234}, {551,17353}, {908,25361}, {1268,17359}, {1738,3634}, {3624,4687}, {3629,29589}, {3644,31183}, {3731,17305}, {3973,17387}, {4358,31205}, {4360,17338}, {4370,26806}, {4431,6666}, {4473,17245}, {4681,29607}, {6687,17319}, {15492,17312}, {16668,29625}, {16669,29575}, {16676,17370}, {16677,17383}, {16706,25600}, {16814,17266}, {16885,29572}, {17045,31332}, {17160,17337}, {17239,17260}, {17248,17279}, {17261,27191}, {17267,17271}, {17295,17335}, {17342,19875}, {17347,29627}, {17351,29626}, {17368,28640}, {27037,27164}, {27073,27111}, {28604,31285}, {29609,31350}

X(31333) = These triangles are triple-perspective.
X(31333) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (344, 18230, 17233), (17233, 18230, 17277)


X(31334) = PERSPECTOR OF THESE TRIANGLES: GEMINI 19 AND GEMINI 111

Barycentrics    15*a^2+17*(b+c)*a+3*b^2+9*b*c+3*c^2 : :

These triangles are triple-perspective.

X(31334) lies on these lines: {2,4445}, {142,29586}, {368,24656}, {1001,20059}, {1278,3622}, {1449,29592}, {3616,20090}, {17014,31352}, {25361,31292}


X(31335) = PERSPECTOR OF THESE TRIANGLES: GEMINI 23 AND GEMINI 63

Barycentrics    3*(b+c)*a^3+(4*b^2+13*b*c+4*c^2)*a^2+(b+c)*(2*b^2+11*b*c+2*c^2)*a+(4*b^2+7*b*c+4*c^2)*b*c : :

X(31335) lies on these lines: {1,31329}, {2,726}, {75,29609}, {86,239}, {142,17238}, {1125,31308}, {3797,31336}, {3946,4699}, {4384,20145}, {4698,31350}, {4751,17291}, {16826,27474}, {17292,27475}, {24325,27495}, {29580,31342}, {31238,31323}

X(31335) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31347, 27481), (3739, 31306, 27483), (27483, 31306, 239)


X(31336) = PERSPECTOR OF THESE TRIANGLES: GEMINI 25 AND GEMINI 63

Barycentrics    (a^2+2*(b+c)*a+b*c)*(3*(b+c)*a+b^2+4*b*c+c^2) : :
X(31336) = 3*X(2)+X(31308) = X(37)+2*X(6707) = X(86)+5*X(4687) = X(1213)-4*X(4698) = 2*X(3842)+X(5625) = X(4733)+2*X(15569) = X(27481)-3*X(31331)

X(31336) lies on these lines: {1,31322}, {2,740}, {9,86}, {10,31342}, {37,4472}, {335,31310}, {524,16590}, {1125,17755}, {1213,3912}, {1654,5308}, {3161,27268}, {3797,31335}, {3842,4649}, {4370,4755}, {4733,15569}, {4789,6544}, {6651,29578}, {17308,31248}, {17322,27487}, {24325,27481}, {25354,29571}, {29592,31314}

X(31336) = midpoint of X(i) and X(j) for these {i,j}: {335, 31310}, {27483, 31308}
X(31336) = complement of X(27483)
X(31336) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31308, 27483), (16826, 20142, 5625)


X(31337) = PERSPECTOR OF THESE TRIANGLES: GEMINI 26 AND GEMINI 57

Barycentrics    (b^3+c^3)*a^2-(b^2+b*c+c^2)*b*c*a+(b+c)*b^2*c^2 : :

X(31337) lies on these lines: {2,18170}, {10,75}, {1213,25624}, {1964,27091}, {3122,17786}, {3661,21238}, {3688,27076}, {3778,30473}, {17065,18040}, {17072,21143}, {17233,21257}, {17234,20340}, {17248,28593}, {18044,24478}, {20352,27095}, {21278,27044}, {24517,29712}, {25292,26963}, {25350,30989}, {29679,31090}

X(31337) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10, 25121, 5224), (20340, 25140, 17234)


X(31338) = PERSPECTOR OF THESE TRIANGLES: GEMINI 26 AND GEMINI 68

Barycentrics
(b+c)*(b^2+3*b*c+c^2)*a^5+(b^2+b*c+c^2)*(3*b^2+7*b*c+3*c^2)*a^4+(b+c)*(3*b^4+3*c^4+4*(2*b^2+3*b*c+2*c^2)*b*c)*a^3+(b^2+b*c+c^2)*(b^2+3*b*c+c^2)^2*a^2+2*(b^2+b*c+c^2)*(b+c)^3*b*c*a+(b+c)^4*b^2*c^2 : :

X(31338) lies on the line {25631,27624}


X(31339) = PERSPECTOR OF THESE TRIANGLES: GEMINI 26 AND GEMINI 80

Barycentrics    (b^2+3*b*c+c^2)*a^2+(b+c)^3*a+(b+c)^2*b*c : :

X(31339) lies on these lines: {1,2}, {5,25960}, {9,26035}, {12,5743}, {31,1010}, {55,19283}, {56,5737}, {65,3739}, {75,2292}, {171,16454}, {226,959}, {238,964}, {333,1468}, {355,13731}, {377,28287}, {388,966}, {442,25760}, {748,13740}, {756,4385}, {958,13738}, {984,4968}, {986,4359}, {992,2295}, {993,4225}, {1042,27339}, {1211,25466}, {1213,2277}, {1215,3876}, {1220,17277}, {1329,5241}, {1334,2345}, {1402,9552}, {1469,15985}, {1655,17248}, {1837,21321}, {1909,5224}, {2049,16466}, {2051,10887}, {2274,27164}, {2352,28265}, {2476,3847}, {2551,28270}, {2886,3142}, {2887,4197}, {2901,27785}, {2975,5235}, {3295,19282}, {3780,10371}, {3846,5836}, {3868,24325}, {3877,14011}, {3915,5263}, {4042,19730}, {4147,17166}, {4357,23536}, {4388,26051}, {4418,12514}, {4424,28612}, {4429,24438}, {4438,28267}, {4474,20316}, {4643,4754}, {4761,27647}, {5192,17123}, {5247,5278}, {5252,24735}, {5295,6051}, {5296,27523}, {5363,16424}, {5484,26044}, {5687,19518}, {5710,27623}, {5711,16458}, {5745,27621}, {5791,28258}, {5837,21246}, {6210,15971}, {6533,24046}, {8476,9666}, {8728,25957}, {9565,22076}, {10106,16878}, {10436,17137}, {10448,11110}, {10456,17183}, {10472,10480}, {11109,25885}, {12435,24220}, {12526,20245}, {13741,17125}, {14005,27644}, {14621,16926}, {14636,18481}, {17239,24656}, {17306,26978}, {17337,25992}, {17529,25961}, {17533,21935}, {19513,26446}, {19804,24443}, {21422,24547}, {24174,24589}, {25253,31025}, {25591,25917}, {25624,27641}, {26040,28272}, {26061,28242}, {26063,28266}, {26066,27622}, {26077,28279}, {27648,29066}

X(31339) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1698, 19858), (2, 20036, 3616), (978, 1698, 2)


X(31340) = PERSPECTOR OF THESE TRIANGLES: GEMINI 26 AND GEMINI 81

Barycentrics
(b+c)*(b^2+3*b*c+c^2)*a^5+(3*b^4+3*c^4+(5*b^2+3*b*c+5*c^2)*b*c)*a^4+(b+c)*(3*b^4+3*c^4-(b^2-8*b*c+c^2)*b*c)*a^3+(b^2+b*c+c^2)*(b^4+c^4+2*(b^2+3*b*c+c^2)*b*c)*a^2+2*(b^3+c^3)*(b+c)^2*b*c*a+(b+c)^4*b^2*c^2 : :

X(31340) lies on these lines: {10,1423}, {25624,28366}


X(31341) = PERSPECTOR OF THESE TRIANGLES: GEMINI 26 AND GEMINI 104

Barycentrics    (b+c)*(b^2+3*b*c+c^2)*a^3+(b^4+c^4+(8*b^2+13*b*c+8*c^2)*b*c)*a^2+2*(b+c)*(b^2+3*b*c+c^2)*b*c*a+(b^2+b*c+c^2)*b^2*c^2 : :

X(31341) lies on these lines: {2,3780}, {10,31008}, {3925,5224}, {17149,26037}, {21238,25624}, {28653,30966}


X(31342) = PERSPECTOR OF THESE TRIANGLES: GEMINI 29 AND GEMINI 63

Barycentrics    3*(b+c)*a^3+2*(b+2*c)*(2*b+c)*a^2-(b^2-c^2)*(b-c)*a-2*(b^2+b*c+c^2)*b*c : :
X(31342) = 3*X(17389)+X(27481) = X(27478)-3*X(29574)

X(31342) lies on these lines: {1,27474}, {2,4891}, {8,31322}, {10,31336}, {37,319}, {75,29585}, {145,27484}, {192,20059}, {335,29619}, {354,31348}, {518,17389}, {536,27494}, {740,27478}, {984,29605}, {2550,17316}, {3661,15569}, {3696,16826}, {3706,17032}, {3739,17393}, {3797,29588}, {3846,29612}, {3886,20131}, {3912,4085}, {4360,27487}, {4663,6651}, {4664,11160}, {4698,17240}, {4702,14621}, {7381,27491}, {29580,31335}, {29592,31238}

X(31342) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 27474, 31306), (6542, 31308, 27495), (27495, 31308, 37)


X(31343) = PERSPECTOR OF THESE TRIANGLES: GEMINI 29 AND GEMINI 71

Barycentrics    a*(-a+b+c)*(-a+c)*(a-3*b+c)*(a-b)*(a+b-3*c) : :

X(31343) lies on these lines: {8,1120}, {9,31316}, {78,1320}, {100,1293}, {200,244}, {404,16945}, {517,1339}, {519,22942}, {1897,17780}, {2415,3952}, {3158,24151}, {3699,25268}, {5524,17958}, {6556,7080}, {6557,14942}, {8706,30610}, {30198,30236}

X(31343) = trilinear pole of the line {9, 3057}


X(31344) = PERSPECTOR OF THESE TRIANGLES: GEMINI 31 AND GEMINI 63

Barycentrics    (b^2+b*c+c^2)*((2*b^2+b*c+2*c^2)*a^4+(b+c)*(b^2+3*b*c+c^2)*a^3+(b^2+7*b*c+c^2)*b*c*a^2+2*(b+c)*b^2*c^2*a-b^3*c^3) : :

X(31344) lies on these lines: {75,27482}, {871,10009}, {984,3661}, {7777,27488}


X(31345) = PERSPECTOR OF THESE TRIANGLES: GEMINI 32 AND GEMINI 62

Barycentrics
((b^3-c^3)*(b-c)*b*c*a^6+(b+c)*(b^6+c^6-2*(b^3-c^3)*(b-c)*b*c)*a^5-(b^6+c^6-(2*b^4+2*c^4-(b^2+5*b*c+c^2)*b*c)*b*c)*b*c*a^4+(b+c)*(b^4+c^4-(4*b^2-9*b*c+4*c^2)*b*c)*b^2*c^2*a^3+(b^2-7*b*c+c^2)*b^4*c^4*a^2+2*(b+c)*b^5*c^5*a-(b^2+b*c+c^2)*b^5*c^5)*((b-c)*a+b*c)*((b-c)*a-b*c) : :

X(31345) lies on the line {76,27447}


X(31346) = PERSPECTOR OF THESE TRIANGLES: GEMINI 38 AND GEMINI 63

Barycentrics    2*(b^2+b*c+c^2)*a^4-3*(b^3+c^3)*a^3-3*(b^2+b*c+c^2)*b*c*a^2+(b^2-c^2)*(b-c)*(b^2+3*b*c+c^2)*a-(b^3-c^3)*(b-c)*b*c : :

X(31346) lies on these lines: {9,192}, {19,27472}, {273,26023}, {7777,27488}


X(31347) = PERSPECTOR OF THESE TRIANGLES: GEMINI 40 AND GEMINI 63

Barycentrics    3*(b+c)*a^3+(2*b^2+11*b*c+2*c^2)*a^2+(b+c)*(b^2+10*b*c+c^2)*a+(5*b^2+8*b*c+5*c^2)*b*c : :

X(31347) lies on these lines: {2,726}, {7,1654}, {8,31314}, {75,4470}, {192,31319}, {239,4772}, {1278,4021}, {1992,4688}, {3616,31308}, {3739,31322}, {4393,24342}, {4704,29609}, {17236,24199}, {17316,24325}, {24349,27495}, {27475,29579}, {29610,31302}

X(31347) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4699, 27484, 27483), (27481, 31335, 2), (27483, 31317, 27484)


X(31348) = PERSPECTOR OF THESE TRIANGLES: GEMINI 63 AND GEMINI 104

Barycentrics    (2*b^2-b*c+2*c^2)*a^2+(b^2-c^2)*(b-c)*a-(b^2+b*c+c^2)*b*c : :

X(31348) lies on these lines: {2,726}, {38,27495}, {42,31314}, {75,27482}, {239,514}, {244,3797}, {310,1921}, {354,31342}, {596,27020}, {982,27474}, {1266,24318}, {1575,3807}, {2228,24413}, {3210,17027}, {3661,4392}, {3666,31306}, {3720,31308}, {4850,31317}, {6381,29576}, {6542,17449}, {6651,7292}, {8720,17692}, {16819,24176}, {17011,17187}, {17490,27484}, {19804,31322}, {21214,25270}, {24166,25264}, {24589,31323}, {28606,31319}, {31329,31330}

X(31348) = reflection of X(3807) in X(1575)
X(31348) = {X(244), X(3797)}-harmonic conjugate of X(30967)


X(31349) = PERSPECTOR OF THESE TRIANGLES: GEMINI 63 AND GEMINI 107

Barycentrics    2*(b+c)*a^3-(2*b^2-b*c+2*c^2)*a^2-(b+c)*(b^2+b*c+c^2)*a+(2*b^2+b*c+2*c^2)*b*c : :
X(31349) = X(335)-4*X(17755) = 3*X(27487)-2*X(31138)

X(31349) lies on these lines: {1,27949}, {2,38}, {37,24625}, {44,190}, {75,545}, {518,3799}, {519,3797}, {528,29617}, {551,31331}, {597,4370}, {673,15481}, {740,31310}, {903,4688}, {1086,17250}, {1386,6651}, {3679,7924}, {3994,17029}, {4366,16468}, {4384,24821}, {4393,4432}, {4422,17397}, {4437,29577}, {4473,26626}, {4740,6172}, {4755,31306}, {9041,17389}, {11329,24826}, {16367,24820}, {16593,29582}, {16826,24841}, {16833,17738}, {27487,31138}, {31322,31351}

X(31349) = midpoint of X(4740) and X(17487)
X(31349) = reflection of X(i) in X(j) for these (i,j): (2, 17755), (335, 2), (903, 4688), (4664, 4370)


X(31350) = PERSPECTOR OF THESE TRIANGLES: GEMINI 63 AND GEMINI 109

Barycentrics    6*(b+c)*a^3+(14*b^2+29*b*c+14*c^2)*a^2+(b+c)*(7*b^2+19*b*c+7*c^2)*a+(b+2*c)*(2*b+c)*b*c : :
X(31350) = 2*X(37)+X(1268)

X(31350) lies on these lines: {2,28516}, {37,1268}, {75,31351}, {239,31311}, {335,31310}, {4687,27481}, {4971,31322}, {5852,27475}, {15481,16826}, {18230,26626}, {27478,31331}, {27480,31352}, {27495,29574}, {29609,31333}


X(31351) = PERSPECTOR OF THESE TRIANGLES: GEMINI 63 AND GEMINI 110

Barycentrics    6*(b+c)*a^3+2*(7*b^2+19*b*c+7*c^2)*a^2+(b+c)*(7*b^2+37*b*c+7*c^2)*a+(11*b^2+23*b*c+11*c^2)*b*c : :
X(31351) = 7*X(4751)-X(30598)

X(31351) lies on these lines: {2,28522}, {75,31350}, {142,17238}, {239,31312}, {3739,27481}, {4725,31306}, {4751,6707}, {29612,31329}, {31238,31319}, {31322,31349}


X(31352) = PERSPECTOR OF THESE TRIANGLES: GEMINI 63 AND GEMINI 111

Barycentrics    9*(b+c)*a^3-(14*b^2+23*b*c+14*c^2)*a^2-(b+c)*(7*b^2+52*b*c+7*c^2)*a-(11*b^2+38*b*c+11*c^2)*b*c : :

X(31352) lies on these lines: {2,4891}, {239,31312}, {1278,27481}, {4678,27475}, {4704,31322}, {4772,20059}, {17014,31334}, {20090,27483}, {27480,31350}






leftri  Conics associated to pairs of orthologic or parallelogic triangles: X(31353) - X(31375)  rightri

This preamble and centers X(31353)-X(31375) were contributed by César Eliud Lozada, January 27, 2019.

I) Conics associated to a pair of orthologic triangles:

  1. Let T'=A'B'C' and T"=A"B"C" be two orthologic triangles. Denote a'b and a'c the perpendicular lines from A' to A"C" and A"B", respectively, and build b'c, b'a, c'a,c'b cyclically. Then these six lines are tangent to a conic Φ't, here named the orthologic tangential-conic T' to T".

    Swap T' and T", repeat the above construction and name the respective lines a"b, a"c, b"c, b"a, c"a, c"b. Then these six lines are tangent to another conic Φ"t, here named the orthologic tangential-conic T" to T'.

  2. Let A'b=a'b ∩ B'C' and A'c=a'c ∩ B'C' and define B'c, B'a, C'a, C'b cyclically. Then these six points lie on a conic Φ'p, here named the orthologic conic T' to T".

    Swap T' and T", repeat the above construction and name the respective points A"b, A"c, B"c, B"a, C"a, C"b. Then these six points lie on another conic Φ"p, here named the orthologic conic T" to T'.

II) Conics associated to a pair of parallelogic triangles:

  1. Let T'=A'B'C' and T"=A"B"C" be two parallelogic triangles. Denote a'b and a'c the parallel lines from A' to A"C" and A"B", respectively, and build b'c, b'a, c'a,c'b cyclically. Then these six lines are tangent to a conic Ψ't, here named the parallelogic tangential-conic T' to T".

    Swap T' and T", repeat the above construction and name the respective lines a"b, a"c, b"c, b"a, c"a, c"b. Then these six lines are tangent to another conic Ψ"t, here named the parallelogic tangential-conic T" to T'.

  2. Let A'b=a'b ∩ B'C' and A'c=a'c ∩ B'C' and define B'c, B'a, C'a, C'b cyclically. Then these six points lie on a conic Ψ'p, here named the parallelogic conic T' to T".

    Swap T' and T", repeat the above construction and name the respective points A"b, A"c, B"c, B"a, C"a, C"b. Then these six points lie on another conic Ψ"p, here named the parallelogic conic T" to T'.

If T' and T" are orthologic (parallelogic) triangles and P is the orthologic (parallelogic) center T' to T", then the orthologic (parallelogic) conic T' to T" does not depend on T". Moreover, the orthologic conic T' to T" and the parallelogic conic T' to T" coincide. Therefore, when T' are P are given, a more convenient name for this conic Φ'p= Ψ'p is P-orthoparallelogic conic of T'. A similar coincidence occurs for the tangential conics Φ't= Ψ't and therefore a better name for this conic is P-orthoparallelogic tangential-conic of T'.

If P = x:y:z (barycentrics) then the center of the P-orthoparallelogic conic of ABC is:

  O’p = x^2*(y+z)*((y+z)*(x^4-(y-z)^2*y*z)-x*(y^2+z^2)*(-x^2+y^2+z^2)-(y^3+z^3)*x^2) : :

and its perspector is:

  Q’p = x*(y+z)*F(x,y,z)*F(x,z,y) : :, where F(x,y,z) = (x^2*(2*x*y+x*z+2*z^2)-(y-z)*(2*y*((x+z)*y+z^2)+x*z*(y+z)))

The center of the P-orthoparallelogic tangential-conic of ABC is O't=complement-of-P and its perspector is:

  Q’t = (x*(2*z+y)+y*(y+z))*(x*(2*y+z)+z*(y+z)) : :

The appearance of (i, j) in the following list means that X(j)=center of the X(i)-orthoparallelogic conic of ABC:
(1,3588), (2,2), (3,31353), (4,17807), (5,31354), (6,31355), (7,7955), (30,3163), (511,11672), (512,1084), (513,1015), (514,1086), (515,23986), (516,23972), (517,23980), (518,6184), (519,4370), (522,1146), (523,115), (524,2482), (525,15526), (526,18334), (536,13466), (542,23967), (690,23992), (726,20532), (826,15449), (1503,23976), (3566,15525), (3906,17416) , (7927,15527), (17430,17429)

The appearance of (i, j) in the following list means that X(j)=perspector of the X(i)-orthoparallelogic conic of ABC:
(1,31356), (2,2), (3,31357), (4,17808), (6,31358)

The appearance of (i, j) in the following list means that X(j)=perspector of the X(i)-orthoparallelogic tangential-conic of ABC:
(1,31359), (2,2), (3,13599), (4,15740), (5,22268), (6,31360), (8,7320), (20,31361), (69,17040), (75,17038), (95,17041), (99,9293), (264,17039), (550,3521), (668,9267), (3244,5559), (3629,13622)

Note: For P in the infinity, O’p lies in the Steiner inellipse, Q’p = P and Q’t = P.

underbar

X(31353) = CENTER OF THE X(3)-ORTHOPARALLELOGIC CONIC OF ABC

Barycentrics    SA^2*(SB+SC)^2*(S^2+SB*SC)*((SA-2*R^2)*S^2+SA*(8*(2*R^2-SW)*R^2+SW^2)) : :

X(31353) lies on these lines: {185,216}, {217,31357}


X(31354) = CENTER OF THE X(5)-ORTHOPARALLELOGIC CONIC OF ABC

Barycentrics    (3*S^2-SB*SC)*(4*S^2+(SB+SC)*(4*R^2-3*SA-SW))*(S^4-(R^2*(12*R^2+8*SA-7*SW)-2*SA^2+SW^2)*S^2+(4*R^2-SW)*(4*R^2-SA)*R^2*SA) : :

X(31354) lies on these lines: {}


X(31355) = CENTER OF THE X(6)-ORTHOPARALLELOGIC CONIC OF ABC

Barycentrics    (SA+SW)*(SB+SC)^2*(S^4*SA-(2*(SW^2+2*SB*SC)*R^2+(SB+SC)*(SA^2-SB*SC))*S^2+SB*SC*SW^3) : :

X(31355) lies on these lines: {39,6467}, {69,15270}, {3051,14820}, {7794,23208}, {20775,31358}


X(31356) = PERSPECTOR OF THE X(1)-ORTHOPARALLELOGIC CONIC OF ABC

Barycentrics    a*(b+c)*((2*b+c)*a^3+2*c^2*a^2-(b-c)*(2*b^2+b*c+c^2)*a-2*(b^2-c^2)*b*c)*((b+2*c)*a^3+2*b^2*a^2+(b-c)*(b^2+b*c+2*c^2)*a+2*(b^2-c^2)*b*c) : :

X(31356) lies on these lines: {8,10435}, {42,10474}, {55,10448}, {210,22299}, {1334,3588}


X(31357) = PERSPECTOR OF THE X(3)-ORTHOPARALLELOGIC CONIC OF ABC

Barycentrics    SA^2*(SB+SC)*(S^2+SB*SC)*(S^2+16*R^2*(2*R^2-SW)-SB^2+2*SW^2)*(S^2+16*R^2*(2*R^2-SW)-SC^2+2*SW^2) : :

X(31357) lies on these lines: {51,8799}, {184,26897}, {217,31353}


X(31358) = PERSPECTOR OF THE X(6)-ORTHOPARALLELOGIC CONIC OF ABC

Barycentrics    (SA^2-SW^2)*((8*R^2*SB-(2*SB-SW)*SW)*S^2-(2*SW-SB)*SB*SW^2)*((8*R^2*SC-(2*SC-SW)*SW)*S^2-(2*SW-SC)*SC*SW^2) : :

X(31358) lies on the line {20775,31355}


X(31359) = PERSPECTOR OF THE X(1)-ORTHOPARALLELOGIC TANGENTIAL CONIC OF ABC

Barycentrics    ((b+2*c)*a+b^2+b*c)*((2*b+c)*a+b*c+c^2) : :

X(31359) lies on these lines: {1,333}, {2,65}, {8,37}, {9,1220}, {10,312}, {19,29}, {21,1610}, {31,1098}, {45,5793}, {75,2292}, {82,3915}, {85,3668}, {86,969}, {92,225}, {145,4981}, {257,17248}, {341,756}, {377,24723}, {388,17257}, {392,25490}, {517,19853}, {740,17038}, {759,931}, {964,4676}, {968,1043}, {984,7275}, {986,19804}, {994,3878}, {997,19270}, {1010,12514}, {1213,3959}, {1621,2218}, {1654,10371}, {1697,14942}, {1698,4674}, {1836,26051}, {1938,25511}, {2176,5275}, {2214,2303}, {2652,5794}, {3085,28807}, {3486,13736}, {3617,3714}, {3646,25531}, {3679,4102}, {3683,4195}, {3868,13476}, {3876,26115}, {3931,9534}, {4181,20682}, {4386,28631}, {4389,23536}, {4511,16342}, {4518,21711}, {4642,26037}, {5233,5530}, {5257,5837}, {5278,17016}, {5730,16343}, {5902,25512}, {5903,16828}, {6051,10449}, {6557,9780}, {6646,10404}, {6682,21214}, {8421,24463}, {10405,27288}, {10436,12526}, {12709,27339}, {15254,17697}, {16062,23604}, {16824,19732}, {17149,18298}, {17250,20955}, {17260,17743}, {17338,25992}, {18359,23541}, {19874,22299}, {21674,25760}, {24627,25524}, {25466,27184}

X(31359) = isogonal conjugate of X(1468)
X(31359) = isotomic conjugate of X(10436)
X(31359) = polar conjugate of X(5307)
X(31359) = trilinear pole of the line {522, 661}
X(31359) = {X(1), X(2258)}-harmonic conjugate of X(5331)


X(31360) = PERSPECTOR OF THE X(6)-ORTHOPARALLELOGIC TANGENTIAL CONIC OF ABC

Barycentrics    (SA*SB+SW^2)*(SA*SC+SW^2) : :

X(31360) lies on these lines: {2,1843}, {6,1799}, {39,69}, {76,23642}, {95,11285}, {141,305}, {264,6656}, {287,19459}, {306,21035}, {2373,26206}, {3589,30489}, {3619,6340}, {7876,9229}, {7950,14977}, {8797,14064}

X(31360) = isotomic conjugate of X(7770)
X(31360) = trilinear pole of the line {525, 3005}


X(31361) = PERSPECTOR OF THE X(20)-ORTHOPARALLELOGIC TANGENTIAL CONIC OF ABC

Barycentrics    (S^2-2*(16*R^2-SB-3*SW)*SB)*(S^2-2*(-3*SW+16*R^2-SC)*SC) : :

X(31361) lies on these lines: {4,14572}, {20,5893}, {253,5895}, {1249,3146}, {3091,6716}, {3543,14249}

X(31361) = isogonal conjugate of X(8567)


X(31362) = CENTER OF THE ORTHOLOGIC CONIC OF THESE TRIANGLES: EULER TO ABC

Barycentrics    SB*SC*((2*R^2*(12*R^2-SA-5*SW)+SW^2)*S^2-(16*R^2*(4*R^2-SW)+SW^2)*(8*R^2*SW+SB*SC-2*SW^2)) : :
X(31362) = 5*X(3091)-X(31369)

The center of the reciprocal orthologic conic of these triangles is X(17807)

X(31362) lies on these lines: {3,15259}, {4,17807}, {5,31367}, {3091,31369}, {5480,13488}, {10002,18537}

X(31362) = midpoint of X(4) and X(17807)
X(31362) = reflection of X(31367) in X(5)


X(31363) = PERSPECTOR OF THE ORTHOLOGIC TANGENTIAL-CONIC OF THESE TRIANGLES: EULER TO ABC

Barycentrics    (S^2-2*(4*R^2-SW)*SB)*(S^2-2*(4*R^2-SW)*SC) : :

The perspector of the reciprocal orthologic tangential-conic of these triangles is X(15740)

X(31363) lies on the Kiepert hyperbola and these lines: {2,9786}, {5,459}, {20,275}, {1498,3424}, {2052,3091}, {3316,6810}, {3317,6809}, {3832,8796}, {5056,16080}, {7395,18841}, {7399,18840}

X(31363) = antigonal conjugate of the antitomic conjugate of X(31363)
X(31363) = antitomic conjugate of the antigonal conjugate of X(31363)
X(31363) = isogonal conjugate of X(11425)


X(31364) = CENTER OF THE ORTHOLOGIC CONIC OF THESE TRIANGLES: ORTHIC TO ABC

Barycentrics    (S^2-SB*SC)*(S^4+(2*(SA-3*SW)*R^2+SB*SC+SW^2)*S^2+SB*SC*(8*R^2*(2*R^2-SW)+SW^2)) : :

The center of the reciprocal orthologic conic of these triangles is X(31353)

X(31364) lies on these lines: {6,14773}, {185,216}, {570,15231}, {1885,3003}, {6467,23195}


X(31365) = PERSPECTOR OF THE ORTHOLOGIC CONIC OF THESE TRIANGLES: ORTHIC TO ABC

Barycentrics
SA*(S^4-(4*R^2*(24*R^2-15*SW-SC)-9*SC^2+8*SW^2)*S^2+(32*R^2*(3*R^2-2*SW)+9*SW^2)*(SA+SC)*(SB+SC))*(S^4-(4*R^2*(24*R^2-15*SW-SB)-9*SB^2+8*SW^2)*S^2+(32*R^2*(3*R^2-2*SW)+9*SW^2)*(SA+SB)*(SB+SC)) : :

The perspector of the reciprocal orthologic conic of these triangles is X(31357)

X(31365) lies on these lines: {}


X(31366) = PERSPECTOR OF THE ORTHOLOGIC TANGENTIAL-CONIC OF THESE TRIANGLES: ORTHIC TO ABC

Barycentrics    SB^2*SC^2*((4*R^2-SB)*S^2+(8*R^2-SW)*SA*SC)*((4*R^2-SC)*S^2+(8*R^2-SW)*SA*SB) : :

The perspector of the reciprocal orthologic tangential-conic of these triangles is X(13599)

X(31366) lies on the Jerabek hyperbola and the line {2052,14457}


X(31367) = CENTER OF THE ORTHOLOGIC CONIC OF THESE TRIANGLES: MEDIAL TO ABC

Barycentrics    SA*(6*R^2*S^4+(32*R^4*(8*R^2-SA-3*SW)+2*(7*SW^2+3*SA^2-SB*SC)*R^2-SW^3)*S^2-(8*R^2-SW)^2*SB*SC*SW) : :
X(31367) = 3*X(2)+X(31369)

The center of the reciprocal orthologic conic of these triangles is X(17807)

X(31367) lies on these lines: {2,17807}, {5,31362}, {141,5894}, {5020,15259}, {13567,31368}

X(31367) = midpoint of X(17807) and X(31369)
X(31367) = reflection of X(31362) in X(5)
X(31367) = complement of X(17807)
X(31367) = {X(2), X(31369)}-harmonic conjugate of X(17807)


X(31368) = PERSPECTOR OF THE ORTHOLOGIC CONIC OF THESE TRIANGLES: MEDIAL TO ABC

Barycentrics    SA*((8*R^2-3*SW)*S^2-32*R^2*SB*(4*R^2-SW)+SW*(2*SA*SC-SB^2))*((8*R^2-3*SW)*S^2-32*R^2*SC*(4*R^2-SW)+SW*(2*SA*SB-SC^2)) : :

The perspector of the reciprocal orthologic conic of these triangles is X(17808)

X(31368) lies on these lines: {1368,2883}, {13567,31367}


X(31369) = CENTER OF THE ORTHOLOGIC CONIC OF THESE TRIANGLES: ANTICOMPLEMENTARY TO ABC

Barycentrics    2*(10*R^2-3*SA-2*SW)*R^2*S^2-(256*R^6-32*R^4*(SA+3*SW)+2*R^2*(3*SA^2-3*SB*SC+7*SW^2)-SW^3)*SA : :
X(31369) = 3*X(2)-4*X(31367) = 5*X(3091)-4*X(31362)

The center of the reciprocal orthologic conic of these triangles is X(17807)

X(31369) lies on these lines: {2,17807}, {253,6995}, {1370,14615}, {1995,15259}, {3091,31362}

X(31369) = reflection of X(17807) in X(31367)
X(31369) = anticomplement of X(17807)
X(31369) = {X(17807), X(31367)}-harmonic conjugate of X(2)


X(31370) = PERSPECTOR OF THE ORTHOLOGIC CONIC OF THESE TRIANGLES: ANTICOMPLEMENTARY TO ABC

Barycentrics
(2*(62*R^2-7*SW)*R^2*S^2-32*R^4*(8*R^2*SB+3*SA*SC-4*SB^2)+2*R^2*SW*(SA*SC-8*SB^2)+SA*SC*SW^2)*(2*(62*R^2-7*SW)*R^2*S^2-32*R^4*(8*SC*R^2+3*SA*SB-4*SC^2)+2*R^2*SW*(SA*SB-8*SC^2)+SA*SB*SW^2) : :

The perspector of the reciprocal orthologic conic of these triangles is X(17808)

X(31370) lies on these lines: {}


X(31371) = PERSPECTOR OF THE ORTHOLOGIC TANGENTIAL-CONIC OF THESE TRIANGLES: ANTICOMPLEMENTARY TO ABC

Barycentrics    SA*(3*S^2-4*(SA+SC)*(SB+SC))*(3*S^2-4*(SB+SC)*(SA+SB)) : :

The perspector of the reciprocal orthologic tangential-conic of these triangles is X(15740)

X(31371) lies on these lines: {2,3532}, {4,15010}, {6,3146}, {20,14528}, {54,3529}, {64,3091}, {65,5225}, {66,6225}, {74,3090}, {185,15077}, {265,18909}, {546,3426}, {1899,15749}, {3431,17538}, {3525,11270}, {3527,3627}, {3531,5076}, {3544,13452}, {3618,5895}, {6145,12324}, {7394,16620}, {11433,22466}, {11541,13472}, {14861,18531}, {14865,18532}, {15022,15751}, {18918,21400}

X(31371) = isogonal conjugate of X(3516)
X(31371) = isotomic conjugate of the anticomplement of X(15851)


X(31372) = CENTER OF THE PARALLELOGIC CONIC OF THESE TRIANGLES: ABC TO 1st ANTI-BROCARD

Barycentrics    3*a^8-6*(b^2+c^2)*a^6-3*(b^4-8*b^2*c^2+c^4)*a^4+6*(b^4-3*b^2*c^2+c^4)*(b^2+c^2)*a^2-b^8-(2*b^4-9*b^2*c^2+2*c^4)*b^2*c^2-c^8 : :

The center of the reciprocal parallelogic conic of these triangles is X(31374)

X(31372) lies on these lines: {148,31373}, {523,20094}, {858,7779}

X(31372) = anticomplementary conjugate of the anticomplement of X(20998)
X(31372) = isotomic conjugate of X(31373)
X(31372) = anticomplement of the antitomic conjugate of X(4590)
X(31372) = anticomplement of the cyclocevian conjugate of X(670)
X(31372) = anticomplement of the isogonal conjugate of X(20998)
X(31372) = anticomplement of the isotomic conjugate of X(148)


X(31373) = PERSPECTOR OF THE PARALLELOGIC CONIC OF THESE TRIANGLES: ABC TO 1st ANTI-BROCARD

Barycentrics
(a^8+2*(b^2-3*c^2)*a^6-3*(3*b^4-4*b^2*c^2-c^4)*a^4+2*(b^6+3*c^6+6*(b^2-2*c^2)*b^2*c^2)*a^2+b^8-3*(2*b^4-b^2*c^2-2*c^4)*b^2*c^2-3*c^8)*(a^8-2*(3*b^2-c^2)*a^6+3*(b^4+4*b^2*c^2-3*c^4)*a^4+2*(3*b^6+c^6-6*(2*b^2-c^2)*b^2*c^2)*a^2-3*b^8+3*(2*b^4+b^2*c^2-2*c^4)*b^2*c^2+c^8) : :

The perspector of the reciprocal parallelogic conic of these triangles is X(31375)

X(31373) lies on the line {148,31372}

X(31373) = isotomic conjugate of X(31372)


X(31374) = CENTER OF THE PARALLELOGIC CONIC OF THESE TRIANGLES: 1st ANTI-BROCARD TO ABC

Barycentrics
3*a^12-5*(b^2+c^2)*a^10-5*(2*b^2-c^2)*(b^2-2*c^2)*a^8+(b^2+c^2)*(7*b^4-4*b^2*c^2+7*c^4)*a^6+(14*b^8+14*c^8-3*(11*b^4-b^2*c^2+11*c^4)*b^2*c^2)*a^4-(b^2+c^2)*(5*b^8+5*c^8+3*(2*b^4-11*b^2*c^2+2*c^4)*b^2*c^2)*a^2-(b^2+3*b*c+c^2)*(b^2-3*b*c+c^2)*(b^4-b^2*c^2-c^4)*(b^4+b^2*c^2-c^4) : :

The center of the reciprocal parallelogic conic of these triangles is X(31372)

X(31374) lies on the line {2396,20094}


X(31375) = PERSPECTOR OF THE PARALLELOGIC CONIC OF THESE TRIANGLES: 1st ANTI-BROCARD TO ABC

Barycentrics
(3*S^6-(384*R^2-8*SC-69*SW)*SW*S^4+(64*(12*SA*SB-3*SC^2+SW^2)*R^2-(144*SA*SB-16*SC^2+15*SW^2)*SW)*SW*S^2+(8*SC-SW)*SW^5)*(3*S^6-(384*R^2-8*SB-69*SW)*SW*S^4+(64*(12*SA*SC-3*SB^2+SW^2)*R^2-(144*SA*SC-16*SB^2+15*SW^2)*SW)*SW*S^2+(8*SB-SW)*SW^5) : :

The perspector of the reciprocal parallelogic conic of these triangles is X(31373)

X(31375) lies on these lines: {}


X(31376) = COMPLEMENT OF X(252)

Barycentrics    (S^2+SB*SC)*(3*S^2-SA^2)*(4*S^2+(SB+SC)*(2*R^2-SB-SC)) : :

See Tran Quang Hung and César Lozada, Hyacinthos 28829.

