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This is PART 15: Centers X(28001) - X(30000)

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(28001) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 75

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

X(28001) lies on these lines:


X(28002) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 75

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

X(28002) lies on these lines:


X(28003) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 75

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

X(28003) lies on these lines:


X(28004) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 75

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

X(28004) lies on these lines:


X(28005) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 75

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

X(28005) lies on these lines:


X(28006) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 75

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

X(28006) lies on these lines:


X(28007) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = GEMINI TRIANGLE 75

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

X(28007) lies on these lines:


X(28008) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 75

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

X(28008) lies on these lines:


X(28009) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 75

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

X(28009) lies on these lines:


X(28010) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 75

Barycentrics    (a^2 + b c) (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(28010) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 76: X(28011) - X(28010)  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 76, as in centers X(27954)-X(28010). Then

m(X) = bc(a^4-b^2c^2)x - ac(a^2+bc)(c^2+ab)y - ab(a^2+bc)(b^2+ac)z : :

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 15, 2018)

underbar




X(28011) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 76

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

X(28011) lies on these lines:


X(28012) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 76

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

X(28012) lies on these lines:


X(28013) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 76

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

X(28013) lies on these lines:


X(28014) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 76

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

X(28014) lies on these lines:


X(28015) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 76

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

X(28015) lies on these lines:


X(28016) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 76

Barycentrics    a^4 + 2 a^3 b + 4 a^2 b^2 + 2 a b^3 - b^4 + 2 a^3 c - 12 a^2 b c - 2 a b^2 c + 4 a^2 c^2 - 2 a b c^2 + 2 b^2 c^2 + 2 a c^3 - c^4 : :

X(28016) lies on these lines:


X(28017) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 76

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

X(28017) lies on these lines:


X(28018) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 76

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

X(28018) lies on these lines:


X(28019) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 76

Barycentrics    2 a^5 b + a^4 b^2 - 4 a^3 b^3 + 2 a b^5 - b^6 + 2 a^5 c - 10 a^4 b c + 6 a^3 b^2 c - 4 a b^4 c + 2 b^5 c + a^4 c^2 + 6 a^3 b c^2 + 2 a b^3 c^2 + b^4 c^2 - 4 a^3 c^3 + 2 a b^2 c^3 - 4 b^3 c^3 - 4 a b c^4 + b^2 c^4 + 2 a c^5 + 2 b c^5 - c^6 : :

X(28019) lies on these lines:


X(28020) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 76

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

X(28020) lies on these lines:


X(28021) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 76

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

X(28021) lies on these lines:


X(28022) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 76

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

X(28022) lies on these lines:


X(28023) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 76

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

X(28023) lies on these lines:


X(28024) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 76

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

X(28024) lies on these lines:


X(28025) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 76

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

X(28025) lies on these lines:


X(28026) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 76

Barycentrics    a (a^5 + a^4 b + a^3 b^2 + a^2 b^3 + a^4 c - 3 a^3 b c - a b^3 c + b^4 c + a^3 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(28026) lies on these lines:


X(28027) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 76

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

X(28027) lies on these lines:


X(28028) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 76

Barycentrics    3 a^7 + 5 a^6 b + 3 a^5 b^2 - 3 a^4 b^3 - 7 a^3 b^4 - a^2 b^5 + a b^6 - b^7 + 5 a^6 c - 10 a^5 b c - a^4 b^2 c + 4 a^3 b^3 c - 5 a^2 b^4 c + 6 a b^5 c + b^6 c + 3 a^5 c^2 - a^4 b c^2 + 6 a^3 b^2 c^2 + 6 a^2 b^3 c^2 - a b^4 c^2 + 3 b^5 c^2 - 3 a^4 c^3 + 4 a^3 b c^3 + 6 a^2 b^2 c^3 - 12 a b^3 c^3 - 3 b^4 c^3 - 7 a^3 c^4 - 5 a^2 b c^4 - a b^2 c^4 - 3 b^3 c^4 - a^2 c^5 + 6 a b c^5 + 3 b^2 c^5 + a c^6 + b c^6 - c^7 : :

X(28028) lies on these lines:


X(28029) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 76

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

X(28029) lies on these lines:


X(28030) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 76

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

X(28030) lies on these lines:


X(28031) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 76

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

X(28031) lies on these lines:


X(28032) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 76

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

X(28032) lies on these lines:


X(28033) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 76

Barycentrics    5 a^7 + 7 a^6 b + 3 a^5 b^2 - 3 a^4 b^3 - 9 a^3 b^4 - 3 a^2 b^5 + a b^6 - b^7 + 7 a^6 c - 14 a^5 b c - a^4 b^2 c + 4 a^3 b^3 c - 7 a^2 b^4 c + 10 a b^5 c + b^6 c + 3 a^5 c^2 - a^4 b c^2 + 10 a^3 b^2 c^2 + 10 a^2 b^3 c^2 - a b^4 c^2 + 3 b^5 c^2 - 3 a^4 c^3 + 4 a^3 b c^3 + 10 a^2 b^2 c^3 - 20 a b^3 c^3 - 3 b^4 c^3 - 9 a^3 c^4 - 7 a^2 b c^4 - a b^2 c^4 - 3 b^3 c^4 - 3 a^2 c^5 + 10 a b c^5 + 3 b^2 c^5 + a c^6 + b c^6 - c^7 : :

X(28033) lies on these lines:


X(28034) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 76

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

X(28034) lies on these lines:


X(28035) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 76

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

X(28035) lies on these lines:


X(28036) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 76

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

X(28036) lies on these lines:


X(28037) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 76

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

X(28037) lies on these lines:


X(28038) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 76

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

X(28038) lies on these lines:


X(28039) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 76

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

X(28039) lies on these lines:


X(28040) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 76

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

X(28040) lies on these lines:


X(28041) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 76

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

X(28041) lies on these lines:


X(28042) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 76

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

X(28042) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 77: X(28043) - X(28073)  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 77, as in centers X(28043)-X(28073). Then

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

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 16, 2018)

underbar




X(28043) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 77

Barycentrics    a (a - b - c) (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) : :

X(28043) lies on these lines:


X(28044) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 77

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

X(28044) lies on these lines:


X(28045) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 77

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

X(28045) lies on these lines:


X(28046) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 77

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

X(28046) lies on these lines:


X(28047) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 77

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

X(28047) lies on these lines:


X(28048) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 77

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

X(28048) lies on these lines:


X(28049) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 77

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

X(28049) lies on these lines:


X(28050) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 77

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

X(28050) lies on these lines:


X(28051) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 77

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

X(28051) lies on these lines:


X(28052) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 77

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

X(28052) lies on these lines:


X(28053) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 77

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

X(28053) lies on these lines:


X(28054) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 77

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

X(28054) lies on these lines:


X(28055) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 77

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

X(28055) lies on these lines:


X(28056) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 77

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

X(28056) lies on these lines:


X(28057) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 77

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

X(28057) lies on these lines:


X(28058) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 77

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

X(28058) lies on these lines:


X(28059) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 77

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

X(28059) lies on these lines:


X(28060) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 77

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

X(28060) lies on these lines:


X(28061) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 77

Barycentrics    (a - b - c) (3 a^7 - 5 a^6 b + 3 a^5 b^2 + 3 a^4 b^3 - 7 a^3 b^4 + a^2 b^5 + a b^6 + b^7 - 5 a^6 c - 10 a^5 b c + a^4 b^2 c + 4 a^3 b^3 c + 5 a^2 b^4 c + 6 a b^5 c - b^6 c + 3 a^5 c^2 + a^4 b c^2 + 6 a^3 b^2 c^2 - 6 a^2 b^3 c^2 - a b^4 c^2 - 3 b^5 c^2 + 3 a^4 c^3 + 4 a^3 b c^3 - 6 a^2 b^2 c^3 - 12 a b^3 c^3 + 3 b^4 c^3 - 7 a^3 c^4 + 5 a^2 b c^4 - a b^2 c^4 + 3 b^3 c^4 + a^2 c^5 + 6 a b c^5 - 3 b^2 c^5 + a c^6 - b c^6 + c^7) : :

X(28061) lies on these lines:


X(28062) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 77

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

X(28062) lies on these lines:


X(28063) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 77

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

X(28063) lies on these lines:


X(28064) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 77

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

X(28064) lies on these lines:


X(28065) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 77

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

X(28065) lies on these lines:


X(28066) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 77

Barycentrics    (a - b - c) (a^10 - a^9 b + 2 a^8 b^2 - 8 a^6 b^4 + 2 a^5 b^5 + 6 a^4 b^6 - a^2 b^8 - a b^9 - a^9 c + a^8 b c + 6 a^7 b^2 c - 2 a^6 b^3 c - 8 a^5 b^4 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 + 6 a^7 b c^2 + 8 a^6 b^2 c^2 - 2 a^5 b^3 c^2 - 6 a^4 b^4 c^2 - 6 a^3 b^5 c^2 - 4 a^2 b^6 c^2 + 2 a b^7 c^2 - 2 a^6 b c^3 - 2 a^5 b^2 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 - 8 a^6 c^4 - 8 a^5 b c^4 - 6 a^4 b^2 c^4 + 4 a^3 b^3 c^4 + 10 a^2 b^4 c^4 + 2 a^5 c^5 - 6 a^3 b^2 c^5 - 2 a^2 b^3 c^5 - 6 b^5 c^5 + 6 a^4 c^6 + 2 a^3 b c^6 - 4 a^2 b^2 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(28066) lies on these lines:


X(28067) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 77

Barycentrics    (a - b - c) (5 a^7 - 7 a^6 b + 3 a^5 b^2 + 3 a^4 b^3 - 9 a^3 b^4 + 3 a^2 b^5 + a b^6 + b^7 - 7 a^6 c - 14 a^5 b c + a^4 b^2 c + 4 a^3 b^3 c + 7 a^2 b^4 c + 10 a b^5 c - b^6 c + 3 a^5 c^2 + a^4 b c^2 + 10 a^3 b^2 c^2 - 10 a^2 b^3 c^2 - a b^4 c^2 - 3 b^5 c^2 + 3 a^4 c^3 + 4 a^3 b c^3 - 10 a^2 b^2 c^3 - 20 a b^3 c^3 + 3 b^4 c^3 - 9 a^3 c^4 + 7 a^2 b c^4 - a b^2 c^4 + 3 b^3 c^4 + 3 a^2 c^5 + 10 a b c^5 - 3 b^2 c^5 + a c^6 - b c^6 + c^7) : :

X(28067) lies on these lines:


X(28068) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 77

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

X(28068) lies on these lines:


X(28069) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 77

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

X(28069) lies on these lines:


X(28070) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 77

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

X(28070) lies on these lines:


X(28071) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 77

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

X(28071) lies on these lines:


X(28072) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 77

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

X(28072) lies on these lines:


X(28073) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 77

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

X(28073) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 78: X(28074) - X(28117)  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 78, as in centers X(28074)-X(28117). Then

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

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 17, 2018)

underbar




X(28074) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28074) lies on these lines:


X(28075) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28075) lies on these lines:


X(28076) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28076) lies on these lines:


X(28077) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28077) lies on these lines:


X(28078) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28078) lies on these lines:


X(28079) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28079) lies on these lines:


X(28080) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28080) lies on these lines:


X(28081) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28081) lies on these lines:


X(28082) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28082) lies on these lines:


X(28083) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28083) lies on these lines:


X(28084) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28084) lies on these lines:


X(28085) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28085) lies on these lines:


X(28086) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28086) lies on these lines:


X(28087) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28087) lies on these lines:


X(28088) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28088) lies on these lines:


X(28089) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28089) lies on these lines:


X(28090) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28090) lies on these lines:


X(28091) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28091) lies on these lines:


X(28092) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 78

Barycentrics    5 a^4 + 6 a^2 b^2 + 8 a b^3 - 3 b^4 - 16 a^2 b c - 8 a b^2 c + 6 a^2 c^2 - 8 a b c^2 + 6 b^2 c^2 + 8 a c^3 - 3 c^4 : :

X(28092) lies on these lines:


X(28093) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28093) lies on these lines:


X(28094) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28094) lies on these lines:


X(28095) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28095) lies on these lines:


X(28096) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28096) lies on these lines:


X(28097) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28097) lies on these lines:


X(28098) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28098) lies on these lines:


X(28099) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28099) lies on these lines:


X(28100) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28100) lies on these lines:


X(28101) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28101) lies on these lines:


X(28102) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28102) lies on these lines:


X(28103) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 78

Barycentrics    7 a^7 + 5 a^6 b + 3 a^5 b^2 - 3 a^4 b^3 - 15 a^3 b^4 + 3 a^2 b^5 + 5 a b^6 - 5 b^7 + 5 a^6 c - 10 a^5 b c - 5 a^4 b^2 c + 8 a^3 b^3 c - 5 a^2 b^4 c + 2 a b^5 c + 5 b^6 c + 3 a^5 c^2 - 5 a^4 b c^2 + 14 a^3 b^2 c^2 + 2 a^2 b^3 c^2 - 5 a b^4 c^2 + 15 b^5 c^2 - 3 a^4 c^3 + 8 a^3 b c^3 + 2 a^2 b^2 c^3 - 4 a b^3 c^3 - 15 b^4 c^3 - 15 a^3 c^4 - 5 a^2 b c^4 - 5 a b^2 c^4 - 15 b^3 c^4 + 3 a^2 c^5 + 2 a b c^5 + 15 b^2 c^5 + 5 a c^6 + 5 b c^6 - 5 c^7 : :

X(28103) lies on these lines:


X(28104) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28104) lies on these lines:


X(28105) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28105) lies on these lines:


X(28106) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28106) lies on these lines:


X(28107) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28107) lies on these lines:


X(28108) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28108) lies on these lines:


X(28109) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28109) lies on these lines:


X(28110) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28110) lies on these lines:


X(28111) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28111) lies on these lines:


X(28112) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28112) lies on these lines:


X(28113) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28113) lies on these lines:


X(28114) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28114) lies on these lines:


X(28115) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28115) lies on these lines:


X(28116) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28116) lies on these lines:


X(28117) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(24), WHERE A'B'C' = GEMINI TRIANGLE 78

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

X(28117) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 79: X(28118) - X(28142)  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 79, as in centers X(28118)-X(28142). Then

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

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 17, 2018)

underbar




X(28118) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 79

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

X(28118) lies on these lines:


X(28119) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 79

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

X(28119) lies on these lines:


X(28120) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 79

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

X(28120) lies on these lines:


X(28121) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 79

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

X(28121) lies on these lines:


X(28122) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 79

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

X(28122) lies on these lines:


X(28123) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 79

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

X(28123) lies on these lines:


X(28124) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 79

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

X(28124) lies on these lines:


X(28125) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 79

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

X(28125) lies on these lines:


X(28126) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 79

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

X(28126) lies on these lines:


X(28127) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 79

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

X(28127) lies on these lines:


X(28128) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 79

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

X(28128) lies on these lines:


X(28129) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 79

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

X(28129) lies on these lines:


X(28130) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 79

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

X(28130) lies on these lines:


X(28131) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 79

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

X(28131) lies on these lines:


X(28132) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 79

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

X(28132) lies on these lines:


X(28133) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 79

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

X(28133) lies on these lines:


X(28134) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 79

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

X(28134) lies on these lines:


X(28135) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 79

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

X(28135) lies on these lines:


X(28136) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 79

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

X(28136) lies on these lines:


X(28137) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 79

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

X(28137) lies on these lines:


X(28138) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 79

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

X(28138) lies on these lines:


X(28139) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 79

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

X(28139) lies on these lines:


X(28140) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 79

Barycentrics    (a - b - c) (7 a^7 - 5 a^6 b + 3 a^5 b^2 + 3 a^4 b^3 - 15 a^3 b^4 - 3 a^2 b^5 + 5 a b^6 + 5 b^7 - 5 a^6 c - 10 a^5 b c + 5 a^4 b^2 c + 8 a^3 b^3 c + 5 a^2 b^4 c + 2 a b^5 c - 5 b^6 c + 3 a^5 c^2 + 5 a^4 b c^2 + 14 a^3 b^2 c^2 - 2 a^2 b^3 c^2 - 5 a b^4 c^2 - 15 b^5 c^2 + 3 a^4 c^3 + 8 a^3 b c^3 - 2 a^2 b^2 c^3 - 4 a b^3 c^3 + 15 b^4 c^3 - 15 a^3 c^4 + 5 a^2 b c^4 - 5 a b^2 c^4 + 15 b^3 c^4 - 3 a^2 c^5 + 2 a b c^5 - 15 b^2 c^5 + 5 a c^6 - 5 b c^6 + 5 c^7) : :

X(28140) lies on these lines:


X(28141) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 79

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

X(28141) lies on these lines:


X(28142) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 79

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

X(28142) lies on these lines:


X(28143) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 79

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

X(28143) lies on these lines:


X(28144) =  34TH HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics    a^10 b^2+2 a^8 b^4-9 a^6 b^6+5 a^4 b^8+4 a^2 b^10-3 b^12+a^10 c^2-8 a^8 b^2 c^2+10 a^6 b^4 c^2+7 a^4 b^6 c^2-17 a^2 b^8 c^2+7 b^10 c^2+2 a^8 c^4+10 a^6 b^2 c^4-24 a^4 b^4 c^4+13 a^2 b^6 c^4-b^8 c^4-9 a^6 c^6+7 a^4 b^2 c^6+13 a^2 b^4 c^6-6 b^6 c^6+5 a^4 c^8-17 a^2 b^2 c^8-b^4 c^8+4 a^2 c^10+7 b^2 c^10-3 c^12 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28647.