X(31376) lies on these lines: {2, 252}, {3, 24573}, {5, 128}, {140, 6150}, {233, 5421}, {1209, 1493}, {2072, 10600}, {5501, 6592}, {7575, 15848}, {13372, 21975}

X(31376) = midpoint of X(3) and X(24573)
X(31376) = reflection of X(i) in X(j) for these (i,j): (5, 23281), (23280, 3628)
X(31376) = complement of X(252)
X(31376) = {X(5), X(15345)}-harmonic conjugate of X(137)


X(31377) = COMPLEMENT OF X(6526)

Barycentrics    (3*a^4-2*(b^2+c^2)*a^2-(b^2-c^2)^2)*(a^8-4*(b^2-c^2)^2*a^4+4*(b^4-c^4)*(b^2-c^2)*a^2-(b^2-c^2)^4)*(-a^2+b^2+c^2)^2 : :
Barycentrics    SA^2*(S^2-2*SB*SC)*(2*S^2-(SB+SC)*(8*R^2-SB-SC)) : :

See Tran Quang Hung and César Lozada, Hyacinthos 28829.

X(31377) lies on these lines: {2, 1105}, {3, 1661}, {4, 12096}, {20, 122}, {131, 3548}, {140, 20207}, {216, 631}, {1073, 6696}, {3546, 10600}, {5895, 27089}, {6225, 11589}, {6389, 16196}, {6523, 6716}, {6760, 14216}, {11457, 14385}

X(31377) = complement of X(6526)


X(31378) = COMPLEMENT OF X(5627)

Barycentrics    (S^2-3*SB*SC)*(3*SA^2-S^2)*(4*S^2-3*(SB+SC)*(6*R^2-SB-SC)) : :
X(31378) = 4*X(140)-X(6070), 3*X(549)+X(18285), 5*X(631)+X(14480), 2*X(1511)+X(3258), X(1553)-4*X(10272), 2*X(3154)+X(30714), 2*X(5972)+X(14934), 4*X(5972)-X(25641), 2*X(6699)+X(14611), X(10564)+2*X(16319), X(14508)+5*X(20125), 2*X(14934)+X(25641), 5*X(15034)+X(17511), 5*X(15040)+X(20957)

See Tran Quang Hung and César Lozada, Hyacinthos 28829.

X(31378) lies on the cubics K515, K900 and these lines: {2, 5627}, {30, 113}, {128, 6760}, {140, 6070}, {186, 14920}, {476, 1138}, {541, 15468}, {549, 18285}, {631, 14480}, {3003, 3163}, {3154, 30714}, {5972, 14934}, {6699, 14611}, {14508, 20125}, {14993, 22104}, {15034, 17511}, {15040, 20957}

X(31378) = midpoint of X(476) and X(1138)
X(31378) = reflection of X(14993) in X(22104)
X(31378) = complement of X(5627)
X(31378) = complementary conjugate of X(20304)
X(31378) = {X(5972), X(14934)}-harmonic conjugate of X(25641)


X(31379) = COMPLEMENT OF X(25641)

Barycentrics    S^4-(3*R^2*(90*R^2+3*SA-40*SW)-2*SA^2-SB*SC+13*SW^2)*S^2+(18*R^2-5*SW)*(9*R^2-SW)*SB*SC : :
X(31379) = 3*X(2)+X(477), 3*X(3)+X(20957), X(476)-5*X(631), 5*X(632)-X(18319), X(1553)-3*X(14643), 5*X(3091)-X(14989), 3*X(3258)-X(20957), 7*X(3523)+X(14731), X(6070)-3*X(15061), X(11749)+7*X(14869), 3*X(15035)+X(17511)

See Tran Quang Hung and César Lozada, Hyacinthos 28829.

X(31379) lies on these lines: {2, 477}, {3, 3258}, {30, 5972}, {125, 14934}, {140, 16168}, {476, 631}, {523, 6699}, {620, 15122}, {632, 18319}, {1511, 16340}, {1553, 14643}, {3091, 14989}, {3154, 17702}, {3523, 14731}, {5446, 12052}, {6070, 15061}, {10625, 16978}, {11749, 14869}, {12079, 20397}, {14611, 16003}, {14915, 16319}, {15035, 17511}, {15088, 21316}

X(31379) = midpoint of X(i) and X(j) for these {i,j}: {3, 3258}, {125, 14934}, {477, 25641}, {1511, 16340}, {10625, 16978}, {14611, 16003}
X(31379) = reflection of X(i) in X(j) for these (i,j): (5446, 12052), (12079, 20397), (21316, 15088), (22104, 140)
X(31379) = complement of X(25641)
X(31379) = {X(2), X(477)}-harmonic conjugate of X(25641)


X(31380) = COMPLEMENT OF X(5513)

Barycentrics    2*a^6-2*(b+c)*a^5-2*(b^2-b*c+c^2)*a^4+(b+c)*(5*b^2-6*b*c+5*c^2)*a^3-(3*b^4+3*c^4-(b^2+c^2)*b*c)*a^2-(b^4-c^4)*(b-c)*a+(b^4+c^4+(b-c)^2*b*c)*(b-c)^2 : :
X(31380) = 3*X(2)+X(675)

See Tran Quang Hung and César Lozada, Hyacinthos 28829.

X(31380) lies on these lines: {2, 101}, {3, 25642}, {142, 6718}, {1054, 4859}, {3035, 3739}, {3315, 8458}, {5432, 6025}, {5972, 6707}, {6678, 6720}, {10165, 15746}, {16056, 25468}

X(31380) = midpoint of X(i) and X(j) for these {i,j}: {3, 25642}, {675, 5513}
X(31380) = complement of X(5513)
X(31380) = orthoptic circle of Steiner inellipse-inverse-of X(150)
X(31380) = {X(2), X(675)}-harmonic conjugate of X(5513)


X(31381) = X(3)X(49)∩X(24)X(157)

Barycentrics    (S^2-SB*SC)*(S^2+2*R^2*(4*R^2+SA-4*SW)+SW^2) : :

See Nguyen Dang Khoa and César Lozada, Hyacinthos 28830.

X(31381) lies on these lines: {3, 49}, {4, 19172}, {24, 157}, {25, 8887}, {26, 2351}, {68, 23181}, {418, 6146}, {578, 16035}, {973, 23635}, {1624, 7505}, {2917, 7669}, {2937, 13558}, {3133, 12134}, {3135, 9833}, {6776, 26876}, {7512, 8266}, {7592, 15231}, {8553, 17849}, {9715, 15512}, {18925, 26874}


X(31382) = X(20)X(64)∩X(154)X(160)

Barycentrics    (SB+SC)*(3*S^4-(4*R^2*(8*R^2+4*SA-3*SW)-4*SA^2+2*SB*SC+SW^2)*S^2+2*(4*R^2-SW)*SB*SC*SW) : :
X(31382) = 3*X(154)-4*X(160)

See Nguyen Dang Khoa and César Lozada, Hyacinthos 28830.

X(31382) lies on these lines: {6, 1987}, {20, 64}, {154, 160}, {6748, 17810}, {18445, 22552}


X(31383) = MARTA POINT

Barycentrics    3a^6-a^4(b^2+c^2)-a^2(b^2-c^2)^2-(b^2-c^2)^2(b^2+c^2) : :

X(31383) = 3 X[25] - 2 X[13567],5 X[25] - 3 X[26869],3 X[1899] - 4 X[13567],5 X[1899] - 6 X[26869],10 X[13567] - 9 X[26869]

See Angel Montesdeoca, Hyacinthos 28832 and Centro ortológico y punto fijo de una afinidad.

X(31383) lies on these lines: {2, 1495}, {3, 16655}, {4, 54}, {5, 3796}, {6, 428}, {20, 3917}, {22, 1352}, {23, 11442}, {24, 14216}, {25, 1503}, {30, 394}, {51, 6776}, {66, 20987}, {68, 7517}, {69, 16276}, {110, 7391}, {125, 6353}, {154, 427}, {155, 7553}, {182, 6997}, {185, 7487}, {193, 21969}, {343, 9909}, {373, 7398}, {378, 16658}, {382, 3167}, {393, 8779}, {462, 5868}, {463, 5869}, {468, 1853}, {511, 7500}, {542, 6515}, {1181, 6756}, {1204, 12324}, {1370, 9306}, {1498, 3575}, {1501, 3767}, {1514, 3830}, {1593, 16621}, {1595, 19357}, {1596, 18396}, {1597, 16654}, {1598, 6146}, {1843, 5596}, {1885, 15811}, {1971, 22075}, {1994, 20423}, {2165, 2980}, {2182, 5101}, {2883, 12173}, {2979, 20062}, {3051, 3331}, {3060, 7519}, {3091, 11572}, {3146, 3292}, {3147, 20299}, {3148, 8721}, {3426, 3534}, {3515, 6247}, {3518, 11457}, {3541, 10282}, {3542, 18381}, {3867, 19125}, {4232, 23291}, {4549, 18435}, {5012, 7394}, {5064, 23292}, {5094, 10192}, {5133, 6800}, {5198, 12241}, {5200, 5871}, {5310, 12588}, {5320, 5800}, {5322, 12589}, {5422, 11179}, {5480, 11402}, {5562, 31305}, {5651, 7386}, {5654, 10540}, {5878, 6240}, {6000, 18533}, {6053, 10706}, {6243, 9936}, {6623, 13851}, {6696, 15750}, {7387, 12134}, {7392, 25406}, {7401, 10984}, {7408, 13366}, {7493, 21243}, {7499, 10516}, {7507, 16252}, {7540, 18445}, {7576, 11456}, {7667, 17811}, {7714, 11433}, {7715, 18914}, {7716, 26926}, {7795, 10328}, {8550, 9777}, {8780, 11064}, {9707, 15559}, {9714, 12359}, {9934, 12140}, {10151, 18405}, {10301, 11245}, {10535, 11393}, {10539, 14790}, {11392, 26888}, {11403, 16656}, {12111, 31304}, {12112, 18559}, {12174, 13568}, {12290, 20427}, {13399, 18931}, {13417, 14683}, {13595, 18911}, {14227, 19219}, {14912, 15004}, {15152, 18386}, {15448, 23332}, {15583, 19118}, {16105, 23236}, {16319, 18870}, {16657, 18535}, {17907, 19558}, {18374, 23327}, {18378, 25738}, {21645, 31291}, {21646, 31299}, {22135, 27376}

X(31383) = reflection of X(i) in X(j) for these (i,j): {1370,9306}, {1899,25}, {18396,1596}
X(31383) = crosssum of X(2972) and X(8673)
X(31383) = crossdifference of every pair of points on the line {9210, 17434}
X(31383) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {4, 9833, 19467}, {4, 11206, 184}, {4, 18925, 11424}, {20, 14826, 3917}, {24, 14216, 26937}, {24, 16659, 14216}, {1495, 11550, 2}, {5012, 7394, 14561}, {5064, 26864, 23292}, {6759, 13419, 4}, {6776, 6995, 51}, {9909, 18440, 343}, {10301, 11245, 17810}, {15811, 17845, 1885}


X(31384) = REFLECTION OF X(4) IN X(431)

Barycentrics    a (a^2+b^2-c^2) (a^2-b^2+c^2) (a^8-a^7 b-3 a^6 b^2+3 a^5 b^3+3 a^4 b^4-3 a^3 b^5-a^2 b^6+a b^7-a^7 c-a^6 b c+a^5 b^2 c+a^4 b^3 c+a^3 b^4 c+a^2 b^5 c-a b^6 c-b^7 c-3 a^6 c^2+a^5 b c^2+4 a^4 b^2 c^2-a^2 b^4 c^2-a b^5 c^2+3 a^5 c^3+a^4 b c^3-2 a^2 b^3 c^3+a b^4 c^3+b^5 c^3+3 a^4 c^4+a^3 b c^4-a^2 b^2 c^4+a b^3 c^4-3 a^3 c^5+a^2 b c^5-a b^2 c^5+b^3 c^5-a^2 c^6-a b c^6+a c^7-b c^7) : :
Barycentrics    (8 a R^4-8 b R^4-4 a R^2 SB+4 b R^2 SB-4 a R^2 SC+4 c R^2 SC-2 a R^2 SW+6 b R^2 SW+a SB SW-b SB SW+a SC SW-c SC SW-b SW^2) S^2 +(-8 R^3 SB SC+2 R SB SC SW) S +2 b R^2 SB SC^2-2 c R^2 SB SC^2-2 b R^2 SB SC SW-b SB SC^2 SW+c SB SC^2 SW+b SB SC SW^2 : :

As a point on the Euler line, X(31384) has Shinagawa coefficients {(p - r - 2 R) (r + R) (p + r + 2 R), r^3 + 6 r^2 R + 10 r R^2 + 6 R^3 - p^2 (r + 2 R)}.

See Kadir Altintas and Ercole Suppa, Hyacinthos 28835.

X(31384) lies on these lines: {2,3}, {100,1299}, {108,1300}, {3563,26706}

X(31384) = reflection of X(4) in X(431)
X(31384) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {4,186,21}, {4,3147,6824}, {4,6853,1594}, {4,6874,7547}, {4,6876,378}, {4,6988,3541}, {4,7505,6828}, {25,6985,4}, {3575,6842,4}, {3651,4231,7414}, {6838,7487,4}


X(31385) = EULER LINE INTERCEPT OF X(108)X(1299)

Barycentrics    a (a^2+b^2-c^2) (a^2-b^2+c^2) (a^11-3 a^9 b^2+2 a^7 b^4+2 a^5 b^6-3 a^3 b^8+a b^10+a^9 b c-a^8 b^2 c-4 a^7 b^3 c+4 a^6 b^4 c+6 a^5 b^5 c-6 a^4 b^6 c-4 a^3 b^7 c+4 a^2 b^8 c+a b^9 c-b^10 c-3 a^9 c^2-a^8 b c^2+6 a^7 b^2 c^2+2 a^6 b^3 c^2-4 a^5 b^4 c^2+2 a^3 b^6 c^2-2 a^2 b^7 c^2-a b^8 c^2+b^9 c^2-4 a^7 b c^3+2 a^6 b^2 c^3+4 a^5 b^3 c^3+2 a^4 b^4 c^3-6 a^2 b^6 c^3+2 b^8 c^3+2 a^7 c^4+4 a^6 b c^4-4 a^5 b^2 c^4+2 a^4 b^3 c^4+2 a^3 b^4 c^4+4 a^2 b^5 c^4-2 b^7 c^4+6 a^5 b c^5+4 a^2 b^4 c^5-2 a b^5 c^5+2 a^5 c^6-6 a^4 b c^6+2 a^3 b^2 c^6-6 a^2 b^3 c^6-4 a^3 b c^7-2 a^2 b^2 c^7-2 b^4 c^7-3 a^3 c^8+4 a^2 b c^8-a b^2 c^8+2 b^3 c^8+a b c^9+b^2 c^9+a c^10-b c^10) : :

As a point on the Euler line, X(31385) has Shinagawa coefficients {(p - r - 2 R) (p + r + 2 R) (p^2 - r^2 - 2 r R - 2 R^2), -p^4 - r^4 - 8 r^3 R - 22 r^2 R^2 - 28 r R^3 - 12 R^4 + 2 p^2 (r + R) (r + 3 R)} .

See Kadir Altintas and Ercole Suppa, Hyacinthos 28835.

X(31385) lies on these lines: {2,3}, {108,1299}

X(31385) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {4,186,16049}, {24,403,28}


X(31386) = X(3)X(20032)∩X(4)X(20034)

Barycentrics    (Sin[B] Tan[A] Sqrt[Tan[B]]-Sin[A] Sqrt[Tan[A]] Tan[B]) (Sin[2 A] Sin[C] (Sin[C]-Sqrt[Tan[A] Tan[B]]) Tan[C]-Sin[A] Sin[2 C] Tan[A] (Sin[A]-Sqrt[Tan[B] Tan[C]]))- (Sin[C] Tan[A] Sqrt[Tan[C]]-Sin[A] Sqrt[Tan[A]] Tan[C]) (Sin[2 A] Sin[B] (Sin[B]-Sqrt[Tan[A] Tan[C]]) Tan[B]-Sin[A] Sin[2 B] Tan[A] (Sin[A]-Sqrt[Tan[B] Tan[C]])) : :
Barycentrics    a SB SC (4 a (b^2-c^2) S^2 SA+b (a^6-2 a^4 b^2+a^2 b^4-5 a^4 c^2+b^4 c^2+5 a^2 c^4-c^6) Sqrt[SA SB]-c (a^6-5 a^4 b^2+5 a^2 b^4-b^6-2 a^4 c^2+a^2 c^4+b^2 c^4) Sqrt[SA SC]-4 a b c (b^2-c^2) SA Sqrt[SB SC]) : :

See Kadir Altintas and Peter Moses, Hyacinthos 28838.

X(31386) lies on the cubic K742 and these lines: {3,20032}, {4,20034}, {5,20033}


X(31387) = X(3)X(915)∩X(4)X(8)

Barycentrics    (a^6-(b+c)^2*a^4-(b^2+c^2)*(b^2-4*b*c+c^2)*a^2+(b^2-c^2)^2*(b-c)^2)*(a^2-b^2+c^2)*(a^2+b^2-c^2) : :
Barycentrics    SB*SC*(b*c+2*R^2-SW) : :

See Tran Quang Hung and César Lozada, Hyacinthos 28843.

X(31387) lies on these lines: {3, 915}, {4, 8}, {11, 7040}, {24, 278}, {52, 5905}, {68, 2994}, {242, 3542}, {281, 1594}, {513, 5553}, {1068, 11398}, {1118, 1870}, {2973, 17170}, {3147, 17923}, {7510, 11396}, {10018, 17917}, {17516, 21664}


X(31388) = X(3)X(54)∩X(4)X(11197)

Barycentrics    a^2 (a^2-b^2-c^2)^2 (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (2 a^6-3 a^4 b^2+b^6-3 a^4 c^2-b^4 c^2-b^2 c^4+c^6) : :
Barycentrics    S^4 +(16 R^4+8 R^2 SB+8 R^2 SC-SB SC-16 R^2 SW-2 SB SW-2 SC SW+3 SW^2) S^2-32 R^4 SB SC+20 R^2 SB SC SW-3 SB SC SW^2 : :
X(31388) = 2*X[4]-3*X[11197], 7*X[3090]-6*X[14635], 5*X[3091]-6*X[10184], 7*X[3523]-6*X[12012]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28848.

X(31388) lies on these lines: {3,54}, {4,11197}, {5,14918}, {30,14978}, {160,6293}, {185,20775}, {216,14531}, {417,3917}, {418,5562}, {511,26897}, {577,8565}, {852,11793}, {1216,2972}, {3090,14635}, {3091,10184}, {3523,12012}, {3575,26166}, {6368,14329}, {6638,11444}, {6641,17834}, {11413,23709}, {15905,26216}, {18564,25043}

X(31388) = barycentric product of X(i) and X(j) for these {i,j}: {343, 13367}, {394, 3574}, {418, 26166}, {5562, 23292}
X(31388) = trilinear product of X(i) and X(j) for these {i,j}: {255, 3574}, {418, 17859}, {418, 17859}
X(31388) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {3,11412,13409}, {97,7691,3}


X(31389) = X(2)X(22269)∩X(5)X(23607)

Barycentrics    (a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) (2 a^4-3 a^2 b^2+b^4-3 a^2 c^2-2 b^2 c^2+c^4) (a^6 b^2-3 a^4 b^4+3 a^2 b^6-b^8+a^6 c^2-4 a^4 b^2 c^2-3 a^2 b^4 c^2+6 b^6 c^2-3 a^4 c^4-3 a^2 b^2 c^4-10 b^4 c^4+3 a^2 c^6+6 b^2 c^6-c^8) : :
Barycentrics    3 S^4 + (16 R^2 SB+16 R^2 SC+9 SB SC-12 R^2 SW-4 SB SW-4 SC SW+3 SW^2)S^2 -16 R^4 SB SC+SB SC SW^2 : :
X(31389) = 6*X[2]-5*X[22269], 7*X[3090]-6*X[12013]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28848.

X(31389) lies on these lines: {2,22269}, {5,23607}, {381,2888}, {1209,11197}, {3078,14978}, {3090,12013}


X(31390) = REFLECTION OF X(23642) IN X(6)

Barycentrics    a^2 (b^2+c^2) (2 a^4+a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) : :
Barycentrics (8 R^2 SB+8 R^2 SC-4 R^2 SW-3 SB SW-3 SC SW+2 SW^2) S^2 -2 SB SC SW^2-SB SW^3-SC SW^3 : :

See Kadir Altintas and Ercole Suppa, Hyacinthos 28848.

X(31390) lies on these lines: {6,22}, {32,23635}, {69,8878}, {141,23297}, {193,732}, {217,5052}, {511,8152}, {688,22260}, {1843,3051}, {3124,9969}, {3172,12167}, {3231,9822}, {3313,8041}, {3618,13410}, {9971,16285}, {11574,20965}

X(31390) = reflection of X(23642) in X(6)
X(31390) = barycentric product of X(i) and X(j) for these {i,j}: {31, 23665}, {39, 7745}, {427, 21637}, {1843, 6676}, {2084, 18063}
X(31390) = trilinear product of X(i) and X(j) for these {i,j}: {32, 23665}, {32, 23665}, {688, 18063}, {688, 18063}, {1964, 7745}, {1964, 7745}, {17442, 21637}, {17442, 21637}
X(31390) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {6,23642,11205}


X(31391) = MIDPOINT OF X(20059) AND X(25722)

Barycentrics    a (a^3 b-3 a^2 b^2+3 a b^3-b^4+a^3 c+6 a^2 b c-3 a b^2 c-4 b^3 c-3 a^2 c^2-3 a b c^2+10 b^2 c^2+3 a c^3-4 b c^3-c^4) : :
X(31391) = 4*X[7]-3*X[354], 2*X[144]-3*X[210], 2*X[960]-3*X[10861], 4*X[15008]-5*X[18398]

See Kadir Altintas and Ercole Suppa, Hyacinthos 28848.

X(31391) lies on these lines: {4,10307}, {7,354}, {9,1155}, {37,3000}, {46,5779}, {55,2951}, {56,11372}, {57,3062}, {65,971}, {142,17605}, {144,210}, {226,5918}, {279,10939}, {513,2262}, {516,3057}, {518,1278}, {527,3059}, {960,10861}, {962,9850}, {990,1456}, {1001,8544}, {1122,4014}, {1418,2310}, {1420,24644}, {1445,16112}, {1721,6180}, {1770,5762}, {1864,11246}, {2646,5732}, {2801,17636}, {3304,10384}, {3600,10866}, {3698,5229}, {3748,4326}, {3983,5128}, {4292,12688}, {4295,12680}, {4298,9848}, {4321,20323}, {4336,6610}, {4887,14523}, {4907,7271}, {5221,10398}, {5226,10178}, {5728,17637}, {5784,17768}, {5805,7702}, {5817,24914}, {5843,14872}, {7988,11575}, {8545,11495}, {10442,21334}, {11375,21151}, {12764,18482}, {15008,18398}, {15185,17660}

X(31391) = midpoint of X(20059 ) and X(25722)
X(31391) = reflection of X(i) in X(j) for these {i,j}: {65,4312}, {144,15587}, {3057,8581}, {3059,17668}, {14100,7}
X(31391) = {X(i),X(j)}-harmonic conjugate of X(k) for these {i,j,k}: {7,14100,354}, {144,15587,210}, {4014,12723,1122}, {7354,17634,3057}, {8545,11495,15837}


X(31392) = REFLECTION OF X(5) IN X(14051)

Barycentrics    (a^4+(b^2-c^2)^2-a^2 (b^2+2 c^2)) (a^12-(b^2-c^2)^6-a^10 (4 b^2+3 c^2)+a^8 (5 b^4+5 b^2 c^2+3 c^4)-a^6 (3 b^4 c^2+2 b^2 c^4)+a^2 (b^2-c^2)^2 (4 b^6-6 b^4 c^2-3 b^2 c^4+3 c^6)+a^4 (-5 b^8+9 b^6 c^2+b^4 c^4+4 b^2 c^6-3 c^8)) : :

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28850.

X(31392) lies on these lines: {4,93}, {5,930}, {17,8173}, {18,8172}, {30,252}, {54,1263}, {143,11538}, {195,20414}, {265,6798}, {550,1487}, {1157,28237}, {1879,2937}, {2070,21394}, {3459,6150}, {11671,15345}, {20030,20424}, {20413,27090}, {21230,24306}

X(31392) = reflection of X(i) in X(j) for these {i, j}: {5,14051}, {930,19268}
X(31392) = isogonal conjugate of X(18212)






leftri  Centroids of curvatures: X(31393) - X(31502)  rightri

This preamble and centers X(31393)-X(31502) were contributed by César Eliud Lozada, February 08, 2019.

Let (O1), (O2), (O3) be three circles with distinct and non-collinear centers. Denote I1, I2, I3 the internal centers of similitude of {(O2), (O3)}, {(O3), (O1)} and {(O1), (O2)}, respectively. Then the lines O1I1, O2I2, O3I3 concur.

Assume that exact trilinear coordinates of the centers are Oi = ( Ui, Vi, Wi ) and their radius are Ri (i = 1..3). If M is the trilinear matrix of the centers then the given point of intersection Q is:

   Q = MT.| ρ1 ρ2 ρ3 |T, where ρi = 1/Ri, i.e., ρi is the curvature of the circle (Oi).

In fewer words:

   Q = ∑ ρiUi : ∑ ρiVi : ∑ ρiWi,  i=1..3

Then, for obvious reasons, the point Q is here named here the centroid of curvatures of the circles (O1), (O2), (O3).

Suppose a fourth circle (O4) is added to the above configuration and let Q be the centroid of curvatures of (Oi), (Oj), (Ok). Then the lines OiQi concur at:

   Q = ∑ ρiUi : ∑ ρiVi : ∑ ρiWi,  i=1..4

The last extension provides a geometrical recursive construction of the centroid of curvatures of any number n>=3 of circles with distinct and not collinear centers.

The appearance of (C, n) in the following list means that X(n) is the centroid of curvatures of the three circles C:
(excircles, 1), (excosine, 6), (Johnson, 381), (Lucas(+1), 6221), (Lucas(+1) secondary, 3311), (Lucas(-1), 6398), (Lucas(-1) secondary, 3312), (Malfatti, 31495), (McCay, 7611), (mixtilinear incircles, 31393), (mixtilinear excircles, 1), (Neuberg, 31394), (2nd Neuberg, 31395), (Soddy, 7) (Yff internal, 10056), (Yff external, 10072).

The appearance of (C1,C2,C3,n) in the following list means that X(n) is the centroid of curvatures of the three circles C1, C2, C3:
(anticomplementary, Bevan, circumcircle, 6684), (anticomplementary, Bevan, half-Moses, 31396), (anticomplementary, Bevan, incircle, 31397), (anticomplementary, Bevan, 2nd Lemoine, 13883), (anticomplementary, Bevan, Moses, 31398), (anticomplementary, Bevan, nine-points, 10175), (anticomplementary, Bevan, Stammler, 26446), (anticomplementary, Bevan, Steiner, 31399), (anticomplementary, circumcircle, half-Moses, 31400), (anticomplementary, circumcircle, incircle, 3085), (anticomplementary, circumcircle, 2nd Lemoine, 3068), (anticomplementary, circumcircle, Moses, 31401), (anticomplementary, circumcircle, Spieker, 19843), (anticomplementary, half-Moses, incircle, 31402), (anticomplementary, half-Moses, 2nd Lemoine, 31403), (anticomplementary, half-Moses, nine-points, 31404), (anticomplementary, half-Moses, Spieker, 31405), (anticomplementary, half-Moses, Stammler, 31406), (anticomplementary, half-Moses, Steiner, 31407), (anticomplementary, incircle, 2nd Lemoine, 31408), (anticomplementary, incircle, Moses, 31409), (anticomplementary, incircle, nine-points, 10590), (anticomplementary, incircle, Spieker, 443), (anticomplementary, incircle, Stammler, 495), (anticomplementary, incircle, Steiner, 31410), (anticomplementary, 2nd Lemoine, Moses, 31411), (anticomplementary, 2nd Lemoine, nine-points, 31412), (anticomplementary, 2nd Lemoine, Spieker, 31413), (anticomplementary, 2nd Lemoine, Stammler, 7583), (anticomplementary, 2nd Lemoine, Steiner, 31414), (anticomplementary, Moses, nine-points, 31415), (anticomplementary, Moses, Spieker, 31416), (anticomplementary, Moses, Stammler, 3815), (anticomplementary, Moses, Steiner, 31417), (anticomplementary, nine-points, Spieker, 31418), (anticomplementary, Spieker, Stammler, 31419), (anticomplementary, Spieker, Steiner, 31420), (Bevan, circumcircle, half-Moses, 31421), (Bevan, circumcircle, 2nd Lemoine, 9616), (Bevan, circumcircle, Moses, 31422), (Bevan, circumcircle, nine-points, 31423), (Bevan, circumcircle, Spieker, 31424), (Bevan, circumcircle, Steiner, 31425), (Bevan, half-Moses, incircle, 31426), (Bevan, half-Moses, 2nd Lemoine, 31427), (Bevan, half-Moses, nine-points, 31428), (Bevan, half-Moses, Spieker, 31429), (Bevan, half-Moses, Stammler, 31430), (Bevan, half-Moses, Steiner, 31431), (Bevan, incircle, 2nd Lemoine, 31432), (Bevan, incircle, Moses, 31433), (Bevan, incircle, nine-points, 31434), (Bevan, incircle, Spieker, 31435), (Bevan, incircle, Steiner, 31436), (Bevan, 2nd Lemoine, Moses, 31437), (Bevan, 2nd Lemoine, nine-points, 13893), (Bevan, 2nd Lemoine, Spieker, 31438), (Bevan, 2nd Lemoine, Stammler, 31439), (Bevan, 2nd Lemoine, Steiner, 31440), (Bevan, Moses, nine-points, 31441), (Bevan, Moses, Spieker, 31442), (Bevan, Moses, Stammler, 31443), (Bevan, Moses, Steiner, 31444), (Bevan, nine-points, Spieker, 5705), (Bevan, nine-points, Stammler, 11231), (Bevan, Spieker, Stammler, 31445), (Bevan, Spieker, Steiner, 31446), (Bevan, Stammler, Steiner, 31447), (circumcircle, half-Moses, incircle, 31448), (circumcircle, half-Moses, nine-points, 31401), (circumcircle, half-Moses, Spieker, 31449), (circumcircle, half-Moses, Steiner, 31450), (circumcircle, incircle, 2nd Lemoine, 2066), (circumcircle, incircle, Moses, 31451), (circumcircle, incircle, nine-points, 498), (circumcircle, incircle, Spieker, 405), (circumcircle, incircle, Steiner, 31452), (circumcircle, 2nd Lemoine, nine-points, 590), (circumcircle, 2nd Lemoine, Spieker, 31453), (circumcircle, 2nd Lemoine, Steiner, 31454), (circumcircle, Moses, nine-points, 31455), (circumcircle, Moses, Spieker, 31456), (circumcircle, Moses, Steiner, 31457), (circumcircle, nine-points, Spieker, 26363), (circumcircle, Spieker, Steiner, 31458), (half-Moses, incircle, 2nd Lemoine, 31459), (half-Moses, incircle, nine-points, 31460), (half-Moses, incircle, Spieker, 5283), (half-Moses, incircle, Stammler, 31461), (half-Moses, incircle, Steiner, 31462), (half-Moses, 2nd Lemoine, nine-points, 31463), (half-Moses, 2nd Lemoine, Spieker, 31464), (half-Moses, 2nd Lemoine, Steiner, 31465), (half-Moses, nine-points, Spieker, 31466), (half-Moses, nine-points, Stammler, 31467), (half-Moses, Spieker, Stammler, 31468), (half-Moses, Spieker, Steiner, 31469), (half-Moses, Stammler, Steiner, 31470), (incircle, 2nd Lemoine, Moses, 31471), (incircle, 2nd Lemoine, nine-points, 31472), (incircle, 2nd Lemoine, Spieker, 31473), (incircle, 2nd Lemoine, Stammler, 31474), (incircle, 2nd Lemoine, Steiner, 31475), (incircle, Moses, nine-points, 31476), (incircle, Moses, Spieker, 16589), (incircle, Moses, Stammler, 31477), (incircle, Moses, Steiner, 31478), (incircle, nine-points, Spieker, 442), (incircle, nine-points, Stammler, 31479), (incircle, Spieker, Stammler, 11108), (incircle, Spieker, Steiner, 17529), (incircle, Stammler, Steiner, 31480), (2nd Lemoine, Moses, nine-points, 31481), (2nd Lemoine, Moses, Spieker, 31482), (2nd Lemoine, Moses, Steiner, 31483), (2nd Lemoine, nine-points, Spieker, 31484), (2nd Lemoine, nine-points, Stammler, 8976), (2nd Lemoine, Spieker, Stammler, 31485), (2nd Lemoine, Spieker, Steiner, 31486), (2nd Lemoine, Stammler, Steiner, 31487), (Moses, nine-points, Spieker, 31488), (Moses, nine-points, Stammler, 31489), (Moses, Spieker, Stammler, 31490), (Moses, Spieker, Steiner, 31491), (Moses, Stammler, Steiner, 31492), (nine-points, Spieker, Stammler, 31493), (Spieker, Stammler, Steiner, 31494).

The appearance of (C, n) in the following list means that X(n) is the centroid of curvatures of the circles { C, circumcircle, incircle, nine-points }:
(Apollonius, 31496), (extangents, 31498), (half-Moses, 31497), (2nd Johnson-Yff, 55), (2nd Lemoine, 9646), (radical circle of Lucas circles, 31499), (inner Lucas, 31500), (Moses, 31501), (Parry, 31502), (Spieker, 2)

For definitions of all these circles, see Triangle circles at Wolfram Mathworld. Note: Only some few circles in this list are treated here.

underbar

X(31393) = CENTROID OF CURVATURES OF THE MIXTILINEAR INCIRCLES

Barycentrics    a*(a^3+(b+c)*a^2-(b^2+10*b*c+c^2)*a-(b^2-c^2)*(b-c)) : :
X(31393) = 5*X(1)-2*X(1159) = 3*X(1)-2*X(15934) = 3*X(1)-X(18421) = 3*X(40)-4*X(3587) = X(1000)+2*X(30331) = 4*X(1001)-X(11525) = X(4900)-6*X(16857) = 2*X(7966)+X(11372) = 2*X(14563)-3*X(15933) = 4*X(15935)-X(16236)

X(31393) lies on these lines: {1,3}, {2,3895}, {4,12575}, {7,28194}, {8,3305}, {9,519}, {10,1058}, {12,9614}, {63,3241}, {78,3890}, {84,4313}, {90,13606}, {145,3219}, {191,15174}, {200,392}, {226,30305}, {355,15172}, {376,4315}, {380,2256}, {381,5726}, {388,10624}, {390,515}, {405,4853}, {495,1699}, {496,1698}, {497,5587}, {516,1056}, {551,5437}, {758,3243}, {908,11239}, {936,3913}, {938,11362}, {943,3680}, {944,4314}, {946,5226}, {950,5881}, {952,10384}, {956,4512}, {958,12629}, {960,6765}, {962,21620}, {997,3158}, {1001,3880}, {1015,9574}, {1125,1706}, {1158,13607}, {1334,16572}, {1376,10179}, {1387,5541}, {1419,1480}, {1445,14563}, {1449,2267}, {1453,3915}, {1478,9580}, {1479,9578}, {1500,9575}, {1512,10596}, {1616,4646}, {1702,3298}, {1703,3297}, {1768,12735}, {1953,2270}, {2346,3577}, {2800,7675}, {2809,7174}, {3058,3586}, {3085,6964}, {3244,6762}, {3476,4304}, {3487,4301}, {3555,12526}, {3584,23708}, {3624,11373}, {3655,7171}, {3656,5719}, {3679,4863}, {3740,8168}, {3753,10582}, {3811,3884}, {3813,5705}, {3817,8164}, {3870,3877}, {3871,19861}, {3878,11523}, {3885,19860}, {3902,5271}, {3982,4295}, {4294,10106}, {4298,6361}, {4307,28881}, {4312,28174}, {4326,6001}, {4342,5603}, {4420,4917}, {4642,28011}, {4653,18163}, {4737,30568}, {4857,10827}, {4859,20328}, {4882,5044}, {4900,16857}, {4915,9708}, {5084,6736}, {5180,31164}, {5219,10056}, {5234,11519}, {5261,18483}, {5274,10175}, {5281,10165}, {5290,12699}, {5438,8715}, {5493,12577}, {5542,28228}, {5552,25522}, {5559,7162}, {5657,11019}, {5687,8583}, {5691,15171}, {5703,13464}, {5727,12647}, {5731,10860}, {5777,9848}, {5790,18527}, {5844,8275}, {6198,7713}, {6284,9613}, {6326,15558}, {6684,14986}, {6738,12245}, {7190,23839}, {7284,13602}, {7686,12260}, {7701,10543}, {7743,7988}, {7989,9669}, {8077,10968}, {8236,28234}, {9581,10039}, {9593,16781}, {9612,12701}, {9624,13411}, {9851,12684}, {10072,31231}, {10197,21630}, {10386,18481}, {10914,25893}, {11037,20070}, {11218,11374}, {11496,12650}, {13463,28628}, {15935,16236}, {18528,30294}, {18530,30286}, {18540,28204}, {18541,28198}, {19843,21627}, {24590,29624}, {27065,31145}, {28150,30332}

X(31393) = midpoint of X(i) and X(j) for these {i,j}: {1, 9819}, {1000, 3488}
X(31393) = reflection of X(i) in X(j) for these (i,j): (1, 6767), (3488, 30331), (4915, 9708), (9623, 1001), (11525, 9623), (11529, 1), (18421, 15934)
X(31393) = (Aquila)-isogonal conjugate of-X(11034)
X(31393) = X(1597)-of-excentral triangle
X(31393) = X(9818)-of-Hutson intouch triangle
X(31393) = X(9819)-of-anti-Aquila triangle
X(31393) = X(11529)-of-5th mixtilinear triangle
X(31393) = X(18494)-of-excenters-reflections triangle
d X(31393) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (40, 7982, 6766), (3576, 12703, 40), (5128, 11010, 40)


X(31394) = CENTROID OF CURVATURES OF THE 1st NEUBERG CIRCLES

Barycentrics    a*((b+c)*a^4-b*c*a^3-(b^3+c^3)*a^2-(b^2+c^2)*b*c*a+(b^2-c^2)*(b-c)*b*c) : :
X(31394) = 2*X(37)-3*X(7611) = X(1742)-3*X(3576) = 3*X(5603)-X(10446) = 3*X(5886)-2*X(24220) = 3*X(7611)-X(31395)

X(31394) lies on these lines: {1,256}, {3,142}, {4,17913}, {5,4026}, {6,15953}, {9,15507}, {21,17202}, {25,30687}, {37,517}, {38,11203}, {40,13731}, {55,17720}, {75,2783}, {98,30670}, {104,13396}, {165,25502}, {182,238}, {226,23853}, {228,3434}, {344,5657}, {390,7390}, {528,15624}, {575,16468}, {576,4649}, {631,25492}, {956,4416}, {991,1279}, {993,2792}, {995,28358}, {999,3664}, {1006,29105}, {1064,4116}, {1361,24806}, {1402,26098}, {1403,24239}, {1482,29311}, {1503,15976}, {1621,4220}, {1699,4192}, {1742,3576}, {1757,7609}, {1836,16678}, {2187,25885}, {2788,4455}, {2886,3185}, {3073,13323}, {3624,19514}, {3705,11688}, {3753,25099}, {3817,19540}, {3870,21319}, {3932,5690}, {4078,11362}, {4184,17174}, {4423,16434}, {4512,8731}, {4847,20760}, {4851,15571}, {5092,15485}, {5143,17717}, {5259,13732}, {5263,6998}, {5272,28364}, {5284,19649}, {5603,10446}, {5794,23846}, {6842,15666}, {7988,19546}, {8227,19513}, {8299,9746}, {8666,17770}, {8692,12017}, {9779,19647}, {9955,19543}, {10246,29353}, {10527,22345}, {10882,11522}, {10902,19548}, {11230,17384}, {11231,17357}, {11248,19547}, {13624,29229}, {13724,19860}, {14636,31162}, {16419,25893}, {16589,20606}, {16825,24257}, {17279,26446}, {17594,21321}, {20368,26102}, {20991,25514}, {23844,26066}, {24325,29057}, {24331,24728}, {24357,29069}, {24541,28348}, {29073,30273}

X(31394) = midpoint of X(1) and X(6210)
X(31394) = reflection of X(i) in X(j) for these (i,j): (991, 1385), (31395, 37)
X(31394) = X(6210)-of-anti-Aquila triangle
X(31394) = X(30258)-of-2nd circumperp triangle
X(31394) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 1756, 1469), (7611, 31395, 37)


X(31395) = CENTROID OF CURVATURES OF THE 2nd NEUBERG CIRCLES

Barycentrics    a*(b*c*a^3-(b+c)*(b^2+b*c+c^2)*a^2+(b^2+c^2)*b*c*a+(b^2-c^2)*(b^3-c^3)) : :
X(31395) = 4*X(37)-3*X(7611) = 3*X(7611)-2*X(31394)

X(31395) lies on these lines: {1,182}, {3,20990}, {4,29073}, {5,3932}, {37,517}, {192,2783}, {262,4518}, {344,5603}, {355,29016}, {392,25099}, {511,984}, {516,20430}, {576,1757}, {946,4078}, {1001,1482}, {1279,24680}, {1284,5903}, {1351,5220}, {1486,10679}, {2177,21326}, {2810,18161}, {2937,11849}, {3098,18788}, {3434,21807}, {3870,21318}, {4026,5690}, {4657,26446}, {5657,17321}, {5886,17279}, {5988,12837}, {8931,12497}, {11230,17357}, {11231,17384}, {13405,20254}, {14853,27549}, {16434,17599}, {16577,23853}, {16593,20330}, {17594,21333}, {24206,29674}

X(31395) = reflection of X(31394) in X(37)
X(31395) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 6211, 182), (37, 31394, 7611)


X(31396) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, BEVAN, HALF-MOSES}

Barycentrics    (b+c)*a^3-(5*b^2+2*b*c+5*c^2)*a^2-(b^2-c^2)*(b-c)*a+(b^2-c^2)^2 : :

X(31396) lies on these lines: {1,31400}, {2,9593}, {4,9574}, {6,6684}, {8,9592}, {10,39}, {20,31421}, {30,31430}, {32,10164}, {37,9843}, {40,7736}, {57,31402}, {226,31460}, {355,5024}, {497,31426}, {515,5013}, {516,1571}, {517,31406}, {519,9619}, {574,4297}, {946,3815}, {950,31448}, {1125,9620}, {1210,2276}, {1500,11019}, {1506,3817}, {1588,31427}, {1698,5286}, {1699,31404}, {1703,31403}, {1706,31405}, {2275,31397}, {2549,19925}, {2551,31429}, {3501,24239}, {3634,3767}, {3828,7739}, {3947,31476}, {4292,9596}, {4298,31409}, {4301,9698}, {4314,31451}, {5218,16780}, {5254,10175}, {5283,8582}, {5305,11231}, {5530,17754}, {5587,7738}, {5657,9575}, {5722,31461}, {5795,31449}, {5886,31467}, {6421,13883}, {6422,13936}, {6734,17756}, {6736,16975}, {7735,31423}, {7737,12512}, {7745,31443}, {9589,31407}, {9599,10624}, {9605,26446}, {9606,11362}, {9956,15048}, {10172,13881}, {12571,31415}, {12575,31433}, {18250,31442}, {19862,31455}

X(31396) = midpoint of X(1571) and X(2548)
X(31396) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31398, 10), (9593, 31428, 2)


X(31397) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, BEVAN, INCIRCLE}

Barycentrics    (b+c)*a^3-(b^2+6*b*c+c^2)*a^2-(b^2-c^2)*(b-c)*a+(b^2-c^2)^2 : :
X(31397) = X(1)-3*X(10056) = 3*X(2)+X(12648) = X(8)+3*X(11239) = X(1836)-3*X(11237) = X(2099)-3*X(17718) = X(4304)+2*X(5252) = 3*X(10956)-X(12831)

X(31397) lies on these lines: {1,2}, {3,4311}, {4,1697}, {5,7743}, {7,2093}, {9,3421}, {11,5919}, {12,946}, {20,9613}, {35,4297}, {36,4315}, {37,1146}, {38,1735}, {40,388}, {46,4298}, {55,515}, {56,6684}, {57,1056}, {65,11362}, {72,5837}, {80,2346}, {92,1785}, {119,15558}, {140,24928}, {142,1145}, {153,29007}, {165,4293}, {226,495}, {255,5264}, {322,4357}, {355,950}, {390,3586}, {392,3452}, {405,5795}, {442,10914}, {443,1706}, {496,9956}, {497,5587}, {516,1478}, {518,8255}, {529,4640}, {549,5126}, {553,3654}, {595,3074}, {611,5847}, {631,1420}, {726,10063}, {908,3877}, {942,4848}, {944,3601}, {952,24929}, {954,3419}, {956,5745}, {960,12607}, {962,5261}, {993,8069}, {999,3911}, {1000,1512}, {1006,2078}, {1015,31398}, {1058,5818}, {1064,4551}, {1074,1111}, {1124,13936}, {1155,5434}, {1167,1220}, {1317,10265}, {1319,5432}, {1335,13883}, {1479,6957}, {1482,11374}, {1496,1771}, {1497,1724}, {1572,31409}, {1588,31432}, {1621,5176}, {1656,11373}, {1699,5726}, {1703,31408}, {1739,24175}, {1770,5270}, {1788,3333}, {1836,11237}, {1837,3303}, {1864,18908}, {2067,13912}, {2098,11375}, {2099,17718}, {2136,5082}, {2269,10445}, {2275,31396}, {2292,17874}, {2476,3885}, {2549,31433}, {2550,24409}, {2551,31435}, {2646,5882}, {2784,10053}, {2796,10054}, {2800,10956}, {2801,18801}, {2802,3822}, {2886,3880}, {3091,9614}, {3245,30424}, {3297,13973}, {3298,13911}, {3304,24914}, {3338,12577}, {3340,3487}, {3361,9588}, {3434,3895}, {3436,5250}, {3475,11529}, {3476,3576}, {3485,7982}, {3486,5881}, {3488,5727}, {3523,4308}, {3543,30332}, {3555,24391}, {3579,18990}, {3600,15803}, {3612,6966}, {3671,5903}, {3672,24213}, {3680,6856}, {3681,18397}, {3692,17355}, {3695,10396}, {3710,4696}, {3717,4737}, {3744,5724}, {3746,4314}, {3812,8256}, {3814,3898}, {3816,5123}, {3817,4342}, {3820,5316}, {3833,18240}, {3873,30274}, {3874,13750}, {3878,10954}, {3884,21616}, {3890,11681}, {3913,5794}, {3947,4301}, {4002,17529}, {4294,5691}, {4295,5290}, {4299,12512}, {4302,28164}, {4642,23536}, {4652,20076}, {4679,31141}, {4711,5572}, {4999,11260}, {5010,21578}, {5048,15950}, {5056,7320}, {5083,10202}, {5183,11246}, {5225,18492}, {5248,11508}, {5281,5731}, {5296,31325}, {5433,20323}, {5450,22759}, {5534,10393}, {5542,5902}, {5559,11009}, {5570,5883}, {5686,10398}, {5692,21060}, {5710,5717}, {5719,5844}, {5722,5790}, {5728,24393}, {5817,10384}, {5828,8165}, {5836,25466}, {5886,31479}, {6260,12672}, {6361,9579}, {6502,13975}, {6706,24181}, {6796,11501}, {6842,23340}, {6847,12650}, {6974,28236}, {7288,31423}, {7738,31426}, {7967,13384}, {7989,10591}, {8068,21630}, {8071,25440}, {8227,10588}, {8275,11224}, {8666,22766}, {8715,11507}, {8983,9646}, {9575,31402}, {9589,31410}, {9654,12699}, {9803,30284}, {9948,12680}, {9955,10592}, {10057,10087}, {10058,12749}, {10064,17766}, {10122,12670}, {10171,23708}, {10523,25639}, {10895,12701}, {10955,20117}, {11023,11024}, {11231,15325}, {11236,24703}, {12019,15170}, {12115,12686}, {12513,26066}, {12514,12527}, {12608,26482}, {12619,12735}, {12667,12705}, {12711,14872}, {12758,21635}, {12763,16140}, {12943,28150}, {15171,18480}, {15172,18357}, {16120,17646}, {16777,24005}, {17527,20789}, {17606,31399}, {21031,25917}, {21290,25101}, {21627,24390}, {23529,24010}, {23675,24171}, {24178,24440}, {24215,24464}

X(31397) = midpoint of X(i) and X(j) for these {i,j}: {1, 12647}, {8, 3870}, {55, 5252}, {1478, 5119}, {2292, 17874}, {3434, 3895}, {10057, 10087}, {10058, 12749}
X(31397) = reflection of X(i) in X(j) for these (i,j): (1, 13405), (226, 495), (956, 5745), (4304, 55), (4847, 10)
X(31397) = complement of the complement of X(12648)
X(31397) = X(10)-of-inner-Yff triangle
X(31397) = X(946)-of-1st Johnson-Yff triangle
X(31397) = X(1210)-of-inner-Yff tangents triangle
X(31397) = X(4847)-of-outer-Garcia triangle
X(31397) = X(12647)-of-anti-Aquila triangle
X(31397) = X(13405)-of-Aquila triangle
X(31397) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10, 10915, 6736), (145, 5703, 1), (18395, 21625, 1210)


X(31398) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, BEVAN, MOSES}

Barycentrics    (b+c)*a^3-(3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)*(b-c)*a+(b^2-c^2)^2 : :

X(31398) lies on these lines: {1,31401}, {2,9620}, {4,1571}, {6,26446}, {8,9619}, {10,39}, {20,31422}, {32,6684}, {40,2548}, {46,9596}, {57,31409}, {65,31460}, {115,10175}, {165,7737}, {187,10164}, {226,31476}, {230,11231}, {355,5013}, {497,31433}, {515,574}, {516,5475}, {517,3815}, {946,1506}, {950,31451}, {962,31404}, {980,25007}, {1015,31397}, {1210,1500}, {1482,31467}, {1504,13936}, {1505,13883}, {1572,5657}, {1588,31437}, {1698,3767}, {1699,31415}, {1703,31411}, {1706,31416}, {1737,2276}, {1788,31402}, {1837,31448}, {2242,3911}, {2275,10039}, {2549,5587}, {2551,31442}, {3055,11230}, {3579,7745}, {3634,7746}, {3679,9592}, {3817,7603}, {4292,9650}, {5024,5790}, {5034,5847}, {5058,13912}, {5062,13975}, {5119,9599}, {5164,10440}, {5254,9956}, {5280,5445}, {5283,24982}, {5286,9780}, {5530,17750}, {5690,31406}, {5691,31421}, {5722,31477}, {5795,31456}, {5818,7738}, {5836,31466}, {5881,31450}, {5886,31489}, {6421,13911}, {6422,13973}, {6735,16975}, {7739,19875}, {7748,19925}, {7765,31399}, {8582,16589}, {9581,31426}, {9589,31417}, {9597,10827}, {9598,10826}, {9665,10624}, {9698,11362}, {10915,17448}, {10916,20691}, {15815,18481}, {18480,31430}, {21965,25066}, {26100,26676}

X(31398) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 31428, 31401), (9620, 31441, 2)


X(31399) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, BEVAN, STEINER}

Barycentrics    3*(b+c)*a^3-(5*b^2+6*b*c+5*c^2)*a^2-3*(b^2-c^2)*(b-c)*a+5*(b^2-c^2)^2 : :
X(31399) = 3*X(1)-13*X(5067) = X(1)-6*X(10172) = 9*X(2)+X(5881) = 6*X(2)-X(5882) = X(3)-6*X(3828) = 3*X(4)+7*X(9588) = X(4)+9*X(19875) = 2*X(5)+3*X(10) = 8*X(5)-3*X(946) = 14*X(5)-9*X(3817) = 6*X(5)-X(4301) = 7*X(5)+3*X(5690) = 11*X(5)-6*X(9955) = X(5)-6*X(9956) = 4*X(5)-9*X(10175) = 4*X(5)+X(11362) = 13*X(5)-3*X(22791) = 4*X(10)+X(946) = 7*X(10)+3*X(3817) = 9*X(10)+X(4301) = 7*X(10)-2*X(5690) = 11*X(10)+4*X(9955) = X(10)+4*X(9956) = 2*X(10)+3*X(10175) = 6*X(10)-X(11362) = 13*X(10)+2*X(22791) = 7*X(946)-12*X(3817) = 13*X(5067)-18*X(10172) = 2*X(5881)+3*X(5882)

X(31399) lies on these lines: {1,5067}, {2,5881}, {3,3828}, {4,9588}, {5,10}, {8,7486}, {20,5587}, {30,31447}, {40,3832}, {57,31410}, {119,3841}, {226,18395}, {355,3526}, {382,19925}, {497,31436}, {515,631}, {516,3843}, {519,1656}, {546,5493}, {547,4669}, {548,10164}, {551,3628}, {632,28204}, {950,31452}, {952,19862}, {1125,5070}, {1210,15888}, {1385,16239}, {1482,4691}, {1483,15808}, {1512,6845}, {1572,31417}, {1588,31440}, {1703,31414}, {1706,31420}, {1737,18398}, {2549,31444}, {2551,31446}, {2800,3698}, {3090,3679}, {3091,28194}, {3244,11230}, {3525,19876}, {3528,5691}, {3530,4297}, {3533,30389}, {3545,7991}, {3576,19877}, {3579,3853}, {3617,5734}, {3624,13607}, {3625,5901}, {3626,5886}, {3636,12645}, {3654,3851}, {3656,5079}, {3671,10592}, {3753,20117}, {3855,5657}, {3856,22793}, {3858,28198}, {3859,28174}, {3911,4317}, {3918,5887}, {3919,5694}, {4015,24474}, {4197,15016}, {4292,9656}, {4309,10826}, {4314,12019}, {4342,10593}, {4413,5450}, {4668,10595}, {4678,16200}, {4701,10247}, {4731,12672}, {4745,5055}, {4848,7951}, {5056,7982}, {5068,31162}, {5071,11522}, {5258,6946}, {5325,6917}, {5722,31480}, {5745,6885}, {5795,6970}, {6256,26040}, {6260,6937}, {6738,31479}, {6861,12437}, {6881,24391}, {7705,24987}, {7738,31431}, {7765,31398}, {7988,12245}, {8582,10265}, {9575,31407}, {9607,31396}, {9657,24914}, {9671,10624}, {9948,18242}, {10246,19878}, {10915,24386}, {11374,14563}, {12005,18908}, {12368,15057}, {12512,17800}, {12571,12702}, {12616,12671}, {12640,24387}, {13605,20396}, {15178,19883}, {15696,28164}, {15712,28208}, {15717,31423}, {17575,17619}, {17578,18492}, {17606,31397}, {28236,31253}, {31441,31450}

X(31399) = midpoint of X(i) and X(j) for these {i,j}: {1698, 5818}, {3617, 8227}, {4668, 10595}
X(31399) = reflection of X(16189) in X(13464)
X(31399) = X(3091)-of-4th Euler triangle
X(31399) = X(3843)-of-Wasat triangle
X(31399) = X(15040)-of-K798i triangle
X(31399) = X(17538)-of-3rd Euler triangle
X(31399) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 10, 11362), (5, 11362, 946), (10, 3817, 5690)


X(31400) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, CIRCUMCIRCLE, HALF-MOSES}

Barycentrics    a^4-6*(b^2+c^2)*a^2+(b^2-c^2)^2 : :

X(31400) lies on these lines: {1,31396}, {2,39}, {3,7736}, {4,3815}, {5,5024}, {6,631}, {8,9619}, {10,9592}, {20,574}, {24,3087}, {32,3523}, {56,31402}, {69,11285}, {115,5056}, {140,7735}, {187,15717}, {193,1078}, {217,18913}, {230,3525}, {232,3088}, {325,16043}, {372,31403}, {376,7745}, {388,31460}, {390,31451}, {393,570}, {439,3972}, {496,31461}, {497,31448}, {516,31421}, {549,30435}, {550,15484}, {597,10542}, {946,9574}, {962,1571}, {1007,6656}, {1058,31477}, {1125,9593}, {1285,5023}, {1376,31405}, {1384,3530}, {1504,7586}, {1505,7585}, {1506,2549}, {1575,19843}, {1587,31463}, {1656,15048}, {2241,5281}, {2242,5265}, {2275,3085}, {2276,3086}, {2550,31466}, {2551,31449}, {3053,3524}, {3055,5067}, {3068,6421}, {3069,6422}, {3090,5254}, {3094,14853}, {3146,5475}, {3329,16925}, {3452,31429}, {3518,9609}, {3522,7737}, {3526,5305}, {3533,14482}, {3543,7756}, {3547,14961}, {3589,14069}, {3600,31409}, {3616,9620}, {3618,7807}, {3619,7881}, {3620,7796}, {3785,7774}, {3820,31468}, {3832,7748}, {4293,9596}, {4294,9599}, {4301,31431}, {5007,14930}, {5063,10312}, {5068,7603}, {5110,5802}, {5206,15692}, {5261,31476}, {5275,17567}, {5276,6921}, {5304,7772}, {5306,15702}, {5319,7616}, {6337,7770}, {6361,31443}, {6459,9600}, {6684,9575}, {6700,16517}, {7080,16975}, {7400,22401}, {7486,7765}, {7512,9608}, {7667,15437}, {7750,11163}, {7753,10304}, {7758,7815}, {7764,7800}, {7776,8359}, {7777,7791}, {7783,16924}, {7789,16045}, {7833,23334}, {7839,17008}, {7854,10513}, {7858,14907}, {7864,17005}, {7891,16898}, {7905,20080}, {7906,16990}, {7907,16989}, {7920,16923}, {8367,11165}, {8589,21734}, {8889,27376}, {9597,10590}, {9598,10591}, {9744,9873}, {9778,31422}, {9780,31441}, {9785,31433}, {10200,25092}, {10527,17756}, {11174,14001}, {11307,11489}, {11308,11488}, {12053,31426}, {12699,31430}, {14535,19697}, {16784,31452}, {17578,31417}, {18228,31442}

X(31400) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 39, 5286), (6683, 7795, 2), (7763, 7786, 2)


X(31401) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, CIRCUMCIRCLE, MOSES}

Barycentrics    a^4-4*(b^2+c^2)*a^2+(b^2-c^2)^2 : :

X(31401) lies on these lines: {1,31398}, {2,39}, {3,2548}, {4,574}, {5,2549}, {6,140}, {10,9619}, {20,5475}, {30,15815}, {32,631}, {35,9599}, {36,9596}, {37,10200}, {56,31409}, {69,5034}, {83,16925}, {99,16924}, {114,10356}, {115,3090}, {183,7758}, {187,3523}, {193,7780}, {216,3546}, {217,26937}, {230,3526}, {232,3541}, {315,7777}, {325,7800}, {372,31411}, {376,7747}, {388,31476}, {496,31477}, {497,31451}, {498,2275}, {499,2276}, {516,31422}, {549,3053}, {566,18281}, {590,6421}, {615,6422}, {620,7808}, {626,1007}, {632,5305}, {858,15880}, {946,1571}, {1015,3085}, {1078,7774}, {1107,26364}, {1125,9620}, {1153,5032}, {1329,31449}, {1376,31416}, {1384,15720}, {1500,3086}, {1504,3069}, {1505,3068}, {1569,14651}, {1572,6684}, {1574,19843}, {1575,26363}, {1587,31481}, {1656,3055}, {1698,9592}, {1699,31421}, {2023,15561}, {2165,13351}, {2241,5218}, {2242,7288}, {2493,14787}, {2550,31488}, {2551,31456}, {3071,9600}, {3088,3199}, {3091,7603}, {3094,11272}, {3147,10311}, {3329,7907}, {3452,31442}, {3517,6748}, {3518,9700}, {3522,8589}, {3524,5206}, {3525,5368}, {3528,6781}, {3530,5023}, {3533,7755}, {3547,22401}, {3549,14961}, {3589,6387}, {3618,5028}, {3619,7869}, {3620,7895}, {3624,9593}, {3628,13881}, {3734,6337}, {3785,7759}, {3820,31490}, {3850,18584}, {4045,7862}, {4293,9650}, {4294,9665}, {4301,31444}, {5007,10303}, {5025,17005}, {5038,10104}, {5041,5304}, {5052,10519}, {5054,9300}, {5058,9540}, {5059,15602}, {5062,13935}, {5067,7765}, {5068,18424}, {5070,9607}, {5071,11648}, {5094,27376}, {5111,14693}, {5210,15712}, {5275,13747}, {5276,17566}, {5277,6921}, {5306,15694}, {5503,9167}, {5552,16975}, {6292,7888}, {6389,28407}, {6459,9674}, {6722,7902}, {7506,9609}, {7512,9699}, {7525,15109}, {7542,23115}, {7618,8370}, {7741,9598}, {7752,7791}, {7760,17008}, {7762,11163}, {7767,9766}, {7771,7858}, {7773,8356}, {7775,7830}, {7778,8362}, {7782,14035}, {7783,11185}, {7784,8359}, {7785,14907}, {7789,15491}, {7796,16990}, {7806,16923}, {7807,11174}, {7810,7903}, {7812,8182}, {7814,7831}, {7820,16045}, {7835,16898}, {7839,17004}, {7847,14063}, {7855,15589}, {7857,16989}, {7872,16041}, {7876,7925}, {7889,14069}, {7947,16986}, {7951,9597}, {8227,9574}, {8367,12040}, {8588,10299}, {8721,13860}, {8889,27371}, {9575,31423}, {9651,10590}, {9664,10591}, {9725,9726}, {9771,11318}, {9955,31430}, {10024,15075}, {10198,16604}, {10304,14537}, {11432,22270}, {12053,31433}, {12108,22331}, {12699,31443}, {14067,16987}, {15513,15717}, {30827,31429}

X(31401) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 31428, 31398), (2, 3926, 3934), (2, 7763, 7795)


X(31402) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, HALF-MOSES, INCIRCLE}

Barycentrics    a^4+4*(b^2+b*c+c^2)*a^2-(b^2-c^2)^2 : :

X(31402) lies on these lines: {1,7736}, {2,31460}, {4,2276}, {6,3085}, {8,31405}, {9,5530}, {11,31404}, {12,5286}, {20,31448}, {30,31461}, {32,5218}, {37,5084}, {39,388}, {43,26036}, {56,31400}, {57,31396}, {69,27020}, {172,631}, {192,16924}, {217,18922}, {226,9593}, {377,17756}, {443,1575}, {495,9605}, {497,1500}, {498,5280}, {499,16785}, {516,31426}, {999,31406}, {1056,2275}, {1058,9599}, {1107,3421}, {1335,31403}, {1478,7738}, {1506,10589}, {1571,3474}, {1574,26040}, {1588,31459}, {1788,31398}, {2241,31478}, {2242,7288}, {2549,5229}, {2551,5283}, {3086,3815}, {3087,11398}, {3476,9619}, {3485,9620}, {3501,26098}, {3767,10588}, {3911,31428}, {4292,9574}, {4293,5013}, {4294,7745}, {4426,6857}, {5024,18990}, {5082,20691}, {5225,5475}, {5254,10590}, {5276,5552}, {5291,30478}, {5299,10056}, {5305,31479}, {5712,17750}, {5725,25066}, {6421,31408}, {6933,17737}, {7031,31452}, {7737,31451}, {7753,10385}, {9300,16781}, {9592,10106}, {9607,31410}, {9654,15048}, {12527,31429}, {13405,16780}, {15171,15484}, {15325,31467}, {16517,21075}, {19767,26074}, {21956,31418}

X(31402) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31409, 388), (2276, 9596, 4)


X(31403) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, HALF-MOSES, 2nd LEMOINE}