X(28144) lies on these lines: {2,3}, {523,23332}, {2452,23291}, {9530,24930}, {11550,16319}






leftri  Points on circumcircle and line at infinity: X(28145) - X(28236)  rightri

Suppose that X = x : y : z is a point on the line at infinity. All the lines that meet in X are parallel, so that X can be regarded as a direction in the plane of the reference triangle ABC. Let X' be the isogonal conjugate of X, so that X' lies on the circumcircle. Let X'' be the circumcircle-antipode of X', and let X''' be its isogonal conjugate, on the line at infinity. As a direction, X''' is perpendicular to X. In this section, X is given by the form (b - c)(h a + k(b + c)) : : , where h and k are constants. (Clark Kimberling, November 17, 2018)

In the table below, Columns 1 and 2 show h and k.

Column 3. (b - c)(h a + k(b + c)) : : , on infinity line, referenced below as x : y : z

Column 4. (isogonal conjugate of x : y : z) = a^2/x + b^2/y + c^2/z : : on circumcircle, referenced below as u : v : w

Column 5. (antipode of u : v : w) = (a^2+b^2-c^2)(a^2-b^2+c^2)u + 2a^2(a^2-b^2-c^2)v + 2a^2(a^2-b^2-c^2)w : : on circumcircle, referenced below as u1 : v1 : w1

Column 6. (isogonal conjugate of u1 : v1 : w1) = a^2/u1 + b^2/v1 + c^2/w1

For each row, let X be the point in Column 3 and X' the point in Column 6. Let U be any point in the finite plane of ABC. Then the lines UX and UX' are perpendicular.

In the table below, the points in Column 3 are here given names of the form Point Pollux(h,k).

>
h k Column 3 Column 4Column 5 Column 6
11514101103516
12480286522814528146
1328147281482814928150
1428151281522815328154
1528155281562815728158
1-1522109102515
1-2477745882815928160
1-328161281622816328164
1-428165281662816728168
1-528169281702817128172
21497787012817328174
2328175281762817728178
2528179281802818128182
2-1900901102515
2-328183381842818528186
2-528187281882818928190
31477886942819328194
3228195281962819728198
3428199282002820128202
3-136671293106519
3-2492686972820328204
3-428205282062820728208
4128209282102821128212
4328213282142821528216
4-12821728218282195844
4-28221282222822328224
5128225282262822728228
3328229282302823128232
5-1600660142823328234
3-3496286992823528236
underbar




X(28145) =  CIRCUMCIRCLE-ANTIPODE OF X(8652)

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

X(28145) lies on the circumcircle and these lines:

X(28145) = isogonal conjugate of X(28146)
X(28145) = circumcircle-antipoade of X(8652)


X(28146) =  ISOGONAL CONJUGATE OF X(28145)

Barycentrics    1/((2 a^4 - 4 a^2 b^2 + 2 b^4 + a^3 c - a^2 b c - a b^2 c + b^3 c + 2 a^2 c^2 + 2 a b c^2 + 2 b^2 c^2 - a c^3 - b c^3 - 4 c^4) (2 a^4 + a^3 b + 2 a^2 b^2 - a b^3 - 4 b^4 - a^2 b c + 2 a b^2 c - b^3 c - 4 a^2 c^2 - a b c^2 + 2 b^2 c^2 + b c^3 + 2 c^4)) : :

X(28146) lies on these lines: {1, 1657}, {3, 1699}, {4, 2355}, {5, 10164}, {10, 3627}, {11, 5122}, {20, 1385}, {30, 511}, {36, 7743}, {40, 382}, {46, 12953}, {56, 4333}, {57, 9668}, {79, 4330}, {80, 5183}, {140, 10171}, {165, 381}, {355, 3146}, {376, 5886}, {392, 17579}, {546, 6684}, {548, 1125}, {549, 3817}, {550, 946}, {551, 15686}, {910, 5134}, {942, 1770}, {944, 5059}, {962, 3529}, {999, 9580}, {1155, 3583}, {1319, 4316}, {1479, 17728}, {1482, 9589}, {1538, 6905}, {1697, 9655}, {1698, 3843}, {1829, 18560}, {1836, 4302}, {1902, 6240}, {2077, 15017}, {2646, 4324}, {2951, 18443}, {3057, 10483}, {3058, 5049}, {3090, 10248}, {3097, 22728}, {3295, 9579}, {3338, 9670}, {3474, 5722}, {3475, 4294}, {3524, 9779}, {3526, 16192}, {3534, 3576}, {3543, 5657}, {3587, 11372}, {3616, 17538}, {3628, 12571}, {3634, 3850}, {3648, 5178}, {3654, 15682}, {3655, 15683}, {3656, 5731}, {3679, 15684}, {3746, 16118}, {3753, 11114}, {3824, 5248}, {3828, 14893}, {3830, 5587}, {3845, 10175}, {3853, 19925}, {4292, 5045}, {4297, 15178}, {4298, 15172}, {4299, 12701}, {4309, 10404}, {4312, 15934}, {4314, 6147}, {4325, 20323}, {4347, 8144}, {4512, 17528}, {5010, 17605}, {5054, 7988}, {5057, 5440}, {5066, 10172}, {5073, 5691}, {5076, 18492}, {5119, 12943}, {5126, 15326}, {5266, 24851}, {5493, 5690}, {5537, 18524}, {5550, 21735}, {5658, 5812}, {5659, 16113}, {5694, 12688}, {5818, 17578}, {5885, 9943}, {5899, 9590}, {5901, 12103}, {5918, 10202}, {6244, 18491}, {6407, 13888}, {6408, 13942}, {6583, 13369}, {6840, 10225}, {6851, 10525}, {6881, 7965}, {6883, 11495}, {6909, 23961}, {6985, 26285}, {6996, 29607}, {7354, 9957}, {7489, 7688}, {7686, 13145}, {7964, 18406}, {7987, 15696}, {7991, 18525}, {7994, 18528}, {8703, 10165}, {8976, 9582}, {9574, 15484}, {9616, 13665}, {9669, 15803}, {9904, 12902}, {9911, 12085}, {10246, 15681}, {10247, 15685}, {10283, 19710}, {10386, 21620}, {10624, 18990}, {10721, 12778}, {10724, 12515}, {10738, 11219}, {10902, 16117}, {10980, 18530}, {11500, 22792}, {11531, 18526}, {11541, 12245}, {11699, 12121}, {12047, 15338}, {12108, 19878}, {12261, 16111}, {12611, 24466}, {13199, 16128}, {13605, 14677}, {13911, 22644}, {13973, 22615}, {15689, 25055}, {15712, 19862}, {17573, 25522}, {17606, 18514}, {21578, 25405}

X(28146) = isogonal conjugate of X(28145)


X(28147) =  POINT POLLUX(1,3)

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

X(28147) lies on these lines: {30, 511}, {693, 4086}, {1635, 4458}, {2517, 4404}, {3239, 4841}, {3251, 5592}, {3676, 4608}, {3737, 4449}, {3762, 7650}, {3835, 4824}, {4036, 4823}, {4041, 23800}, {4088, 21297}, {4163, 20294}, {4391, 4815}, {4397, 4801}, {4462, 4985}, {4468, 14779}, {4521, 4893}, {4818, 21104}, {4928, 23770}, {7661, 21102}, {14413, 21180}, {14837, 23752}

X(28147) = isogonal conjugate of X(28148)


X(28148) =  ISOGONAL CONJUGATE OF X(28147)

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

X(28148) lies on the circumcircle and these lines:

X(28148) = isogonal conjugate of X(28147)
X(28148) = circumcircle-antipode of X(28149)


X(28149) =  CIRCUMCIRCLE-ANTIPODE OF X(28148)

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

X(28149) lies on the circumcircle and these lines:

X(28149) = isogonal conjugate of X(28150)
X(28149) = circumcircle-antipoade of X(28148)


X(28150) =  ISOGONAL CONJUGATE OF X(28149)

Barycentrics    1/((3 a^4 - 6 a^2 b^2 + 3 b^4 + a^3 c - a^2 b c - a b^2 c + b^3 c + 3 a^2 c^2 + 2 a b c^2 + 3 b^2 c^2 - a c^3 - b c^3 - 6 c^4) (3 a^4 + a^3 b + 3 a^2 b^2 - a b^3 - 6 b^4 - a^2 b c + 2 a b^2 c - b^3 c - 6 a^2 c^2 - a b c^2 + 3 b^2 c^2 + b c^3 + 3 c^4)) : :

X(28150) lies on these lines: {1, 3529}, {3, 3817}, {4, 165}, {5, 12512}, {10, 382}, {20, 946}, {30, 511}, {40, 3146}, {140, 12571}, {226, 4302}, {354, 4292}, {355, 5073}, {376, 1699}, {381, 10164}, {484, 1776}, {546, 3634}, {548, 9955}, {549, 10171}, {550, 1125}, {551, 15681}, {944, 9589}, {950, 1770}, {962, 3623}, {1210, 12953}, {1385, 15704}, {1479, 4333}, {1657, 4297}, {1737, 15228}, {1836, 4304}, {2071, 9625}, {3090, 16192}, {3474, 3586}, {3488, 4312}, {3520, 9591}, {3522, 8227}, {3523, 10248}, {3524, 7988}, {3530, 19878}, {3534, 5886}, {3543, 5587}, {3560, 12511}, {3579, 3627}, {3583, 3911}, {3656, 15685}, {3740, 12572}, {3828, 15687}, {3830, 26446}, {3845, 11231}, {3848, 12436}, {3853, 9956}, {4293, 9580}, {4294, 9579}, {4298, 5049}, {4299, 12053}, {4301, 10247}, {4311, 12701}, {4324, 12047}, {4330, 13407}, {4668, 5691}, {4816, 7991}, {5225, 15803}, {5542, 18541}, {5603, 11001}, {5657, 15682}, {5717, 17592}, {5731, 15683}, {5758, 16127}, {5763, 18243}, {5790, 15684}, {5881, 20052}, {5919, 7354}, {5927, 11826}, {6253, 18908}, {6705, 6851}, {6744, 24470}, {6840, 24042}, {6869, 12608}, {6912, 7688}, {6913, 11495}, {6999, 17266}, {7987, 17538}, {8703, 11230}, {9616, 23249}, {9626, 12087}, {9668, 11019}, {9779, 10304}, {10106, 10483}, {12103, 13624}, {12575, 18990}, {12577, 15172}, {13411, 15338}, {13605, 20127}, {13893, 23253}, {13912, 23251}, {13947, 23263}, {13975, 23261}, {15688, 19883}, {17578, 18492}, {21164, 26333}

X(28150) = isogonal conjugate of X(28149)


X(28151) =  POINT POLLUX(1,4)

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

X(28151) lies on these lines: {30, 511}, {650, 23758}, {693, 21606}, {2529, 4765}, {4024, 24078}, {4379, 4948}, {4411, 21433}, {4776, 4824}, {4828, 20906}, {7653, 21196}

X(28151) = isogonal conjugate of X(28152)


X(28152) =  ISOGONAL CONJUGATE OF X(28151)

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

X(28152) lies on the circumcircle and these lines:

X(28152) = isogonal conjugate of X(28151)
X(28152) = circumcircle-antipode of X(28153)


X(28153) =  CIRCUMCIRCLE-ANTIPODE OF X(28152)

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

X(28153) lies on the circumcircle and these lines:

X(28153) = isogonal conjugate of X(28154)
X(28153) = circumcircle-antipoade of X(28152)


X(28154) =  ISOGONAL CONJUGATE OF X(28143)

Barycentrics    1/((4 a^4 - 8 a^2 b^2 + 4 b^4 + a^3 c - a^2 b c - a b^2 c + b^3 c + 4 a^2 c^2 + 2 a b c^2 + 4 b^2 c^2 - a c^3 - b c^3 - 8 c^4) (4 a^4 + a^3 b + 4 a^2 b^2 - a b^3 - 8 b^4 - a^2 b c + 2 a b^2 c - b^3 c - 8 a^2 c^2 - a b c^2 + 4 b^2 c^2 + b c^3 + 4 c^4)) : :

X(28154) lies on these lines: {1, 17800}, {3, 7988}, {4, 11231}, {8, 11541}, {20, 5886}, {30, 511}, {40, 5073}, {165, 3830}, {376, 9779}, {382, 3579}, {546, 10172}, {548, 18483}, {550, 9955}, {946, 15704}, {1125, 12103}, {1385, 1657}, {1698, 5076}, {1699, 3534}, {3146, 5657}, {3528, 10248}, {3529, 5731}, {3530, 12571}, {3543, 26446}, {3576, 15681}, {3583, 5122}, {3627, 9956}, {3634, 3861}, {3654, 15640}, {3817, 8703}, {3845, 10164}, {3851, 16192}, {3853, 6684}, {4297, 10283}, {4302, 17718}, {4316, 5126}, {4333, 12953}, {5045, 6284}, {5059, 18481}, {5603, 15683}, {6985, 26086}, {7743, 15326}, {8227, 15696}, {9589, 11278}, {9625, 18859}, {9778, 15682}, {9812, 11001}, {9957, 10483}, {10171, 12100}, {10175, 15687}, {10246, 15685}, {10431, 18407}

X(28154) = isogonal conjugate of X(28153)


X(28155) =  POINT POLLUX(1,5)

Barycentrics    (b - c) (a + 5 b + 5 c) : :

X(28155) lies on these lines: {30, 511}, {693, 4404}, {3239, 4988}, {4025, 4608}, {4086, 4823}, {4397, 4978}, {4791, 4815}, {4841, 4944}, {14779, 25259}

X(28155) = isogonal conjugate of X(28156)


X(28156) =  ISOGONAL CONJUGATE OF X(28155)

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

X(28156) lies on the circumcircle and these lines:

X(28156) = isogonal conjugate of X(28155)
X(28156) = circumcircle-antipode of X(28157)


X(28157) =  CIRCUMCIRCLE-ANTIPODE OF X(28156)

Barycentrics    a^2 (5 a^4 - 10 a^2 b^2 + 5 b^4 + a^3 c - a^2 b c - a b^2 c + b^3 c + 5 a^2 c^2 + 2 a b c^2 + 5 b^2 c^2 - a c^3 - b c^3 - 10 c^4) (5 a^4 + a^3 b + 5 a^2 b^2 - a b^3 - 10 b^4 - a^2 b c + 2 a b^2 c - b^3 c - 10 a^2 c^2 - a b c^2 + 5 b^2 c^2 + b c^3 + 5 c^4) : :

X(28157) lies on the circumcircle and these lines:

X(28157) = isogonal conjugate of X(28158)
X(28157) = circumcircle-antipode of X(28156)


X(28158) =  ISOGONAL CONJUGATE OF X(28147)

Barycentrics    1/((5 a^4 - 10 a^2 b^2 + 5 b^4 + a^3 c - a^2 b c - a b^2 c + b^3 c + 5 a^2 c^2 + 2 a b c^2 + 5 b^2 c^2 - a c^3 - b c^3 - 10 c^4) (5 a^4 + a^3 b + 5 a^2 b^2 - a b^3 - 10 b^4 - a^2 b c + 2 a b^2 c - b^3 c - 10 a^2 c^2 - a b c^2 + 5 b^2 c^2 + b c^3 + 5 c^4)) : :

X(28158) lies on these lines: {1, 5059}, {3, 10171}, {4, 3634}, {10, 3146}, {20, 1125}, {30, 511}, {40, 4691}, {165, 3543}, {376, 3817}, {382, 19925}, {550, 11230}, {551, 9812}, {946, 1657}, {950, 11246}, {962, 3635}, {1210, 4333}, {1698, 17578}, {1770, 6738}, {3244, 9589}, {3475, 4314}, {3522, 19862}, {3529, 3636}, {3534, 10165}, {3576, 11001}, {3624, 10248}, {3625, 20070}, {3626, 5493}, {3627, 6684}, {3830, 10175}, {3832, 16192}, {3833, 10178}, {3845, 10172}, {3854, 19872}, {4084, 9961}, {4292, 6744}, {4298, 6284}, {4301, 7967}, {4302, 13405}, {4315, 9580}, {4324, 13411}, {4701, 7991}, {4745, 15640}, {4746, 6361}, {5073, 5790}, {5587, 15682}, {5883, 5918}, {5886, 15681}, {6838, 20104}, {6890, 20107}, {7354, 12575}, {7464, 9625}, {7988, 10304}, {8227, 17538}, {9582, 23253}, {9591, 12086}, {9779, 19883}, {9955, 12103}, {10483, 10624}, {10724, 11219}, {11231, 15687}, {12565, 30147}, {12577, 15171}, {12699, 17800}, {12953, 17728}, {13912, 22644}, {13975, 22615}, {15686, 17502}, {15704, 22793}

X(28158) = isogonal conjugate of X(28157)


X(28159) =  CIRCUMCIRCLE-ANTIPODE OF X(4588)

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

X(28159) lies on the circumcircle and these lines:

X(28159) = isogonal conjugate of X(28160)
X(28159) = circumcircle-antipode of X(4588)


X(28160) =  ISOGONAL CONJUGATE OF X(28159)

Barycentrics    1/((2 a^4 - 4 a^2 b^2 + 2 b^4 - a^3 c + a^2 b c + a b^2 c - b^3 c + 2 a^2 c^2 - 2 a b c^2 + 2 b^2 c^2 + a c^3 + b c^3 - 4 c^4) (2 a^4 - a^3 b + 2 a^2 b^2 + a b^3 - 4 b^4 + a^2 b c - 2 a b^2 c + b^3 c - 4 a^2 c^2 + a b c^2 + 2 b^2 c^2 - b c^3 + 2 c^4)) : :

X(28160) lies on these lines: {1, 382}, {2, 17502}, {3, 1698}, {4, 1385}, {5, 4297}, {8, 3529}, {10, 550}, {11, 5126}, {20, 355}, {30, 511}, {40, 1657}, {65, 10483}, {80, 1155}, {140, 10172}, {165, 3534}, {214, 5087}, {376, 26446}, {381, 3576}, {392, 11114}, {495, 4304}, {496, 4311}, {546, 1125}, {548, 6684}, {549, 10175}, {551, 15687}, {908, 10609}, {942, 7354}, {944, 3146}, {946, 3627}, {950, 5045}, {962, 11278}, {999, 3586}, {1056, 8236}, {1159, 4312}, {1319, 3583}, {1420, 9669}, {1478, 17718}, {1479, 24928}, {1482, 5073}, {1483, 4301}, {1512, 12619}, {1532, 18857}, {1539, 11720}, {1656, 7987}, {1699, 3830}, {1737, 5122}, {1768, 12747}, {1770, 10950}, {1829, 6240}, {1837, 4299}, {1902, 18560}, {2077, 18524}, {2646, 3585}, {3109, 18653}, {3245, 9897}, {3295, 9613}, {3428, 18519}, {3488, 11038}, {3522, 5818}, {3526, 7989}, {3530, 3634}, {3543, 3655}, {3601, 9654}, {3612, 10895}, {3624, 3851}, {3653, 3839}, {3654, 9778}, {3656, 7967}, {3679, 15681}, {3753, 17579}, {3817, 3845}, {3834, 24261}, {3843, 8227}, {3844, 14810}, {3853, 5901}, {3855, 5550}, {3861, 12571}, {3911, 12019}, {3916, 5086}, {3935, 9963}, {4188, 17619}, {4293, 5722}, {4298, 12433}, {4302, 5252}, {4305, 5229}, {4816, 5881}, {4857, 20323}, {5044, 17647}, {5046, 17614}, {5049, 5434}, {5057, 6224}, {5059, 6361}, {5066, 10171}, {5076, 18493}, {5080, 5440}, {5134, 6603}, {5183, 15228}, {5196, 6740}, {5204, 10826}, {5217, 10827}, {5225, 11373}, {5270, 5441}, {5537, 12331}, {5542, 15935}, {5560, 15079}, {5660, 10742}, {5690, 15704}, {5694, 14110}, {5709, 10864}, {5770, 5787}, {5806, 13373}, {5817, 6987}, {5882, 22791}, {5885, 7686}, {5899, 9625}, {6261, 22792}, {6265, 10728}, {6282, 18528}, {6284, 9957}, {6449, 13893}, {6450, 13947}, {6583, 12675}, {6738, 24470}, {6745, 9945}, {6796, 26086}, {6827, 18516}, {6847, 26487}, {6848, 26492}, {6850, 18517}, {6851, 10526}, {6905, 23961}, {6923, 18407}, {6985, 12114}, {6996, 17266}, {6999, 29590}, {7280, 17606}, {7489, 15931}, {7497, 15942}, {7580, 22758}, {7728, 11699}, {7982, 18526}, {7991, 12645}, {8148, 9589}, {8582, 17563}, {8703, 10164}, {8976, 9615}, {9583, 13665}, {9592, 15484}, {9798, 12085}, {9833, 12779}, {9940, 20420}, {9943, 13145}, {10039, 15338}, {10106, 15171}, {10113, 11709}, {10157, 28459}, {10247, 15684}, {10269, 19541}, {10299, 19877}, {10310, 18518}, {10431, 12115}, {10543, 13407}, {10620, 12407}, {10724, 12737}, {10902, 13743}, {11012, 26321}, {11015, 20060}, {11227, 28452}, {11500, 26285}, {11529, 18541}, {11710, 22515}, {11711, 22505}, {11715, 22938}, {11722, 19160}, {12083, 15177}, {12103, 12512}, {12121, 12368}, {12261, 12295}, {12262, 18381}, {12265, 19163}, {12573, 15008}, {12680, 24474}, {12690, 26015}, {12743, 18839}, {13211, 20127}, {13902, 23253}, {13959, 23263}, {14269, 25055}, {15688, 19875}, {15700, 19876}, {17605, 18513}, {18514, 21842}, {22835, 24042}

X(28160) = isogonal conjugate of X(28159)


X(28161) =  POINT POLLUX(1,-3)

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

X(28161) lies on these lines: {1, 17418}, {30, 511}, {693, 17894}, {1022, 1219}, {1577, 4397}, {1635, 4765}, {1734, 4017}, {2517, 4768}, {3239, 4024}, {3762, 4811}, {3835, 4804}, {4025, 17161}, {4041, 21189}, {4086, 4791}, {4391, 4404}, {4449, 21173}, {4500, 4928}, {4516, 24224}, {4560, 5214}, {4818, 23770}, {4820, 4841}, {4913, 7662}, {6129, 14838}, {7317, 23838}, {7649, 17734}, {7658, 21186}, {21201, 23757}

X(28161) = isogonal conjugate of X(28162)


X(28162) =  ISOGONAL CONJUGATE OF X(28161)

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

X(28162) lies on the circumcircle and these lines:

X(28162) = isogonal conjugate of X(28161)
X(28162) = circumcircle-antipode of X(28163)


X(28163) =  CIRCUMCIRCLE-ANTIPODE OF X(28162)

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

X(28163) lies on the circumcircle and these lines:

X(28163) = isogonal conjugate of X(28164)
X(28163) = circumcircle-antipode of X(28162)


X(28164) =  ISOGONAL CONJUGATE OF X(28163)

Barycentrics    1/((3 a^4 - 6 a^2 b^2 + 3 b^4 - a^3 c + a^2 b c + a b^2 c - b^3 c + 3 a^2 c^2 - 2 a b c^2 + 3 b^2 c^2 + a c^3 + b c^3 - 6 c^4) (3 a^4 - a^3 b + 3 a^2 b^2 + a b^3 - 6 b^4 + a^2 b c - 2 a b^2 c + b^3 c - 6 a^2 c^2 + a b c^2 + 3 b^2 c^2 - b c^3 + 3 c^4)) : :

X(28164) lies on these lines: {1, 3146}, {3, 3634}, {4, 1125}, {5, 17502}, {8, 5059}, {10, 20}, {30, 511}, {40, 3529}, {145, 9589}, {226, 12943}, {354, 950}, {355, 1657}, {376, 3828}, {381, 10165}, {382, 946}, {388, 4314}, {411, 5267}, {497, 4315}, {546, 13624}, {548, 9956}, {549, 10172}, {550, 6684}, {551, 1699}, {631, 18492}, {910, 5199}, {944, 3635}, {958, 12511}, {962, 3244}, {993, 7580}, {997, 1750}, {1131, 13888}, {1132, 13942}, {1210, 4299}, {1323, 4872}, {1385, 3627}, {1420, 5225}, {1478, 4304}, {1479, 4311}, {1490, 22836}, {1698, 3522}, {1737, 4316}, {1742, 30116}, {2071, 9590}, {2951, 9623}, {3008, 6999}, {3091, 7987}, {3241, 16191}, {3474, 5727}, {3476, 4342}, {3486, 3671}, {3488, 5542}, {3520, 9626}, {3523, 7989}, {3534, 26446}, {3579, 15704}, {3583, 21578}, {3585, 6895}, {3586, 4293}, {3600, 21625}, {3601, 3947}, {3616, 17578}, {3622, 10248}, {3624, 3832}, {3625, 7991}, {3632, 20070}, {3654, 15685}, {3655, 15684}, {3679, 9778}, {3681, 6743}, {3740, 18250}, {3751, 14927}, {3753, 5918}, {3754, 9943}, {3822, 8727}, {3830, 5886}, {3833, 11227}, {3839, 7988}, {3845, 11230}, {3853, 9955}, {3874, 12680}, {3878, 12688}, {3884, 9856}, {3911, 15326}, {4015, 9947}, {4067, 12528}, {4084, 15071}, {4190, 8582}, {4292, 5902}, {4294, 9613}, {4305, 9612}, {4313, 5290}, {4324, 10039}, {4333, 10573}, {4701, 5881}, {4745, 5657}, {4746, 11362}, {4880, 13243}, {5049, 12577}, {5073, 5882}, {5080, 6745}, {5122, 12019}, {5251, 7411}, {5274, 13462}, {5281, 5726}, {5441, 13407}, {5450, 6985}, {5586, 18221}, {5603, 15682}, {5790, 15681}, {5794, 18249}, {5818, 17538}, {5883, 10167}, {5903, 9961}, {5919, 6284}, {5927, 10176}, {6245, 6869}, {6256, 6851}, {6700, 6836}, {6833, 20104}, {6834, 20107}, {6840, 15017}, {6868, 12617}, {6912, 15931}, {6951, 18406}, {7406, 29571}, {7424, 18653}, {7671, 12573}, {7982, 11541}, {8185, 11413}, {8703, 11231}, {8983, 23251}, {9583, 23249}, {9591, 12087}, {9655, 21620}, {9708, 11495}, {9779, 25055}, {9780, 16192}, {9940, 16616}, {10430, 18391}, {10439, 10454}, {10722, 21636}, {10723, 11599}, {10724, 21630}, {10727, 15735}, {10728, 12119}, {10730, 11814}, {10733, 13605}, {10884, 30143}, {10902, 21669}, {11192, 12580}, {11194, 24386}, {11195, 12582}, {11203, 12579}, {11211, 12578}, {11217, 12581}, {11222, 12574}, {11223, 12576}, {11709, 12295}, {11720, 13202}, {11827, 18908}, {12053, 12953}, {12082, 15177}, {12103, 18357}, {12244, 12407}, {12520, 30147}, {12558, 25466}, {12650, 22837}, {12767, 20085}, {12779, 17845}, {13464, 22793}, {13607, 22791}, {13971, 23261}, {15680, 24987}, {17539, 25982}, {17690, 25967}, {17706, 24470}, {19645, 20106}, {19877, 21734}, {20067, 26015}

X(28164) = isogonal conjugate of X(28163)


X(28165) =  POINT POLLUX(1,-4)

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

X(28165) lies on these lines: {30, 511}, {650, 4838}, {2516, 6590}, {4024, 4944}, {4820, 4988}, {6129, 8043}

X(28165) = isogonal conjugate of X(28166)


X(28166) =  ISOGONAL CONJUGATE OF X(28165)

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

X(28166) lies on the circumcircle and these lines:

X(28166) = isogonal conjugate of X(28165)
X(28166) = circumcircle-antipode of X(28167)


X(28167) =  CIRCUMCIRCLE-ANTIPODE OF X(28167)

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

X(28167) lies on the circumcircle and these lines:

X(28167) = isogonal conjugate of X(28168)
X(28167) = circumcircle-antipode of X(28166)


X(28168) =  ISOGONAL CONJUGATE OF X(28167)

Barycentrics    1/((4 a^4 - 8 a^2 b^2 + 4 b^4 - a^3 c + a^2 b c + a b^2 c - b^3 c + 4 a^2 c^2 - 2 a b c^2 + 4 b^2 c^2 + a c^3 + b c^3 - 8 c^4) (4 a^4 - a^3 b + 4 a^2 b^2 + a b^3 - 8 b^4 + a^2 b c - 2 a b^2 c + b^3 c - 8 a^2 c^2 + a b c^2 + 4 b^2 c^2 - b c^3 + 4 c^4)) : :

X(28168) lies on these lines: {1, 5073}, {3, 7989}, {4, 5550}, {10, 15704}, {20, 5818}, {30, 511}, {40, 17800}, {165, 15681}, {355, 3529}, {376, 11231}, {381, 17502}, {382, 1385}, {546, 10171}, {548, 19925}, {550, 9956}, {942, 10483}, {962, 11541}, {1125, 3853}, {1657, 3579}, {1698, 15696}, {3146, 5603}, {3534, 5587}, {3543, 5886}, {3576, 3830}, {3583, 5126}, {3627, 4297}, {3653, 9779}, {3655, 9812}, {3656, 15640}, {3817, 15687}, {3828, 15691}, {3843, 7987}, {3845, 10165}, {3858, 19862}, {4293, 18527}, {4299, 17728}, {4316, 5122}, {5045, 7354}, {5076, 8227}, {5657, 15683}, {5731, 15682}, {5734, 7967}, {6684, 12103}, {6925, 18407}, {7743, 21578}, {7988, 14269}, {8703, 10175}, {9589, 18526}, {9590, 18859}, {10172, 12100}, {10246, 15684}, {10572, 11246}, {10721, 11699}, {10728, 22935}, {12102, 12571}, {12512, 18357}, {12811, 19878}, {12943, 24929}, {12953, 24928}

X(28168) = isogonal conjugate of X(28167)


X(28169) =  POINT POLLUX(1,-5)

Barycentrics    (b - c) (a - 5 (b + c)) : :

X(28169) lies on these lines: {30, 511}, {693, 4768}, {2529, 14351}, {4397, 4815}, {4404, 7650}, {4521, 4838}, {4763, 7662}, {4776, 4804}, {4789, 21180}, {4828, 20907}, {8689, 17494}

X(28169) = isogonal conjugate of X(28170)


X(28170) =  ISOGONAL CONJUGATE OF X(28169)

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

X(28170) lies on the circumcircle and these lines:

X(28170) = isogonal conjugate of X(28169)
X(28170) = circumcircle-antipode of X(28171)


X(28171) =  CIRCUMCIRCLE-ANTIPODE OF X(28170)

Barycentrics    a^2 (5 a^4 - 10 a^2 b^2 + 5 b^4 - a^3 c + a^2 b c + a b^2 c - b^3 c + 5 a^2 c^2 - 2 a b c^2 + 5 b^2 c^2 + a c^3 + b c^3 - 10 c^4) (5 a^4 - a^3 b + 5 a^2 b^2 + a b^3 - 10 b^4 + a^2 b c - 2 a b^2 c + b^3 c - 10 a^2 c^2 + a b c^2 + 5 b^2 c^2 - b c^3 + 5 c^4) : :

X(28171) lies on the circumcircle and these lines:

X(28171) = isogonal conjugate of X(28172)
X(28171) = circumcircle-antipode of X(28170)


X(28172) =  ISOGONAL CONJUGATE OF X(28171)

Barycentrics    1/((5 a^4 - 10 a^2 b^2 + 5 b^4 - a^3 c + a^2 b c + a b^2 c - b^3 c + 5 a^2 c^2 - 2 a b c^2 + 5 b^2 c^2 + a c^3 + b c^3 - 10 c^4) (5 a^4 - a^3 b + 5 a^2 b^2 + a b^3 - 10 b^4 + a^2 b c - 2 a b^2 c + b^3 c - 10 a^2 c^2 + a b c^2 + 5 b^2 c^2 - b c^3 + 5 c^4)) : :

X(28172) lies on these lines: {3, 10172}, {4, 3624}, {10, 1657}, {20, 5587}, {30, 511}, {40, 4678}, {165, 11001}, {355, 17800}, {376, 10175}, {382, 4297}, {548, 3634}, {550, 11231}, {551, 15684}, {944, 11541}, {946, 3146}, {950, 10483}, {1125, 3627}, {1698, 17538}, {1699, 15682}, {3522, 18492}, {3529, 5657}, {3534, 10164}, {3543, 3576}, {3817, 3830}, {3828, 15686}, {3843, 19862}, {3845, 10171}, {3850, 19878}, {3853, 12571}, {3911, 4316}, {4292, 17706}, {4304, 12943}, {4311, 12953}, {4312, 14563}, {4314, 9655}, {4315, 9668}, {4333, 4848}, {5073, 13464}, {5493, 18525}, {5660, 10728}, {5790, 15685}, {5882, 20057}, {6705, 6869}, {6999, 29607}, {7354, 17609}, {7464, 9590}, {8227, 17578}, {8983, 22644}, {9615, 23253}, {9624, 10248}, {9626, 12086}, {9812, 15640}, {9956, 12103}, {10283, 22793}, {11230, 15687}, {12511, 18761}, {12512, 15704}, {12699, 13607}, {13971, 22615}, {15681, 26446}, {16173, 21578}

X(28172) = isogonal conjugate of X(28171)


X(28173) =  CIRCUMCIRCLE-ANTIPODE OF X(8701)

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

X(28173) lies on the circumcircle and these lines:

X(28173) = isogonal conjugate of X(28174)
X(28173) = circumcircle-antipode of X(8701)


X(28174) =  ISOGONAL CONJUGATE OF X(28173)

Barycentrics    1/((a^4 - 2 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 + 4 a b c^2 + b^2 c^2 - 2 a c^3 - 2 b c^3 - 2 c^4) (a^4 + 2 a^3 b + a^2 b^2 - 2 a b^3 - 2 b^4 - 2 a^2 b c + 4 a b^2 c - 2 b^3 c - 2 a^2 c^2 - 2 a b c^2 + b^2 c^2 + 2 b c^3 + c^4)) : :

X(28174) lies on these lines: {1, 550}, {3, 962}, {4, 3617}, {5, 40}, {7, 6767}, {8, 382}, {10, 546}, {11, 484}, {12, 11010}, {20, 1482}, {26, 9911}, {30, 511}, {36, 1387}, {46, 496}, {55, 5719}, {65, 12433}, {79, 15888}, {100, 5180}, {140, 946}, {145, 3529}, {165, 549}, {354, 15170}, {355, 3627}, {376, 10246}, {381, 5657}, {390, 15934}, {411, 11849}, {495, 1836}, {547, 3817}, {548, 1385}, {551, 17502}, {553, 5049}, {631, 18493}, {632, 8227}, {942, 10624}, {944, 1657}, {999, 3474}, {1056, 18541}, {1058, 5708}, {1125, 3530}, {1145, 5080}, {1146, 5134}, {1155, 15325}, {1159, 3488}, {1319, 24465}, {1320, 20067}, {1483, 7982}, {1484, 5535}, {1537, 6905}, {1565, 5195}, {1572, 15048}, {1702, 19117}, {1703, 19116}, {1737, 5183}, {1770, 3057}, {1788, 9669}, {1829, 13488}, {1902, 6756}, {2077, 11729}, {2093, 5722}, {2098, 4299}, {2099, 4302}, {2448, 23477}, {2449, 23517}, {3035, 11813}, {3058, 5902}, {3146, 12245}, {3241, 15681}, {3245, 3583}, {3295, 3475}, {3428, 6914}, {3522, 10595}, {3528, 3622}, {3534, 5731}, {3576, 3656}, {3587, 5805}, {3624, 14869}, {3628, 6684}, {3649, 3746}, {3654, 3845}, {3655, 15686}, {3679, 15687}, {3748, 11551}, {3820, 24703}, {3828, 11737}, {3843, 5818}, {3850, 9956}, {3851, 9780}, {3853, 11362}, {3857, 7989}, {3861, 19925}, {3877, 11112}, {3881, 26201}, {3884, 26200}, {3911, 7743}, {3927, 5082}, {3940, 17784}, {4292, 9957}, {4297, 10222}, {4300, 5453}, {4316, 12735}, {4318, 18455}, {4324, 11009}, {4330, 10543}, {4338, 10404}, {4511, 9945}, {4816, 5691}, {4973, 21630}, {5010, 15950}, {5011, 17747}, {5045, 12575}, {5048, 21578}, {5055, 9779}, {5057, 17757}, {5066, 10175}, {5073, 12645}, {5079, 19877}, {5128, 9614}, {5131, 5298}, {5250, 8728}, {5432, 18393}, {5445, 7173}, {5538, 6265}, {5541, 16128}, {5550, 15720}, {5658, 5758}, {5659, 16139}, {5687, 11415}, {5697, 7354}, {5698, 9708}, {5709, 10943}, {5720, 7994}, {5734, 15696}, {5759, 6913}, {5763, 6985}, {5771, 8727}, {5812, 10942}, {5874, 12698}, {5875, 12697}, {5882, 11278}, {5887, 7957}, {5903, 6284}, {6197, 15763}, {6240, 12135}, {6244, 6911}, {6449, 13902}, {6450, 13959}, {6713, 10225}, {6840, 10738}, {6909, 22765}, {6924, 10310}, {6996, 29590}, {7373, 9785}, {7491, 25413}, {7526, 8193}, {7575, 9625}, {7580, 10679}, {7973, 9833}, {7978, 12121}, {7984, 20127}, {7988, 15699}, {8666, 13463}, {9441, 15251}, {9620, 18907}, {9624, 16192}, {9668, 18391}, {10109, 10172}, {10165, 12100}, {10483, 10944}, {10573, 12953}, {10593, 24914}, {10724, 19914}, {11024, 16853}, {11224, 19710}, {11373, 15803}, {11495, 20330}, {11522, 15712}, {11529, 15935}, {11544, 13407}, {12085, 12410}, {12512, 13464}, {12571, 12811}, {12647, 12943}, {12672, 20420}, {12705, 26921}, {12732, 17484}, {13375, 16142}, {13912, 13925}, {13975, 13993}, {15177, 18570}, {17044, 17729}, {17504, 25055}, {17563, 19861}, {17682, 26790}, {17800, 18526}, {19907, 24466}

X(28174) = isogonal conjugate of X(28173)


X(28175) =  POINT POLLUX(2,3)

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

X(28175) lies on these lines: {30, 511}, {676, 21102}, {1022, 1224}, {1635, 2527}, {2977, 3004}, {3837, 4824}, {3960, 8043}, {4122, 23729}, {4608, 18004}, {4841, 4893}, {6545, 28602}, {14315, 17420}, {14413, 21112}

X(28175) = isogonal conjugate of X(28176)


X(28176) =  ISOGONAL CONJUGATE OF X(28175)

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

X(28176) lies on the circumcircle and these lines:

X(28176) = isogonal conjugate of X(28175)
X(28176) = circumcircle-antipode of X(28177)


X(28177) =  CIRCUMCIRCLE-ANTIPODE OF X(28176)

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

X(28177) lies on the circumcircle and these lines:

X(28177) = isogonal conjugate of X(28178)
X(28177) = circumcircle-antipode of X(28176)


X(28178) =  ISOGONAL CONJUGATE OF X(28171)

Barycentrics    1/((3 a^4 - 6 a^2 b^2 + 3 b^4 + 2 a^3 c - 2 a^2 b c - 2 a b^2 c + 2 b^3 c + 3 a^2 c^2 + 4 a b c^2 + 3 b^2 c^2 - 2 a c^3 - 2 b c^3 - 6 c^4) (3 a^4 + 2 a^3 b + 3 a^2 b^2 - 2 a b^3 - 6 b^4 - 2 a^2 b c + 4 a b^2 c - 2 b^3 c - 6 a^2 c^2 - 2 a b c^2 + 3 b^2 c^2 + 2 b c^3 + 3 c^4)) : :

X(28178) lies on these lines: {1, 15704}, {3, 5284}, {5, 165}, {8, 5073}, {10, 3853}, {11, 5131}, {20, 10246}, {30, 511}, {40, 3627}, {140, 3817}, {354, 1770}, {381, 9778}, {382, 5690}, {390, 18541}, {484, 12019}, {546, 3579}, {547, 10164}, {548, 946}, {549, 1699}, {550, 3576}, {551, 15691}, {944, 17800}, {962, 1657}, {1385, 12103}, {1387, 15326}, {1482, 3529}, {1483, 11224}, {1698, 3858}, {1836, 5719}, {3146, 12702}, {3474, 9668}, {3522, 18493}, {3530, 9955}, {3534, 5603}, {3543, 5790}, {3616, 15696}, {3628, 18483}, {3634, 12811}, {3649, 4330}, {3650, 5178}, {3656, 19710}, {3830, 5657}, {3845, 26446}, {3850, 6684}, {3851, 10248}, {3861, 9956}, {4292, 5049}, {4294, 6147}, {4333, 12701}, {5054, 9779}, {5066, 11231}, {5076, 5818}, {5180, 10609}, {5183, 11545}, {5428, 15911}, {5493, 18480}, {5587, 15687}, {5731, 15681}, {5886, 8703}, {5902, 6284}, {5919, 18990}, {6840, 22938}, {7967, 15683}, {7988, 11539}, {8227, 15712}, {9589, 16200}, {9625, 15646}, {9911, 12084}, {10124, 10171}, {10172, 11737}, {10283, 15686}, {10386, 10389}, {11230, 12100}, {12102, 19925}, {14869, 16192}, {15888, 16118}, {18525, 20070}, {18530, 21454}

X(28178) = isogonal conjugate of X(28177)


X(28179) =  POINT POLLUX(2,5)

Barycentrics    (b - c) (2 a + 5 (b + c)) : :

X(28179) lies on these lines: {30, 511}, {1268, 3004}, {4728, 4824}, {4789, 23770}, {4833, 21343}, {4988, 6544}, {14475, 28602}

X(28179) = isogonal conjugate of X(28180)


X(28180) =  ISOGONAL CONJUGATE OF X(28179)

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

X(28180) lies on the circumcircle and these lines:

X(28180) = isogonal conjugate of X(28179)
X(28180) = circumcircle-antipode of X(28181)


X(28181) =  CIRCUMCIRCLE-ANTIPODE OF X(28180)

Barycentrics    a^2 (5 a^4 - 10 a^2 b^2 + 5 b^4 + 2 a^3 c - 2 a^2 b c - 2 a b^2 c + 2 b^3 c + 5 a^2 c^2 + 4 a b c^2 + 5 b^2 c^2 - 2 a c^3 - 2 b c^3 - 10 c^4) (5 a^4 + 2 a^3 b + 5 a^2 b^2 - 2 a b^3 - 10 b^4 - 2 a^2 b c + 4 a b^2 c - 2 b^3 c - 10 a^2 c^2 - 2 a b c^2 + 5 b^2 c^2 + 2 b c^3 + 5 c^4) : :

X(28181) lies on the circumcircle and these lines:

X(28181) = isogonal conjugate of X(28182)
X(28181) = circumcircle-antipode of X(28180)


X(28182) =  ISOGONAL CONJUGATE OF X(28181)

Barycentrics    1/((5 a^4 - 10 a^2 b^2 + 5 b^4 + 2 a^3 c - 2 a^2 b c - 2 a b^2 c + 2 b^3 c + 5 a^2 c^2 + 4 a b c^2 + 5 b^2 c^2 - 2 a c^3 - 2 b c^3 - 10 c^4) (5 a^4 + 2 a^3 b + 5 a^2 b^2 - 2 a b^3 - 10 b^4 - 2 a^2 b c + 4 a b^2 c - 2 b^3 c - 10 a^2 c^2 - 2 a b c^2 + 5 b^2 c^2 + 2 b c^3 + 5 c^4)) : :

X(28182) lies on these lines: {3, 9779}, {5, 19872}, {20, 5901}, {30, 511}, {165, 3845}, {382, 5657}, {496, 4333}, {546, 11231}, {548, 10165}, {549, 7988}, {550, 5886}, {946, 12103}, {962, 17800}, {1387, 4316}, {1482, 5059}, {1483, 9589}, {1657, 5731}, {1699, 8703}, {1770, 12433}, {3146, 5690}, {3526, 10248}, {3530, 18483}, {3534, 9812}, {3576, 15686}, {3579, 3853}, {3627, 5587}, {3628, 12512}, {3634, 3856}, {3817, 12100}, {3830, 9778}, {3850, 10172}, {3861, 6684}, {4302, 5719}, {4312, 15935}, {5057, 9945}, {5066, 10164}, {5073, 6361}, {5603, 15681}, {5790, 15682}, {6284, 18398}, {9579, 10386}, {9956, 12102}, {10171, 11812}, {10175, 14893}, {10246, 11001}, {10283, 12699}, {11038, 18541}, {11541, 20070}, {12571, 16239}, {15171, 17609}, {15326, 16173}, {15687, 26446}, {15690, 17502}, {17538, 18493}

X(28182) = isogonal conjugate of X(28181)


X(28183) = POINT POLLUX(2,-3)

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

X(28183) lies on these lines: {30, 511}, {656, 14315}, {1635, 4024}, {1639, 4931}, {2490, 4765}, {2527, 6590}, {3004, 17161}, {3700, 4893}, {3837, 4804}, {4036, 4768}, {4500, 17069}, {4528, 4825}, {4560, 25569}, {4820, 14321}, {4928, 21196}, {14421, 24099}, {17420, 24457}, {21121, 23775}

X(28183) = isogonal conjugate of X(28184)


X(28184) =  ISOGONAL CONJUGATE OF X(28183)

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

X(28184) lies on the circumcircle and these lines:

X(28184) = isogonal conjugate of X(28183)
X(28184) = circumcircle-antipode of X(28185)


X(28185) =  CIRCUMCIRCLE-ANTIPODE OF X(28184)

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

X(28185) lies on the circumcircle and these lines:

X(28185) = isogonal conjugate of X(28186)
X(28185) = circumcircle-antipode of X(28184)


X(28186) =  ISOGONAL CONJUGATE OF X(28185)

Barycentrics    1/((3 a^4 - 6 a^2 b^2 + 3 b^4 - 2 a^3 c + 2 a^2 b c + 2 a b^2 c - 2 b^3 c + 3 a^2 c^2 - 4 a b c^2 + 3 b^2 c^2 + 2 a c^3 + 2 b c^3 - 6 c^4) (3 a^4 - 2 a^3 b + 3 a^2 b^2 + 2 a b^3 - 6 b^4 + 2 a^2 b c - 4 a b^2 c + 2 b^3 c - 6 a^2 c^2 + 2 a b c^2 + 3 b^2 c^2 - 2 b c^3 + 3 c^4)) : :