Barycentrics    a^4+4*S*a^2+4*(b^2+c^2)*a^2-(b^2-c^2)^2 : :

X(31403) lies on these lines: {2,6}, {4,6422}, {20,31465}, {32,9540}, {39,1587}, {372,31400}, {376,9600}, {393,3127}, {485,5286}, {486,31404}, {488,18993}, {493,6806}, {497,31459}, {516,31427}, {631,6423}, {1164,3087}, {1335,31402}, {1378,31405}, {1504,1588}, {1575,31413}, {1703,31396}, {2275,31408}, {2549,23249}, {2551,31464}, {3070,7738}, {3103,21737}, {3156,19006}, {3312,31406}, {3399,14244}, {3523,12968}, {3767,31481}, {5013,6460}, {5058,31483}, {5062,13935}, {5254,31412}, {5280,13904}, {5299,13905}, {5305,8976}, {5475,23259}, {6221,18907}, {6420,19103}, {6421,7581}, {6459,7745}, {7583,9605}, {7737,9541}, {7748,23253}, {8981,30435}, {9575,13883}, {9607,31414}, {10577,19102}, {13665,15048}, {13966,31467}, {13975,31428}

X(31403) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 590, 7735), (3068, 5591, 590), (3595, 8974, 590)


X(31404) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, HALF-MOSES, NINE-POINTS}

Barycentrics    a^4+6*(b^2+c^2)*a^2-3*(b^2-c^2)^2 : :

X(31404) lies on these lines: {2,32}, {4,3815}, {5,5286}, {6,3090}, {11,31402}, {20,5475}, {30,31467}, {39,3091}, {50,7558}, {115,5068}, {140,15484}, {187,10303}, {193,7858}, {217,23291}, {230,5067}, {381,7738}, {390,9665}, {393,1594}, {486,31403}, {497,31460}, {516,31428}, {546,5024}, {574,3146}, {631,5023}, {632,1384}, {962,31398}, {1007,7770}, {1015,5261}, {1285,3533}, {1329,31405}, {1500,5274}, {1571,9812}, {1572,9780}, {1573,8165}, {1575,31418}, {1587,12969}, {1588,12962}, {1656,7735}, {1699,31396}, {2207,8889}, {2275,10590}, {2276,10591}, {2549,3832}, {2551,31466}, {2996,7757}, {3053,3055}, {3085,9599}, {3086,9596}, {3087,3542}, {3199,7378}, {3522,7747}, {3523,7737}, {3526,18907}, {3529,11742}, {3544,18584}, {3545,5254}, {3600,9650}, {3613,8801}, {3618,7887}, {3620,5052}, {3628,30435}, {3767,5041}, {3817,9593}, {3839,7748}, {3851,15048}, {3855,9606}, {3861,31470}, {3926,7777}, {3933,9770}, {3972,5395}, {5034,5921}, {5038,6776}, {5055,5305}, {5058,8972}, {5062,13941}, {5071,9300}, {5225,31448}, {5276,6931}, {5283,6919}, {5304,7486}, {5319,14930}, {6337,8370}, {6421,31412}, {6781,21734}, {7484,15437}, {7585,31481}, {7586,31411}, {7756,17578}, {7759,15589}, {7772,15022}, {7773,16043}, {7774,16921}, {7778,16045}, {7784,15491}, {7789,11184}, {7861,8176}, {7903,10513}, {7921,16922}, {7925,16898}, {7941,16990}, {7949,11160}, {8164,16781}, {9574,18483}, {9575,10175}, {9592,19925}, {11174,14064}, {12108,15655}, {14537,15692}, {14986,31409}, {15302,31099}, {15880,30769}, {16925,17005}

X(31404) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 7785, 3785), (1506, 2548, 2)


X(31405) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, HALF-MOSES, SPIEKER}

Barycentrics    a^4+4*(b^2+c^2)*a^2+4*(b+c)*b*c*a-(b^2-c^2)^2 : :

X(31405) lies on these lines: {1,26036}, {2,31466}, {4,1107}, {6,19843}, {8,31402}, {10,7736}, {20,31449}, {30,31468}, {32,30478}, {37,1058}, {39,2550}, {69,17030}, {388,16975}, {443,2275}, {497,5283}, {516,31429}, {631,4386}, {946,16517}, {966,2300}, {1056,17448}, {1329,31404}, {1376,31400}, {1378,31403}, {1573,2548}, {1588,31464}, {1706,31396}, {1914,6857}, {2082,29639}, {2276,5082}, {2280,28246}, {2886,5286}, {3086,5275}, {3421,9596}, {3487,16973}, {3767,31488}, {3926,20172}, {4307,5021}, {5084,9599}, {5254,31418}, {5276,10527}, {5277,7288}, {5299,19854}, {5305,31493}, {5698,31442}, {5712,20963}, {6421,31413}, {7080,31460}, {7735,26363}, {7737,31456}, {7745,31490}, {8817,26134}, {9605,31419}, {9607,31420}, {9711,31407}, {14001,20179}, {16604,17582}, {17784,31448}, {21384,26098}, {21921,28074}

X(31405) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31416, 2550), (1573, 2548, 2551)


X(31406) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, HALF-MOSES, STAMMLER}

Barycentrics    5*(b^2+c^2)*a^2-(b^2-c^2)^2 : :

X(31406) lies on these lines: {2,3933}, {3,7736}, {4,5024}, {5,39}, {6,140}, {20,15484}, {30,2548}, {32,549}, {83,5503}, {141,6683}, {187,15712}, {230,632}, {232,1595}, {315,8359}, {325,3096}, {355,9592}, {381,7738}, {382,31407}, {495,2275}, {496,2276}, {497,31461}, {516,31430}, {517,31396}, {524,7815}, {546,2549}, {547,7739}, {548,7737}, {550,574}, {570,23335}, {597,6680}, {631,30435}, {952,9619}, {999,31402}, {1007,7866}, {1107,3820}, {1285,15717}, {1353,5034}, {1384,3523}, {1504,19116}, {1505,19117}, {1570,14693}, {1571,28174}, {1575,31419}, {1656,5286}, {2551,31468}, {3053,3530}, {3054,7755}, {3055,7746}, {3068,11316}, {3069,11315}, {3087,3517}, {3094,21850}, {3312,31403}, {3329,7807}, {3525,5304}, {3526,7735}, {3589,3788}, {3618,6393}, {3627,5475}, {3628,3767}, {3629,7780}, {3631,7916}, {3665,24786}, {3816,25092}, {3845,7748}, {3850,31415}, {3861,31417}, {3934,15491}, {5007,14869}, {5023,12100}, {5028,20576}, {5041,5306}, {5067,14482}, {5120,19547}, {5206,31457}, {5276,13747}, {5277,17564}, {5280,5433}, {5283,17527}, {5299,5432}, {5309,15699}, {5319,16239}, {5690,31398}, {5886,9593}, {5901,9620}, {6337,11286}, {6390,7770}, {6421,7583}, {6422,7584}, {6431,19104}, {6432,19103}, {6656,7777}, {6661,7891}, {6676,23115}, {6704,7880}, {6748,7715}, {6823,14961}, {7502,9608}, {7542,22120}, {7747,15704}, {7750,7858}, {7752,7918}, {7753,8703}, {7758,15271}, {7759,14929}, {7762,7824}, {7763,7819}, {7767,7774}, {7769,7792}, {7773,8357}, {7776,16043}, {7778,8364}, {7783,8370}, {7785,8356}, {7789,7808}, {7797,17005}, {7800,9766}, {7803,8361}, {7805,13468}, {7813,31239}, {7817,9771}, {7867,22110}, {7894,22329}, {7925,8363}, {8360,11184}, {8770,13361}, {8962,15235}, {9574,12699}, {9575,26446}, {9596,18990}, {9599,15171}, {9709,31405}, {10299,15655}, {10303,14930}, {10336,16987}, {10386,31451}, {12108,21843}, {13846,19105}, {13847,19102}, {14537,15686}, {14910,30537}, {15172,31477}, {15302,30739}, {15513,17504}, {15720,21309}, {17756,24390}, {24239,25066}, {25068,29639}

X(31406) = midpoint of X(2548) and X(5013)
X(31406) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 9605, 5305), (9605, 31467, 2)


X(31407) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, HALF-MOSES, STEINER}

Barycentrics    3*a^4+14*(b^2+c^2)*a^2-5*(b^2-c^2)^2 : :

X(31407) lies on these lines: {2,5007}, {4,9606}, {5,5286}, {6,3316}, {20,574}, {30,31470}, {39,3832}, {382,31406}, {497,31462}, {516,31431}, {548,15484}, {631,1285}, {1506,5319}, {1575,31420}, {1588,31465}, {1906,15433}, {2275,31410}, {2551,31469}, {3090,9300}, {3091,7765}, {3523,7753}, {3526,21309}, {3528,7745}, {3530,15655}, {3843,7738}, {3853,5024}, {3855,9607}, {5056,7772}, {5068,7739}, {5070,7735}, {5309,15022}, {5475,17578}, {6421,31414}, {7737,21734}, {7746,14930}, {7777,16898}, {9575,31399}, {9589,31396}, {9711,31405}, {10513,31239}, {11184,14069}, {15513,15717}, {15702,22331}, {16239,30435}

X(31407) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (20, 9698, 31400), (2548, 9698, 20), (7736, 31404, 5286)


X(31408) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, INCIRCLE, 2nd LEMOINE}

Barycentrics    ((-a+b+c)*(a+b+c)*a^2+(a^2+(b+c)^2)*S)*(a-b+c)*(a+b-c) : :

X(31408) lies on these lines: {1,1587}, {2,6502}, {4,1124}, {6,388}, {7,16232}, {8,2362}, {11,31412}, {12,3069}, {20,2066}, {30,31474}, {36,9540}, {55,6460}, {56,3068}, {57,13883}, {65,19066}, {176,3177}, {189,13389}, {226,18992}, {329,1659}, {371,4293}, {372,3085}, {443,1378}, {485,3086}, {486,10590}, {495,3312}, {496,13665}, {497,3070}, {498,13935}, {516,31432}, {590,7288}, {605,1935}, {615,10588}, {631,9646}, {999,7583}, {1015,31411}, {1056,1335}, {1058,23267}, {1131,5274}, {1152,5218}, {1319,13902}, {1377,3421}, {1420,8983}, {1478,1588}, {1479,23249}, {1505,31409}, {1702,4292}, {1703,31397}, {1788,13911}, {2067,3600}, {2275,31403}, {2549,31471}, {2551,31473}, {3071,5229}, {3072,3076}, {3304,19030}, {3311,18990}, {3476,7969}, {3485,7968}, {3529,9660}, {3583,23253}, {3585,23259}, {3614,13955}, {3911,13893}, {4294,6560}, {4299,9541}, {4311,9583}, {5083,19078}, {5204,13901}, {5219,13971}, {5225,23251}, {5252,19065}, {5261,7586}, {5265,8972}, {5290,19003}, {5433,13897}, {5434,18996}, {5563,13904}, {6421,31402}, {6459,7354}, {6564,10591}, {7584,9654}, {7738,31459}, {7951,13962}, {8976,15325}, {9578,13936}, {9661,13886}, {10106,18991}, {10895,19029}, {10956,19112}, {11237,19027}, {11375,13959}, {12527,31438}, {13966,31479}, {13975,31434}, {15888,19037}, {18924,19349}, {19013,19026}

X(31408) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (12, 18995, 3069), (6502, 31472, 2)


X(31409) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, INCIRCLE, MOSES}

Barycentrics    a^4+2*(b+c)^2*a^2-(b^2-c^2)^2 : :

X(31409) lies on these lines: {1,2548}, {2,2242}, {4,1500}, {6,495}, {8,31416}, {11,31415}, {12,3767}, {20,31451}, {30,31477}, {32,3085}, {37,5725}, {39,388}, {55,7737}, {56,31401}, {57,31398}, {115,10590}, {172,498}, {187,5218}, {192,11185}, {226,9620}, {230,31479}, {443,1574}, {497,5475}, {516,31433}, {529,31449}, {574,4293}, {609,3584}, {999,3815}, {1015,1056}, {1058,9665}, {1335,31411}, {1478,2276}, {1505,31408}, {1506,3086}, {1571,4292}, {1572,31397}, {1573,3421}, {1588,31471}, {1909,7758}, {1914,10056}, {2551,16589}, {3295,7745}, {3361,31428}, {3436,5283}, {3583,9331}, {3585,9598}, {3600,31400}, {3911,31441}, {4294,7747}, {4298,31396}, {4317,31450}, {4426,10198}, {5013,18990}, {5229,7748}, {5254,9654}, {5261,5286}, {5270,9597}, {5275,17757}, {5277,5552}, {5280,5319}, {5290,9593}, {5432,21843}, {6645,7763}, {6767,15484}, {7288,31455}, {7354,31448}, {7603,10589}, {7735,8164}, {7739,11237}, {7746,10588}, {7765,31410}, {7800,27020}, {7951,16785}, {9579,31426}, {9619,10106}, {9655,31461}, {9657,31462}, {10385,14537}, {10592,13881}, {12513,31466}, {12527,31442}, {14986,31404}, {15325,31489}, {17532,21956}

X(31409) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 9596, 2548), (2242, 31476, 2)


X(31410) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, INCIRCLE, STEINER}

Barycentrics    3*a^4+2*(b^2+6*b*c+c^2)*a^2-5*(b^2-c^2)^2 : :

X(31410) lies on these lines: {1,3832}, {2,4317}, {4,3058}, {5,388}, {7,10827}, {8,31420}, {10,9965}, {12,631}, {20,35}, {30,31480}, {56,5067}, {57,31399}, {80,11036}, {226,5881}, {382,495}, {443,9711}, {497,3843}, {498,4325}, {516,31436}, {529,6856}, {548,5218}, {1015,31417}, {1056,3855}, {1058,9671}, {1335,31414}, {1588,31475}, {2275,31407}, {2549,31478}, {2551,17529}, {3090,5434}, {3146,4330}, {3295,3853}, {3304,3545}, {3421,9710}, {3436,4197}, {3475,18480}, {3522,3584}, {3526,10588}, {3528,7354}, {3530,31479}, {3543,3746}, {3582,15022}, {3585,4309}, {3600,7486}, {3627,10385}, {3822,31458}, {3856,9669}, {3861,5225}, {3947,9613}, {4292,5726}, {4295,9578}, {4299,21734}, {4301,9612}, {4338,10039}, {4995,17538}, {5056,5563}, {5068,10072}, {5070,7288}, {5252,5714}, {5290,18391}, {5734,12047}, {5818,10404}, {6845,12115}, {6896,10711}, {7738,31462}, {7765,31409}, {9589,31397}, {9607,31402}, {9624,10106}, {10197,17576}, {10826,11037}, {11271,12946}, {12527,31446}, {17552,25466}, {17582,31141}, {18395,21454}, {19843,20060}, {31450,31476}

X(31410) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9656, 11237, 15888), (9656, 15888, 4)


X(31411) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, 2nd LEMOINE, MOSES}

Barycentrics    a^4+4*S*a^2+2*(b^2+c^2)*a^2-(b^2-c^2)^2 : :

X(31411) lies on these lines: {2,5062}, {4,1504}, {5,6}, {20,31483}, {32,638}, {39,1587}, {76,13707}, {115,31412}, {172,13904}, {187,9540}, {230,8976}, {371,7737}, {372,31401}, {376,9674}, {491,7795}, {497,31471}, {516,31437}, {574,6460}, {590,6423}, {1015,31408}, {1335,31409}, {1378,31416}, {1384,13903}, {1505,7581}, {1506,3069}, {1572,13883}, {1574,31413}, {1588,5475}, {1703,31398}, {1914,13905}, {2549,3070}, {2551,31482}, {3053,8981}, {3299,9599}, {3301,9596}, {3311,7745}, {3312,3815}, {5058,7585}, {5254,13665}, {5418,12968}, {6228,13879}, {6275,13648}, {6395,31467}, {6417,15484}, {6459,7747}, {6561,12962}, {7586,31404}, {7735,13886}, {7738,23267}, {7748,23249}, {7753,19054}, {7765,31414}, {9605,18512}, {13935,31455}, {13966,31489}, {13975,31441}, {19037,31460}

X(31411) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 485, 3767), (486, 19103, 6), (5062, 31481, 2)


X(31412) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, 2nd LEMOINE, NINE-POINTS}

Barycentrics    a^4+4*S*a^2+2*(b^2+c^2)*a^2-3*(b^2-c^2)^2 : :
X(31412) = X(6449)-3*X(8976) = 2*X(6449)-3*X(9540) = 2*X(6449)+3*X(23253) = 2*X(8976)+X(23253)

X(31412) lies on these lines: {2,490}, {3,18538}, {4,371}, {5,1587}, {6,3091}, {11,31408}, {20,590}, {30,6449}, {114,13773}, {115,31411}, {140,6456}, {372,3090}, {376,1327}, {378,8276}, {381,1588}, {382,6407}, {486,3545}, {487,7620}, {488,26361}, {492,12222}, {497,31472}, {515,13902}, {516,13893}, {546,3311}, {550,6496}, {615,5056}, {626,638}, {631,6560}, {632,6450}, {637,5861}, {640,7375}, {642,26620}, {946,19066}, {962,13911}, {1124,10591}, {1132,3854}, {1151,2671}, {1270,23311}, {1329,31413}, {1335,10590}, {1378,31418}, {1505,31415}, {1593,13889}, {1656,13935}, {1699,13883}, {1702,18483}, {1703,10175}, {2066,5225}, {2067,5229}, {2549,31481}, {2551,31484}, {3071,3832}, {3092,6623}, {3127,13051}, {3297,5274}, {3298,5261}, {3365,18582}, {3390,18581}, {3520,9682}, {3523,8253}, {3525,6396}, {3529,6200}, {3533,6487}, {3536,8968}, {3543,6429}, {3544,6420}, {3564,26468}, {3583,13905}, {3585,13904}, {3590,5059}, {3594,13941}, {3614,19037}, {3627,6221}, {3628,6398}, {3817,18992}, {3830,13903}, {3839,6470}, {3850,13785}, {3851,6501}, {3855,6565}, {3857,6427}, {3861,31487}, {4293,9661}, {4294,9646}, {5055,13966}, {5066,19116}, {5067,5420}, {5068,7586}, {5071,10577}, {5072,6418}, {5079,6395}, {5254,31403}, {5410,23047}, {5413,6622}, {5414,10588}, {5448,19062}, {5475,26463}, {5587,19065}, {5591,7389}, {5640,12240}, {5691,8983}, {5893,19088}, {5895,8991}, {6253,13896}, {6256,13906}, {6284,13897}, {6353,8280}, {6410,10303}, {6419,23273}, {6421,31404}, {6428,12811}, {6447,12102}, {6451,12103}, {6452,14869}, {6455,15704}, {6475,15703}, {6497,12108}, {6502,10589}, {6919,31473}, {7173,18995}, {7354,13898}, {7374,13638}, {7486,8252}, {7687,19111}, {7714,18289}, {7728,13915}, {7738,31463}, {7988,13971}, {7989,13936}, {8227,13959}, {8854,8889}, {8980,10722}, {8994,10721}, {8997,10723}, {8998,10733}, {9582,28150}, {9615,28164}, {9975,14853}, {10195,10299}, {10724,13922}, {10728,13913}, {10735,13923}, {10895,19030}, {10896,19028}, {11291,26362}, {11479,19006}, {12111,12239}, {12173,13884}, {12297,13882}, {12510,22646}, {12943,18965}, {12953,13901}, {17578,31454}, {18945,19355}, {18991,19925}, {19055,23514}, {19059,23515}, {19081,23513}, {19087,23332}, {22554,22618}

X(31412) = midpoint of X(9540) and X(23253)
X(31412) = reflection of X(9540) in X(8976)
X(31412) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 1131, 3070), (2, 3070, 6460)


X(31413) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, 2nd LEMOINE, SPIEKER}

Barycentrics    4*S*a^2+(a+b+c)*(a^3-(b+c)*a^2+(b+c)^2*a-(b^2-c^2)*(b-c)) : :

X(31413) lies on these lines: {2,5414}, {4,1377}, {6,2550}, {8,2362}, {10,1587}, {20,31453}, {30,31485}, {372,19843}, {376,9678}, {443,1335}, {486,31418}, {497,31473}, {516,31438}, {958,6460}, {962,30556}, {1124,5082}, {1131,8165}, {1152,30478}, {1329,31412}, {1376,3068}, {1378,7581}, {1505,31416}, {1574,31411}, {1575,31403}, {1706,13883}, {2066,17784}, {2067,6904}, {2549,31482}, {2551,3070}, {2886,3069}, {3312,31419}, {3820,13665}, {3925,19037}, {4413,19030}, {5438,8983}, {5705,13975}, {5794,19065}, {5836,19066}, {6421,31405}, {7080,31472}, {7090,27382}, {7583,9709}, {7738,31464}, {8987,10270}, {9540,25440}, {9661,17567}, {9711,31414}, {13935,26363}, {13958,31245}, {13966,31493}, {19029,31140}, {19087,20306}


X(31414) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, 2nd LEMOINE, STEINER}

Barycentrics    3*a^4+12*S*a^2+2*(b^2+c^2)*a^2-5*(b^2-c^2)^2 : :
X(31414) = 3*X(6447)-5*X(31487)

X(31414) lies on these lines: {2,6426}, {4,1327}, {5,1587}, {6,1131}, {20,1151}, {30,6447}, {371,23269}, {372,5067}, {376,8960}, {382,6199}, {485,631}, {486,6436}, {497,31475}, {516,31440}, {548,6451}, {590,15717}, {1335,31410}, {1378,31420}, {1505,31417}, {1588,3843}, {1703,31399}, {2549,31483}, {2551,31486}, {3090,10194}, {3091,19053}, {3311,3853}, {3522,13846}, {3526,6408}, {3528,6560}, {3530,6497}, {3533,6454}, {3543,3592}, {3545,6420}, {3594,5056}, {3845,6427}, {3850,6428}, {3855,6564}, {3856,6499}, {3859,19116}, {3861,19117}, {4301,19066}, {4325,13904}, {4330,13905}, {5059,6425}, {5070,13935}, {5590,12222}, {6250,10784}, {6398,16239}, {6417,23263}, {6421,31407}, {6453,11001}, {6470,7585}, {6489,10303}, {6522,11539}, {6776,26330}, {7738,31465}, {7745,26463}, {7765,31411}, {7863,11291}, {8972,21734}, {8981,15696}, {9541,17800}, {9589,13883}, {9607,31403}, {9657,19030}, {9670,19028}, {9711,31413}, {10195,15702}, {12322,26339}, {13847,15022}, {31450,31481}

X(31414) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1327, 6419, 4), (1587, 13665, 31412), (1587, 31412, 3069)


X(31415) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, MOSES, NINE-POINTS}

Barycentrics    a^4+4*(b^2+c^2)*a^2-3*(b^2-c^2)^2 : :
X(31415) = X(11742)-9*X(31489)

X(31415) lies on these lines: {2,187}, {3,3055}, {4,574}, {5,6}, {11,31409}, {20,8589}, {30,11742}, {32,3090}, {39,3091}, {69,7775}, {115,3545}, {140,5210}, {141,11173}, {230,5055}, {315,16921}, {381,2549}, {497,31476}, {498,10987}, {516,31441}, {546,5013}, {547,18907}, {549,5585}, {575,14162}, {590,8375}, {615,8376}, {620,14033}, {626,3619}, {631,7747}, {632,5023}, {1007,3734}, {1015,10590}, {1329,31416}, {1384,1656}, {1500,10591}, {1505,31412}, {1571,18483}, {1572,10175}, {1574,31418}, {1588,31481}, {1699,31398}, {1992,7617}, {2241,10588}, {2242,10589}, {2551,31488}, {3053,3628}, {3085,9665}, {3086,9650}, {3524,6781}, {3525,5206}, {3526,15655}, {3529,15515}, {3542,10985}, {3543,15602}, {3544,7772}, {3547,22052}, {3589,11318}, {3618,7844}, {3620,3934}, {3627,15815}, {3631,7776}, {3763,8367}, {3785,7843}, {3793,7610}, {3817,9620}, {3832,7748}, {3843,31450}, {3850,31406}, {3851,5254}, {3855,7738}, {3857,22332}, {3861,31492}, {4045,16041}, {5007,15022}, {5008,5056}, {5033,7808}, {5066,15048}, {5067,7749}, {5068,5286}, {5071,7735}, {5072,9605}, {5079,30435}, {5107,14853}, {5187,5283}, {5225,31451}, {5277,6931}, {5355,18362}, {5461,5477}, {6565,31463}, {6639,18472}, {6643,10979}, {6919,16589}, {7505,10986}, {7577,8744}, {7615,11163}, {7618,11317}, {7694,13860}, {7741,9596}, {7751,11008}, {7752,7795}, {7755,14075}, {7759,20080}, {7763,16044}, {7769,14035}, {7773,7800}, {7777,11185}, {7782,14068}, {7785,14023}, {7786,14063}, {7809,16990}, {7812,17008}, {7813,9770}, {7823,16922}, {7825,16043}, {7845,15589}, {7862,14001}, {7867,16045}, {7899,16898}, {7940,14037}, {7951,9599}, {8556,14929}, {8586,20423}, {9112,16268}, {9113,16267}, {9300,19709}, {9619,19925}, {9743,22682}, {9754,10788}, {9771,11159}, {10153,18842}, {10254,22121}, {10303,15513}, {10592,16781}, {10896,31460}, {11284,24855}, {11287,15491}, {11305,23303}, {11306,23302}, {11361,17005}, {11623,14912}, {11646,22566}, {12571,31396}, {14061,16989}, {14790,14806}, {15534,16509}, {15603,15694}, {17578,31457}

X(31415) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 5475, 7737), (2, 7737, 21843), (11614, 14537, 187)


X(31416) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, MOSES, SPIEKER}

Barycentrics    a^4+2*(b^2+c^2)*a^2+4*(b+c)*b*c*a-(b^2-c^2)^2 : :

X(31416) lies on these lines: {2,2241}, {4,1573}, {6,31419}, {8,31409}, {10,1572}, {20,31456}, {30,31490}, {32,19843}, {39,2550}, {75,7758}, {115,31418}, {187,30478}, {230,31493}, {377,16975}, {443,1015}, {497,16589}, {516,31442}, {958,7737}, {1107,2549}, {1329,31415}, {1378,31411}, {1500,5082}, {1505,31413}, {1574,7736}, {1588,31482}, {1698,9599}, {1706,31398}, {1914,19854}, {2551,5475}, {2886,3767}, {3421,9650}, {3434,5283}, {3679,9596}, {3815,9709}, {3933,20181}, {4340,9346}, {4386,26363}, {4999,21843}, {5084,9665}, {5275,24390}, {5277,10527}, {7080,31476}, {7745,9708}, {7765,31420}, {7795,20172}, {7800,17030}, {8728,16781}, {9711,31417}, {12609,16973}, {17784,31451}

X(31416) = {X(2550), X(31405)}-harmonic conjugate of X(39)


X(31417) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, MOSES, STEINER}

Barycentrics    3*a^4+8*(b^2+c^2)*a^2-5*(b^2-c^2)^2 : :

X(31417) lies on these lines: {2,7843}, {4,9698}, {5,6}, {20,5475}, {30,31492}, {32,5067}, {39,3832}, {381,9607}, {382,3815}, {497,31478}, {516,31444}, {631,1506}, {1015,31410}, {1505,31414}, {1572,31399}, {1574,31420}, {1588,31483}, {2549,3843}, {3053,16239}, {3090,7753}, {3091,7739}, {3523,14537}, {3526,7745}, {3528,7747}, {3530,31489}, {3545,7772}, {3845,22332}, {3851,9300}, {3853,5013}, {3855,7736}, {3859,15048}, {3861,31406}, {5007,5056}, {5068,5309}, {5070,15484}, {5071,7755}, {5079,5306}, {7486,7603}, {7752,16898}, {7759,11160}, {7775,21356}, {7795,7814}, {7796,16924}, {9589,31398}, {9670,31460}, {9671,31462}, {9711,31416}, {9770,17130}, {14023,16921}, {15717,31455}, {17578,31400}, {17800,31467}

X(31417) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2548, 31415, 3767), (5475, 31404, 31401)


X(31418) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, NINE-POINTS, SPIEKER}

Barycentrics    a^4+2*(b^2+c^2)*a^2+4*(b+c)*b*c*a-3*(b^2-c^2)^2 : :

X(31418) lies on these lines: {1,5175}, {2,35}, {4,958}, {5,2550}, {7,10916}, {8,6871}, {9,18483}, {10,962}, {11,443}, {12,5082}, {20,5267}, {30,30478}, {40,6844}, {55,6856}, {79,9965}, {100,6933}, {115,31416}, {346,30172}, {355,6982}, {376,4999}, {377,3086}, {381,2551}, {387,26098}, {388,17532}, {390,10198}, {405,5225}, {442,497}, {452,3583}, {474,10589}, {486,31413}, {496,17528}, {498,17784}, {499,6904}, {516,5705}, {517,6867}, {518,5714}, {519,4323}, {546,9708}, {936,3817}, {938,12609}, {946,6843}, {956,5229}, {993,3146}, {997,6993}, {1000,13463}, {1056,3813}, {1058,11235}, {1125,4208}, {1329,3545}, {1376,3090}, {1378,31412}, {1478,5288}, {1574,31415}, {1575,31404}, {1588,31484}, {1698,6919}, {1706,10175}, {1770,5744}, {2475,4293}, {2476,3085}, {2478,19855}, {2549,31488}, {3035,5067}, {3189,11374}, {3333,24386}, {3419,3485}, {3421,10895}, {3436,17577}, {3487,3838}, {3488,28628}, {3616,17647}, {3617,4015}, {3622,26725}, {3634,30332}, {3811,5226}, {3814,5068}, {3820,3851}, {3826,17559}, {3829,25524}, {3855,9710}, {3861,31494}, {3913,8164}, {3925,5084}, {3947,6765}, {4187,26040}, {4292,5231}, {4295,6734}, {4305,24541}, {4307,5292}, {4340,11269}, {4413,7173}, {4421,6668}, {4847,9612}, {5051,19866}, {5056,26364}, {5141,5552}, {5178,10129}, {5187,9780}, {5254,31405}, {5587,9842}, {5603,5794}, {5687,10588}, {5696,30275}, {5698,5791}, {5704,15299}, {5722,28629}, {5734,16206}, {5784,13374}, {5795,18492}, {5810,21293}, {5818,5836}, {5837,31162}, {5840,6892}, {5842,6988}, {5883,12446}, {6284,6857}, {6361,26066}, {6675,9668}, {6700,7988}, {6824,10525}, {6829,10531}, {6842,18518}, {6846,26333}, {6850,26470}, {6855,11496}, {6859,11248}, {6861,10738}, {6864,7681}, {6866,12699}, {6869,18407}, {6870,9812}, {6881,11928}, {6923,26321}, {6935,11826}, {6937,12116}, {6939,10893}, {6951,10785}, {6956,10310}, {6973,9956}, {7080,7951}, {7283,30741}, {7288,11112}, {7613,24046}, {7738,31466}, {8728,9669}, {9623,19925}, {9671,17552}, {9779,21616}, {9940,17668}, {10200,17580}, {10584,17531}, {10593,16408}, {10599,12245}, {11036,11263}, {11111,12953}, {12529,13750}, {14064,20172}, {14986,24387}, {16041,26558}, {17578,31458}, {21620,24392}, {21956,31402}, {23259,31453}, {24898,30653}, {24987,30305}, {28740,31031}

X(31418) = reflection of X(30478) in X(31493)
X(31418) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (4, 2886, 19843), (8, 6871, 10590)


X(31419) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, SPIEKER, STAMMLER}

Barycentrics    (b^2+c^2)*a^2+4*(b+c)*b*c*a-(b^2-c^2)^2 : :
X(31419) = 3*X(2)+X(5082) = 3*X(10)-X(5837) = X(388)-3*X(17528) = X(1697)-5*X(1698) = X(3340)+3*X(3679) = 7*X(3624)-3*X(10389) = 3*X(3742)-2*X(16216) = X(4294)-3*X(16418)

X(31419) lies on these lines: {1,3925}, {2,496}, {3,1602}, {4,9708}, {5,10}, {6,31416}, {8,442}, {9,12699}, {11,1697}, {12,3340}, {20,31494}, {21,20066}, {30,958}, {36,17563}, {40,5791}, {55,6675}, {72,25006}, {75,3933}, {84,5833}, {100,7483}, {119,11530}, {140,1376}, {141,2140}, {142,5045}, {145,4197}, {149,5047}, {200,11374}, {210,12047}, {281,15763}, {355,1490}, {377,956}, {381,2551}, {382,31420}, {388,17528}, {390,16845}, {405,3434}, {427,29667}, {443,999}, {452,9668}, {474,10527}, {497,11108}, {498,31245}, {499,4413}, {516,31445}, {518,6147}, {519,3841}, {528,5248}, {548,31458}, {549,4999}, {550,993}, {596,7263}, {632,1484}, {908,3697}, {912,18251}, {936,5886}, {942,4847}, {952,5794}, {962,8226}, {984,21926}, {997,5901}, {1056,4208}, {1107,15048}, {1125,3813}, {1377,7584}, {1378,7583}, {1387,19861}, {1479,31140}, {1482,6881}, {1532,5818}, {1573,5254}, {1574,3815}, {1575,31406}, {1588,31485}, {1595,1861}, {1706,5705}, {1714,5710}, {1731,17369}, {1737,3698}, {1834,30116}, {2078,5433}, {2476,3617}, {2549,31490}, {2975,11112}, {3036,11698}, {3058,5259}, {3086,16408}, {3242,24159}, {3293,5718}, {3312,31413}, {3333,6067}, {3419,19860}, {3421,5177}, {3428,20420}, {3436,17532}, {3485,3940}, {3545,8165}, {3555,5249}, {3579,5745}, {3612,9945}, {3616,17529}, {3624,10389}, {3626,3822}, {3628,26364}, {3632,15888}, {3634,3816}, {3649,5904}, {3656,15829}, {3695,29641}, {3703,4647}, {3704,30172}, {3742,16216}, {3753,6734}, {3757,5100}, {3811,5719}, {3812,10916}, {3824,21620}, {3825,3828}, {3831,21242}, {3838,4662}, {3881,25557}, {3884,13463}, {3889,27186}, {3897,10609}, {3913,10198}, {3927,4295}, {3961,24161}, {3983,17605}, {3996,25650}, {4026,19858}, {4187,9780}, {4205,19853}, {4294,16418}, {4323,4930}, {4426,18907}, {4514,16817}, {4714,30171}, {4731,17606}, {4848,15844}, {4972,13728}, {5015,16824}, {5084,9669}, {5250,14022}, {5251,6284}, {5253,26060}, {5258,7354}, {5260,11113}, {5263,17698}, {5265,30312}, {5267,8703}, {5273,6361}, {5274,17559}, {5283,21956}, {5284,17590}, {5288,5434}, {5325,28198}, {5432,26475}, {5439,26015}, {5440,24541}, {5484,17678}, {5533,31235}, {5554,11545}, {5587,15908}, {5657,6831}, {5708,24477}, {5714,5815}, {5787,30503}, {5789,14647}, {5790,6842}, {5795,18480}, {5880,24470}, {5903,21677}, {5905,11544}, {6000,20306}, {6048,17717}, {6174,31260}, {6175,20060}, {6244,6847}, {6260,9947}, {6347,15235}, {6348,15236}, {6585,6924}, {6690,8715}, {6700,11230}, {6763,11246}, {6765,25525}, {6824,10306}, {6826,22770}, {6829,12245}, {6841,12702}, {6845,18231}, {6856,7080}, {6857,17784}, {6861,10679}, {7270,16821}, {7280,31157}, {7288,16417}, {7308,9614}, {7502,9712}, {7535,12410}, {7738,31468}, {7741,19875}, {7795,20181}, {7819,20172}, {7951,21031}, {7958,11522}, {7965,9589}, {8227,8580}, {8362,17030}, {8583,11373}, {9605,31405}, {9607,31491}, {9843,24386}, {10283,30144}, {10707,26127}, {10883,20070}, {10914,24987}, {10957,24914}, {11019,16201}, {11681,17530}, {12447,13464}, {12514,28174}, {12558,28228}, {12572,22793}, {12675,15587}, {13576,16850}, {14019,16823}, {14986,17582}, {15674,20095}, {15935,30143}, {16160,18253}, {16415,23853}, {16829,26561}, {16853,26105}, {17061,30145}, {17355,18257}, {17575,19877}, {17670,26801}, {17687,20533}, {18249,28194}, {18250,18483}, {20299,20307}, {20790,21625}, {21949,23537}, {23708,24954}, {24174,29676}, {24443,29690}, {24988,26094}, {25917,30384}, {26321,28458}

X(31419) = midpoint of X(i) and X(j) for these {i,j}: {3295, 5082}, {3927, 4295}
X(31419) = reflection of X(i) in X(j) for these (i,j): (6147, 12609), (10267, 140), (10386, 5248), (21620, 3824), (25466, 3841)
X(31419) = complement of X(3295)
X(31419) = X(1593)-of-4th Euler triangle
X(31419) = X(1595)-of-Wasat triangle
X(31419) = X(11414)-of-3rd Euler triangle
X(31419) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3925, 8728), (2, 5082, 3295), (2, 24390, 496)


X(31420) = CENTROID OF CURVATURES OF THESE CIRCLES: {ANTICOMPLEMENTARY, SPIEKER, STEINER}

Barycentrics    3*a^4+2*(b^2+c^2)*a^2+12*(b+c)*b*c*a-5*(b^2-c^2)^2 : :

X(31420) lies on these lines: {2,4309}, {4,9710}, {5,2550}, {8,31410}, {10,3832}, {20,993}, {30,31494}, {382,31419}, {390,3841}, {443,31140}, {497,17529}, {516,31446}, {548,30478}, {631,2886}, {1376,5067}, {1378,31414}, {1574,31417}, {1575,31407}, {1588,31486}, {2549,31491}, {2551,3843}, {3421,9656}, {3434,4197}, {3530,31493}, {3853,9708}, {3855,9711}, {3925,9670}, {4330,19854}, {5082,15888}, {5084,9671}, {5758,11362}, {6919,31159}, {7288,17583}, {7486,25639}, {7738,31469}, {7765,31416}, {9607,31405}, {11235,17582}, {15717,26363}, {16126,31145}, {17575,26040}, {17580,24387}, {17784,31452}, {31450,31488}

X(31420) = {X(3925), X(9670)}-harmonic conjugate of X(17552)


X(31421) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, CIRCUMCIRCLE, HALF-MOSES}

Barycentrics    a*(3*a^3+(b+c)*a^2-(7*b^2+2*b*c+7*c^2)*a-(b^2-c^2)*(b-c)) : :

X(31421) lies on these lines: {1,574}, {3,9574}, {4,31428}, {6,9582}, {20,31396}, {32,16192}, {39,165}, {40,5013}, {56,31426}, {57,31448}, {372,31427}, {516,31400}, {936,21879}, {1376,31429}, {1500,3361}, {1574,5234}, {1575,31424}, {1698,2549}, {1699,31401}, {1706,31449}, {2276,15803}, {3333,31477}

X(31421) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 9574, 9593), (3, 31430, 9574), (574, 1571, 1)


X(31422) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, CIRCUMCIRCLE, MOSES}

Barycentrics    a*(3*a^3+(b+c)*a^2-(5*b^2+2*b*c+5*c^2)*a-(b^2-c^2)*(b-c)) : :

X(31422) lies on these lines: {3,1571}, {4,31441}, {6,31430}, {20,31398}, {32,9574}, {39,165}, {40,574}, {56,31433}, {57,31451}, {115,31423}, {187,9593}, {372,31437}, {516,31401}, {517,15815}, {1155,31448}, {1376,31442}, {1500,15803}, {1505,9616}, {1572,3579}, {1574,31424}, {1698,7748}, {1699,31455}, {1706,31456}, {2242,31426}, {2549,6684}

X(31422) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 1571, 9620), (3, 31443, 1571)


X(31423) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, CIRCUMCIRCLE, NINE-POINTS}

Barycentrics    3*a^4+(b+c)*a^3-(5*b^2+2*b*c+5*c^2)*a^2-(b^2-c^2)*(b-c)*a+2*(b^2-c^2)^2 : :
X(31423) = X(1)-8*X(140) = 3*X(1)+4*X(5690) = X(1)+6*X(26446) = 6*X(2)+X(40) = 9*X(2)-2*X(946) = 15*X(2)-X(962) = 3*X(2)+4*X(6684) = 12*X(2)-5*X(8227) = 8*X(2)-X(31162) = 3*X(40)+4*X(946) = 5*X(40)+2*X(962) = X(40)-8*X(6684) = 2*X(40)+5*X(8227) = 9*X(40)-2*X(20070) = 4*X(40)+3*X(31162) = 6*X(140)+X(5690) = 4*X(140)+3*X(26446) = 10*X(946)-3*X(962) = X(946)+6*X(6684) = 8*X(946)-15*X(8227) = 6*X(946)+X(20070) = 16*X(946)-9*X(31162) = 9*X(962)+5*X(20070) = 8*X(962)-15*X(31162) = 2*X(5690)-9*X(26446) = 10*X(8227)-3*X(31162)

X(31423) lies on these lines: {1,140}, {2,40}, {3,1698}, {4,3634}, {5,165}, {6,31428}, {8,10165}, {9,2252}, {10,631}, {12,15803}, {20,10175}, {30,7989}, {32,31441}, {35,6883}, {36,9578}, {46,5219}, {56,31434}, {57,498}, {63,27529}, {72,15016}, {115,31422}, {119,16209}, {191,31142}, {210,9940}, {230,9593}, {355,549}, {371,13947}, {372,13893}, {376,19925}, {405,2077}, {411,9342}, {442,16113}, {451,1753}, {474,11012}, {486,9616}, {495,3361}, {499,1697}, {515,3523}, {516,3090}, {517,3526}, {519,15702}, {550,30315}, {551,12245}, {580,750}, {581,899}, {590,1703}, {615,1702}, {632,5886}, {936,3035}, {952,14869}, {970,29825}, {993,6940}, {1006,25440}, {1040,9610}, {1064,17749}, {1125,3525}, {1155,9612}, {1158,3305}, {1210,5218}, {1329,31424}, {1376,5705}, {1385,3679}, {1420,10039}, {1482,15694}, {1483,3653}, {1512,6977}, {1571,7746}, {1572,31455}, {1656,1699}, {1706,6967}, {1737,3601}, {1788,11529}, {1837,30282}, {2093,5326}, {2948,15061}, {3068,13975}, {3069,13912}, {3071,9582}, {3085,3333}, {3086,31393}, {3091,10172}, {3149,7688}, {3158,10916}, {3336,4654}, {3338,3584}, {3339,11374}, {3359,6863}, {3428,16408}, {3524,3828}, {3528,28164}, {3530,18481}, {3533,5603}, {3541,7713}, {3586,5217}, {3587,6861}, {3612,5727}, {3616,11362}, {3617,5882}, {3622,28234}, {3626,7967}, {3628,7988}, {3632,10246}, {3633,15178}, {3654,5901}, {3655,11812}, {3656,10124}, {3681,12005}, {3683,31246}, {3697,12675}, {3767,9574}, {3814,6937}, {3817,5067}, {3820,5234}, {3822,21165}, {3826,6831}, {3832,28150}, {3841,6830}, {3844,5085}, {3851,28146}, {3876,5884}, {3901,5885}, {3928,21077}, {4292,10588}, {4298,8164}, {4652,11681}, {4668,30392}, {4999,9623}, {5010,10826}, {5012,9622}, {5044,5693}, {5049,31480}, {5122,9654}, {5128,12047}, {5204,9613}, {5231,5687}, {5254,31421}, {5258,10269}, {5260,5450}, {5281,5704}, {5288,16203}, {5290,31479}, {5418,18991}, {5420,18992}, {5435,21620}, {5437,10198}, {5535,25525}, {5703,18221}, {5708,18217}, {5720,16132}, {5726,18990}, {5744,21075}, {5771,17718}, {5790,13624}, {5791,8580}, {5904,10202}, {5919,31436}, {5955,18229}, {6264,6713}, {6667,14217}, {6705,10864}, {7280,10827}, {7288,31397}, {7294,11376}, {7735,31396}, {7741,9580}, {7749,9620}, {7815,12197}, {7951,9579}, {8715,24392}, {8981,19004}, {9540,13936}, {9575,31401}, {9583,13973}, {9819,11373}, {9860,15561}, {9902,11171}, {9904,14643}, {10200,12703}, {10222,11415}, {10283,16189}, {10389,31452}, {10573,13384}, {10589,10624}, {11010,23708}, {11227,14872}, {12407,15035}, {12515,15017}, {12619,15015}, {12751,21154}, {12779,23328}, {12782,15819}, {13881,31443}, {13883,13935}, {13951,31439}, {13966,19003}, {14986,31188}, {15805,16472}, {16150,22937}, {17286,24257}, {24954,31235}, {26878,28609}, {28292,31207}

X(31423) = midpoint of X(i) and X(j) for these {i,j}: {3523, 9780}, {3624, 9588}, {7989, 16192}
X(31423) = reflection of X(i) in X(j) for these (i,j): (3624, 3526), (9624, 3624)
X(31423) = X(15056)-of-hexyl triangle
X(31423) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (140, 26446, 1), (5432, 24914, 1)


X(31424) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, CIRCUMCIRCLE, SPIEKER}

Barycentrics    a*(3*a^3+(b+c)*a^2-(3*b^2+2*b*c+3*c^2)*a-(b+c)^3) : :

X(31424) lies on these lines: {1,21}, {2,4292}, {3,9}, {4,5705}, {6,31429}, {7,1125}, {8,3977}, {10,20}, {30,5791}, {32,16517}, {33,22361}, {35,200}, {36,8583}, {37,4252}, {40,958}, {44,4255}, {46,5251}, {55,6765}, {56,3683}, {57,405}, {72,3601}, {78,3219}, {90,6512}, {140,30827}, {142,16845}, {144,5703}, {197,15592}, {210,5217}, {214,13243}, {223,1935}, {224,18232}, {226,6857}, {238,988}, {326,1098}, {329,13411}, {372,31438}, {376,5325}, {377,1698}, {379,16832}, {380,4267}, {386,1743}, {392,1420}, {404,3305}, {411,1750}, {442,9579}, {452,1210}, {474,7308}, {515,10268}, {516,5833}, {519,4313}, {527,3487}, {551,11036}, {553,17561}, {572,15479}, {580,1741}, {631,2096}, {908,6910}, {938,11106}, {942,3928}, {944,5837}, {946,5698}, {950,11111}, {956,1697}, {975,3731}, {978,27640}, {991,3682}, {997,5267}, {1001,3333}, {1006,8726}, {1010,19859}, {1011,22345}, {1158,30503}, {1214,1394}, {1247,8769}, {1329,31423}, {1376,5302}, {1378,9616}, {1385,15829}, {1453,3666}, {1466,16293}, {1479,5231}, {1572,31456}, {1574,31422}, {1575,31421}, {1699,6837}, {1703,31453}, {1706,3579}, {1709,12565}, {1722,17596}, {1724,2999}, {1728,11344}, {1729,3496}, {1770,19854}, {1836,24953}, {2093,19860}, {2551,6256}, {2886,16113}, {2951,12511}, {3085,12527}, {3194,8765}, {3218,16865}, {3295,6762}, {3306,5047}, {3338,5259}, {3423,17742}, {3428,12705}, {3436,31434}, {3488,24391}, {3522,10430}, {3523,6700}, {3555,10389}, {3560,5709}, {3586,6734}, {3612,5692}, {3616,9965}, {3624,5249}, {3634,4208}, {3646,15254}, {3712,10371}, {3753,5128}, {3811,5223}, {3814,4197}, {3876,4855}, {3911,5084}, {3923,10444}, {3927,11523}, {3973,4256}, {3984,17574}, {4015,9859}, {4187,31231}, {4188,27065}, {4294,4847}, {4297,9799}, {4305,6737}, {4312,12609}, {4421,4662}, {4426,9593}, {4641,19765}, {4654,15670}, {4668,11015}, {4679,5433}, {4853,5119}, {4862,24159}, {4882,8715}, {4999,8227}, {5122,16408}, {5129,5435}, {5204,25917}, {5218,21075}, {5219,7483}, {5247,17594}, {5266,7174}, {5268,26264}, {5272,25494}, {5281,5815}, {5285,13730}, {5290,10198}, {5316,17567}, {5328,10303}, {5437,11108}, {5440,19535}, {5584,10860}, {5587,26066}, {5657,5795}, {5708,16866}, {5715,6824}, {5730,13384}, {5794,18253}, {5832,12699}, {5904,16465}, {6061,17104}, {6173,24470}, {6245,6987}, {6260,6988}, {6282,6906}, {6666,17582}, {6675,25525}, {6692,17559}, {6705,6865}, {6839,7989}, {6875,18446}, {6884,7988}, {6914,26921}, {6950,26878}, {6985,18540}, {6986,10857}, {7193,13323}, {7283,11679}, {7411,8580}, {7992,12520}, {8171,11035}, {8720,16825}, {8822,10436}, {9575,31449}, {9580,24390}, {9581,11113}, {9583,30557}, {9589,31458}, {9613,24987}, {9614,10527}, {9678,18991}, {9711,31425}, {9963,15863}, {10085,15931}, {10164,18250}, {10176,11220}, {10434,10463}, {10479,14058}, {10882,15654}, {10883,25639}, {11374,28609}, {11376,31157}, {11415,24541}, {11512,17123}, {11518,19526}, {12513,31393}, {12579,29635}, {12650,22758}, {13369,28466}, {13738,22060}, {13747,20196}, {14021,17284}, {15015,18254}, {15171,24392}, {15674,31019}, {15676,17483}, {16020,24171}, {16058,23085}, {16209,26364}, {16287,23206}, {16289,16574}, {16347,26223}, {16552,19763}, {16850,17754}, {16859,27003}, {17064,24851}, {17272,18650}, {17306,17698}, {17527,31190}, {17579,19875}, {17580,18230}, {17697,24627}, {17768,28628}, {18229,19645}, {18421,30147}, {18443,24467}, {18444,30144}, {18655,25590}, {19278,27064}, {21982,25083}, {22076,26892}, {22793,31493}, {25513,27339}, {26357,30223}, {27385,31018}

X(31424) = reflection of X(i) in X(j) for these (i,j): (5290, 10198), (5715, 6824)
X(31424) = X(18925)-of-2nd circumperp triangle
X(31424) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 16570, 1046), (968, 1468, 1), (3901, 5426, 1)


X(31425) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, CIRCUMCIRCLE, STEINER}

Barycentrics    9*a^4+3*(b+c)*a^3-(11*b^2+6*b*c+11*c^2)*a^2-3*(b^2-c^2)*(b-c)*a+2*(b^2-c^2)^2 : :
X(31425) = 3*X(1)-16*X(3530) = 10*X(3)+3*X(3679) = 12*X(3)+X(5881) = 6*X(3)+7*X(9588) = 3*X(3)+10*X(31447) = 4*X(5)+9*X(165) = 6*X(10)+7*X(3528) = 4*X(20)+9*X(5587) = X(20)+12*X(6684) = 3*X(20)+10*X(31399) = 3*X(40)+10*X(631) = 5*X(40)+8*X(1125) = 9*X(40)+4*X(4301) = 7*X(40)+6*X(5603) = 6*X(40)+7*X(9624) = X(40)+12*X(10164) = 6*X(165)+7*X(31423) = 25*X(631)-12*X(1125) = 15*X(631)-2*X(4301) = 20*X(631)-7*X(9624) = 5*X(631)-18*X(10164) = 18*X(1125)-5*X(4301) = 28*X(1125)-15*X(5603) = 2*X(1125)-15*X(10164) = 18*X(3679)-5*X(5881) = 3*X(5587)-16*X(6684) = X(5603)-14*X(10164) = X(5881)-14*X(9588) = 20*X(6684)-7*X(9780) = 18*X(6684)-5*X(31399) = 7*X(9588)-20*X(31447)

X(31425) lies on these lines: {1,3530}, {3,3679}, {5,165}, {6,31431}, {10,3528}, {20,5587}, {32,31444}, {40,631}, {56,31436}, {57,31452}, {140,31162}, {145,3576}, {372,31440}, {382,1698}, {515,21734}, {516,5067}, {519,10299}, {546,19876}, {548,16192}, {549,7991}, {550,19875}, {944,4746}, {1376,31446}, {1572,31457}, {1699,5070}, {1703,31454}, {1706,31458}, {3523,7982}, {3525,5493}, {3526,3579}, {3529,3828}, {3625,5657}, {3627,30315}, {3632,17502}, {3634,3855}, {3653,16189}, {3654,15712}, {3656,12108}, {3843,11231}, {3853,7989}, {4325,9578}, {4330,9581}, {4338,5219}, {4512,17575}, {4669,15715}, {4677,17504}, {4745,15710}, {5054,11522}, {5079,28202}, {5319,9574}, {5691,15696}, {5734,10165}, {5882,15692}, {7486,9778}, {7765,31422}, {9575,31450}, {9607,31421}, {9657,31434}, {9680,18991}, {9711,31424}, {9956,17800}, {10175,17578}, {10303,28194}, {12512,18492}, {12699,16239}, {15693,24680}, {15720,25055}, {15803,15888}, {19872,22793}, {19877,28150}

X(31425) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 9588, 5881), (3, 31447, 9588)


X(31426) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, HALF-MOSES, INCIRCLE}

Barycentrics    a*(a^3+(b+c)*a^2-(5*b^2+6*b*c+5*c^2)*a-(b^2-c^2)*(b-c)) : :

X(31426) lies on these lines: {1,5013}, {8,31429}, {9,6048}, {11,31428}, {39,1697}, {40,2276}, {55,9593}, {56,31421}, {57,1500}, {497,31396}, {516,31402}, {517,31461}, {574,1420}, {988,3208}, {999,31430}, {1335,31427}, {1453,17735}, {1574,7308}, {1575,31435}, {1699,31460}, {1703,31459}, {1706,5283}, {2136,16975}, {2242,31422}, {2548,9580}, {2549,9578}, {2999,14974}, {3057,9592}, {3338,9331}, {3501,17594}, {3601,9620}, {3815,9614}, {4255,21872}, {4853,31449}, {5024,9957}, {5119,9575}, {5250,17756}, {5254,31434}, {5587,9598}, {5687,16517}, {5705,21956}, {6421,31432}, {7736,10624}, {7738,31397}, {7743,31467}, {7962,9619}, {9579,31409}, {9581,31398}, {9589,31462}, {9607,31436}, {12053,31400}, {15803,31443}

X(31426) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31433, 1697), (55, 9593, 16780)


X(31427) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, HALF-MOSES, 2nd LEMOINE}

Barycentrics    a*(a^3+(b+c)*a^2-4*S*a-(5*b^2+2*b*c+5*c^2)*a-(b^2-c^2)*(b-c)) : :

X(31427) lies on these lines: {6,165}, {32,9582}, {39,1702}, {40,6422}, {57,31459}, {371,9593}, {372,31421}, {486,31428}, {516,31403}, {1335,31426}, {1378,31429}, {1504,1571}, {1575,31438}, {1588,31396}, {1699,31463}, {1706,31464}, {2275,31432}, {3312,31430}

X(31427) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31437, 1702), (1504, 1571, 1703)


X(31428) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, HALF-MOSES, NINE-POINTS}

Barycentrics    a^4+(b+c)*a^3-(7*b^2+2*b*c+7*c^2)*a^2-(b^2-c^2)*(b-c)*a+2*(b^2-c^2)^2 : :

X(31428) lies on these lines: {1,31398}, {2,9593}, {4,31421}, {5,9574}, {6,31423}, {10,9592}, {11,31426}, {39,1698}, {40,3815}, {57,31460}, {165,2548}, {381,31430}, {486,31427}, {516,31404}, {517,31467}, {574,5691}, {1329,31429}, {1506,1571}, {1572,9588}, {1575,5705}, {1703,31463}, {1706,31466}, {2275,31434}, {2549,7989}, {3361,31409}, {3624,9620}, {3634,5286}, {3679,9619}, {3911,31402}, {5013,5587}, {5024,9956}, {5290,31476}, {5432,16780}, {5475,31422}, {5881,31492}, {6421,13893}, {6422,13947}, {6684,7736}, {7738,10175}, {7739,19876}, {7746,19872}, {8227,31489}, {9575,26446}, {9581,31448}, {9589,31444}, {9596,15803}, {9605,11231}, {13975,31403}, {16517,26364}

X(31428) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31396, 9593), (31398, 31401, 1)


X(31429) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, HALF-MOSES, SPIEKER}

Barycentrics    a*(a^3+(b+c)*a^2-(5*b^2+2*b*c+5*c^2)*a-(b+c)^3) : :

X(31429) lies on these lines: {1,5021}, {3,16517}, {6,31424}, {8,31426}, {9,39}, {10,7738}, {21,16780}, {37,3333}, {40,1107}, {57,5283}, {200,31448}, {516,31405}, {517,31468}, {574,5438}, {936,5013}, {958,9593}, {960,9592}, {968,1475}, {975,5030}, {1329,31428}, {1376,31421}, {1378,31427}, {1500,6762}, {1571,1573}, {1697,16975}, {1699,31466}, {1703,31464}, {2082,4414}, {2136,31433}, {2275,31435}, {2551,31396}, {3158,31451}, {3452,31400}, {3646,16604}, {4652,5276}, {5024,5044}, {5254,5705}, {5275,15803}, {5286,5745}, {5437,16589}, {5791,15048}, {6421,31438}, {6765,31477}, {7736,12572}, {9575,12514}, {9589,31469}, {9607,31446}, {9619,15829}, {9620,31456}, {9623,31490}, {9709,31430}, {9711,31431}, {12527,31402}, {17448,31393}, {24627,27523}

X(31429) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31442, 9), (1571, 1573, 1706)


X(31430) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, HALF-MOSES, STAMMLER}

Barycentrics    a*(2*a^3+(b+c)*a^2-2*(3*b^2+b*c+3*c^2)*a-(b^2-c^2)*(b-c)) : :
X(31430) = 3*X(1571)+X(9619) = 3*X(5013)-X(9619)

X(31430) lies on these lines: {3,9574}, {6,31422}, {30,31396}, {39,3579}, {40,5024}, {57,31461}, {165,9605}, {381,31428}, {382,31431}, {516,31406}, {517,1571}, {574,1385}, {942,31448}, {999,31426}, {1384,16192}, {1572,22332}, {1575,31445}, {1699,31467}, {1706,31468}, {2548,28146}, {2549,9956}, {3312,31427}

X(31430) = midpoint of X(1571) and X(5013)
X(31430) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31443, 3579), (9574, 31421, 3)


X(31431) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, HALF-MOSES, STEINER}

Barycentrics    3*a^4+3*(b+c)*a^3-(17*b^2+6*b*c+17*c^2)*a^2-3*(b^2-c^2)*(b-c)*a+2*(b^2-c^2)^2 : :

X(31431) lies on these lines: {1,31450}, {5,9574}, {6,31425}, {20,31396}, {39,9588}, {40,9606}, {57,31462}, {382,31430}, {516,31407}, {517,31470}, {631,9593}, {1571,9589}, {1575,31446}, {1698,7765}, {1703,31465}, {1706,31469}, {2275,31436}, {4301,31400}, {5013,5881}, {6421,31440}, {7738,31399}, {9592,11362}, {9605,31447}, {9620,31457}, {9624,31492}, {9711,31429}

X(31431) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31444, 9588), (1571, 9698, 9589)


X(31432) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, INCIRCLE, 2nd LEMOINE}

Barycentrics    a*(4*S*a+(a^2+2*(b+c)*a+(b-c)^2)*(-a+b+c)) : :

X(31432) lies on these lines: {1,371}, {6,1697}, {8,31438}, {11,13893}, {36,9582}, {40,1124}, {55,18992}, {56,9616}, {57,3297}, {165,6502}, {200,15892}, {485,9614}, {486,31434}, {497,13883}, {516,31408}, {517,31474}, {944,19068}, {950,19066}, {999,31439}, {1015,31437}, {1151,1420}, {1319,9615}, {1335,31393}, {1378,31435}, {1505,31433}, {1572,31471}, {1587,10624}, {1588,31397}, {1699,31472}, {1703,3299}, {1706,31473}, {2275,31427}, {2362,7991}, {3057,18991}, {3068,12053}, {3070,9580}, {3071,9578}, {3086,13912}, {3311,9957}, {3601,7968}, {4311,9541}, {4853,31453}, {4866,7133}, {5126,6449}, {5218,13971}, {5414,19003}, {5919,18996}, {6221,24928}, {6421,31426}, {6459,10106}, {6561,9613}, {7585,9785}, {7743,8976}, {7962,7969}, {8227,9646}, {8981,11373}, {9575,31459}, {9581,13911}, {9589,31475}, {9819,19004}, {11376,13901}, {12701,19028}, {13905,30384}, {13947,19029}, {15558,19113}

X(31432) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3057, 19038, 18991), (3299, 5119, 1703)


X(31433) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, INCIRCLE, MOSES}

Barycentrics    a*(a^3+(b+c)*a^2-3*(b+c)^2*a-(b^2-c^2)*(b-c)) : :

X(31433) lies on these lines: {1,574}, {8,31442}, {11,31441}, {39,1697}, {40,1500}, {55,9620}, {56,31422}, {115,31434}, {165,2242}, {484,9331}, {497,31398}, {516,31409}, {517,31477}, {999,31443}, {1015,9574}, {1335,31437}, {1505,31432}, {1506,9614}, {1572,2276}, {1574,31435}, {1699,31476}, {1703,31471}, {1706,16589}, {2136,31429}, {2241,9593}, {2271,21872}, {2548,10624}, {2549,31397}, {3057,9619}, {3880,31449}, {3895,16975}, {4646,14974}, {4853,31456}, {4868,16972}, {5013,9957}, {5475,9580}, {5587,9664}, {7743,31489}, {7748,9578}, {7756,9613}, {7765,31436}, {9589,31478}, {9592,9819}, {9598,10039}, {9785,31400}, {12053,31401}, {12514,20691}, {12575,31396}, {12701,31460}, {15815,24928}, {16973,25439}

X(31433) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1697, 31426, 39), (9574, 31393, 1015)


X(31434) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, INCIRCLE, NINE-POINTS}

Barycentrics    a^4+(b+c)*a^3-3*(b+c)^2*a^2-(b^2-c^2)*(b-c)*a+2*(b^2-c^2)^2 : :

X(31434) lies on these lines: {1,2}, {3,9578}, {5,1697}, {9,17757}, {11,31393}, {12,40}, {35,1012}, {46,5290}, {55,3586}, {56,31423}, {57,495}, {115,31433}, {140,1420}, {165,1478}, {191,16152}, {210,18397}, {226,2093}, {281,1785}, {355,3601}, {380,26063}, {381,9580}, {388,6684}, {392,30827}, {442,1706}, {484,4312}, {486,31432}, {497,10175}, {515,5218}, {516,10590}, {517,5219}, {518,30274}, {631,10106}, {946,6969}, {950,5818}, {952,13384}, {984,1735}, {999,11231}, {1001,5123}, {1015,31441}, {1056,3911}, {1124,13947}, {1155,11237}, {1329,31435}, {1335,13893}, {1467,6989}, {1479,6939}, {1490,10786}, {1532,1699}, {1572,31476}, {1656,9957}, {1703,31472}, {1728,21031}, {1739,4859}, {1788,21620}, {2078,6883}, {2098,9624}, {2136,24390}, {2275,31428}, {2646,5881}, {3057,8227}, {3074,5264}, {3090,12053}, {3091,10624}, {3097,10063}, {3158,3419}, {3247,21933}, {3295,9581}, {3303,17606}, {3333,15888}, {3336,4355}, {3338,5445}, {3339,13407}, {3340,5690}, {3421,5745}, {3436,31424}, {3476,10165}, {3485,11362}, {3487,4848}, {3523,4311}, {3526,24928}, {3576,5252}, {3579,9579}, {3583,6957}, {3585,6925}, {3614,12701}, {3628,11373}, {3681,18389}, {3683,31141}, {3746,10826}, {3753,25525}, {3817,30305}, {3820,7308}, {3839,30332}, {3877,30852}, {3885,7504}, {3895,11680}, {3921,5728}, {3947,4295}, {3983,10399}, {4292,5261}, {4293,10164}, {4299,16192}, {4304,5281}, {4308,10303}, {4342,10171}, {4540,12564}, {4640,11236}, {4652,20060}, {5010,6909}, {5054,5126}, {5055,7743}, {5056,9785}, {5250,11681}, {5251,8069}, {5254,31426}, {5258,22766}, {5259,11508}, {5269,5725}, {5443,30323}, {5541,8068}, {5697,7686}, {5722,10389}, {5727,5790}, {5795,6857}, {5815,18231}, {5832,5856}, {5886,7962}, {5904,13750}, {6284,18492}, {6600,17057}, {6763,17700}, {6833,12650}, {6932,9589}, {7330,10942}, {7741,15845}, {7982,11375}, {7988,9819}, {7991,12047}, {8256,28628}, {9575,31460}, {9597,31421}, {9646,18991}, {9651,31422}, {9657,31425}, {10057,15015}, {10172,10589}, {10391,18908}, {10572,31452}, {10591,12575}, {10592,12699}, {10902,11501}, {11529,17718}, {12260,15079}, {12436,26062}, {12526,21077}, {12607,26066}, {12705,18242}, {12758,15017}, {13905,19004}, {13963,19003}, {13975,31408}, {14647,30304}, {15950,16200}, {16601,23058}, {17605,31162}

X(31434) = reflection of X(i) in X(j) for these (i,j): (5219, 31479), (30282, 5218)
X(31434) = X(5231)-of-inner-Yff triangle
X(31434) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1737, 10056, 1), (6734, 10528, 6765), (10915, 26363, 4853)


X(31435) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, INCIRCLE, SPIEKER}

Barycentrics    a*(a^3+(b+c)*a^2-(b^2+6*b*c+c^2)*a-(b+c)^3) : :

X(31435) lies on these lines: {1,6}, {2,40}, {3,4512}, {8,3305}, {10,497}, {11,5705}, {12,4679}, {19,7498}, {20,11372}, {21,84}, {35,5438}, {38,28011}, {46,3624}, {55,936}, {56,3683}, {57,1125}, {63,3333}, {65,4423}, {78,1621}, {140,3359}, {142,4295}, {144,11037}, {145,27065}, {165,474}, {169,19868}, {191,3338}, {200,3295}, {210,3303}, {329,21620}, {377,24564}, {380,965}, {388,12572}, {406,1848}, {442,1699}, {443,516}, {452,515}, {496,5231}, {498,30827}, {517,11108}, {551,3929}, {581,25941}, {595,975}, {602,2328}, {612,3915}, {614,2292}, {758,11518}, {846,988}, {859,10882}, {942,10582}, {968,1193}, {978,17594}, {986,5272}, {993,1420}, {997,3601}, {999,31445}, {1009,2944}, {1010,12717}, {1015,31442}, {1056,12527}, {1058,4847}, {1158,10165}, {1210,26105}, {1329,31434}, {1378,31432}, {1385,7330}, {1467,12709}, {1482,16857}, {1490,13615}, {1512,6898}, {1519,6889}, {1572,16589}, {1574,31433}, {1575,31426}, {1698,1706}, {1703,31473}, {1709,7987}, {1722,17123}, {1730,16844}, {1754,19523}, {1764,16343}, {1788,9843}, {2093,3812}, {2136,3679}, {2270,5257}, {2275,31429}, {2478,5587}, {2550,10624}, {2551,31397}, {2646,30223}, {2821,25926}, {2950,21154}, {2999,3931}, {3057,9623}, {3085,3452}, {3086,5745}, {3149,10268}, {3158,3746}, {3219,3622}, {3306,5550}, {3339,5439}, {3340,3878}, {3361,3916}, {3421,18250}, {3474,12436}, {3488,6737}, {3579,16408}, {3586,5794}, {3587,8728}, {3600,8545}, {3617,3895}, {3652,3653}, {3670,5573}, {3680,7162}, {3697,4882}, {3702,5271}, {3740,3913}, {3749,5293}, {3753,7991}, {3757,19582}, {3811,10176}, {3816,26066}, {3817,6856}, {3868,4666}, {3869,5284}, {3870,3876}, {3873,3951}, {3877,5047}, {3884,7962}, {3885,11525}, {3886,9534}, {3897,16858}, {3925,12701}, {3927,5045}, {4292,5698}, {4308,29007}, {4314,10384}, {4357,17170}, {4385,30568}, {4414,28352}, {4640,15803}, {4642,17125}, {4652,5253}, {4659,24424}, {4673,17277}, {4853,9708}, {4915,30337}, {5128,19862}, {5218,6700}, {5219,10198}, {5249,11415}, {5255,5268}, {5273,14986}, {5285,11365}, {5327,25526}, {5432,24954}, {5534,16202}, {5584,21153}, {5686,6764}, {5687,8580}, {5704,18231}, {5709,5886}, {5711,17022}, {5720,10267}, {5731,10864}, {5732,8273}, {5785,14100}, {5799,21363}, {5837,18391}, {5881,7966}, {5887,18443}, {5901,26921}, {5903,25542}, {5919,12629}, {6001,8726}, {6282,11496}, {6666,19855}, {6690,25681}, {6743,30331}, {7091,13462}, {7171,13624}, {7743,31493}, {7992,10167}, {9612,24703}, {9624,12704}, {9711,31436}, {9785,18230}, {9819,10914}, {10085,30389}, {10200,31231}, {10436,17753}, {10439,18180}, {10476,17185}, {10587,31018}, {10595,26878}, {10866,15837}, {11010,31262}, {11019,18249}, {11235,17619}, {11376,24953}, {11512,17596}, {12047,25525}, {12053,19843}, {12608,15239}, {12672,30503}, {12703,24982}, {13384,30144}, {13407,28609}, {16878,19762}, {17306,18589}, {18164,28619}, {18228,21075}, {18481,18540}, {19854,30384}, {19919,24467}, {20196,26364}, {21077,31142}, {24178,24248}, {24627,26093}, {25430,27784}, {25591,29828}, {25924,28292}, {26102,30979}, {26128,28039}