X(28186) lies on these lines: {1, 3627}, {3, 5260}, {4, 3622}, {5, 3576}, {8, 1657}, {10, 548}, {20, 4678}, {30, 511}, {36, 12019}, {40, 15704}, {80, 5131}, {140, 4297}, {165, 355}, {354, 10572}, {376, 5790}, {381, 5731}, {382, 944}, {411, 26321}, {546, 1385}, {547, 10165}, {549, 5587}, {551, 14893}, {632, 7987}, {946, 3853}, {950, 5049}, {962, 5073}, {1125, 3850}, {1155, 11545}, {1387, 3583}, {1478, 5719}, {1482, 3146}, {1483, 12678}, {1698, 15712}, {1699, 3655}, {3241, 15684}, {3476, 9668}, {3486, 6147}, {3529, 12702}, {3530, 9956}, {3534, 5657}, {3543, 7967}, {3579, 12103}, {3612, 10592}, {3616, 3843}, {3617, 17538}, {3628, 13624}, {3634, 12108}, {3653, 7988}, {3654, 19710}, {3679, 15686}, {3740, 17647}, {3828, 14891}, {3830, 5603}, {3845, 5886}, {3858, 8227}, {3861, 9955}, {4305, 9654}, {4315, 18527}, {4881, 17533}, {5059, 12245}, {5066, 11230}, {5072, 5550}, {5080, 10609}, {5270, 10543}, {5441, 15888}, {5538, 12738}, {5882, 22793}, {5902, 7354}, {5919, 15171}, {6361, 12645}, {6840, 10742}, {6909, 18524}, {7580, 18519}, {8144, 21147}, {8703, 26446}, {9590, 15646}, {9613, 10389}, {9778, 15681}, {9779, 14269}, {9798, 12084}, {9897, 15228}, {9945, 17757}, {9961, 25413}, {10106, 15172}, {10124, 10172}, {10171, 11737}, {10202, 20420}, {10483, 10950}, {10575, 16980}, {10595, 17578}, {10721, 12898}, {10741, 15735}, {10864, 24467}, {11231, 12100}, {11698, 12119}, {11709, 11801}, {12102, 15178}, {12135, 18560}, {12248, 12747}, {12680, 24475}, {12812, 19862}, {13211, 14677}, {13373, 16616}, {13407, 15174}, {14892, 19883}, {15325, 21578}, {15950, 18513}, {23046, 25055}

X(28186) = isogonal conjugate of X(28185)


X(28187) =  POINT POLLUX(1,-5)

Barycentrics    (b - c) (2 a - 5 (b + c)) : :

X(28187) lies on these lines: {30, 511}, {1639, 4024}, {2527, 4838}, {4453, 17161}, {4958, 4988}

X(28187) = isogonal conjugate of X(28188)


X(28188) =  ISOGONAL CONJUGATE OF X(28187)

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

X(28188) lies on the circumcircle and these lines:

X(28188) = isogonal conjugate of X(28187)
X(28188) = circumcircle-antipode of X(28189)


X(28189) =  CIRCUMCIRCLE-ANTIPODE OF X(28188)

Barycentrics    a^2 (5 a^4 - 10 a^2 b^2 + 5 b^4 - 2 a^3 c + 2 a^2 b c + 2 a b^2 c - 2 b^3 c + 5 a^2 c^2 - 4 a b c^2 + 5 b^2 c^2 + 2 a c^3 + 2 b c^3 - 10 c^4) (5 a^4 - 2 a^3 b + 5 a^2 b^2 + 2 a b^3 - 10 b^4 + 2 a^2 b c - 4 a b^2 c + 2 b^3 c - 10 a^2 c^2 + 2 a b c^2 + 5 b^2 c^2 - 2 b c^3 + 5 c^4) : :

X(28189) lies on the circumcircle and these lines:

X(28189) = isogonal conjugate of X(28190)
X(28189) = circumcircle-antipode of X(28188)


X(28190) =  ISOGONAL CONJUGATE OF X(28189)

Barycentrics    1/((5 a^4 - 10 a^2 b^2 + 5 b^4 - 2 a^3 c + 2 a^2 b c + 2 a b^2 c - 2 b^3 c + 5 a^2 c^2 - 4 a b c^2 + 5 b^2 c^2 + 2 a c^3 + 2 b c^3 - 10 c^4) (5 a^4 - 2 a^3 b + 5 a^2 b^2 + 2 a b^3 - 10 b^4 + 2 a^2 b c - 4 a b^2 c + 2 b^3 c - 10 a^2 c^2 + 2 a b c^2 + 5 b^2 c^2 - 2 b c^3 + 5 c^4)) : :

X(28190) lies on these lines: {3, 9342}, {5, 7987}, {8, 17800}, {10, 12103}, {20, 5790}, {30, 511}, {145, 11541}, {165, 15686}, {355, 15704}, {382, 5603}, {546, 4297}, {547, 17502}, {548, 10164}, {550, 5691}, {632, 18492}, {944, 5073}, {1125, 3861}, {1385, 3853}, {1483, 16189}, {1657, 5690}, {1699, 3627}, {3146, 7967}, {3475, 9655}, {3529, 18525}, {3530, 19925}, {3543, 10246}, {3576, 3845}, {3616, 5076}, {3624, 3857}, {3817, 14893}, {3830, 5731}, {3850, 10171}, {5059, 12702}, {5066, 10165}, {5080, 9945}, {5587, 8703}, {5657, 15681}, {5818, 15696}, {5886, 15687}, {6840, 22799}, {7354, 12433}, {7988, 23046}, {7989, 14869}, {9613, 10386}, {9812, 15684}, {9955, 12102}, {10172, 11812}, {10175, 12100}, {10483, 11246}, {10572, 24470}, {12019, 15326}, {12690, 20067}, {17578, 18493}, {17609, 18990}

X(28190) = isogonal conjugate of X(28189)


X(28191) =  POINT POLLUX(3,5)

Barycentrics    (b - c) (3 a + 5 b + 5 c) : :

X(28191) lies on these lines: {30, 511}, {4086, 4801}, {4462, 4815}, {4521, 4841}


X(28192) =  POINT POLLUX(3,-5)

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

X(28192) lies on these lines:


X(28193) =  CIRCUMCIRCLE-ANTIPODE OF X(28192)

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

X(28193) lies on the circumcircle and these lines:

X(28193) = isogonal conjugate of X(28194)
X(28193) = circumcircle-antipode of X(28192)


X(28194) =  ISOGONAL CONJUGATE OF X(28193)

Barycentrics    1/((a^4 - 2 a^2 b^2 + b^4 + 3 a^3 c - 3 a^2 b c - 3 a b^2 c + 3 b^3 c + a^2 c^2 + 6 a b c^2 + b^2 c^2 - 3 a c^3 - 3 b c^3 - 2 c^4) (a^4 + 3 a^3 b + a^2 b^2 - 3 a b^3 - 2 b^4 - 3 a^2 b c + 6 a b^2 c - 3 b^3 c - 2 a^2 c^2 - 3 a b c^2 + b^2 c^2 + 3 b c^3 + c^4)) : :

X(28194) lies on these lines: {1, 376}, {2, 40}, {3, 551}, {4, 3679}, {5, 3828}, {8, 3543}, {10, 381}, {11, 5183}, {20, 3241}, {30, 511}, {46, 10072}, {65, 3058}, {84, 6766}, {142, 3587}, {145, 15683}, {165, 3524}, {226, 5119}, {355, 3830}, {390, 11529}, {428, 1902}, {484, 3582}, {497, 2093}, {547, 3634}, {548, 15178}, {549, 1125}, {550, 10222}, {631, 11522}, {908, 5180}, {942, 12575}, {944, 11001}, {950, 5903}, {993, 28444}, {999, 4342}, {1056, 4312}, {1058, 3339}, {1155, 5298}, {1158, 3928}, {1210, 11238}, {1385, 8703}, {1387, 5122}, {1479, 4848}, {1480, 3946}, {1482, 3534}, {1483, 19710}, {1519, 15017}, {1537, 6174}, {1572, 7739}, {1616, 24171}, {1621, 7688}, {1651, 12696}, {1697, 4295}, {1698, 5071}, {1699, 3545}, {1702, 19054}, {1703, 19053}, {1737, 3245}, {1770, 5697}, {1788, 9614}, {1836, 11237}, {2077, 13587}, {2094, 10860}, {2098, 4311}, {2099, 4304}, {2136, 16127}, {3057, 4292}, {3086, 5128}, {3090, 9588}, {3146, 5881}, {3244, 8148}, {3295, 3671}, {3333, 9785}, {3340, 4294}, {3428, 16370}, {3488, 14563}, {3522, 5734}, {3523, 9624}, {3576, 9778}, {3584, 11010}, {3616, 15692}, {3617, 18492}, {3624, 15702}, {3625, 15684}, {3626, 15687}, {3635, 11278}, {3636, 13624}, {3651, 3746}, {3817, 5055}, {3839, 5587}, {3845, 4745}, {3878, 28452}, {3881, 13369}, {3884, 16004}, {3895, 5905}, {3902, 4001}, {3929, 12705}, {3982, 11552}, {4293, 7962}, {4298, 9957}, {4302, 25415}, {4324, 11280}, {4421, 6796}, {4512, 17561}, {4665, 18505}, {4677, 5691}, {4691, 14893}, {4857, 6903}, {4870, 4995}, {5048, 15326}, {5054, 5886}, {5057, 6735}, {5059, 20049}, {5066, 9956}, {5082, 12526}, {5195, 9436}, {5248, 28466}, {5258, 21669}, {5325, 12514}, {5450, 11194}, {5537, 6905}, {5542, 6767}, {5550, 15721}, {5655, 12778}, {5698, 9623}, {5708, 21625}, {5709, 6705}, {5731, 16200}, {5735, 6916}, {5758, 6260}, {5790, 14269}, {5812, 11236}, {5836, 12572}, {5860, 12698}, {5861, 12697}, {5884, 24473}, {5901, 12100}, {5919, 11246}, {6054, 9881}, {6055, 12258}, {6172, 11372}, {6175, 24987}, {6244, 16417}, {6261, 6769}, {6738, 15171}, {6744, 15172}, {6851, 24391}, {6869, 12437}, {6985, 8715}, {6999, 17310}, {7811, 12497}, {7957, 12672}, {7967, 11224}, {7970, 12117}, {7987, 10595}, {8074, 17747}, {8158, 12114}, {8169, 9709}, {8724, 21636}, {9441, 13635}, {9569, 19648}, {9580, 18391}, {9582, 13902}, {9856, 20117}, {9860, 12243}, {9909, 9911}, {9943, 12005}, {10031, 12119}, {10124, 19878}, {10171, 11231}, {10246, 15688}, {10247, 15689}, {10267, 12511}, {10283, 17502}, {10310, 16371}, {10445, 17281}, {10707, 14217}, {10914, 12527}, {11011, 15338}, {11012, 17549}, {11111, 12651}, {11112, 14110}, {11207, 12458}, {11208, 12459}, {11230, 11539}, {11235, 12616}, {11239, 12703}, {11240, 12704}, {11415, 21075}, {11496, 16418}, {11599, 11632}, {12150, 12197}, {12152, 22841}, {12153, 22842}, {12248, 12653}, {12515, 21630}, {12577, 24470}, {13199, 13253}, {13605, 20126}, {13846, 13912}, {13847, 13975}, {15228, 21578}, {15678, 16113}, {15694, 18493}, {15698, 16192}, {15700, 15808}, {15908, 17530}, {15941, 18589}, {16189, 17538}, {16483, 24177}, {16616, 18250}, {21163, 22475}, {21168, 24644}, {24466, 25485}

X(28194) = isogonal conjugate of X(28193)


X(28195) =  POINT POLLUX(3,2)

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

X(28195) lies on these lines: {30, 511}, {650, 2457}, {656, 23738}, {659, 21115}, {693, 18158}, {2605, 4724}, {3733, 4960}, {4024, 4949}, {4057, 4378}, {4394, 4841}, {4408, 7199}, {4411, 20949}, {4782, 7192}, {4790, 4988}, {4813, 4931}, {6129, 21103}, {16507, 21140}

X(28195) = isogonal conjugate of X(28196)


X(28196) =  ISOGONAL CONJUGATE OF X(28195)

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

X(28196) lies on the circumcircle and these lines:

X(28196) = isogonal conjugate of X(28195)
X(28196) = circumcircle-antipode of X(28197)


X(28197) =  CIRCUMCIRCLE-ANTIPODE OF X(28196)

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

X(28197) lies on the circumcircle and these lines:

X(28197) = isogonal conjugate of X(28198)
X(28197) = circumcircle-antipode of X(28196)


X(28198) =  ISOGONAL CONJUGATE OF X(28197)

Barycentrics    1/((2 a^4 - 4 a^2 b^2 + 2 b^4 + 3 a^3 c - 3 a^2 b c - 3 a b^2 c + 3 b^3 c + 2 a^2 c^2 + 6 a b c^2 + 2 b^2 c^2 - 3 a c^3 - 3 b c^3 - 4 c^4) (2 a^4 + 3 a^3 b + 2 a^2 b^2 - 3 a b^3 - 4 b^4 - 3 a^2 b c + 6 a b^2 c - 3 b^3 c - 4 a^2 c^2 - 3 a b c^2 + 2 b^2 c^2 + 3 b c^3 + 2 c^4)) : :

X(28198) lies on these lines: {1, 3534}, {2, 3579}, {3, 9589}, {4, 3654}, {5, 5493}, {8, 15682}, {10, 3845}, {20, 3655}, {30, 511}, {35, 4870}, {40, 381}, {46, 11238}, {165, 5054}, {355, 3543}, {376, 962}, {382, 7991}, {547, 6684}, {548, 13464}, {549, 946}, {550, 4301}, {551, 8703}, {553, 5045}, {573, 16590}, {942, 3058}, {944, 15683}, {1125, 12100}, {1155, 3582}, {1319, 15228}, {1327, 13911}, {1328, 13973}, {1482, 15681}, {1647, 14000}, {1657, 7982}, {1698, 19709}, {1699, 5055}, {1770, 5434}, {1836, 10056}, {1902, 7576}, {2093, 9668}, {2098, 4333}, {3241, 11001}, {3295, 4654}, {3303, 4338}, {3428, 28444}, {3524, 5886}, {3545, 9812}, {3576, 15688}, {3583, 5183}, {3616, 19708}, {3624, 15701}, {3627, 11362}, {3634, 10109}, {3653, 5603}, {3671, 10386}, {3679, 3830}, {3746, 16117}, {3748, 11552}, {3817, 15699}, {3828, 5066}, {3839, 5657}, {3844, 25561}, {3851, 9588}, {4295, 10385}, {4297, 15686}, {4312, 6767}, {4316, 5048}, {4324, 11011}, {4421, 6985}, {4677, 18525}, {4745, 12101}, {4995, 12047}, {5049, 11246}, {5073, 5881}, {5119, 11237}, {5122, 5298}, {5128, 9669}, {5180, 5440}, {5195, 17078}, {5550, 15719}, {5587, 14269}, {5690, 15687}, {5691, 15684}, {5694, 7957}, {5734, 17538}, {5882, 15704}, {5901, 12512}, {6033, 9881}, {6174, 12611}, {6583, 9943}, {7987, 14093}, {8148, 15685}, {8227, 15694}, {9580, 18527}, {9911, 14070}, {10032, 16138}, {10072, 12701}, {10164, 11539}, {10165, 17504}, {10246, 15689}, {10706, 12778}, {10707, 12515}, {11496, 28466}, {11540, 19878}, {11737, 12571}, {12042, 12258}, {12575, 24470}, {14893, 19925}, {15326, 25405}, {15693, 18493}, {15700, 16192}, {15713, 19862}, {24715, 27637}, {26200, 28458}

X(28198) = isogonal conjugate of X(28197)


X(28199) =  POINT POLLUX(3,4)