X(31435) = X(10982)-of-excentral triangle
X(31435) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 5223, 3555), (1, 5904, 3243), (10179, 12513, 1)


X(31436) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, INCIRCLE, STEINER}

Barycentrics    3*a^4+3*(b+c)*a^3-(5*b^2+18*b*c+5*c^2)*a^2-3*(b^2-c^2)*(b-c)*a+2*(b^2-c^2)^2 : :

X(31436) lies on these lines: {1,631}, {5,1697}, {8,31446}, {20,9613}, {40,10404}, {55,5881}, {56,31425}, {165,4317}, {382,9578}, {405,3679}, {497,31399}, {498,9819}, {499,30337}, {516,31410}, {517,31480}, {999,31447}, {1015,31444}, {1145,5436}, {1335,31440}, {1420,3530}, {1572,31478}, {1698,3816}, {1703,31475}, {1706,17529}, {2275,31431}, {3057,9624}, {3085,4301}, {3526,9957}, {3528,10106}, {3584,6834}, {3586,4309}, {3624,5836}, {3654,11518}, {3680,7483}, {3832,10624}, {3843,9580}, {3895,5705}, {4311,21734}, {4330,5691}, {4338,5290}, {4512,10915}, {4853,31458}, {4857,6898}, {5067,12053}, {5119,9589}, {5587,9670}, {5690,10389}, {5734,13411}, {5919,31423}, {6857,12640}, {6908,7991}, {7320,10303}, {7486,9785}, {7765,31433}, {9575,31462}, {9607,31426}, {9711,31435}, {11373,16239}, {11525,24953}, {13747,25055}, {17559,19875}

X(31436) = {X(1697), X(31434)}-harmonic conjugate of X(9614)


X(31437) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, 2nd LEMOINE, MOSES}

Barycentrics    a*(a^3+(b+c)*a^2-4*S*a-(3*b^2+2*b*c+3*c^2)*a-(b^2-c^2)*(b-c)) : :

X(31437) lies on these lines: {6,1571}, {32,9616}, {39,1702}, {40,1504}, {57,31471}, {115,13893}, {165,5062}, {187,9582}, {371,9620}, {372,31422}, {486,31441}, {516,31411}, {574,18992}, {1015,31432}, {1335,31433}, {1378,31442}, {1505,9574}, {1572,6422}, {1574,31438}, {1588,31398}, {1699,31481}, {1706,31482}, {2549,13883}, {3312,31443}

X(31437) = {X(1702), X(31427)}-harmonic conjugate of X(39)


X(31438) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, 2nd LEMOINE, SPIEKER}

Barycentrics    a*((a+b+c)*S+(a-b+c)*(a+b-c)*a)*(-a+b+c) : :

X(31438) lies on these lines: {1,6}, {3,19068}, {8,31432}, {10,1588}, {40,1377}, {57,31473}, {200,2066}, {210,19038}, {371,936}, {372,31424}, {391,30412}, {486,5705}, {590,30827}, {997,9583}, {1151,5438}, {1329,13893}, {1376,9616}, {1572,31482}, {1574,31437}, {1575,31427}, {1587,12572}, {1703,12514}, {2362,12526}, {2551,13883}, {3068,3452}, {3069,5745}, {3311,5044}, {3312,31445}, {3686,7090}, {5273,7586}, {5325,19053}, {5328,8972}, {5791,7584}, {5795,19066}, {5837,19065}, {6421,31429}, {6700,9540}, {7330,19067}, {7585,18228}, {9582,25440}, {9709,31439}, {9711,31440}, {12527,31408}, {13947,26066}, {13971,30478}

X(31438) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9, 1449, 30557), (30556, 31453, 1)


X(31439) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, 2nd LEMOINE, STAMMLER}

Barycentrics    a*(2*a^3+(b+c)*a^2-4*S*a-2*(b^2+b*c+c^2)*a-(b^2-c^2)*(b-c)) : :
X(31439) = 3*X(371)-X(7969)

X(31439) lies on these lines: {1,6221}, {3,1702}, {5,13912}, {6,1571}, {30,13883}, {40,3311}, {46,19038}, {57,31474}, {165,3312}, {355,6459}, {371,517}, {381,13893}, {382,31440}, {485,22793}, {486,11231}, {516,7583}, {549,13971}, {582,605}, {590,9955}, {942,2066}, {999,31432}, {1151,1385}, {1155,3299}, {1378,31445}, {1482,9583}, {1588,26446}, {1698,13785}, {1699,8976}, {1703,6417}, {1706,31485}, {1770,19028}, {1836,13905}, {1902,10880}, {2067,9957}, {3068,12699}, {3070,28146}, {3071,9956}, {3316,9779}, {3576,6449}, {3622,9542}, {3634,18762}, {3654,19065}, {3656,13902}, {5119,18996}, {5122,6502}, {5418,11230}, {5886,9540}, {6199,12702}, {6200,7968}, {6361,7585}, {6398,19003}, {6407,9615}, {6409,17502}, {6421,31430}, {6425,24680}, {6437,11278}, {6447,7982}, {6450,16192}, {6455,7987}, {6519,9618}, {6561,13911}, {6684,7584}, {7581,9778}, {7743,9661}, {8983,22791}, {8988,22938}, {8994,12261}, {9541,18481}, {9584,30389}, {9589,31487}, {9605,31427}, {9648,15950}, {9679,30556}, {9709,31438}, {9780,23273}, {9812,13886}, {10164,13966}, {10819,11699}, {12047,13901}, {12515,19113}, {12611,13922}, {12701,13904}, {12778,19060}, {13888,31162}, {13947,18510}, {13951,31423}, {13975,19116}, {16232,24929}, {18483,18538}, {18965,30384}

X(31439) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1702, 9582, 18992), (1702, 9616, 3), (9582, 18992, 3)


X(31440) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, 2nd LEMOINE, STEINER}

Barycentrics    3*a^4-12*S*a^2+3*(b+c)*a^3-(5*b^2+6*b*c+5*c^2)*a^2-3*(b^2-c^2)*(b-c)*a+2*(b^2-c^2)^2 : :

X(31440) lies on these lines: {1,31454}, {5,1702}, {6,9588}, {20,9616}, {57,31475}, {371,5881}, {372,31425}, {382,31439}, {516,31414}, {517,31487}, {548,9582}, {631,13912}, {1335,31436}, {1378,31446}, {1505,31444}, {1572,31483}, {1588,31399}, {1706,31486}, {3068,4301}, {3312,31447}, {3576,9680}, {3592,3679}, {5731,9692}, {5734,8983}, {6421,31431}, {6447,28204}, {7765,31437}, {8960,31162}, {9575,31465}, {9607,31427}, {9615,19066}, {9711,31438}, {11362,18991}, {11522,13846}


X(31441) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, MOSES, NINE-POINTS}

Barycentrics    a^4+(b+c)*a^3-(5*b^2+2*b*c+5*c^2)*a^2-(b^2-c^2)*(b-c)*a+2*(b^2-c^2)^2 : :

X(31441) lies on these lines: {2,9620}, {4,31422}, {5,1571}, {6,11231}, {10,9619}, {11,31433}, {32,31423}, {39,1698}, {40,1506}, {57,31476}, {115,9574}, {165,5475}, {381,31443}, {486,31437}, {516,31415}, {574,5587}, {1015,31434}, {1329,31442}, {1505,13893}, {1572,3815}, {1574,5705}, {1699,7603}, {1703,31481}, {1706,31488}, {2242,31231}, {2548,6684}, {2549,10175}, {3055,5886}, {3634,3767}, {3911,31409}, {5013,9956}, {5123,31449}, {5286,19877}, {5309,19876}, {7737,10164}, {7746,9593}, {7748,7989}, {7756,18492}, {9575,9698}, {9581,31451}, {9592,19875}, {9650,15803}, {9780,31400}, {13975,31411}, {15815,18480}, {17606,31448}, {24914,31460}, {31399,31450}

X(31441) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31398, 9620), (10, 31401, 9619)


X(31442) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, MOSES, SPIEKER}

Barycentrics    a*(a^3+(b+c)*a^2-(3*b^2+2*b*c+3*c^2)*a-(b+c)^3) : :

X(31442) lies on these lines: {1,9346}, {6,31445}, {8,31433}, {9,39}, {10,1571}, {32,16517}, {37,5021}, {40,1573}, {45,5022}, {57,16589}, {63,5283}, {115,5705}, {200,31451}, {210,31448}, {516,31416}, {517,31490}, {574,936}, {846,21384}, {958,9620}, {960,9619}, {968,20963}, {997,21879}, {1015,31435}, {1107,1572}, {1329,31441}, {1376,31422}, {1378,31437}, {1505,31438}, {1574,9574}, {1699,31488}, {1703,31482}, {2241,4512}, {2548,12572}, {2551,31398}, {3452,31401}, {3691,4414}, {3767,5745}, {3916,5275}, {4652,5277}, {5013,5044}, {5234,9593}, {5248,16973}, {5250,16975}, {5254,5791}, {5273,5286}, {5325,7739}, {5698,31405}, {7765,31446}, {8580,31421}, {9589,31491}, {9709,31443}, {12527,31409}, {18228,31400}, {18250,31396}, {24703,31466}, {30827,31455}

X(31442) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9, 31429, 39), (16517, 31424, 32)


X(31443) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, MOSES, STAMMLER}

Barycentrics    a*(2*a^3+(b+c)*a^2-2*(2*b^2+b*c+2*c^2)*a-(b^2-c^2)*(b-c)) : :

X(31443) lies on these lines: {1,15815}, {3,1571}, {6,165}, {37,17122}, {39,3579}, {40,5013}, {46,31448}, {57,31477}, {115,11231}, {230,10164}, {381,31441}, {382,31444}, {516,3815}, {517,574}, {942,31451}, {999,31433}, {1100,3550}, {1155,2276}, {1386,5116}, {1506,22793}, {1572,5024}, {1574,31445}, {1575,4640}, {1699,31489}, {1706,31490}, {1770,31460}, {2108,15254}, {2242,5122}, {2549,26446}, {3053,9593}, {3055,3817}, {3097,4663}, {3312,31437}, {4095,8720}, {4421,16973}

X(31443) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (165, 9574, 6), (1571, 31422, 3)


X(31444) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, MOSES, STEINER}

Barycentrics    3*a^4+3*(b+c)*a^3-(11*b^2+6*b*c+11*c^2)*a^2-3*(b^2-c^2)*(b-c)*a+2*(b^2-c^2)^2 : :

X(31444) lies on these lines: {1,31457}, {5,1571}, {6,31447}, {20,31398}, {32,31425}, {39,9588}, {40,9698}, {57,31478}, {382,31443}, {516,31417}, {517,31492}, {574,5881}, {631,9620}, {1015,31436}, {1505,31440}, {1572,9606}, {1574,31446}, {1703,31483}, {1706,31491}, {2549,31399}, {4301,31401}, {5319,6684}, {7765,9574}, {9589,31428}, {9607,26446}, {9619,11362}, {9711,31442}

X(31444) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9588, 31431, 39), (11362, 31450, 9619)


X(31445) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, SPIEKER, STAMMLER}

Barycentrics    a*(2*a^3+(b+c)*a^2-2*(b^2+b*c+c^2)*a-(b+c)^3) : :
X(31445) = X(1)+3*X(3929) = X(1)-3*X(16418) = X(8)+3*X(11111) = X(10)-3*X(5325) = 5*X(1698)-X(9579) = 5*X(1698)-3*X(17528) = X(3295)-3*X(4512) = 5*X(3616)-9*X(17561) = 7*X(3624)-3*X(4654) = X(3927)-3*X(3929) = X(3927)+3*X(16418) = X(6850)-3*X(26446) = X(9579)-3*X(17528) = 2*X(13624)-3*X(28466)

X(31445) lies on these lines: {1,3683}, {2,3824}, {3,9}, {4,5273}, {5,5745}, {6,31442}, {7,16845}, {8,11111}, {10,30}, {21,72}, {28,1868}, {35,210}, {36,25917}, {37,58}, {40,5234}, {44,386}, {45,975}, {57,11108}, {63,405}, {65,191}, {78,16370}, {100,3697}, {140,3452}, {142,24470}, {144,3487}, {165,9709}, {226,6675}, {228,17524}, {329,6857}, {333,5295}, {345,5814}, {354,5259}, {355,6868}, {381,5705}, {382,31446}, {392,2975}, {404,27065}, {411,5927}, {452,5722}, {474,3305}, {484,3698}, {495,12527}, {499,4679}, {500,3682}, {515,18249}, {516,31419}, {517,958}, {518,5248}, {527,1125}, {549,6700}, {579,16848}, {580,1212}, {631,18228}, {672,16850}, {846,3931}, {894,11110}, {908,7483}, {912,960}, {946,5762}, {952,5837}, {956,5250}, {984,5266}, {997,13624}, {999,31435}, {1001,5045}, {1155,1698}, {1214,1935}, {1329,11231}, {1378,31439}, {1437,26885}, {1468,6051}, {1572,31490}, {1574,31443}, {1575,31430}, {1621,3555}, {1699,31493}, {1703,31485}, {1707,5711}, {1709,5584}, {1724,3666}, {1770,3925}, {1776,12711}, {1836,19854}, {1867,14016}, {1998,13615}, {2003,22136}, {2074,11363}, {2355,19822}, {2551,6850}, {2646,5692}, {2886,22793}, {3057,5258}, {3074,17102}, {3149,10157}, {3191,17194}, {3218,5047}, {3246,30148}, {3293,4689}, {3295,4512}, {3296,3616}, {3306,16842}, {3312,31438}, {3337,25542}, {3338,4423}, {3361,3646}, {3419,6872}, {3428,9856}, {3474,19855}, {3488,11106}, {3525,5328}, {3526,30827}, {3601,3940}, {3624,4654}, {3648,20292}, {3678,15481}, {3715,5217}, {3739,14377}, {3740,25440}, {3753,5260}, {3811,5220}, {3820,6684}, {3868,16865}, {3876,4189}, {3878,24680}, {3884,11260}, {3899,11011}, {3911,17527}, {3928,5708}, {3951,19526}, {3976,15485}, {4018,11684}, {4255,16885}, {4256,15492}, {4257,16814}, {4292,8728}, {4309,4863}, {4357,17698}, {4420,4533}, {4662,8715}, {4847,15171}, {4855,19535}, {4973,19862}, {4999,11230}, {5084,5744}, {5123,10225}, {5126,19861}, {5231,9669}, {5267,10176}, {5282,16601}, {5285,20831}, {5288,5919}, {5289,15178}, {5294,13728}, {5314,20833}, {5435,17559}, {5436,15934}, {5437,16853}, {5690,5795}, {5698,12699}, {5703,6172}, {5709,5806}, {5787,6987}, {5794,28160}, {5805,6846}, {5811,6988}, {5812,6824}, {5818,18231}, {5843,10165}, {5853,10386}, {5886,30478}, {5956,21892}, {6048,17601}, {6259,6908}, {6666,12436}, {6690,21077}, {6734,11113}, {6762,6767}, {6883,9940}, {6906,26878}, {6907,22792}, {6910,31018}, {6917,9956}, {6986,10167}, {7066,20122}, {7069,22361}, {7082,26357}, {7085,13730}, {7308,15803}, {7489,24474}, {7688,7701}, {7743,10527}, {8273,10085}, {9275,21873}, {9352,19877}, {9575,31468}, {9589,31494}, {9605,31429}, {9623,12702}, {9678,30557}, {9711,31447}, {9776,17552}, {9947,11500}, {9955,24703}, {10246,15829}, {10436,16844}, {10572,21677}, {10855,16410}, {10902,14872}, {10916,18527}, {11015,15677}, {11491,18908}, {11517,20835}, {11520,19538}, {12047,24953}, {12127,31393}, {12433,24391}, {12511,15726}, {12609,17768}, {12649,31156}, {12680,15931}, {13723,25083}, {13725,26065}, {13741,24627}, {15670,17781}, {15674,17484}, {16058,20805}, {16091,17095}, {16113,18406}, {16138,17668}, {16192,30393}, {16286,23169}, {16287,22345}, {16288,16574}, {16342,26223}, {16374,22344}, {16453,22060}, {16517,30435}, {16846,17754}, {16858,24473}, {17276,24159}, {17536,27003}, {17570,23958}, {17582,18230}, {18254,22935}, {19270,27064}, {19277,19859}, {19518,21371}

X(31445) = midpoint of X(i) and X(j) for these {i,j}: {1, 3927}, {3, 7330}, {355, 6868}, {958, 12514}, {3560, 26921}, {3929, 16418}
X(31445) = reflection of X(i) in X(j) for these (i,j): (6147, 1125), (6917, 9956)
X(31445) = anticomplement of X(3824)
X(31445) = X(3927)-of-anti-Aquila triangle
X(31445) = X(12161)-of-2nd Zaniah triangle
X(31445) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 3929, 3927), (3927, 16418, 1)


X(31446) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, SPIEKER, STEINER}

Barycentrics    3*a^4+3*(b+c)*a^3-(5*b^2+6*b*c+5*c^2)*a^2-3*(b+c)^3*a+2*(b^2-c^2)^2 : :

X(31446) lies on these lines: {1,24597}, {2,3951}, {4,5325}, {5,9}, {7,3634}, {8,31436}, {10,20}, {21,3679}, {40,9710}, {57,17529}, {63,1698}, {84,26446}, {191,4338}, {200,31452}, {377,19875}, {382,31445}, {442,3929}, {516,31420}, {517,31494}, {519,17558}, {631,936}, {958,5881}, {960,9624}, {1259,5251}, {1376,31425}, {1378,31440}, {1572,31491}, {1574,31444}, {1575,31431}, {1703,31486}, {2551,31399}, {3219,9612}, {3452,5067}, {3526,5044}, {3530,5438}, {3617,4304}, {3624,3868}, {3626,4313}, {3654,11530}, {3681,10122}, {3683,9670}, {3697,10391}, {3729,25446}, {3731,5292}, {3828,4208}, {3832,12572}, {3841,4312}, {3876,18389}, {3927,25525}, {3928,8728}, {4292,9780}, {4301,18249}, {4309,4512}, {4669,12536}, {5070,30827}, {5223,10198}, {5302,5587}, {5319,16517}, {5708,20195}, {5732,6684}, {6245,21153}, {6675,11523}, {6769,9623}, {6837,7991}, {6838,30326}, {6884,11522}, {6993,30315}, {7308,17575}, {7486,18228}, {7765,31442}, {9575,31469}, {9589,10883}, {9607,31429}, {9708,11248}, {9709,31447}, {9711,26066}, {9799,10164}, {9843,18230}, {9960,15064}, {9965,19877}, {11036,19862}, {11520,25055}, {12526,19854}, {12527,31410}, {12625,16418}, {16845,24391}, {24046,31183}, {26364,30393}, {31164,31254}

X(31446) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9, 5791, 5705), (10, 5273, 31424)


X(31447) = CENTROID OF CURVATURES OF THESE CIRCLES: {BEVAN, STAMMLER, STEINER}

Barycentrics    6*a^4+3*(b+c)*a^3-2*(4*b^2+3*b*c+4*c^2)*a^2-3*(b^2-c^2)*(b-c)*a+2*(b^2-c^2)^2 : :
X(31447) = 7*X(3)+3*X(3679) = 9*X(3)+X(5881) = 3*X(3)+7*X(9588) = 3*X(3)-13*X(31425) = 2*X(5)+3*X(3579) = 7*X(5)-12*X(3634) = X(5)-6*X(6684) = 13*X(5)-18*X(10172) = 4*X(5)-9*X(11231) = 17*X(5)-12*X(12571) = 11*X(5)-6*X(18483) = 8*X(5)-3*X(22793) = 7*X(3579)+8*X(3634) = X(3579)+4*X(6684) = 2*X(3579)+3*X(11231) = 11*X(3579)+4*X(18483) = 4*X(3579)+X(22793) = 2*X(3634)-7*X(6684) = 17*X(3634)-7*X(12571) = 22*X(3634)-7*X(18483) = 27*X(3679)-7*X(5881) = 13*X(6684)-3*X(10172) = 8*X(6684)-3*X(11231) = 17*X(6684)-2*X(12571) = 11*X(6684)-X(18483) = 7*X(9588)+13*X(31425)

X(31447) lies on these lines: {3,3679}, {5,516}, {6,31444}, {10,548}, {20,5818}, {30,31399}, {40,3526}, {57,31480}, {140,4301}, {165,382}, {355,3528}, {517,631}, {519,15712}, {549,24680}, {551,12108}, {632,28194}, {942,31452}, {946,16239}, {999,31436}, {1385,3244}, {1572,31492}, {1656,28198}, {1657,19875}, {1698,3843}, {1703,31487}, {1706,31494}, {3091,28202}, {3312,31440}, {3522,28208}, {3523,3654}, {3621,5657}, {3627,3828}, {3628,5493}, {3655,10299}, {3656,10303}, {3855,9778}, {3859,28178}, {3861,10175}, {4309,18527}, {4317,5122}, {4330,5445}, {4669,14891}, {4677,15706}, {4701,5690}, {5054,7991}, {5067,12699}, {5070,9589}, {5072,19876}, {5442,5919}, {5587,17800}, {5790,16192}, {5882,12100}, {6361,7486}, {7765,31443}, {7982,15720}, {9575,31470}, {9605,31431}, {9607,31430}, {9624,12702}, {9681,13973}, {9709,31446}, {9711,31445}, {10165,11278}, {11522,15694}, {12778,15057}, {13464,14869}, {15696,28160}, {17552,26062}, {17578,28154}, {17583,24987}, {18481,21734}, {19862,28212}, {28232,31253}

X(31447) = midpoint of X(40) and X(18493)
X(31447) = reflection of X(i) in X(j) for these (i,j): (18480, 5818), (18492, 9956)
X(31447) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3579, 6684, 11231), (3579, 11231, 22793), (9588, 31425, 3)


X(31448) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, HALF-MOSES, INCIRCLE}

Barycentrics    a^2*(a^2-3*b^2-2*b*c-3*c^2) : :

X(31448) lies on these lines: {1,5013}, {3,172}, {4,31460}, {5,9598}, {6,35}, {8,31449}, {12,2549}, {20,31402}, {21,17756}, {30,9596}, {32,5217}, {36,15815}, {37,474}, {39,55}, {42,5021}, {46,31443}, {56,574}, {57,31421}, {65,1571}, {183,25264}, {192,7824}, {200,31429}, {210,31442}, {213,4255}, {218,18755}, {219,5110}, {350,11285}, {372,31459}, {405,1575}, {495,9597}, {497,31400}, {498,5254}, {609,5023}, {611,3094}, {672,2271}, {942,31430}, {950,31396}, {956,20691}, {1015,3303}, {1107,5687}, {1155,31422}, {1193,14974}, {1376,5283}, {1384,7296}, {1475,2177}, {1479,3815}, {1504,19037}, {1505,19038}, {1506,9664}, {1697,9592}, {1837,31398}, {1914,9605}, {1975,27020}, {2023,10086}, {2066,6421}, {2067,9600}, {2242,5204}, {2275,3295}, {2334,9346}, {2345,19270}, {2548,6284}, {2646,9620}, {3053,5010}, {3057,9619}, {3085,7738}, {3269,19349}, {3434,31466}, {3496,17601}, {3601,9593}, {3666,21477}, {3679,31490}, {3730,4256}, {3746,16781}, {3760,15271}, {3767,5432}, {3913,16975}, {4261,16287}, {4294,7736}, {4302,7745}, {4309,9606}, {4354,9595}, {4400,22253}, {4413,16589}, {4426,16370}, {4995,7739}, {5022,20963}, {5038,10801}, {5218,5286}, {5225,31404}, {5275,25092}, {5414,6422}, {5475,12953}, {5563,9331}, {6181,6554}, {7280,16785}, {7354,31409}, {7737,15338}, {7741,31489}, {7748,10895}, {7756,9650}, {7763,26590}, {7803,26629}, {8588,9341}, {9581,31428}, {9599,15171}, {9651,11237}, {9657,31478}, {9665,9670}, {9669,31467}, {11507,13006}, {15668,25599}, {16343,17303}, {16466,17735}, {17606,31441}, {17684,17759}, {17750,19765}, {17784,31405}, {18995,31471}, {21859,22759}, {21956,26363}, {25520,26042}, {31140,31488}

X(31448) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 31461, 2276), (5013, 31477, 1)


X(31449) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, HALF-MOSES, SPIEKER}

Barycentrics    a*(a^3-3*(b^2+c^2)*a-2*(b+c)*b*c) : :

X(31449) lies on these lines: {1,5021}, {3,1107}, {4,31466}, {6,993}, {8,31448}, {9,9592}, {10,5013}, {20,31405}, {36,5275}, {37,999}, {39,958}, {45,9259}, {55,11998}, {56,5283}, {99,20172}, {372,31464}, {405,2275}, {442,9597}, {519,31477}, {529,31409}, {574,1376}, {956,2276}, {960,9619}, {988,16583}, {1001,1015}, {1003,20179}, {1146,6181}, {1329,31401}, {1475,10448}, {1500,12513}, {1571,5836}, {1572,4640}, {1575,5024}, {1682,23630}, {1706,31421}, {1914,16370}, {1975,17030}, {2170,4414}, {2242,11194}, {2271,21384}, {2549,2886}, {2551,31400}, {3053,5267}, {3125,17595}, {3295,17448}, {3436,31460}, {3576,16517}, {3767,4999}, {3814,31489}, {3880,31433}, {3913,31451}, {4426,9605}, {4853,31426}, {5022,17750}, {5030,30116}, {5123,31441}, {5204,5277}, {5248,16781}, {5254,26363}, {5286,30478}, {5795,31396}, {6376,11285}, {6381,15271}, {6421,31453}, {7738,19843}, {7748,31488}, {7763,26558}, {7786,26687}, {8666,25092}, {8716,20181}, {9574,9623}, {9575,31424}, {9598,24390}, {9599,11113}, {9607,31458}, {9664,11235}, {9711,31450}, {11108,16604}, {11236,31476}, {15482,27076}, {15815,25440}, {16589,25524}, {16973,24929}, {17684,21226}, {18967,20616}, {20691,31461}, {24215,25500}

X(31449) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 31468, 1107), (5013, 31490, 10)


X(31450) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, HALF-MOSES, STEINER}

Barycentrics    3*a^4-10*(b^2+c^2)*a^2+(b^2-c^2)^2 : :

X(31450) lies on these lines: {1,31431}, {2,7765}, {3,9300}, {5,2549}, {6,3530}, {20,574}, {32,14930}, {39,631}, {56,31462}, {115,7486}, {140,7739}, {372,31465}, {382,3815}, {384,7618}, {548,7737}, {1352,12055}, {1376,31469}, {1506,3832}, {1571,4301}, {1575,31458}, {2275,31452}, {3054,3526}, {3522,7753}, {3523,7772}, {3524,5007}, {3525,5309}, {3528,7736}, {3843,31415}, {3855,7748}, {3926,15482}, {4317,31409}, {4325,9596}, {4330,9599}, {5056,11648}, {5067,7738}, {5070,5254}, {5306,15720}, {5881,31398}, {6337,6683}, {6421,31454}, {7622,7829}, {7745,15696}, {7755,10303}, {7756,17578}, {7758,7824}, {7763,7876}, {7786,16898}, {7791,7814}, {7796,7800}, {7849,16043}, {7873,9770}, {7891,16896}, {8357,11184}, {8364,12040}, {9574,9624}, {9575,31425}, {9588,9592}, {9589,31421}, {9605,21843}, {9619,11362}, {9657,31460}, {9711,31449}, {12100,22331}, {14537,17538}, {15048,16239}, {15515,21734}, {31399,31441}, {31410,31476}, {31414,31481}, {31420,31488}

X(31450) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 31470, 9606), (5, 31492, 31401), (5013, 31401, 2549)


X(31451) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, INCIRCLE, MOSES}

Barycentrics    a^2*(a^2-2*b^2-2*b*c-2*c^2) : :

X(31451) lies on these lines: {1,574}, {3,1500}, {4,31476}, {5,9664}, {6,24047}, {8,31456}, {11,31455}, {12,7748}, {20,31409}, {30,9650}, {32,35}, {36,15515}, {37,25440}, {39,55}, {57,31422}, {100,5283}, {101,28509}, {115,498}, {172,5010}, {187,5217}, {192,1078}, {200,31442}, {350,7815}, {372,31471}, {386,17735}, {390,31400}, {405,1574}, {495,9651}, {497,31401}, {528,31466}, {942,31443}, {950,31398}, {993,20691}, {999,15815}, {1015,3295}, {1107,8715}, {1376,16589}, {1478,7756}, {1479,1506}, {1504,5414}, {1505,2066}, {1569,10053}, {1573,5687}, {1575,5248}, {1697,9619}, {1909,7781}, {1914,7772}, {2067,9674}, {2176,4256}, {2177,20963}, {2275,3746}, {2548,4294}, {2549,3085}, {3055,10593}, {3056,5034}, {3158,31429}, {3298,9600}, {3434,31488}, {3584,11648}, {3601,9620}, {3614,18424}, {3730,18755}, {3734,27020}, {3767,5218}, {3788,26590}, {3815,9665}, {3871,16975}, {3913,31449}, {3954,4414}, {4189,5291}, {4255,14974}, {4302,7747}, {4309,9599}, {4314,31396}, {4354,9636}, {4366,7786}, {4386,25092}, {4995,5309}, {5023,9341}, {5024,16781}, {5062,31459}, {5204,8589}, {5225,31415}, {5281,5286}, {5299,10987}, {5432,7746}, {5475,6284}, {6645,7782}, {7280,9331}, {7483,21956}, {7603,10896}, {7737,31402}, {7751,25264}, {7765,31452}, {7834,26629}, {7913,30104}, {8588,16785}, {9351,21008}, {9581,31441}, {9597,10056}, {9669,31489}, {10386,31406}, {15482,26959}, {16783,20331}, {17143,17684}, {17448,25439}, {17784,31416}, {21888,30147}, {26100,31020}

X(31451) = isogonal conjugate of the isotomic conjugate of X(4445)
X(31451) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 1500, 2242), (3, 31477, 1500)


X(31452) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, INCIRCLE, STEINER}

Barycentrics    3*a^4-2*(2*b^2+3*b*c+2*c^2)*a^2+(b^2-c^2)^2 : :

X(31452) lies on these lines: {1,631}, {2,3746}, {3,4317}, {4,3584}, {5,55}, {6,31462}, {8,31458}, {11,5070}, {12,382}, {20,35}, {30,9656}, {32,31478}, {36,15717}, {40,5761}, {46,13405}, {56,3530}, {57,31425}, {63,7162}, {79,9778}, {80,4313}, {100,4197}, {140,3303}, {165,13407}, {191,25568}, {200,31446}, {226,4338}, {372,31475}, {376,5270}, {377,10197}, {388,3528}, {390,7486}, {405,9711}, {442,4421}, {484,3487}, {495,548}, {496,16239}, {497,5067}, {499,3295}, {519,6910}, {529,19535}, {549,3304}, {550,11237}, {551,6921}, {612,7493}, {632,15170}, {938,5445}, {942,31447}, {943,6937}, {993,10528}, {1001,17575}, {1015,31457}, {1056,7280}, {1253,5733}, {1335,31454}, {1376,17529}, {1621,26364}, {1656,3058}, {1697,6970}, {1698,17552}, {1907,11398}, {2066,13963}, {2067,9680}, {2177,5292}, {2241,9698}, {2275,31450}, {2276,5319}, {2548,10987}, {2646,12647}, {3090,4857}, {3241,5559}, {3336,3475}, {3338,10164}, {3411,7005}, {3412,7006}, {3485,11010}, {3488,18395}, {3523,5563}, {3525,3582}, {3579,17718}, {3583,3855}, {3585,8164}, {3601,5881}, {3612,6966}, {3614,9668}, {3616,3833}, {3628,11238}, {3632,30478}, {3679,6857}, {3689,5791}, {3828,31259}, {3832,4294}, {3843,6284}, {3853,10895}, {3861,10592}, {3871,26363}, {3913,7483}, {4187,4428}, {4293,21734}, {4301,5119}, {4304,10827}, {4314,10826}, {4324,5229}, {4354,9644}, {5128,11551}, {5175,17057}, {5188,22729}, {5248,5552}, {5251,7080}, {5298,15720}, {5414,13905}, {5433,6767}, {5443,30305}, {5537,6908}, {5550,9802}, {5687,6690}, {5697,5734}, {5703,5903}, {5790,10543}, {5882,6977}, {6174,16408}, {6681,10586}, {6845,11491}, {6880,13464}, {6927,11522}, {6954,7982}, {6988,7991}, {7031,31402}, {7354,15696}, {7765,31451}, {8666,11239}, {9589,12047}, {9607,31448}, {9654,15338}, {10053,14981}, {10054,10992}, {10065,15063}, {10070,20398}, {10088,16003}, {10389,31423}, {10572,31434}, {10573,24929}, {10578,18398}, {10590,17578}, {11849,26487}, {12575,23708}, {12607,16370}, {12904,20396}, {13901,31487}, {13958,31474}, {15079,19877}, {15621,16455}, {16418,21031}, {16781,31492}, {16784,31400}, {16845,19875}, {16898,27020}, {17567,25055}, {17784,31420}, {20075,25639}, {21155,22770}, {24953,31494}

X(31452) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 15888, 4317), (3, 31480, 15888), (15888, 31480, 10056)


X(31453) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, 2nd LEMOINE, SPIEKER}

Barycentrics    a*(2*S*a+(-a+b+c)*(a^2+(b+c)*a+2*b*c)) : :

X(31453) lies on these lines: {1,6}, {2,2067}, {3,1377}, {4,31484}, {8,2066}, {10,371}, {20,31413}, {21,5414}, {32,31482}, {56,31473}, {63,2362}, {333,7090}, {372,993}, {486,26363}, {590,1329}, {615,4999}, {936,9583}, {1151,1376}, {1220,14121}, {1378,3311}, {1505,31456}, {1573,5058}, {1588,19843}, {1702,9623}, {1703,31424}, {1706,9616}, {1861,11473}, {2550,6459}, {2551,3068}, {2886,3071}, {2975,6502}, {3069,30478}, {3436,31472}, {3452,8983}, {3814,10576}, {3820,8981}, {4187,9661}, {4853,31432}, {5415,19065}, {5418,26364}, {5438,9615}, {6200,25440}, {6221,9679}, {6429,9689}, {8165,8972}, {8991,20307}, {9646,17757}, {13897,31141}, {13901,21031}, {18966,31157}

X(31453) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 31438, 30556), (9, 18991, 30557)


X(31454) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, 2nd LEMOINE, STEINER}

Barycentrics    3*a^4-6*S*a^2-4*(b^2+c^2)*a^2+(b^2-c^2)^2 : :

X(31454) lies on these lines: {1,31440}, {2,3591}, {3,9680}, {4,6425}, {5,371}, {6,631}, {20,1151}, {30,6453}, {32,31483}, {56,31475}, {140,6419}, {230,12962}, {372,3530}, {381,6447}, {382,485}, {395,2042}, {396,2041}, {486,5070}, {548,6200}, {549,6420}, {615,3311}, {1131,9543}, {1152,7585}, {1327,5073}, {1328,5072}, {1335,31452}, {1376,31486}, {1378,31458}, {1505,31457}, {1587,3528}, {1588,5067}, {1657,6519}, {1702,9624}, {1703,31425}, {1906,11473}, {1907,5412}, {1991,11292}, {2066,18965}, {2067,13901}, {2548,8375}, {3069,6431}, {3316,23259}, {3364,3411}, {3389,3412}, {3523,3594}, {3524,6426}, {3525,13847}, {3832,6437}, {3843,6561}, {3853,6564}, {3855,23261}, {3861,18538}, {4301,8983}, {4309,13904}, {4317,13905}, {5024,19105}, {5054,6427}, {5055,10195}, {5058,9698}, {5319,6422}, {5420,6417}, {5734,13902}, {5881,9583}, {6247,11241}, {6278,13882}, {6396,19117}, {6407,13665}, {6410,7581}, {6411,6460}, {6421,31450}, {6428,15720}, {6429,9541}, {6433,23267}, {6441,13941}, {6449,6560}, {6454,15712}, {6455,18512}, {6468,23249}, {6470,7582}, {6522,15700}, {7584,16239}, {7969,11362}, {8276,9714}, {8909,9936}, {8991,12964}, {8994,16003}, {8997,14981}, {8998,15063}, {9588,18991}, {9589,9616}, {9648,19030}, {9656,13897}, {9657,31472}, {9663,19028}, {9671,13898}, {9711,31453}, {10141,11001}, {10147,17538}, {10303,19053}, {10819,23236}, {11294,13637}, {11541,14241}, {12239,14531}, {12305,26516}, {15057,19111}, {15765,16963}, {16962,18585}, {17578,31412}

X(31454) = midpoint of X(6453) and X(8960)
X(31454) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (371, 590, 3071), (371, 8981, 590), (6425, 13846, 4)


X(31455) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, MOSES, NINE-POINTS}

Barycentrics    a^4-3*(b^2+c^2)*a^2+(b^2-c^2)^2 : :

X(31455) lies on these lines: {2,39}, {3,1506}, {4,7603}, {5,574}, {6,3411}, {11,31451}, {20,8589}, {30,15515}, {32,140}, {35,9665}, {36,9650}, {56,31476}, {83,7907}, {99,16921}, {115,1656}, {141,5034}, {183,7764}, {187,631}, {216,3548}, {230,632}, {325,7815}, {372,31481}, {381,7756}, {498,1015}, {499,1500}, {547,11648}, {549,5206}, {577,7542}, {590,1505}, {615,1504}, {620,7770}, {625,7791}, {626,11285}, {1007,7800}, {1078,7759}, {1329,31456}, {1376,31488}, {1570,3618}, {1571,8227}, {1572,31423}, {1573,26364}, {1574,26363}, {1698,9619}, {1699,31422}, {2241,5432}, {2242,5433}, {2549,3090}, {2896,7814}, {2937,15109}, {3035,31466}, {3053,5054}, {3054,5305}, {3071,9674}, {3096,7925}, {3199,3541}, {3329,7857}, {3523,7737}, {3524,14537}, {3525,5007}, {3530,8588}, {3533,5041}, {3549,22401}, {3589,5028}, {3624,9620}, {3628,5254}, {3785,7845}, {3818,5116}, {3832,15602}, {3843,18584}, {4045,7887}, {5023,15484}, {5024,5070}, {5058,5418}, {5062,5420}, {5067,7738}, {5277,17566}, {5306,10124}, {5319,11614}, {5471,22236}, {5472,22238}, {5569,7812}, {5972,14901}, {6292,7778}, {6390,17130}, {6421,8253}, {6422,8252}, {6639,14961}, {6656,7862}, {6680,11174}, {6722,7851}, {7230,28808}, {7288,31409}, {7506,9700}, {7516,9608}, {7608,11170}, {7622,8370}, {7741,9664}, {7750,7775}, {7752,7761}, {7755,9605}, {7760,17004}, {7767,7903}, {7771,7785}, {7773,7830}, {7774,7780}, {7776,7810}, {7782,16044}, {7783,16922}, {7787,10631}, {7793,7858}, {7794,15271}, {7804,16925}, {7805,17008}, {7807,7808}, {7809,7904}, {7811,7941}, {7816,16924}, {7819,15491}, {7825,8356}, {7826,9766}, {7831,7912}, {7833,8176}, {7838,11163}, {7839,17006}, {7843,14907}, {7853,16043}, {7864,14061}, {7867,8362}, {7876,7899}, {7889,31274}, {7890,8667}, {7895,16990}, {7909,16986}, {7913,8361}, {7935,8359}, {7951,9651}, {7988,31421}, {8366,22247}, {8960,12969}, {9167,16508}, {9300,11539}, {9603,13353}, {9955,31443}, {10018,10311}, {10303,21843}, {10314,16238}, {13935,31411}, {14064,31275}, {14869,18907}, {15694,30435}, {15699,18362}, {15717,31417}, {16975,27529}, {17683,24918}, {19102,31465}, {19862,31396}, {20107,25092}, {30827,31442}

X(31455) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 3788, 7822), (2, 7763, 3934), (3934, 7763, 7801)


X(31456) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, MOSES, SPIEKER}

Barycentrics    a*(a^3-2*(b^2+c^2)*a-2*(b+c)*b*c) : :

X(31456) lies on these lines: {1,9346}, {3,1573}, {4,31488}, {6,31468}, {8,31451}, {9,9619}, {10,574}, {20,31416}, {21,2241}, {32,993}, {37,5042}, {39,958}, {45,5053}, {56,16589}, {75,7781}, {115,26363}, {372,31482}, {405,1015}, {442,9651}, {668,17684}, {956,1500}, {1329,31455}, {1505,31453}, {1571,9623}, {1572,31424}, {1574,5013}, {1706,31422}, {1759,21332}, {2242,2975}, {2275,5251}, {2276,5258}, {2549,19843}, {2551,31401}, {2886,7748}, {3294,9351}, {3436,31476}, {3734,17030}, {3767,30478}, {3788,26558}, {4386,5206}, {4426,7772}, {4853,31433}, {4877,21769}, {4999,7746}, {5058,9678}, {5062,31464}, {5234,9592}, {5475,31466}, {5795,31398}, {6376,7815}, {6683,26687}, {7737,31405}, {7765,31458}, {7816,20172}, {7935,20541}, {9597,19854}, {9620,31429}, {9664,24390}, {9665,11113}, {9709,15815}, {9711,31457}, {10448,20963}, {11285,27076}, {15482,27091}, {15515,25440}, {16418,16781}, {17130,21264}, {21879,30144}

X(31456) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 31490, 1573), (21, 16975, 2241)


X(31457) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, MOSES, STEINER}

Barycentrics    3*a^4-7*(b^2+c^2)*a^2+(b^2-c^2)^2 : :

X(31457) lies on these lines: {1,31444}, {3,7753}, {5,574}, {6,31470}, {20,5475}, {32,3530}, {39,631}, {56,31478}, {115,5070}, {140,5309}, {187,15717}, {372,31483}, {382,1506}, {548,3815}, {549,7772}, {1015,31452}, {1376,31491}, {1505,31454}, {1571,9624}, {1572,31425}, {1574,31458}, {2242,31462}, {2548,3528}, {2549,5067}, {3522,14537}, {3523,5007}, {3526,5013}, {3628,11648}, {3788,7876}, {3832,7603}, {3843,7756}, {4325,9650}, {4330,9665}, {5024,7749}, {5041,21843}, {5054,7755}, {5058,9680}, {5062,31465}, {5206,31406}, {5254,16239}, {5306,12108}, {5569,6179}, {6337,31239}, {6683,16898}, {7622,7807}, {7736,15513}, {7737,21734}, {7739,10303}, {7747,15696}, {7759,9939}, {7761,7814}, {7763,7849}, {7769,7933}, {7796,7824}, {7801,11285}, {7822,15482}, {7835,16896}, {7888,8359}, {9300,15712}, {9588,9619}, {9589,31422}, {9620,31431}, {9657,31476}, {9711,31456}, {11742,17800}, {17578,31415}

X(31457) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 31492, 9698), (574, 31455, 7748)


X(31458) = CENTROID OF CURVATURES OF THESE CIRCLES: {CIRCUMCIRCLE, SPIEKER, STEINER}

Barycentrics    3*a^4-4*(b^2+c^2)*a^2-6*(b+c)*b*c*a+(b^2-c^2)^2 : :

X(31458) lies on these lines: {1,24597}, {2,5258}, {3,9710}, {5,958}, {6,31469}, {8,31452}, {9,9624}, {10,631}, {20,993}, {21,4309}, {32,31491}, {56,17529}, {372,31486}, {377,4325}, {382,2886}, {442,9657}, {452,24387}, {474,31157}, {499,5260}, {519,6857}, {528,17571}, {535,5177}, {548,31419}, {551,11523}, {956,10198}, {962,3647}, {1107,5319}, {1125,3475}, {1329,5070}, {1376,3530}, {1378,31454}, {1482,18253}, {1574,31457}, {1575,31450}, {1706,31425}, {2550,3528}, {2551,5067}, {2975,4197}, {3058,19526}, {3241,15674}, {3303,15670}, {3434,4330}, {3526,4999}, {3612,25006}, {3616,3881}, {3626,5218}, {3634,4315}, {3679,6910}, {3754,5744}, {3813,16418}, {3814,7486}, {3820,16239}, {3822,31410}, {3828,17567}, {3832,25639}, {3841,4293}, {3843,31493}, {3878,5273}, {3968,26062}, {4015,27383}, {4301,12514}, {4853,31436}, {4857,31156}, {5047,10072}, {5204,17583}, {5234,21616}, {5251,10527}, {5259,10529}, {5298,16862}, {5302,5886}, {5325,13464}, {5443,31018}, {5534,10165}, {5705,6962}, {5745,6892}, {5791,24299}, {5795,6970}, {6675,12513}, {6690,31480}, {6921,19875}, {6966,9588}, {6974,7991}, {7765,31456}, {8728,11194}, {9589,31424}, {9607,31449}, {9656,31245}, {9670,24390}, {9671,11113}, {10199,17559}, {10200,17575}, {10586,25542}, {11373,15254}, {12577,19862}, {15717,25440}, {15862,20050}, {16898,17030}, {17578,31418}, {24477,30143}, {25055,31259}

X(31458) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3, 31494, 9710), (956, 24953, 10198)


X(31459) = CENTROID OF CURVATURES OF THESE CIRCLES: {HALF-MOSES, INCIRCLE, 2nd LEMOINE}

Barycentrics    a^2*(b^2+b*c+c^2+S) : :

X(31459) lies on these lines: {1,6422}, {6,31}, {8,31464}, {9,8941}, {11,31463}, {35,6423}, {36,9600}, {37,493}, {39,1124}, {57,31427}, {172,1151}, {372,31448}, {486,31460}, {491,26590}, {497,31403}, {1335,1500}, {1378,5283}, {1575,31473}, {1588,31402}, {1703,31426}, {2275,3297}, {3070,9598}, {3071,9596}, {3299,6421}, {3312,31461}, {3767,9646}, {5013,6502}, {5062,31451}, {5217,12968}, {5277,9679}, {5280,6424}, {5291,9678}, {7737,9660}, {7738,31408}, {9575,31432}, {9605,31474}, {12962,18996}, {21956,31484}

X(31459) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6, 31477, 5414), (39, 31471, 1124)


X(31460) = CENTROID OF CURVATURES OF THESE CIRCLES: {HALF-MOSES, INCIRCLE, NINE-POINTS}

Barycentrics    (3*b^2+2*b*c+3*c^2)*a^2-(b^2-c^2)^2 : :

X(31460) lies on these lines: {1,3815}, {2,31402}, {3,9596}, {4,31448}, {5,2276}, {6,498}, {8,31466}, {11,1500}, {12,39}, {32,5432}, {35,7745}, {37,4187}, {55,2548}, {56,31401}, {57,31428}, {65,31398}, {83,26629}, {115,3614}, {140,172}, {192,16921}, {217,26956}, {226,31396}, {230,5280}, {325,27020}, {381,9598}, {388,31400}, {442,1575}, {486,31459}, {495,2275}, {497,31404}, {499,31489}, {574,7354}, {594,30171}, {999,31467}, {1015,9698}, {1086,24786}, {1107,17757}, {1329,5283}, {1335,31463}, {1478,5013}, {1479,31477}, {1504,19027}, {1505,19028}, {1571,1836}, {1573,21031}, {1574,3925}, {1699,31426}, {1770,31443}, {2242,5433}, {2476,17756}, {2549,10895}, {3035,5277}, {3055,16785}, {3058,9665}, {3085,7736}, {3295,9599}, {3436,31449}, {3501,17717}, {3584,5299}, {3589,30104}, {3814,25092}, {4299,15815}, {4317,31492}, {4426,7483}, {4995,7753}, {4999,5291}, {5024,9597}, {5058,13901}, {5062,13958}, {5217,7737}, {5219,9593}, {5252,9619}, {5254,7951}, {5275,26364}, {5286,10588}, {5326,7749}, {5475,6284}, {5718,17750}, {7080,31405}, {7173,7603}, {7504,17737}, {7738,10590}, {7747,15338}, {7752,26590}, {7769,26686}, {7786,26561}, {9560,10406}, {9574,9612}, {9575,31434}, {9578,9592}, {9579,31421}, {9605,31479}, {9620,11375}, {9657,31450}, {9670,31417}, {10056,16781}, {10592,15048}, {10896,31415}, {10955,11998}, {10957,21859}, {12607,16975}, {12701,31433}, {17243,30122}, {19029,31471}, {19030,31481}, {19037,31411}, {20691,24390}, {21956,25639}, {24914,31441}

X(31460) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31476, 12), (1500, 1506, 11)


X(31461) = CENTROID OF CURVATURES OF THESE CIRCLES: {HALF-MOSES, INCIRCLE, STAMMLER}

Barycentrics    a^2*(a^2-5*b^2-4*b*c-5*c^2) : :

X(31461) lies on these lines: {1,5024}, {3,172}, {6,24047}, {8,31468}, {11,31467}, {30,31402}, {35,30435}, {37,16408}, {39,3295}, {55,5299}, {57,31430}, {192,11285}, {220,4256}, {381,9598}, {382,9596}, {405,17756}, {495,7738}, {496,31400}, {497,31406}, {517,31426}, {942,9574}, {968,25068}, {988,3991}, {999,1500}, {1015,22332}, {1376,25092}, {1384,5217}, {1575,11108}, {2242,15815}, {2275,6767}, {2345,19273}, {2548,9668}, {2549,9654}, {3085,15048}, {3304,9331}, {3312,31459}, {3666,21526}, {3730,4255}, {3815,9669}, {3970,17595}, {4261,16286}, {4426,17571}, {5110,20818}, {5204,16785}, {5218,5305}, {5254,31479}, {5283,9709}, {5722,31396}, {6284,15484}, {6421,31474}, {7736,15171}, {8162,9336}, {8572,9327}, {9592,9957}, {9593,24929}, {9607,31480}, {9651,31478}, {9655,31409}, {10987,22246}, {16457,17303}, {16549,19765}, {17594,25066}, {20691,31449}, {21956,31493}

X(31461) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31477, 3295), (2276, 31448, 3)


X(31462) = CENTROID OF CURVATURES OF THESE CIRCLES: {HALF-MOSES, INCIRCLE, STEINER}

Barycentrics    (7*b^2+6*b*c+7*c^2)*a^2-(b^2-c^2)^2 : :

X(31462) lies on these lines: {1,9606}, {5,2276}, {6,31452}, {8,31469}, {12,7765}, {20,31402}, {37,17575}, {39,15888}, {56,31450}, {57,31431}, {172,3530}, {382,9596}, {497,31407}, {999,31470}, {1335,31465}, {1575,17529}, {2242,31457}, {2548,9670}, {2549,9656}, {3746,9300}, {3843,9598}, {4197,17756}, {4309,31477}, {4317,5013}, {4330,7745}, {4995,5007}, {5283,9711}, {6421,31475}, {7738,31410}, {7814,26590}, {9575,31436}, {9589,31426}, {9657,31409}, {9671,31417}

X(31462) = {X(39), X(31478)}-harmonic conjugate of X(15888)


X(31463) = CENTROID OF CURVATURES OF THESE CIRCLES: {HALF-MOSES, 2nd LEMOINE, NINE-POINTS}

Barycentrics    -(b^2-c^2)^2+2*S*a^2+3*(b^2+c^2)*a^2 : :

X(31463) lies on these lines: {2,6}, {5,6422}, {11,31459}, {30,9600}, {32,5418}, {39,485}, {53,3127}, {140,6423}, {371,2548}, {372,31401}, {486,1504}, {493,1592}, {574,6560}, {631,12968}, {1151,7745}, {1329,31464}, {1335,31460}, {1378,31466}, {1505,9698}, {1575,31484}, {1587,31400}, {1588,12962}, {1699,31427}, {1703,31428}, {2066,9599}, {2067,9596}, {2275,31472}, {2549,6564}, {3070,5013}, {3094,6813}, {3312,31467}, {3767,10576}, {5024,13665}, {5062,5420}, {5200,6748}, {5309,13711}, {5475,6561}, {6200,7737}, {6221,15484}, {6421,7583}, {6424,8981}, {6565,31415}, {7581,12969}, {7738,31412}, {7747,9674}, {7753,9675}, {7755,10195}, {7756,22644}, {8976,9605}, {9540,12963}, {9575,13893}, {13889,19448}, {15048,18538}

X(31463) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31403, 6), (6, 8253, 230), (3068, 7736, 6)


X(31464) = CENTROID OF CURVATURES OF THESE CIRCLES: {HALF-MOSES, 2nd LEMOINE, SPIEKER}

Barycentrics    a*((b+c)*b*c+S*a+(b^2+c^2)*a) : :

X(31464) lies on these lines: {1,6}, {8,31459}, {10,6422}, {32,9678}, {39,1377}, {368,27343}, {372,31449}, {486,31466}, {491,26558}, {493,6348}, {993,6423}, {1151,4386}, {1329,31463}, {1378,1504}, {1588,31405}, {1703,31429}, {1706,31427}, {2067,5275}, {2275,31473}, {2551,31403}, {3312,31468}, {5062,31456}, {5254,31484}, {7738,31413}, {9600,25440}, {9605,31485}, {9607,31486}, {9711,31465}

X(31464) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31482, 1377), (1504, 1573, 1378)


X(31465) = CENTROID OF CURVATURES OF THESE CIRCLES: {HALF-MOSES, 2nd LEMOINE, STEINER}

Barycentrics    -(b^2-c^2)^2+6*S*a^2+7*(b^2+c^2)*a^2 : :

X(31465) lies on these lines: {5,6422}, {6,631}, {20,31403}, {32,9680}, {39,31483}, {372,31450}, {485,7765}, {491,7876}, {548,9600}, {1335,31462}, {1378,31469}, {1504,9698}, {1575,31486}, {1588,31407}, {1703,31431}, {1991,16043}, {2275,31475}, {3312,31470}, {3530,6423}, {3592,9300}, {5062,31457}, {7375,13846}, {7736,12962}, {7738,31414}, {7739,8960}, {9575,31440}, {9589,31427}, {9605,31487}, {9711,31464}, {12968,15717}, {19102,31455}


X(31466) = CENTROID OF CURVATURES OF THESE CIRCLES: {HALF-MOSES, NINE-POINTS, SPIEKER}

Barycentrics    3*(b^2+c^2)*a^2+2*(b+c)*b*c*a-(b^2-c^2)^2 : :

X(31466) lies on these lines: {2,31405}, {4,31449}, {5,1107}, {6,26363}, {8,31460}, {10,3815}, {11,5283}, {12,16975}, {32,4999}, {37,496}, {39,2886}, {140,4386}, {325,17030}, {381,31468}, {405,9599}, {442,2275}, {486,31464}, {495,17448}, {499,5275}, {528,31451}, {529,9650}, {956,9596}, {958,2548}, {993,7745}, {1015,25466}, {1329,1506}, {1376,31401}, {1378,31463}, {1500,3813}, {1572,26066}, {1574,9698}, {1575,31406}, {1699,31429}, {1706,31428}, {1914,7483}, {2241,6690}, {2276,24390}, {2300,5742}, {2550,31400}, {2551,31404}, {3035,31455}, {3434,31448}, {3816,16589}, {3933,21264}, {4884,22036}, {5021,26098}, {5254,25639}, {5277,5433}, {5475,31456}, {5626,23112}, {5705,9575}, {5718,20963}, {5794,9619}, {5836,31398}, {6421,31484}, {7736,19843}, {7738,31418}, {7752,26558}, {7763,20172}, {7786,26582}, {7807,20179}, {8227,16517}, {8362,20541}, {8728,16604}, {9597,17532}, {9605,31493}, {9709,31467}, {9711,31491}, {10198,16781}, {11374,16973}, {12513,31409}, {12607,31476}, {16583,24239}, {17045,25598}, {17717,21384}, {21921,28096}, {24387,25092}, {24703,31442}, {26364,31489}

X(31466) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31488, 2886), (1506, 1573, 1329)


X(31467) = CENTROID OF CURVATURES OF THESE CIRCLES: {HALF-MOSES, NINE-POINTS, STAMMLER}

Barycentrics    a^4-7*(b^2+c^2)*a^2+2*(b^2-c^2)^2 : :

X(31467) lies on these lines: {2,3933}, {3,2548}, {5,5024}, {6,3411}, {11,31461}, {30,31404}, {32,5054}, {39,1656}, {83,11288}, {115,5079}, {140,7736}, {183,7905}, {381,1506}, {382,574}, {517,31428}, {590,13934}, {615,13882}, {626,11184}, {631,1384}, {632,7735}, {999,31460}, {1007,8362}, {1078,11163}, {1329,31468}, {1482,31398}, {1575,31493}, {1657,5475}, {1699,31430}, {1975,11165}, {2275,31479}, {2549,3851}, {2896,7776}, {3053,15720}, {3054,5319}, {3055,3767}, {3090,15048}, {3312,31463}, {3523,18907}, {3530,15655}, {3533,5304}, {3618,10008}, {3628,5286}, {3763,7888}, {3788,6704}, {3843,31415}, {5023,7753}, {5034,11898}, {5041,15723}, {5055,5254}, {5072,7603}, {5076,7756}, {5094,15302}, {5790,9619}, {5886,31396}, {6395,31411}, {6421,8976}, {6422,13951}, {6683,7778}, {7542,15905}, {7608,11257}, {7610,7805}, {7739,15703}, {7743,31426}, {7747,15696}, {7752,11287}, {7764,15271}, {7769,7846}, {7770,7891}, {7773,7910}, {7782,11159}, {7784,15482}, {7786,7866}, {7795,15491}, {7815,7882}, {7824,7900}, {7855,8556}, {7868,31268}, {7887,7923}, {9300,15694}, {9574,9955}, {9575,11231}, {9592,9956}, {9593,11230}, {9669,31448}, {9709,31466}, {10542,25555}, {13966,31403}, {14001,14535}, {14093,14537}, {15325,31402}, {15513,15700}, {17800,31417}, {18512,31481}, {22793,31421}

X(31467) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31406, 9605), (3815, 31401, 3)


X(31468) = CENTROID OF CURVATURES OF THESE CIRCLES: {HALF-MOSES, SPIEKER, STAMMLER}

Barycentrics    a*(a^3-5*(b^2+c^2)*a-4*(b+c)*b*c) : :

X(31468) lies on these lines: {1,4520}, {3,1107}, {6,31456}, {8,31461}, {10,5024}, {30,31405}, {37,7373}, {39,9708}, {381,31466}, {382,31469}, {517,31429}, {958,9605}, {993,30435}, {999,5283}, {1329,31467}, {1385,16517}, {1573,5013}, {1574,22332}, {1706,31430}, {1914,17571}, {2275,11108}, {2551,31406}, {3312,31464}, {3820,31400}, {5022,30116}, {5044,9592}, {5254,31493}, {5305,30478}, {6421,31485}, {6767,17448}, {7738,31419}, {7781,20181}, {9575,31445}, {9597,17528}, {9607,31494}, {9711,31470}, {12513,25092}, {15048,19843}, {16345,23632}, {16604,16853}

X(31468) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31490, 9708), (1107, 31449, 3)


X(31469) = CENTROID OF CURVATURES OF THESE CIRCLES: {HALF-MOSES, SPIEKER, STEINER}

Barycentrics    7*(b^2+c^2)*a^2+6*(b+c)*b*c*a-(b^2-c^2)^2 : :

X(31469) lies on these lines: {5,1107}, {6,31458}, {8,31462}, {10,9606}, {20,31405}, {39,9710}, {382,31468}, {1376,31450}, {1378,31465}, {1573,9698}, {1706,31431}, {2275,17529}, {2551,31407}, {2886,7765}, {3530,4386}, {6421,31486}, {7738,31420}, {7814,26558}, {9575,31446}, {9589,31429}, {9605,31494}, {9624,16517}, {9709,31470}, {15888,16975}

X(31469) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (39, 31491, 9710), (1573, 9698, 9711)


X(31470) = CENTROID OF CURVATURES OF THESE CIRCLES: {HALF-MOSES, STAMMLER, STEINER}

Barycentrics    3*a^4-17*(b^2+c^2)*a^2+2*(b^2-c^2)^2 : :

X(31470) lies on these lines: {3,9300}, {5,5024}, {6,31457}, {20,15484}, {30,31407}, {39,3526}, {382,5013}, {517,31431}, {548,7736}, {631,5304}, {999,31462}, {1384,15717}, {1575,31494}, {1656,7765}, {2275,31480}, {2548,17800}, {3312,31465}, {3530,30435}, {3815,3843}, {3861,31404}, {5007,15693}, {5067,15048}, {5070,9607}, {5286,16239}, {6421,31487}, {7772,15720}, {7814,11287}, {9575,31447}, {9589,31430}, {9709,31469}, {9711,31468}, {10335,16987}, {12040,14069}, {15700,22331}, {18907,21734}

X(31470) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5024, 31400, 31467), (9606, 31450, 3)


X(31471) = CENTROID OF CURVATURES OF THESE CIRCLES: {INCIRCLE, 2nd LEMOINE, MOSES}

Barycentrics    a^2*(2*S+(b+c)^2) : :

X(31471) lies on these lines: {1,1504}, {6,595}, {8,31482}, {11,31481}, {32,2066}, {36,9674}, {39,1124}, {55,5062}, {57,31437}, {115,31472}, {172,9675}, {371,2242}, {372,31451}, {486,31476}, {497,31411}, {574,6502}, {1015,3297}, {1378,16589}, {1505,2276}, {1572,31432}, {1574,31473}, {1588,31409}, {1703,31433}, {2549,31408}, {3070,9664}, {3071,9650}, {3312,31477}, {5058,19038}, {7746,9646}, {18995,31448}, {19029,31460}

X(31471) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1124, 31459, 39), (3297, 6422, 1015)


X(31472) = CENTROID OF CURVATURES OF THESE CIRCLES: {INCIRCLE, 2nd LEMOINE, NINE-POINTS}

Barycentrics    2*S*a^2+(b+c)^2*(a-b+c)*(a+b-c) : :

X(31472) lies on these lines: {1,485}, {2,6502}, {3,9646}, {4,2066}, {5,1124}, {6,12}, {8,31484}, {10,2362}, {11,3297}, {35,6560}, {36,5418}, {55,3070}, {56,590}, {57,13893}, {65,13911}, {92,1585}, {115,31471}, {226,13883}, {371,1478}, {372,498}, {381,31474}, {382,9660}, {388,2067}, {390,1131}, {442,1378}, {486,3299}, {491,1909}, {495,1335}, {496,18538}, {497,31412}, {499,10576}, {615,18995}, {908,30556}, {999,8976}, {1015,31481}, {1056,13886}, {1151,7354}, {1152,5432}, {1329,31473}, {1377,17757}, {1479,6564}, {1505,31476}, {1587,3085}, {1588,10590}, {1699,31432}, {1702,9612}, {1703,31434}, {1773,6204}, {2275,31463}, {3069,10588}, {3071,10895}, {3103,10063}, {3295,13665}, {3298,15888}, {3304,13898}, {3311,9654}, {3312,31479}, {3436,31453}, {3476,13902}, {3485,19066}, {3585,6561}, {3594,13958}, {3600,8972}, {3614,19029}, {4292,13912}, {4293,9540}, {4294,23249}, {4299,6200}, {4325,9680}, {5058,9650}, {5083,8988}, {5218,6460}, {5229,6459}, {5252,7969}, {5254,31459}, {5261,7585}, {5412,11392}, {5433,8253}, {5434,13846}, {5726,19004}, {6221,9647}, {6284,23251}, {6348,13389}, {6409,15326}, {6420,13963}, {6429,9649}, {7080,31413}, {7581,8164}, {7584,10592}, {7968,11375}, {8252,18966}, {8831,26040}, {8909,18970}, {8960,13904}, {8981,18990}, {8983,10106}, {9578,18991}, {9579,9616}, {9583,9613}, {9657,31454}, {9679,11112}, {10055,10665}, {10577,13962}, {10819,18968}, {11237,18996}, {13882,18988}, {18513,22615}, {19048,26482}, {19050,26481}, {24987,30557}

X(31472) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31408, 6502), (12, 19028, 6)


X(31473) = CENTROID OF CURVATURES OF THESE CIRCLES: {INCIRCLE, 2nd LEMOINE, SPIEKER}

Barycentrics    a*((a+b+c)*b*c+S*a) : :

X(31473) lies on these lines: {1,1377}, {2,6}, {8,3297}, {9,13389}, {10,1124}, {11,31484}, {21,1152}, {36,9678}, {37,494}, {56,31453}, {57,31438}, {65,6203}, {220,30412}, {371,474}, {372,405}, {377,3071}, {404,1151}, {406,3093}, {442,486}, {443,1588}, {452,6460}, {475,3092}, {485,4187}, {497,31413}, {517,8231}, {572,16433}, {573,16432}, {958,6502}, {960,2362}, {999,31485}, {1001,5414}, {1015,31482}, {1030,1600}, {1100,3084}, {1125,1335}, {1172,3536}, {1329,31472}, {1376,2066}, {1378,1698}, {1505,16589}, {1574,31471}, {1575,31459}, {1587,5084}, {1599,5124}, {1703,31435}, {1706,31432}, {2067,25524}, {2256,6351}, {2264,7348}, {2275,31464}, {2475,23261}, {2478,3070}, {2551,31408}, {3232,31221}, {3298,3616}, {3301,3624}, {3311,16408}, {3312,11108}, {3592,17531}, {3594,5047}, {3812,16232}, {3925,19029}, {4188,6409}, {4189,6410}, {4413,19038}, {4423,19037}, {5046,23251}, {5069,8962}, {5277,6424}, {5283,6421}, {5416,13940}, {5418,13747}, {5420,7483}, {6200,16371}, {6221,16417}, {6347,17275}, {6348,17303}, {6395,16857}, {6396,16370}, {6398,16418}, {6411,13587}, {6412,17549}, {6417,16863}, {6418,16853}, {6419,16862}, {6420,16842}, {6425,17572}, {6426,16865}, {6428,16855}, {6431,17535}, {6432,17536}, {6438,16858}, {6449,17573}, {6450,17571}, {6454,19526}, {6459,6904}, {6471,17570}, {6485,19539}, {6560,11113}, {6561,11112}, {6564,17556}, {6565,17532}, {6675,13966}, {6857,13935}, {6919,31412}, {7581,17559}, {7582,17582}, {7583,17527}, {7584,8728}, {7968,19860}, {7969,19861}, {8582,13883}, {8583,18991}, {9540,17567}, {9646,26364}, {9661,10200}, {9681,17583}, {9711,31475}, {11513,25947}, {11514,25907}, {13785,17528}, {13887,25893}, {13893,26459}, {13897,31246}, {13911,19047}, {13947,26464}, {13955,31245}, {13962,19854}, {13973,19050}, {18966,24953}, {19023,25973}, {19049,24541}, {25917,30557}