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

X(28199) lies on these lines: {30, 511}, {3669, 8043}, {4036, 4801}, {4106, 4608}, {4380, 14779}, {4394, 4988}, {9508, 21115}

X(28199) = isogonal conjugate of X(28200)


X(28200) =  ISOGONAL CONJUGATE OF X(28199)

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

X(28200) lies on the circumcircle and these lines:

X(28200) = isogonal conjugate of X(28199)
X(28200) = circumcircle-antipode of X(28201)


X(28201) =  CIRCUMCIRCLE-ANTIPODE OF X(28200)

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

X(28201) lies on the circumcircle and these lines:

X(28201) = isogonal conjugate of X(28202)
X(28201) = circumcircle-antipode of X(28201)


X(28202) =  ISOGONAL CONJUGATE OF X(28201)

Barycentrics    1/((4 a^4 - 8 a^2 b^2 + 4 b^4 + 3 a^3 c - 3 a^2 b c - 3 a b^2 c + 3 b^3 c + 4 a^2 c^2 + 6 a b c^2 + 4 b^2 c^2 - 3 a c^3 - 3 b c^3 - 8 c^4) (4 a^4 + 3 a^3 b + 4 a^2 b^2 - 3 a b^3 - 8 b^4 - 3 a^2 b c + 6 a b^2 c - 3 b^3 c - 8 a^2 c^2 - 3 a b c^2 + 4 b^2 c^2 + 3 b c^3 + 4 c^4)) : :

X(28202) lies on these lines: {1, 15681}, {2, 22793}, {10, 15687}, {20, 3656}, {30, 511}, {40, 3830}, {165, 5055}, {355, 15682}, {376, 3616}, {381, 1698}, {382, 3679}, {546, 3828}, {547, 18483}, {549, 9955}, {550, 551}, {553, 15171}, {946, 8703}, {962, 3655}, {1385, 3534}, {1482, 15685}, {1657, 9589}, {1699, 5054}, {1770, 3058}, {1902, 18559}, {3241, 3529}, {3474, 18527}, {3524, 9812}, {3543, 3617}, {3545, 9778}, {3576, 15689}, {3582, 5122}, {3623, 15683}, {3624, 15700}, {3627, 5493}, {3634, 11737}, {3817, 11539}, {3839, 26446}, {3845, 9956}, {3851, 19876}, {4292, 15170}, {4297, 19710}, {4301, 15704}, {4316, 25405}, {4333, 24928}, {4668, 12702}, {4816, 18525}, {5066, 6684}, {5073, 7991}, {5298, 7743}, {5550, 15715}, {5886, 10304}, {5901, 15690}, {7982, 17800}, {7987, 15695}, {8227, 15693}, {9779, 15709}, {9860, 12355}, {10109, 12571}, {10164, 15699}, {10172, 14892}, {10175, 23046}, {11522, 15696}, {12100, 12512}, {12101, 19925}, {12103, 13464}, {14093, 18493}, {14269, 19875}, {15640, 20070}, {15686, 22791}, {15688, 17502}, {15701, 16192}, {17504, 19883}

X(28202) = isogonal conjugate of X(28201)


X(28203) =  CIRCUMCIRCLE-ANTIPODE OF X(8697)

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

X(28203) lies on the circumcircle and these lines:

X(28203) = isogonal conjugate of X(28204)
X(28203) = circumcircle-antipode of X(8697)


X(28204) =  ISOGONAL CONJUGATE OF X(28203)

Barycentrics    1/((2 a^4 - 4 a^2 b^2 + 2 b^4 - 3 a^3 c + 3 a^2 b c + 3 a b^2 c - 3 b^3 c + 2 a^2 c^2 - 6 a b c^2 + 2 b^2 c^2 + 3 a c^3 + 3 b c^3 - 4 c^4) (2 a^4 - 3 a^3 b + 2 a^2 b^2 + 3 a b^3 - 4 b^4 + 3 a^2 b c - 6 a b^2 c + 3 b^3 c - 4 a^2 c^2 + 3 a b c^2 + 2 b^2 c^2 - 3 b c^3 + 2 c^4)) : :

X(28204) lies on these lines: {1, 381}, {2, 355}, {3, 3679}, {4, 1392}, {5, 551}, {8, 376}, {10, 549}, {11, 25405}, {30, 511}, {35, 26321}, {36, 9897}, {40, 3534}, {55, 18519}, {56, 18518}, {80, 1319}, {104, 13587}, {119, 17533}, {140, 3828}, {145, 3543}, {150, 17078}, {165, 15688}, {214, 5123}, {382, 7982}, {388, 18517}, {497, 18516}, {500, 10459}, {546, 13464}, {547, 1125}, {550, 11362}, {553, 18990}, {942, 5434}, {946, 1483}, {950, 15170}, {958, 28466}, {962, 15682}, {999, 5727}, {1056, 15933}, {1317, 12611}, {1386, 5476}, {1388, 10826}, {1478, 18407}, {1482, 3830}, {1538, 10707}, {1644, 19515}, {1657, 7991}, {1698, 15694}, {1699, 10247}, {1737, 5126}, {1829, 7576}, {1837, 10072}, {2077, 12331}, {2646, 3584}, {3058, 9957}, {3146, 20049}, {3244, 15687}, {3295, 18761}, {3340, 9655}, {3476, 5722}, {3488, 8232}, {3524, 5731}, {3526, 19876}, {3545, 5886}, {3560, 4428}, {3576, 5054}, {3583, 5048}, {3585, 11011}, {3616, 5071}, {3617, 15692}, {3621, 6361}, {3624, 15703}, {3625, 15686}, {3627, 4301}, {3632, 12702}, {3633, 8148}, {3634, 10124}, {3635, 14893}, {3636, 11737}, {3652, 15678}, {3746, 13743}, {3812, 26089}, {3814, 11698}, {3817, 10283}, {3824, 30147}, {3829, 10943}, {3839, 5603}, {3843, 11522}, {3851, 9624}, {3860, 12571}, {3911, 11545}, {4297, 4669}, {4316, 5183}, {4330, 5559}, {4421, 12114}, {4511, 12738}, {4654, 9613}, {4668, 14093}, {4691, 14891}, {4701, 15691}, {4745, 6684}, {4930, 28609}, {4995, 10039}, {5010, 18515}, {5045, 10106}, {5055, 5587}, {5066, 5901}, {5073, 9589}, {5076, 16189}, {5122, 21578}, {5176, 5440}, {5252, 10056}, {5441, 26202}, {5450, 26086}, {5493, 15704}, {5534, 12650}, {5655, 11699}, {5657, 10304}, {5694, 14872}, {5777, 11113}, {5836, 13145}, {5885, 12675}, {5887, 11114}, {6054, 9884}, {6246, 22835}, {6261, 11235}, {6264, 12747}, {6265, 10031}, {6583, 7686}, {6735, 10609}, {6740, 7478}, {6797, 18838}, {6841, 15888}, {6985, 12513}, {6996, 17310}, {7962, 9668}, {7987, 15693}, {8227, 19709}, {8724, 9864}, {9780, 15702}, {9798, 14070}, {9834, 11208}, {9835, 11207}, {9845, 19706}, {9856, 26200}, {10164, 17504}, {10165, 11539}, {10175, 15699}, {10225, 12247}, {10267, 16418}, {10269, 16417}, {10284, 12672}, {10525, 12667}, {10543, 22798}, {10698, 23960}, {10902, 28443}, {11001, 12245}, {11112, 26201}, {11194, 11500}, {11274, 19907}, {11366, 18497}, {11367, 18495}, {11491, 17549}, {11499, 16371}, {11632, 13178}, {12119, 13528}, {12512, 15690}, {12515, 12531}, {12619, 18857}, {12688, 23340}, {12751, 22935}, {12943, 18499}, {13211, 20126}, {13463, 18243}, {14831, 16980}, {14848, 16475}, {15071, 25413}, {15623, 19254}, {15670, 24987}, {15677, 22936}, {15862, 19919}, {16370, 22758}, {16496, 18440}, {17530, 26470}, {17532, 18446}, {17577, 21740}, {17606, 21842}, {18421, 18541}, {22799, 24042}, {22937, 28460}

X(28204) = isogonal conjugate of X(28203)


X(28205) =  POINT POLLUX(3,-4)

Barycentrics    (b - c) (3 a - 4 b - 4 c) : :

X(28205) lies on these lines: {30, 511}, {650, 4931}, {2516, 4976}, {4024, 4394}, {4106, 17161}, {4120, 4820}, {4765, 14425}, {4773, 6590}, {4790, 4838}, {4841, 4949}

X(28205) = isogonal conjugate of X(28206)


X(28206) =  ISOGONAL CONJUGATE OF X(28205)

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

X(28206) lies on the circumcircle and these lines:

X(28206) = isogonal conjugate of X(28205)
X(28206) = circumcircle-antipode of X(28207)


X(28207) =  CIRCUMCIRCLE-ANTIPODE OF X(28206)

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

X(28207) lies on the circumcircle and these lines:

X(28207) = isogonal conjugate of X(28208)
X(28207) = circumcircle-antipode of X(28206)


X(28208) =  ISOGONAL CONJUGATE OF X(28207)

Barycentrics    1/((4 a^4 - 8 a^2 b^2 + 4 b^4 - 3 a^3 c + 3 a^2 b c + 3 a b^2 c - 3 b^3 c + 4 a^2 c^2 - 6 a b c^2 + 4 b^2 c^2 + 3 a c^3 + 3 b c^3 - 8 c^4) (4 a^4 - 3 a^3 b + 4 a^2 b^2 + 3 a b^3 - 8 b^4 + 3 a^2 b c - 6 a b^2 c + 3 b^3 c - 8 a^2 c^2 + 3 a b c^2 + 4 b^2 c^2 - 3 b c^3 + 4 c^4)) : :

X(28208) lies on these lines: {1, 3830}, {2, 13624}, {3, 19875}, {4, 3655}, {5, 19883}, {8, 11001}, {10, 8703}, {20, 3654}, {30, 511}, {40, 15681}, {80, 5122}, {145, 15640}, {165, 15689}, {355, 376}, {381, 1385}, {382, 10222}, {547, 19925}, {549, 4297}, {551, 3845}, {944, 3543}, {946, 15687}, {1125, 5066}, {1482, 15684}, {1657, 5881}, {1698, 15693}, {1829, 18559}, {3058, 26088}, {3241, 12699}, {3524, 11231}, {3534, 3579}, {3545, 3653}, {3576, 5055}, {3582, 5126}, {3583, 25405}, {3585, 4870}, {3617, 15697}, {3627, 5882}, {3634, 11812}, {3817, 23046}, {3828, 12100}, {3839, 5886}, {3853, 13464}, {4654, 9655}, {4669, 19710}, {4677, 12702}, {4745, 15690}, {5045, 5434}, {5054, 5587}, {5073, 7982}, {5076, 11522}, {5183, 9897}, {5258, 16117}, {5298, 21578}, {5690, 15686}, {5790, 15688}, {5818, 15692}, {5901, 14893}, {6261, 11567}, {6985, 11194}, {7686, 26201}, {7987, 15694}, {7989, 15703}, {7991, 17800}, {9780, 15698}, {9875, 12188}, {9884, 10722}, {9940, 28452}, {10031, 10728}, {10106, 15170}, {10165, 15699}, {10171, 14892}, {10175, 11539}, {10246, 14269}, {10304, 26446}, {10706, 11699}, {10711, 22935}, {10902, 28453}, {11237, 24929}, {11238, 24928}, {11278, 18526}, {11362, 15704}, {11500, 26086}, {11711, 22566}, {12101, 18483}, {12258, 22515}, {12512, 15691}, {13145, 17579}, {13178, 14830}, {15678, 26202}, {15701, 19876}, {16417, 18491}, {16418, 18761}, {17525, 22798}, {18492, 19709}

X(28208) = isogonal conjugate of X(28207)


X(28209) =  POINT POLLUX(4,1)

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

X(28209) lies on these lines: {30, 511}, {86, 4833}, {659, 20142}, {661, 1213}, {1022, 24857}, {1638, 4724}, {1769, 23738}, {2487, 4932}, {2526, 2977}, {2529, 4521}, {2533, 21714}, {3766, 4828}, {3837, 4776}, {4367, 4491}, {4369, 6707}, {4374, 21606}, {4448, 14475}, {4728, 4806}, {4733, 4761}, {4813, 14321}, {4841, 4979}, {5592, 24099}, {9182, 17930}, {13602, 23838}, {20949, 21433}, {21104, 23728}

X(28209) = isogonal conjugate of X(28206)


X(28210) =  ISOGONAL CONJUGATE OF X(28209)

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

X(28210) lies on the circumcircle and these lines:

X(28210) = isogonal conjugate of X(28209)
X(28210) = circumcircle-antipode of X(28211)


X(28211) =  CIRCUMCIRCLE-ANTIPODE OF X(28210)

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

X(28211) lies on the circumcircle and these lines:

X(28221) = isogonal conjugate of X(28212)
X(28221) = circumcircle-antipode of X(28210)


X(28212) =  ISOGONAL CONJUGATE OF X(28211)

Barycentrics    1/((a^4 - 2 a^2 b^2 + b^4 + 4 a^3 c - 4 a^2 b c - 4 a b^2 c + 4 b^3 c + a^2 c^2 + 8 a b c^2 + b^2 c^2 - 4 a c^3 - 4 b c^3 - 2 c^4) (a^4 + 4 a^3 b + a^2 b^2 - 4 a b^3 - 2 b^4 - 4 a^2 b c + 8 a b^2 c - 4 b^3 c - 2 a^2 c^2 - 4 a b c^2 + b^2 c^2 + 4 b c^3 + c^4)) : :

X(28212) lies on these lines: {1, 548}, {3, 3622}, {4, 4678}, {5, 962}, {8, 3627}, {10, 3850}, {11, 3245}, {20, 1483}, {30, 511}, {40, 140}, {65, 15172}, {145, 1657}, {165, 3656}, {355, 3853}, {376, 10247}, {382, 12245}, {390, 1159}, {484, 15325}, {546, 5587}, {547, 7988}, {549, 5603}, {550, 1482}, {551, 14891}, {632, 18493}, {944, 15704}, {946, 3628}, {1125, 12108}, {1145, 5057}, {1155, 1387}, {1317, 4316}, {1385, 5493}, {1697, 6147}, {1698, 12812}, {1699, 3654}, {3146, 12645}, {3241, 15686}, {3428, 7508}, {3529, 18526}, {3530, 3579}, {3534, 7967}, {3583, 11545}, {3616, 15712}, {3617, 3843}, {3623, 17538}, {3655, 11224}, {3679, 14893}, {3746, 16137}, {3817, 10109}, {3845, 5790}, {3856, 18483}, {3858, 5818}, {3861, 11362}, {3935, 12732}, {4297, 11278}, {4424, 17726}, {4867, 6154}, {5119, 17718}, {5128, 11373}, {5180, 17757}, {5536, 12515}, {5559, 16118}, {5697, 18990}, {5708, 9785}, {5902, 15170}, {5903, 15171}, {6684, 16239}, {6767, 11038}, {6913, 21168}, {7982, 12103}, {7984, 14677}, {8236, 15934}, {8703, 9778}, {9590, 12105}, {9911, 17714}, {9955, 10172}, {9956, 12811}, {9957, 24470}, {10124, 11230}, {10164, 11812}, {10175, 11737}, {10624, 12433}, {11009, 15338}, {11276, 11281}, {11531, 18481}, {11544, 15888}, {12084, 12410}, {12102, 18480}, {12512, 15178}, {12735, 21578}, {14890, 19883}, {14892, 19875}, {15690, 16200}, {15759, 17502}, {19512, 29607}

X(28212) = isogonal conjugate of X(28211)


X(28213) =  POINT POLLUX(4,3)

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

X(28213) lies on these lines: {30, 511}, {1635, 4841}, {2490, 4893}, {5592, 9269}, {14588, 17930}, {18014, 21135}, {21106, 21120}

X(28213) = isogonal conjugate of X(28214)


X(28214) =  ISOGONAL CONJUGATE OF X(28213)

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

X(28214) lies on the circumcircle and these lines:

X(28214) = isogonal conjugate of X(28213)
X(28214) = circumcircle-antipode of X(28215)


X(28215) =  CIRCUMCIRCLE-ANTIPODE OF X(28214)

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

X(28215) lies on the circumcircle and these lines:

X(28215) = isogonal conjugate of X(28216)
X(28215) = circumcircle-antipode of X(28214)


X(28216) =  ISOGONAL CONJUGATE OF X(28215)

Barycentrics    1/((3 a^4 - 6 a^2 b^2 + 3 b^4 + 4 a^3 c - 4 a^2 b c - 4 a b^2 c + 4 b^3 c + 3 a^2 c^2 + 8 a b c^2 + 3 b^2 c^2 - 4 a c^3 - 4 b c^3 - 6 c^4) (3 a^4 + 4 a^3 b + 3 a^2 b^2 - 4 a b^3 - 6 b^4 - 4 a^2 b c + 8 a b^2 c - 4 b^3 c - 6 a^2 c^2 - 4 a b c^2 + 3 b^2 c^2 + 4 b c^3 + 3 c^4)) : :

X(28216) lies on these lines: {1, 12103}, {5, 6361}, {10, 3861}, {20, 10247}, {30, 511}, {40, 546}, {140, 165}, {145, 17800}, {354, 15172}, {376, 10283}, {382, 20070}, {547, 1699}, {548, 3576}, {549, 9778}, {550, 962}, {946, 3530}, {1482, 15704}, {1483, 1657}, {1658, 9911}, {1770, 5919}, {3245, 11545}, {3529, 8148}, {3579, 3628}, {3617, 5076}, {3621, 11541}, {3627, 12702}, {3654, 12101}, {3656, 15690}, {3746, 11544}, {3845, 5657}, {3850, 5493}, {3853, 5690}, {3856, 9956}, {3857, 9780}, {4295, 10386}, {4330, 15174}, {5049, 10624}, {5059, 18526}, {5066, 26446}, {5073, 12245}, {5131, 15325}, {5183, 12019}, {5587, 14893}, {5603, 8703}, {5731, 15686}, {5790, 15687}, {5886, 12100}, {5901, 17502}, {5902, 15171}, {6147, 10389}, {7967, 15681}, {9779, 15699}, {9955, 16239}, {10109, 11231}, {10124, 10164}, {10165, 14891}, {10595, 15696}, {11224, 18481}, {11230, 11812}, {11246, 15170}, {12102, 18357}, {12811, 18483}, {15712, 18493}

X(28216) = isogonal conjugate of X(28215)


X(28217) =  POINT POLLUX(4,-1)

Barycentrics    (b - c) (4 a - b - c) : :

X(28217) lies on these lines: {21, 3733}, {30, 511}, {79, 23838}, {649, 1639}, {661, 4984}, {2254, 4806}, {2487, 3835}, {2527, 3239}, {3649, 4017}, {3700, 4958}, {3798, 4940}, {4057, 27086}, {4106, 21183}, {4406, 23794}, {4453, 4897}, {4773, 4893}, {4782, 4925}, {4790, 4944}, {4813, 4976}, {4840, 7253}, {4874, 7659}, {4905, 4992}, {5441, 14812}, {6701, 23808}, {10543, 14284}, {13250, 14610}, {14315, 23800}, {21143, 21834}

X(28217) = isogonal conjugate of X(28218)


X(28218) =  ISOGONAL CONJUGATE OF X(28217)

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

X(28218) lies on the circumcircle and these lines:

X(28218) = isogonal conjugate of X(28217)
X(28218) = circumcircle-antipode of X(28219)


X(28219) =  CIRCUMCIRCLE-ANTIPODE OF X(28218)

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

X(28219) lies on the circumcircle and these lines:

X(28219) = isogonal conjugate of X(28220)
X(28219) = circumcircle-antipode of X(28218)


X(28220) =  POINT POLLUX(5,2)

Barycentrics    (b - c) (5 a + 2 b + 2 c) : :

X(28220) lies on these lines: {30, 511}, {4378, 4491}, {4406, 21433}, {4408, 4828}, {4411, 21606}, {4448, 6548}, {4776, 21146}, {4833, 4960}, {23345, 23598}


X(28221) =  POINT POLLUX(4,-3)

Barycentrics    (b - c) (4 a - 3 b - 3 c) : :

X(28221) lies on these lines: {30, 511}, {656, 24457}, {1635, 2490}, {4467, 21297}, {4893, 4976}, {4928, 17069}, {4931, 4984}, {4944, 14425}, {6615, 21714}, {9269, 24099}, {13251, 14610}

X(28221) = isogonal conjugate of X(28220)


X(28222) =  ISOGONAL CONJUGATE OF X(28221)

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

X(28222) lies on the circumcircle and these lines:

X(28222) = isogonal conjugate of X(28221)
X(28222) = circumcircle-antipode of X(28223)


X(28223) =  CIRCUMCIRCLE-ANTIPODE OF X(28222)

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

X(28223) lies on the circumcircle and these lines:

X(28223) = isogonal conjugate of X(28224)
X(28223) = circumcircle-antipode of X(28222)


X(28224) =  ISOGONAL CONJUGATE OF X(28223)

Barycentrics    1/((3 a^4 - 6 a^2 b^2 + 3 b^4 - 4 a^3 c + 4 a^2 b c + 4 a b^2 c - 4 b^3 c + 3 a^2 c^2 - 8 a b c^2 + 3 b^2 c^2 + 4 a c^3 + 4 b c^3 - 6 c^4) (3 a^4 - 4 a^3 b + 3 a^2 b^2 + 4 a b^3 - 6 b^4 + 4 a^2 b c - 8 a b^2 c + 4 b^3 c - 6 a^2 c^2 + 4 a b c^2 + 3 b^2 c^2 - 4 b c^3 + 3 c^4)) : :

X(28224) lies on these lines: {1, 546}, {3, 3617}, {4, 1483}, {5, 944}, {8, 550}, {10, 3530}, {20, 12645}, {30, 511}, {36, 11545}, {40, 4816}, {80, 15325}, {104, 18524}, {140, 355}, {145, 382}, {165, 548}, {381, 7967}, {547, 3655}, {549, 5731}, {551, 11737}, {632, 5818}, {946, 3861}, {1056, 15935}, {1317, 3583}, {1319, 12019}, {1385, 3628}, {1388, 10593}, {1482, 3627}, {1484, 1532}, {1657, 12245}, {1658, 9798}, {1699, 14893}, {3146, 8148}, {3241, 15687}, {3244, 22793}, {3529, 3621}, {3614, 24926}, {3622, 3851}, {3654, 15690}, {3656, 12101}, {3754, 26201}, {3817, 3850}, {3843, 10595}, {3845, 5603}, {3853, 5691}, {3856, 9955}, {3858, 18493}, {3859, 18492}, {5049, 10106}, {5066, 5886}, {5131, 9897}, {5176, 10609}, {5270, 16137}, {5554, 17563}, {5657, 8703}, {5902, 10950}, {5919, 10572}, {6102, 16980}, {6147, 9613}, {6224, 17757}, {6735, 9945}, {6767, 8543}, {6882, 11698}, {6905, 12773}, {6909, 12331}, {6914, 18519}, {6924, 18518}, {7508, 22758}, {7526, 8192}, {7555, 15177}, {7966, 18540}, {7988, 14892}, {9778, 15686}, {9779, 23046}, {9780, 14869}, {9956, 16239}, {10109, 11230}, {10124, 10165}, {10164, 14891}, {10222, 12102}, {10902, 12104}, {10944, 15171}, {11041, 18541}, {11224, 12699}, {11231, 11812}, {11491, 26321}, {12006, 23841}, {12100, 26446}, {12108, 13624}, {12135, 13488}, {12702, 15704}, {12811, 15178}, {13375, 17637}, {15174, 15888}, {17266, 19512}, {17800, 20070}

X(28224) = isogonal conjugate of X(28223)


X(28225) =  POINT POLLUX(5,1)

Barycentrics    (b - c) (5 a + b + c) : :

X(28225) lies on these lines: {30, 511}, {1459, 4794}, {3239, 4813}, {4453, 13246}, {4724, 4932}, {4765, 4979}, {4811, 4978}, {4823, 4985}, {4905, 17420}, {4960, 7253}, {6615, 23738}, {7192, 17218}, {14812, 21201}, {23774, 24224}

X(28225) = isogonal conjugate of X(28226)


X(28226) =  ISOGONAL CONJUGATE OF X(28225)

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

X(28226) lies on the circumcircle and these lines:

X(28226) = isogonal conjugate of X(28225)
X(28226) = circumcircle-antipode of X(28227)


X(28227) =  CIRCUMCIRCLE-ANTIPODE OF X(28226)

Barycentrics    a^2 (a^4 - 2 a^2 b^2 + b^4 + 5 a^3 c - 5 a^2 b c - 5 a b^2 c + 5 b^3 c + a^2 c^2 + 10 a b c^2 + b^2 c^2 - 5 a c^3 - 5 b c^3 - 2 c^4) (a^4 + 5 a^3 b + a^2 b^2 - 5 a b^3 - 2 b^4 - 5 a^2 b c + 10 a b^2 c - 5 b^3 c - 2 a^2 c^2 - 5 a b c^2 + b^2 c^2 + 5 b c^3 + c^4) : :

X(28227) lies on the circumcircle and these lines:

X(28227) = isogonal conjugate of X(28228)
X(28227) = circumcircle-antipode of X(28226)


X(28228) =  ISOGONAL CONJUGATE OF X(28227)

Barycentrics    1/((a^4 - 2 a^2 b^2 + b^4 + 5 a^3 c - 5 a^2 b c - 5 a b^2 c + 5 b^3 c + a^2 c^2 + 10 a b c^2 + b^2 c^2 - 5 a c^3 - 5 b c^3 - 2 c^4) (a^4 + 5 a^3 b + a^2 b^2 - 5 a b^3 - 2 b^4 - 5 a^2 b c + 10 a b^2 c - 5 b^3 c - 2 a^2 c^2 - 5 a b c^2 + b^2 c^2 + 5 b c^3 + c^4)) : :

X(28228) lies on these lines: {1, 3522}, {3, 3636}, {4, 3626}, {7, 9819}, {8, 9589}, {10, 962}, {20, 3244}, {30, 511}, {40, 631}, {57, 4342}, {65, 6744}, {144, 4915}, {165, 551}, {355, 4746}, {376, 16200}, {390, 18421}, {550, 11278}, {553, 5919}, {632, 6684}, {946, 1656}, {997, 7994}, {1482, 15696}, {1697, 3475}, {2093, 11019}, {3057, 4298}, {3146, 3632}, {3295, 12511}, {3339, 9785}, {3340, 4314}, {3474, 4315}, {3523, 15808}, {3576, 19708}, {3579, 13464}, {3622, 16192}, {3625, 5691}, {3635, 4297}, {3654, 10175}, {3656, 10165}, {3678, 9856}, {3679, 9812}, {3817, 3828}, {3843, 4691}, {3858, 5690}, {3869, 6743}, {3878, 7957}, {3881, 9943}, {3884, 12436}, {3892, 10167}, {3911, 5183}, {3918, 5806}, {3956, 10157}, {4134, 15104}, {4292, 5697}, {4304, 25415}, {4345, 13462}, {4678, 10248}, {4701, 12245}, {4745, 5587}, {4848, 12701}, {4973, 17613}, {5049, 10178}, {5059, 20050}, {5119, 13405}, {5180, 6735}, {5199, 17747}, {5731, 11224}, {5734, 7987}, {5758, 10915}, {5836, 18250}, {5882, 8148}, {5886, 15694}, {5903, 6738}, {6736, 11415}, {6767, 11495}, {6769, 22836}, {7580, 25439}, {7673, 12573}, {8158, 8666}, {9779, 19875}, {9802, 12767}, {9955, 12812}, {9957, 12577}, {10246, 14093}, {10247, 15695}, {10283, 15711}, {11010, 13411}, {11219, 21630}, {11522, 19862}, {12053, 17728}, {12527, 14923}, {15172, 17706}, {15714, 17502}

X(28228) = isogonal conjugate of X(28227)


X(28229) =  POINT POLLUX(5,3)

Barycentrics    (b - c) (5 a + 3 b + 3 c) : :

X(28229) lies on these lines: {30, 511}, {4801, 4985}, {5592, 14421}, {20316, 23789}, {23738, 23800}, {24720, 25627}

X(28229) = isogonal conjugate of X(28230)


X(28232) =  ISOGONAL CONJUGATE OF X(28229)

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

X(28230) lies on the circumcircle and these lines:

X(28230) = isogonal conjugate of X(28229)
X(28230) = circumcircle-antipode of X(28231)


X(28231) =  CIRCUMCIRCLE-ANTIPODE OF X(28230)

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

X(28231) lies on the circumcircle and these lines:

X(28231) = isogonal conjugate of X(28232)
X(28231) = circumcircle-antipode of X(28230)


X(28232) =  ISOGONAL CONJUGATE OF X(28231)

Barycentrics    1/((3 a^4 - 6 a^2 b^2 + 3 b^4 + 5 a^3 c - 5 a^2 b c - 5 a b^2 c + 5 b^3 c + 3 a^2 c^2 + 10 a b c^2 + 3 b^2 c^2 - 5 a c^3 - 5 b c^3 - 6 c^4) (3 a^4 + 5 a^3 b + 3 a^2 b^2 - 5 a b^3 - 6 b^4 - 5 a^2 b c + 10 a b^2 c - 5 b^3 c - 6 a^2 c^2 - 5 a b c^2 + 3 b^2 c^2 + 5 b c^3 + 3 c^4)) : :

X(28232) lies on these lines: {1, 4114}, {3, 15808}, {10, 3843}, {20, 13607}, {30, 511}, {40, 3091}, {165, 631}, {354, 10624}, {548, 3636}, {551, 14093}, {632, 3579}, {962, 3522}, {1125, 15712}, {1656, 3817}, {1657, 3244}, {1699, 5071}, {3529, 11531}, {3626, 3627}, {3634, 12812}, {3656, 15695}, {3858, 22793}, {3859, 9956}, {4292, 5919}, {4295, 10389}, {4297, 10247}, {4301, 10246}, {4678, 11362}, {5049, 12575}, {5076, 12702}, {5603, 19708}, {5882, 11224}, {5886, 15693}, {7967, 16191}, {9778, 10165}, {9911, 16195}, {10164, 15694}, {10386, 12563}, {11230, 15713}, {11278, 15704}, {12512, 17502}, {15171, 17706}, {19709, 26446}

X(28232) = isogonal conjugate of X(28231)


X(28233) =  CIRCUMCIRCLE-ANTIPODE OF X(6014)

Barycentrics    a^2 (a^4 - 2 a^2 b^2 + b^4 - 5 a^3 c + 5 a^2 b c + 5 a b^2 c - 5 b^3 c + a^2 c^2 - 10 a b c^2 + b^2 c^2 + 5 a c^3 + 5 b c^3 - 2 c^4) (a^4 - 5 a^3 b + a^2 b^2 + 5 a b^3 - 2 b^4 + 5 a^2 b c - 10 a b^2 c + 5 b^3 c - 2 a^2 c^2 + 5 a b c^2 + b^2 c^2 - 5 b c^3 + c^4) : :

X(28233) lies on the circumcircle and these lines:

X(28233) = isogonal conjugate of X(28234)
X(28233) = circumcircle-antipode of X(6014)


X(28234) =  ISOGONAL CONJUGATE OF X(28233)

Barycentrics    1/((a^4 - 2 a^2 b^2 + b^4 - 5 a^3 c + 5 a^2 b c + 5 a b^2 c - 5 b^3 c + a^2 c^2 - 10 a b c^2 + b^2 c^2 + 5 a c^3 + 5 b c^3 - 2 c^4) (a^4 - 5 a^3 b + a^2 b^2 + 5 a b^3 - 2 b^4 + 5 a^2 b c - 10 a b^2 c + 5 b^3 c - 2 a^2 c^2 + 5 a b c^2 + b^2 c^2 - 5 b c^3 + c^4)) : :