X(31473) = {X(1698), X(3299)}-harmonic conjugate of X(1378)


X(31474) = CENTROID OF CURVATURES OF THESE CIRCLES: {INCIRCLE, 2nd LEMOINE, STAMMLER}

Barycentrics    a^2*(-a^2+b^2+4*b*c+c^2+4*S) : :

X(31474) lies on these lines: {1,3311}, {3,1124}, {6,595}, {8,31485}, {11,8976}, {12,13785}, {30,31408}, {35,6398}, {36,6449}, {55,3299}, {56,6221}, {57,31439}, {371,999}, {381,31472}, {382,31475}, {390,7581}, {485,9669}, {486,31479}, {495,1588}, {496,3068}, {497,7583}, {498,13951}, {499,13901}, {517,31432}, {942,1702}, {1058,7585}, {1335,6417}, {1378,11108}, {1387,13902}, {1479,13665}, {1504,16781}, {1505,31477}, {1587,15171}, {1656,9646}, {1657,9660}, {2067,6199}, {2362,12702}, {3070,9668}, {3071,9654}, {3085,7584}, {3086,8981}, {3298,6419}, {3301,3303}, {3584,13954}, {3746,6428}, {3940,30556}, {5010,6456}, {5122,9582}, {5126,9615}, {5204,6455}, {5217,6450}, {5218,13966}, {5261,23273}, {5274,13886}, {5298,9648}, {5410,6198}, {5414,6418}, {5432,13962}, {5563,6447}, {5722,13883}, {6421,31461}, {6451,7280}, {6459,18990}, {6561,9655}, {7741,13897}, {8983,11373}, {9540,15325}, {9583,24928}, {9605,31459}, {9661,13903}, {9679,16417}, {9709,31473}, {9957,18991}, {10056,19027}, {10072,18965}, {10588,18762}, {10591,18538}, {12735,19082}, {13904,31487}, {13958,31452}, {15170,19054}, {15172,19117}, {15934,16232}, {18992,24929}, {18999,26464}, {19000,26459}, {19004,31393}

X(31474) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 19038, 3311), (1124, 2066, 3)


X(31475) = CENTROID OF CURVATURES OF THESE CIRCLES: {INCIRCLE, 2nd LEMOINE, STEINER}

Barycentrics    -(b^2-c^2)^2+6*S*a^2+(b^2+6*b*c+c^2)*a^2 : :

X(31475) lies on these lines: {5,1124}, {6,15888}, {8,31486}, {20,2066}, {36,9680}, {56,31454}, {57,31440}, {371,4317}, {372,31452}, {382,31474}, {497,31414}, {631,6502}, {999,31487}, {1378,17529}, {1505,31478}, {1588,31410}, {1703,31436}, {2275,31465}, {2362,11362}, {3070,9670}, {3071,9656}, {3297,19028}, {3312,31480}, {3526,9646}, {3592,5434}, {4325,9681}, {4330,6560}, {4995,6426}, {6420,10056}, {6421,31462}, {8960,10072}, {9589,31432}, {9657,19038}, {9660,17800}, {9679,17583}, {9711,31473}, {16785,19105}


X(31476) = CENTROID OF CURVATURES OF THESE CIRCLES: {INCIRCLE, MOSES, NINE-POINTS}

Barycentrics    2*(b^2+b*c+c^2)*a^2-(b^2-c^2)^2 : :

X(31476) lies on these lines: {1,1506}, {2,2242}, {3,9650}, {4,31451}, {5,1500}, {6,17734}, {8,31488}, {11,7603}, {12,39}, {32,498}, {35,7747}, {37,3814}, {55,5475}, {56,31455}, {57,31441}, {115,2276}, {172,7749}, {187,5432}, {226,31398}, {381,9664}, {388,31401}, {442,1574}, {486,31471}, {495,1015}, {497,31415}, {574,1478}, {625,26590}, {626,27020}, {999,31489}, {1213,10469}, {1329,16589}, {1335,31481}, {1505,31472}, {1571,9612}, {1572,31434}, {1573,17757}, {1575,3822}, {1699,31433}, {1909,7764}, {1914,3584}, {2241,2548}, {2275,9698}, {2549,10590}, {3295,9665}, {3436,31456}, {3585,7756}, {3761,7813}, {3767,10588}, {3947,31396}, {4299,15515}, {4400,7890}, {4995,14537}, {5010,6781}, {5013,9651}, {5034,12588}, {5058,9646}, {5218,7737}, {5219,9620}, {5254,10592}, {5261,31400}, {5277,27529}, {5280,7755}, {5283,11681}, {5290,31428}, {5472,7127}, {5726,9592}, {6645,7769}, {6683,26561}, {7080,31416}, {7736,8164}, {7748,10895}, {7804,26629}, {7889,30104}, {8589,15326}, {9578,9619}, {9579,31422}, {9599,10056}, {9655,15815}, {9657,31457}, {11236,31449}, {12607,31466}, {17530,21956}, {20691,25639}, {21021,30171}, {31410,31450}

X(31476) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31409, 2242), (12, 31460, 39)


X(31477) = CENTROID OF CURVATURES OF THESE CIRCLES: {INCIRCLE, MOSES, STAMMLER}

Barycentrics    a^2*(a^2-3*b^2-4*b*c-3*c^2) : :

X(31477) lies on these lines: {1,5013}, {2,20181}, {3,1500}, {6,31}, {8,31490}, {11,31489}, {12,9598}, {30,31409}, {35,609}, {36,9331}, {37,1376}, {39,3295}, {45,3693}, {56,15815}, {57,31443}, {100,5275}, {115,31479}, {172,5023}, {183,192}, {220,18755}, {230,5218}, {232,7071}, {345,594}, {346,26244}, {350,15271}, {381,9664}, {382,9650}, {386,14974}, {390,7736}, {495,2549}, {496,31401}, {497,3815}, {498,13881}, {517,31433}, {519,31449}, {574,999}, {942,1571}, {958,20691}, {967,1796}, {1001,1575}, {1015,5024}, {1058,31400}, {1100,3749}, {1107,3913}, {1479,31460}, {1506,9669}, {1574,11108}, {1621,17756}, {2176,4255}, {2223,16523}, {2241,9605}, {2256,5110}, {2271,3730}, {2275,3303}, {2286,21794}, {2295,19765}, {2548,15171}, {2933,3207}, {3055,10589}, {3058,9599}, {3085,5254}, {3158,16517}, {3247,17122}, {3306,3666}, {3312,31471}, {3333,31421}, {3509,17601}, {3726,17595}, {3744,16884}, {3750,17754}, {3930,4414}, {4294,7745}, {4309,31462}, {4366,11174}, {4386,4421}, {4646,16968}, {4704,16999}, {5010,5210}, {5021,24047}, {5045,31430}, {5220,20693}, {5281,7735}, {5283,5687}, {5291,16370}, {5475,9668}, {6284,9596}, {6765,31429}, {7748,9654}, {7756,9655}, {7765,31480}, {7778,26590}, {8715,25092}, {9300,10385}, {9592,31393}, {9597,15888}, {9619,9957}, {9620,24929}, {9709,16589}, {14829,17314}, {15172,31406}, {16992,17759}, {17118,24326}, {17450,17599}

X(31477) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1, 31448, 5013), (1500, 31451, 3)


X(31478) = CENTROID OF CURVATURES OF THESE CIRCLES: {INCIRCLE, MOSES, STEINER}

Barycentrics    2*(2*b^2+3*b*c+2*c^2)*a^2-(b^2-c^2)^2 : :

X(31478) lies on these lines: {1,9698}, {5,1500}, {6,31480}, {8,31491}, {20,31409}, {32,31452}, {39,15888}, {57,31444}, {382,9650}, {495,9607}, {497,31417}, {574,4317}, {631,2242}, {999,31492}, {1015,9606}, {1335,31483}, {1505,31475}, {1572,31436}, {1574,17529}, {2241,31402}, {2276,7765}, {2549,31410}, {3085,5319}, {3584,7755}, {3746,7753}, {3843,9664}, {4309,9596}, {4330,7747}, {5475,9670}, {7748,9656}, {7749,16785}, {7772,10056}, {9589,31433}, {9651,31461}, {9657,31448}, {9711,16589}

X(31478) = {X(15888), X(31462)}-harmonic conjugate of X(39)


X(31479) = CENTROID OF CURVATURES OF THESE CIRCLES: {INCIRCLE, NINE-POINTS, STAMMLER}

Barycentrics    a^4-(3*b^2+4*b*c+3*c^2)*a^2+2*(b^2-c^2)^2 : :

X(31479) lies on these lines: {1,1656}, {2,495}, {3,12}, {4,5281}, {5,497}, {6,17734}, {8,31493}, {10,3940}, {11,5055}, {30,5218}, {35,382}, {36,5054}, {55,381}, {56,3526}, {57,11231}, {100,17532}, {115,31477}, {119,6913}, {140,388}, {145,7504}, {202,16645}, {203,16644}, {210,942}, {226,26446}, {230,31409}, {355,13411}, {390,3545}, {392,30852}, {403,7071}, {405,11681}, {442,1260}, {474,27529}, {486,31474}, {496,3090}, {517,5219}, {546,4294}, {547,10589}, {549,4293}, {550,5229}, {631,5261}, {632,7288}, {954,6829}, {993,11236}, {1001,3814}, {1015,31489}, {1058,5056}, {1060,5268}, {1068,7140}, {1124,13951}, {1155,18541}, {1329,10198}, {1335,8976}, {1376,3822}, {1385,9578}, {1479,3614}, {1482,10039}, {1500,13881}, {1506,16781}, {1621,17556}, {1657,3585}, {1697,9955}, {1737,15934}, {1788,6147}, {1870,5094}, {1914,15484}, {2066,13785}, {2098,5443}, {2275,31467}, {2330,18440}, {2476,5687}, {2551,6675}, {2646,10827}, {3035,16417}, {3053,9650}, {3057,18493}, {3058,19709}, {3086,3628}, {3091,15171}, {3167,10055}, {3297,10577}, {3298,10576}, {3299,13954}, {3301,13897}, {3303,5079}, {3311,9646}, {3312,31472}, {3434,17530}, {3436,7483}, {3485,5690}, {3486,18357}, {3487,9780}, {3488,12019}, {3517,11392}, {3525,3600}, {3530,31410}, {3533,5265}, {3534,5010}, {3576,5726}, {3579,9612}, {3601,18480}, {3624,24928}, {3632,31262}, {3634,21620}, {3679,31245}, {3746,5072}, {3748,18530}, {3753,31266}, {3830,4302}, {3843,6284}, {3850,5225}, {3871,5141}, {3913,25639}, {3920,7539}, {3927,21077}, {3947,6684}, {4245,19721}, {4316,15688}, {4417,5774}, {4870,25415}, {5050,12588}, {5066,10385}, {5067,14986}, {5071,5274}, {5073,15338}, {5080,16370}, {5119,17605}, {5204,5270}, {5221,5445}, {5226,5657}, {5251,31141}, {5254,31461}, {5290,31423}, {5305,31402}, {5326,5434}, {5414,13665}, {5587,24929}, {5703,5818}, {5708,13407}, {5719,18391}, {5722,10175}, {5789,14872}, {5791,21075}, {5886,31397}, {5919,23708}, {6199,13901}, {6244,6907}, {6395,13958}, {6417,13905}, {6418,13963}, {6455,9647}, {6668,12607}, {6690,16418}, {6738,31399}, {6796,10894}, {6831,10786}, {6842,10306}, {6856,7080}, {6861,10321}, {6862,10942}, {6863,22770}, {6931,10587}, {6933,10528}, {6969,7956}, {6971,16202}, {6980,10679}, {7082,17699}, {7280,9657}, {7489,8069}, {7506,10831}, {7545,9673}, {7571,29815}, {7680,19541}, {7743,7988}, {7866,27020}, {8068,12331}, {8165,16845}, {8227,9957}, {8540,14848}, {8666,20104}, {9596,30435}, {9605,31460}, {9613,13624}, {9651,15815}, {9656,10483}, {9658,13564}, {10072,15703}, {10172,11019}, {10247,12647}, {10320,10954}, {10387,19130}, {10389,18527}, {10742,28444}, {10980,19876}, {11235,25439}, {11286,26629}, {11318,26590}, {11499,15865}, {11529,19875}, {12047,12702}, {12373,15041}, {12577,31253}, {12735,31272}, {13384,28204}, {13465,14526}, {13743,18542}, {13903,18996}, {13961,18995}, {13966,31408}, {15040,18968}, {16137,31254}, {16203,26482}, {16408,25466}, {18510,19038}, {18512,19037}, {18545,22768}, {19322,26231}, {19349,26944}, {19854,21031}, {21696,27577}, {25055,25405}, {28160,30282}, {30116,30858}

X(31479) = midpoint of X(i) and X(j) for these {i,j}: {5218, 10590}, {5219, 31434}
X(31479) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 495, 999), (2, 8164, 495), (2, 17757, 9708)


X(31480) = CENTROID OF CURVATURES OF THESE CIRCLES: {INCIRCLE, STAMMLER, STEINER}

Barycentrics    3*a^4-(5*b^2+12*b*c+5*c^2)*a^2+2*(b^2-c^2)^2 : :

X(31480) lies on these lines: {1,3526}, {3,4317}, {5,497}, {6,31478}, {8,31494}, {12,3843}, {20,495}, {30,31410}, {35,9657}, {55,382}, {57,31447}, {381,3746}, {388,548}, {390,3855}, {496,5067}, {498,5070}, {517,31436}, {546,10385}, {631,999}, {942,9588}, {954,6937}, {1015,31492}, {1056,15717}, {1058,7486}, {1335,31487}, {1656,3303}, {1657,11237}, {2275,31470}, {2894,4197}, {3058,3851}, {3086,16239}, {3090,15170}, {3304,5054}, {3312,31475}, {3528,5281}, {3530,5218}, {3534,5270}, {3624,3893}, {3832,8164}, {3853,4294}, {3856,5225}, {3861,10386}, {3913,10197}, {4301,11374}, {4325,5217}, {4857,5072}, {5049,31423}, {5079,11238}, {5432,7373}, {5552,17575}, {5563,15720}, {5722,31399}, {5881,24929}, {6690,31458}, {7080,17552}, {7483,11239}, {7765,31477}, {7951,9671}, {8715,17528}, {9605,31462}, {9607,31461}, {9624,9957}, {9698,16781}, {9708,10528}, {9709,17529}, {9710,10198}, {9711,11108}, {9956,10389}, {11362,13405}, {12607,16418}, {12702,17718}, {17606,18530}

X(31480) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (10056, 31452, 15888), (15888, 31452, 3)


X(31481) = CENTROID OF CURVATURES OF THESE CIRCLES: {2nd LEMOINE, MOSES, NINE-POINTS}

Barycentrics    -(b^2-c^2)^2+2*S*a^2+2*(b^2+c^2)*a^2 : :

X(31481) lies on these lines: {2,5062}, {5,1504}, {6,17}, {11,31471}, {30,9674}, {32,590}, {39,485}, {115,6422}, {187,5418}, {371,5475}, {372,31455}, {486,7603}, {491,3934}, {574,3070}, {615,6118}, {1015,31472}, {1151,7747}, {1329,31482}, {1335,31476}, {1378,31488}, {1505,3815}, {1572,13893}, {1574,31484}, {1587,31401}, {1588,31415}, {1699,31437}, {1703,31441}, {2066,9665}, {2067,9650}, {2241,9646}, {2242,9661}, {2548,3068}, {2549,31412}, {3055,13966}, {3312,31489}, {3316,7735}, {3767,31403}, {5013,13665}, {5254,18538}, {6409,6781}, {6421,9698}, {6423,7749}, {6424,7753}, {6564,7748}, {6565,12962}, {7736,13886}, {7737,9540}, {7745,8981}, {7756,9600}, {9596,13904}, {9599,13905}, {13903,15484}, {18512,31467}, {19030,31460}, {31414,31450}

X(31481) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 31411, 5062), (6, 10576, 7746)


X(31482) = CENTROID OF CURVATURES OF THESE CIRCLES: {2nd LEMOINE, MOSES, SPIEKER}

Barycentrics    a*(2*(b+c)*b*c+2*S*a+(b^2+c^2)*a) : :

X(31482) lies on these lines: {6,1573}, {8,31471}, {10,1504}, {32,31453}, {39,1377}, {115,31484}, {187,9678}, {372,31456}, {486,31488}, {958,5062}, {1015,31473}, {1107,1505}, {1329,31481}, {1335,16589}, {1572,31438}, {1574,6422}, {1588,31416}, {1703,31442}, {1706,31437}, {2549,31413}, {2551,31411}, {3312,31490}, {4386,9675}, {7765,31486}, {9674,25440}, {9711,31483}

X(31482) = {X(1377), X(31464)}-harmonic conjugate of X(39)


X(31483) = CENTROID OF CURVATURES OF THESE CIRCLES: {2nd LEMOINE, MOSES, STEINER}

Barycentrics    -(b^2-c^2)^2+6*S*a^2+4*(b^2+c^2)*a^2 : :

X(31483) lies on these lines: {5,1504}, {6,3411}, {20,31411}, {32,31454}, {39,31465}, {187,9680}, {372,31457}, {491,7849}, {548,9674}, {631,5062}, {1335,31478}, {1378,31491}, {1505,9606}, {1572,31440}, {1574,31486}, {1588,31417}, {1703,31444}, {1991,7854}, {2549,31414}, {3068,5319}, {3312,31492}, {3592,7753}, {5058,31403}, {5309,8960}, {5475,12962}, {6422,7765}, {7583,9607}, {7755,13846}, {7817,13637}, {9589,31437}, {9711,31482}


X(31484) = CENTROID OF CURVATURES OF THESE CIRCLES: {2nd LEMOINE, NINE-POINTS, SPIEKER}

Barycentrics    2*S*a^2+(a+b+c)*((b^2+c^2)*a-(b+c)*(b-c)^2) : :

X(31484) lies on these lines: {2,5414}, {4,31453}, {5,1377}, {6,2886}, {8,31472}, {10,485}, {11,31473}, {30,9678}, {75,491}, {115,31482}, {142,5393}, {372,26363}, {377,2067}, {381,31485}, {442,1335}, {474,9661}, {486,25639}, {590,1376}, {946,30556}, {958,3070}, {993,6560}, {1124,24390}, {1152,4999}, {1378,7583}, {1505,31488}, {1574,31481}, {1575,31463}, {1587,19843}, {1588,31418}, {1699,31438}, {1703,5705}, {1706,13893}, {2066,3434}, {2362,6734}, {2550,3068}, {2551,31412}, {3035,8253}, {3297,3813}, {3298,25466}, {3312,31493}, {3820,18538}, {3925,19030}, {5254,31464}, {5418,25440}, {5687,9646}, {5709,6213}, {6502,10527}, {8981,9679}, {10576,26364}, {19038,31140}, {21956,31459}

X(31484) = {X(7583), X(31419)}-harmonic conjugate of X(1378)


X(31485) = CENTROID OF CURVATURES OF THESE CIRCLES: {2nd LEMOINE, SPIEKER, STAMMLER}

Barycentrics    a*(a^3-4*(b+c)*b*c-4*S*a-(b^2+c^2)*a) : :

X(31485) lies on these lines: {3,1377}, {6,1573}, {8,31474}, {10,3311}, {30,31413}, {371,9709}, {381,31484}, {382,31486}, {486,31493}, {517,31438}, {958,3312}, {993,6398}, {999,31473}, {1329,8976}, {1335,11108}, {1376,6221}, {1378,6417}, {1482,30556}, {1505,31490}, {1588,31419}, {1698,18996}, {1703,31445}, {1706,31439}, {2067,16408}, {2362,3927}, {2551,7583}, {2886,13785}, {3068,3820}, {3679,19038}, {5044,18991}, {5251,19037}, {5258,18995}, {5267,6456}, {5414,16418}, {5791,13936}, {6421,31468}, {6449,25440}, {7584,19843}, {8165,13886}, {9605,31464}, {9711,31487}, {13905,21031}, {13951,26363}, {13963,24953}, {13966,30478}, {19027,19854}

X(31485) = {X(1377), X(31453)}-harmonic conjugate of X(3)


X(31486) = CENTROID OF CURVATURES OF THESE CIRCLES: {2nd LEMOINE, SPIEKER, STEINER}

Barycentrics    6*S*a^2+6*(b+c)*b*c*a+(b^2+c^2)*a^2-(b^2-c^2)^2 : :

X(31486) lies on these lines: {5,1377}, {6,9710}, {8,31475}, {20,31413}, {372,31458}, {382,31485}, {548,9678}, {1335,17529}, {1376,31454}, {1505,31491}, {1574,31483}, {1575,31465}, {1588,31420}, {1703,31446}, {1706,31440}, {2551,31414}, {3312,31494}, {9607,31464}, {9680,25440}


X(31487) = CENTROID OF CURVATURES OF THESE CIRCLES: {2nd LEMOINE, STAMMLER, STEINER}

Barycentrics    3*a^4-12*S*a^2-5*(b^2+c^2)*a^2+2*(b^2-c^2)^2 : :
X(31487) = 3*X(6447)+2*X(31414)

X(31487) lies on these lines: {2,6427}, {3,9680}, {5,1588}, {6,3411}, {20,6221}, {30,6447}, {140,6428}, {371,382}, {376,9692}, {381,3592}, {485,3843}, {517,31440}, {548,1587}, {550,6519}, {590,5070}, {615,6500}, {631,3312}, {632,19053}, {999,31475}, {1151,15696}, {1335,31480}, {1505,31492}, {1656,6419}, {1657,6425}, {1703,31447}, {3069,16239}, {3070,9681}, {3316,18762}, {3523,6448}, {3524,6522}, {3528,6455}, {3530,6398}, {3534,6453}, {3544,3590}, {3594,15720}, {3832,13886}, {3853,6459}, {3855,18538}, {3856,23259}, {3861,31412}, {4309,19030}, {4317,19028}, {5054,6420}, {5067,7584}, {5410,15559}, {5418,6418}, {5420,6501}, {6407,6560}, {6421,31470}, {6426,15693}, {6450,7581}, {6451,6460}, {6470,6565}, {6496,21734}, {7486,7582}, {9589,31439}, {9605,31465}, {9624,13888}, {9709,31486}, {9711,31485}, {10195,15703}, {11314,13637}, {13901,31452}, {13904,31474}, {13905,15888}, {19111,20379}

X(31487) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1588, 3068, 13925), (3068, 3311, 8976), (3311, 8976, 13785)


X(31488) = CENTROID OF CURVATURES OF THESE CIRCLES: {MOSES, NINE-POINTS, SPIEKER}

Barycentrics    2*(b^2+c^2)*a^2+2*(b+c)*b*c*a-(b^2-c^2)^2 : :

X(31488) lies on these lines: {2,2241}, {4,31456}, {5,1573}, {6,31493}, {8,31476}, {10,1506}, {11,16589}, {32,26363}, {37,24387}, {39,2886}, {75,7764}, {115,1107}, {187,4999}, {381,31490}, {405,9665}, {442,1015}, {486,31482}, {625,26558}, {626,17030}, {956,9650}, {958,5475}, {993,7747}, {1329,7603}, {1376,31455}, {1378,31481}, {1500,24390}, {1505,31484}, {1572,5705}, {1574,3815}, {1575,9698}, {1699,31442}, {1706,31441}, {2242,10527}, {2476,16975}, {2548,19843}, {2549,31418}, {2550,31401}, {2551,31415}, {3434,31451}, {3767,31405}, {3788,20172}, {3822,17448}, {3841,16604}, {3847,6537}, {4386,7749}, {4426,7753}, {5267,6781}, {5283,11680}, {6155,29688}, {6292,20541}, {6547,17062}, {6680,20179}, {6683,26582}, {7737,30478}, {7748,31449}, {7794,21264}, {7813,20888}, {9599,19854}, {9651,17532}, {9709,31489}, {11813,21879}, {31140,31448}, {31420,31450}

X(31488) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1107, 25639, 115), (2886, 31466, 39)


X(31489) = CENTROID OF CURVATURES OF THESE CIRCLES: {MOSES, NINE-POINTS, STAMMLER}

Barycentrics    a^4-5*(b^2+c^2)*a^2+2*(b^2-c^2)^2 : :
X(31489) = X(11742)+8*X(31415)

X(31489) lies on these lines: {2,6}, {3,1506}, {4,8719}, {5,2549}, {11,31477}, {22,15109}, {30,11742}, {32,3526}, {39,1656}, {53,8889}, {98,11669}, {111,11639}, {114,10516}, {115,5024}, {140,2548}, {187,5054}, {194,16922}, {216,30771}, {232,566}, {262,7608}, {381,574}, {498,16781}, {499,31460}, {547,15048}, {549,5210}, {569,9603}, {620,11286}, {625,11287}, {631,5023}, {999,31476}, {1015,31479}, {1030,16434}, {1078,7926}, {1196,5421}, {1285,15709}, {1329,31490}, {1368,15880}, {1384,7753}, {1504,13951}, {1505,8976}, {1571,9955}, {1572,11231}, {1574,31493}, {1609,16419}, {1657,15515}, {1699,31443}, {1975,16921}, {1995,9609}, {2165,11548}, {2453,16316}, {2493,15302}, {3090,5254}, {3291,13337}, {3312,31481}, {3363,7618}, {3524,5585}, {3525,22331}, {3530,31417}, {3534,8589}, {3614,9597}, {3628,3767}, {3814,31449}, {3843,7756}, {3851,7748}, {3934,7908}, {4045,11318}, {5008,15723}, {5038,15069}, {5050,5477}, {5056,7738}, {5063,10314}, {5067,5286}, {5070,5355}, {5077,8176}, {5107,14848}, {5116,13860}, {5124,19544}, {5159,16303}, {5206,15720}, {5309,15703}, {5432,9599}, {5433,9596}, {5471,11485}, {5472,11486}, {5544,6388}, {5651,30516}, {5886,31398}, {6143,8743}, {6353,6748}, {6421,10576}, {6422,10577}, {6459,9601}, {6565,9600}, {6639,23115}, {6683,7862}, {6811,23261}, {6813,23251}, {7173,9598}, {7484,8553}, {7486,9607}, {7509,9608}, {7539,13351}, {7622,11159}, {7739,15699}, {7741,31448}, {7743,31433}, {7749,30435}, {7752,7784}, {7769,7770}, {7773,7824}, {7776,7815}, {7786,7887}, {7787,16923}, {7804,11288}, {7814,7879}, {7867,13357}, {7874,13356}, {7888,31239}, {7913,31275}, {7988,9574}, {8227,31428}, {8588,14537}, {8716,11185}, {8770,9722}, {9112,16963}, {9113,16962}, {9306,9604}, {9619,9956}, {9620,11230}, {9669,31451}, {9709,31488}, {9744,9756}, {9769,16176}, {10011,14561}, {10194,19102}, {10195,19105}, {10418,11284}, {11637,15546}, {13330,15819}, {13966,31411}, {14494,14853}, {14535,31274}, {15325,31409}, {15355,31236}, {15655,15701}, {16308,30745}, {16777,24239}, {18424,19709}, {21448,30537}, {22240,30744}, {22793,31422}, {26364,31466}

X(31489) = complement of the isotomic conjugate of X(14494)
X(31489) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 1007, 141), (7610, 11163, 15534), (7774, 8667, 6144)


X(31490) = CENTROID OF CURVATURES OF THESE CIRCLES: {MOSES, SPIEKER, STAMMLER}

Barycentrics    a*(a^3-3*(b^2+c^2)*a-4*(b+c)*b*c) : :

X(31490) lies on these lines: {1,6}, {3,1573}, {8,31477}, {10,5013}, {30,31416}, {39,9708}, {115,31493}, {230,30478}, {381,31488}, {382,31491}, {517,31442}, {574,9709}, {846,4051}, {988,16605}, {993,3053}, {999,16589}, {1015,11108}, {1030,22654}, {1329,31489}, {1376,15815}, {1505,31485}, {1572,31445}, {1574,5024}, {1575,22332}, {1706,31443}, {1975,20181}, {2241,16418}, {2549,31419}, {2551,3815}, {2975,5275}, {3312,31482}, {3679,31448}, {3780,19765}, {3820,31401}, {3925,9597}, {4386,5023}, {5021,30116}, {5044,9619}, {5210,5267}, {5254,19843}, {5737,24359}, {6376,15271}, {7745,31405}, {7765,31494}, {7778,26558}, {9336,25542}, {9623,31429}, {9651,17528}, {9711,31492}, {13881,26363}, {16992,21226}, {17595,21951}

X(31490) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (405, 16975, 16781), (958, 1107, 6)


X(31491) = CENTROID OF CURVATURES OF THESE CIRCLES: {MOSES, SPIEKER, STEINER}

Barycentrics    4*(b^2+c^2)*a^2+6*(b+c)*b*c*a-(b^2-c^2)^2 : :

X(31491) lies on these lines: {5,1573}, {6,31494}, {8,31478}, {10,9698}, {20,31416}, {32,31458}, {39,9710}, {382,31490}, {1015,17529}, {1107,7765}, {1376,31457}, {1378,31483}, {1505,31486}, {1572,31446}, {1574,9606}, {1706,31444}, {2549,31420}, {4197,16975}, {5319,19843}, {9589,31442}, {9607,31419}, {9709,31492}, {9711,31466}

X(31491) = {X(9710), X(31469)}-harmonic conjugate of X(39)


X(31492) = CENTROID OF CURVATURES OF THESE CIRCLES: {MOSES, STAMMLER, STEINER}

Barycentrics    3*a^4-11*(b^2+c^2)*a^2+2*(b^2-c^2)^2 : :

X(31492) lies on these lines: {2,9607}, {3,7753}, {5,2549}, {6,631}, {20,3815}, {30,31417}, {39,3526}, {140,5319}, {382,574}, {517,31444}, {548,2548}, {599,7796}, {632,7739}, {999,31478}, {1015,31480}, {1078,6144}, {1505,31487}, {1506,3843}, {1572,31447}, {1574,31494}, {1656,18362}, {3053,3530}, {3055,7486}, {3312,31483}, {3523,9300}, {3528,7745}, {3763,7763}, {3767,16239}, {3861,31415}, {4317,31460}, {5007,15720}, {5023,7736}, {5024,5070}, {5054,7772}, {5067,5254}, {5079,11648}, {5306,10303}, {5475,17800}, {5881,31428}, {6337,15491}, {7748,18584}, {7755,15694}, {7778,7876}, {7784,7814}, {7786,24256}, {7791,11184}, {7824,7946}, {7849,15482}, {9589,31443}, {9608,15109}, {9624,31431}, {9709,31491}, {9711,31490}, {10516,12055}, {11165,17130}, {11307,16645}, {11308,16644}, {15484,15515}, {16781,31452}

X(31492) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (5, 31450, 5013), (9698, 31457, 3), (31401, 31450, 5)


X(31493) = CENTROID OF CURVATURES OF THESE CIRCLES: {NINE-POINTS, SPIEKER, STAMMLER}

Barycentrics    a^4-3*(b^2+c^2)*a^2-4*(b+c)*b*c*a+2*(b^2-c^2)^2 : :

X(31493) lies on these lines: {1,31245}, {2,496}, {3,2886}, {5,2551}, {6,31488}, {8,31479}, {9,9955}, {10,1482}, {11,11108}, {21,9668}, {30,30478}, {35,31140}, {56,17528}, {115,31490}, {140,2550}, {230,31416}, {381,958}, {382,993}, {405,9669}, {442,999}, {443,15325}, {474,26060}, {486,31485}, {495,6856}, {497,6675}, {499,3925}, {517,5705}, {595,31187}, {936,11230}, {942,5231}, {946,5791}, {956,2476}, {960,18493}, {1001,24387}, {1125,12437}, {1191,24880}, {1329,5055}, {1376,3526}, {1377,13951}, {1378,8976}, {1479,16418}, {1573,13881}, {1574,31489}, {1575,31467}, {1617,10957}, {1698,3057}, {1699,31445}, {1706,11231}, {2975,9655}, {3086,8728}, {3090,3820}, {3242,24160}, {3312,31484}, {3333,3824}, {3419,24541}, {3421,10592}, {3434,7483}, {3436,17530}, {3452,12864}, {3530,31420}, {3534,5267}, {3555,31266}, {3617,7504}, {3622,31254}, {3656,5837}, {3679,11011}, {3695,30741}, {3697,30852}, {3788,20181}, {3813,6767}, {3814,5079}, {3816,16853}, {3822,12513}, {3826,10200}, {3829,16857}, {3843,31458}, {3927,12047}, {3940,11375}, {4299,31157}, {4413,11508}, {4426,15484}, {4847,11374}, {5044,8227}, {5045,25525}, {5054,25440}, {5070,9710}, {5071,8165}, {5084,10593}, {5177,18990}, {5248,11235}, {5251,10896}, {5254,31468}, {5258,10895}, {5259,11238}, {5260,17556}, {5274,16845}, {5288,11237}, {5305,31405}, {5433,16417}, {5436,18527}, {5708,12609}, {5745,12699}, {5779,12608}, {5784,13373}, {5789,6001}, {5794,10246}, {5827,16821}, {6147,24477}, {6244,6833}, {6284,17571}, {6846,7956}, {6857,15171}, {6862,10306}, {6933,17757}, {7373,25466}, {7489,11928}, {7539,29667}, {7680,8158}, {7743,31435}, {7866,17030}, {9605,31466}, {9623,9956}, {9965,11544}, {10202,18251}, {10584,17575}, {10589,17527}, {10916,15934}, {11318,26558}, {12702,26066}, {12953,31159}, {13785,31453}, {13966,31413}, {16466,24892}, {17542,26127}, {19314,31084}, {19875,30323}, {19877,31272}, {21956,31461}, {22793,31424}, {23708,25917}, {24161,29676}

X(31493) = midpoint of X(30478) and X(31418)
X(31493) = {X(i),X(j)}-