X(28234) lies on these lines: {1, 631}, {2, 16200}, {3, 3244}, {4, 3632}, {5, 3626}, {8, 908}, {10, 1482}, {20, 20050}, {30, 511}, {36, 10087}, {40, 145}, {55, 17010}, {63, 12703}, {80, 24297}, {104, 5537}, {140, 3636}, {165, 7967}, {226, 12647}, {355, 3625}, {551, 10247}, {632, 1125}, {944, 3633}, {950, 5697}, {960, 13600}, {962, 3621}, {993, 10679}, {1056, 18421}, {1145, 6745}, {1155, 1317}, {1158, 6762}, {1159, 5542}, {1210, 2098}, {1320, 10265}, {1385, 3635}, {1483, 3579}, {1512, 4867}, {1698, 10595}, {1699, 4677}, {1737, 16173}, {2077, 13278}, {2093, 3476}, {2099, 17718}, {2325, 4752}, {3036, 5087}, {3146, 20054}, {3241, 3576}, {3245, 7972}, {3340, 21620}, {3421, 26333}, {3427, 3680}, {3488, 9819}, {3523, 20057}, {3526, 15808}, {3555, 5884}, {3617, 5734}, {3634, 5901}, {3654, 10164}, {3655, 14093}, {3656, 3817}, {3679, 5071}, {3689, 13996}, {3811, 12640}, {3814, 23513}, {3828, 11230}, {3858, 4701}, {3859, 12571}, {3871, 11012}, {3872, 6974}, {3878, 5795}, {3889, 15016}, {3892, 10202}, {3913, 6796}, {3935, 6326}, {3940, 7682}, {3982, 6951}, {4031, 6955}, {4084, 25413}, {4292, 10944}, {4297, 12702}, {4316, 13199}, {4342, 5722}, {4668, 5818}, {4691, 9956}, {4745, 10171}, {4746, 9955}, {4915, 5817}, {5057, 12531}, {5076, 12645}, {5080, 24042}, {5082, 26332}, {5121, 26727}, {5126, 12735}, {5267, 11849}, {5288, 6906}, {5330, 24982}, {5450, 10306}, {5493, 18481}, {5536, 5541}, {5705, 16204}, {5709, 12437}, {5730, 6736}, {5758, 12625}, {5770, 6705}, {5903, 10106}, {6260, 11523}, {6261, 6765}, {6603, 8074}, {6700, 8256}, {6737, 10914}, {6738, 9957}, {6743, 7686}, {6769, 12629}, {7743, 11545}, {7962, 18391}, {8158, 11500}, {8666, 11248}, {8715, 11249}, {9624, 9780}, {9778, 15697}, {9948, 11519}, {10039, 11009}, {10107, 12436}, {10445, 17299}, {10573, 12053}, {10624, 10950}, {10680, 25440}, {10703, 16870}, {10912, 12616}, {11011, 13411}, {11038, 11526}, {11280, 12047}, {11715, 25416}, {12247, 12653}, {12410, 16195}, {12447, 13374}, {12454, 12459}, {12455, 12458}, {12736, 18839}, {15711, 17502}, {16191, 19875}, {17753, 25719}, {18492, 20052}, {19914, 21630}, {20014, 20070}

X(28234) = isogonal conjugate of X(28233)


X(28235) =  CIRCUMCIRCLE-ANTIPODE OF X(8699)

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

X(28235) lies on the circumcircle and these lines:

X(28235) = isogonal conjugate of X(28236)
X(28235) = circumcircle-antipode of X(8699)


X(28236) =  ISOGONAL CONJUGATE OF X(28235)

Barycentrics    1/((3 a^4 - 6 a^2 b^2 + 3 b^4 - 5 a^3 c + 5 a^2 b c + 5 a b^2 c - 5 b^3 c + 3 a^2 c^2 - 10 a b c^2 + 3 b^2 c^2 + 5 a c^3 + 5 b c^3 - 6 c^4) (3 a^4 - 5 a^3 b + 3 a^2 b^2 + 5 a b^3 - 6 b^4 + 5 a^2 b c - 10 a b^2 c + 5 b^3 c - 6 a^2 c^2 + 5 a b c^2 + 3 b^2 c^2 - 5 b c^3 + 3 c^4)) : :

X(28236) lies on these lines: {1, 3091}, {3, 3626}, {4, 3244}, {5, 3636}, {8, 165}, {10, 631}, {20, 3632}, {30, 511}, {40, 3625}, {101, 5199}, {145, 4301}, {150, 1323}, {153, 3583}, {354, 6738}, {355, 1125}, {388, 12563}, {392, 15064}, {411, 5288}, {551, 5071}, {632, 1385}, {946, 3635}, {950, 5919}, {962, 3633}, {1012, 25439}, {1149, 5400}, {1482, 5076}, {1483, 3858}, {1519, 21630}, {1699, 3241}, {1706, 9845}, {1709, 3895}, {1737, 9897}, {2136, 10864}, {3090, 15808}, {3146, 11531}, {3476, 5727}, {3486, 10389}, {3586, 4342}, {3617, 7987}, {3621, 7991}, {3622, 7989}, {3623, 11522}, {3627, 11278}, {3654, 15695}, {3655, 3828}, {3671, 9613}, {3679, 5731}, {3681, 6737}, {3740, 5795}, {3754, 12675}, {3811, 12650}, {3832, 20057}, {3859, 9955}, {3878, 14872}, {3881, 7686}, {3884, 5777}, {3918, 9940}, {3935, 5538}, {4298, 5902}, {4304, 12647}, {4311, 10573}, {4315, 18391}, {4511, 5531}, {4669, 5657}, {4677, 15697}, {4678, 9588}, {4691, 6684}, {4701, 11362}, {4745, 15693}, {4746, 5690}, {4853, 12520}, {4856, 10445}, {4915, 5732}, {5049, 6744}, {5059, 20054}, {5126, 11545}, {5131, 9803}, {5176, 6745}, {5252, 13405}, {5267, 11491}, {5493, 12245}, {5534, 22836}, {5660, 10031}, {5697, 12528}, {5768, 21164}, {5787, 12437}, {5818, 19862}, {5839, 10443}, {5884, 10273}, {5886, 19709}, {6224, 6735}, {6244, 8168}, {6245, 10915}, {6261, 22837}, {6681, 20418}, {7406, 29605}, {7743, 12735}, {7972, 21635}, {7993, 20085}, {8666, 11500}, {8715, 12114}, {9798, 16195}, {9948, 12640}, {9956, 19878}, {10157, 10179}, {10176, 18250}, {10202, 12436}, {10222, 18483}, {10572, 12575}, {10595, 18492}, {10724, 26726}, {10742, 24042}, {10914, 12680}, {11231, 15713}, {12019, 25405}, {12812, 15178}, {14923, 15071}, {15674, 24987}, {17355, 24247}, {17857, 30144}, {18242, 24387}, {20053, 20070}

X(28236) = isogonal conjugate of X(28235)


X(28237) =  EULER LINE INTERCEPT OF X(195)X(11671)

Barycentrics    2 a^16-11 a^14 b^2+25 a^12 b^4-29 a^10 b^6+15 a^8 b^8+3 a^6 b^10-9 a^4 b^12+5 a^2 b^14-b^16-11 a^14 c^2+34 a^12 b^2 c^2-35 a^10 b^4 c^2+12 a^8 b^6 c^2-9 a^6 b^8 c^2+26 a^4 b^10 c^2-25 a^2 b^12 c^2+8 b^14 c^2+25 a^12 c^4-35 a^10 b^2 c^4+12 a^8 b^4 c^4-3 a^6 b^6 c^4-16 a^4 b^8 c^4+45 a^2 b^10 c^4-28 b^12 c^4-29 a^10 c^6+12 a^8 b^2 c^6-3 a^6 b^4 c^6-2 a^4 b^6 c^6-25 a^2 b^8 c^6+56 b^10 c^6+15 a^8 c^8-9 a^6 b^2 c^8-16 a^4 b^4 c^8-25 a^2 b^6 c^8-70 b^8 c^8+3 a^6 c^10+26 a^4 b^2 c^10+45 a^2 b^4 c^10+56 b^6 c^10-9 a^4 c^12-25 a^2 b^2 c^12-28 b^4 c^12+5 a^2 c^14+8 b^2 c^14-c^16 : :

See Tran Quang Hung and Peter Moses, Hyacinthos 28651.

X(28237) = lies on these lines: {2,3}, {195,11671}, {930,24573}, {1263,25044}, {6343,20424}, {10627,20327}, {15345,20414}

X(28237)= reflection of X(i) in X(j) for these {i,j}: {3,10285}, {4,20120}, {20,14142}, {10205,5501}, {10627,20327}, {15345,20414}, {27868,20030}






leftri  Collineation mappings involving Gemini triangle 80: X(28238) - X(28290)  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 80, as in centers X(28238)-X(28290). Then

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

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 18, 2018)

underbar




X(28238) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28238) lies on these lines:


X(28239) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28239) lies on these lines:


X(28240) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28240) lies on these lines:


X(28241) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28241) lies on these lines:


X(28242) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 80

Barycentrics    a (a^4 b + a^3 b^2 + a^4 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(28242) lies on these lines:


X(28243) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28243) lies on these lines:


X(28244) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28244) lies on these lines:


X(28245) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28245) lies on these lines:


X(28246) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28246) lies on these lines:


X(28247) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28247) lies on these lines:


X(28248) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28248) lies on these lines:


X(28249) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28249) lies on these lines:


X(28250) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28250) lies on these lines:


X(28251) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28251) lies on these lines:


X(28252) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28252) lies on these lines:


X(28253) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28253) lies on these lines:


X(28254) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28254) lies on these lines:


X(28255) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28255) lies on these lines:


X(28256) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28256) lies on these lines:


X(28257) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28257) lies on these lines:


X(28258) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28258) lies on these lines:


X(28259) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28259) lies on these lines:


X(28260) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28260) lies on these lines:


X(28261) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28261) lies on these lines:


X(28262) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28262) lies on these lines:


X(28263) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28263) lies on these lines:


X(28264) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28264) lies on these lines:


X(28265) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28265) lies on these lines:


X(28266) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(19), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28266) lies on these lines:


X(28267) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28267) lies on these lines:


X(28268) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28268) lies on these lines:


X(28269) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28269) lies on these lines:


X(28270) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(40), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28270) lies on these lines:


X(28271) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28271) lies on these lines:


X(28272) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28272) lies on these lines:


X(28273) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28273) lies on these lines:


X(28274) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28274) lies on these lines:


X(28275) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28275) lies on these lines:


X(28276) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28276) lies on these lines:


X(28277) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28277) lies on these lines:


X(28278) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28278) lies on these lines:


X(28279) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28279) lies on these lines:


X(28280) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28280) lies on these lines:


X(28281) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = GEMINI TRIANGLE 80

Barycentrics    a (2 a^4 b + 3 a^3 b^2 - a b^4 + 2 a^4 c + 4 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 - 11 a b^2 c^2 - 3 b^3 c^2 - 2 a b c^3 - 3 b^2 c^3 - a c^4 - b c^4) : :

X(28281) lies on these lines:


X(28282) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28282) lies on these lines:


X(28283) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28283) lies on these lines:


X(28284) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28284) lies on these lines:


X(28285) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28285) lies on these lines:


X(28286) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28286) lies on these lines:


X(28287) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28287) lies on these lines:


X(28288) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28288) lies on these lines:


X(28289) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28289) lies on these lines:


X(28290) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(24), WHERE A'B'C' = GEMINI TRIANGLE 80

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

X(28290) lies on these lines:






leftri  Points on circumcircle and line at infinity: X(28145) - X(28236)  rightri

Suppose that X = x : y : z is a point on the line at infinity. All the lines that meet in X are parallel, so that X can be regarded as a direction in the plane of the reference triangle ABC. Let X' be the isogonal conjugate of X, so that X' lies on the circumcircle. Let X'' be the circumcircle-antipode of X', and let X''' be its isogonal conjugate, on the line at infinity. As a direction, X''' is perpendicular to X. In this section, X is given by the form h(-2a^2 + b^2 + c^2) + k(-2bc + ab + ac) : : , where h and k are constants, and K is given the name Point Porrima(h,k). See also the preamble just before X(28145). (Clark Kimberling, November 19, 2018)

In the table below, Columns 1 and 2 show h and k.

Column 3. h(-2a^2 + b^2 + c^2) + k(-2bc + ab + ac ) : : , on infinity line, referenced below as x : y : z

Column 4. (isogonal conjugate of x : y : z) = a^2/x + b^2/y + c^2/z : : on circumcircle, referenced below as u : v : w

Column 5. (antipode of u : v : w) = (a^2+b^2-c^2)(a^2-b^2+c^2)u + 2a^2 (a^2-b^2-c^2)v + 2a^2 (a^2-b^2-c^2)w : : on circumcircle, referenced below as u1 : v1 : w1

Column 6. (isogonal conjugate of u1 : v1 : w1) = a^2/u1 + b^2/v1 + c^2/w1

For each row, let X be the point in Column 3 and X' the point in Column 6. Let U be any point in the finite plane of ABC. Then the lines UX and UX' are perpendicular.

>/tr> >/tr> >/tr> >/tr> >/tr> >/tr> >/tr> >/tr> >/tr> >/tr> >/tr> >/tr> >/tr> >/tr> >/tr> >/tr> >/tr>
h k Column 3 Column 4 Column 5 Column 6
1152722912829128292
1254523842829328294
1317132172222829528296
1428297282982829928300
1528301283022830328304
1-151910612933667
1-2497187002830528306
1-317133172232830728308
1-428309283102831128312
1-528313283142831528316
214715283172831828319
23491286962832028321
2528322283232832428325
2-14725283262832728328
2-3283292833028331/td>28332
3228333283342833528336
3-228337283382833928340
underbar




X(28291) =  CIRCUMCIRCLE-ANTIPODE OF X(2291)

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

X(28291) lies on the circumcircle and these lines:

X(28291) = isogonal conjugate of X(28292)
X(28291) = circumcircle-antipode of X(2291)


X(28292) =  ISOGONAL CONJUGATE OF X(28291)

Barycentrics    1/((a - b) (a - c) (a^3 + a^2 b - 5 a b^2 + 3 b^3 - a^2 c + 2 a b c - 5 b^2 c - a c^2 + b c^2 + c^3) (a^3 - a^2 b - a b^2 + b^3 + a^2 c + 2 a b c + b^2 c - 5 a c^2 - 5 b c^2 + 3 c^3)) : :

X(28292) lies on these lines: {1, 3676}, {3, 15599}, {4, 3064}, {8, 4468}, {10, 4521}, {30, 511}, {40, 649}, {581, 23655}, {663, 14837}, {885, 3577}, {944, 23726}, {946, 3835}, {962, 20295}, {1512, 14330}, {4162, 7178}, {4761, 8611}, {4827, 14324}, {5592, 11068}, {6332, 21302}, {6545, 16200}, {11525, 21129}, {20070, 26853}

X(28292) = isogonal conjugate of X(28291)


X(28293) =  CIRCUMCIRCLE-ANTIPODE OF X(2384)

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

X(28293) lies on the circumcircle and these lines:

X(28293) = isogonal conjugate of X(28294)
X(28293) = circumcircle-antipode of X(2384)


X(28294) =  ISOGONAL CONJUGATE OF X(28293)

Barycentrics    1/((a - b) (a - c) (a^3 + 2 a^2 b - 5 a b^2 + 6 b^3 - 3 a^2 c + 4 a b c - 5 b^2 c - 3 a c^2 + 2 b c^2 + c^3) (a^3 - 3 a^2 b - 3 a b^2 + b^3 + 2 a^2 c + 4 a b c + 2 b^2 c - 5 a c^2 - 5 b c^2 + 6 c^3)) : :

X(28294) lies on these lines: {1, 4927}, {10, 14425}, {30, 511}, {676, 4049}, {9123, 24809}, {9185, 24810}

X(28294) = isogonal conjugate of X(28293)


X(28295) =  CIRCUMCIRCLE-ANTIPODE OF X(17222)

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

X(28295) lies on the circumcircle and these lines:

X(28295) = isogonal conjugate of X(28296)
X(28295) = circumcircle-antipode of X(17222)


X(28296) =  ISOGONAL CONJUGATE OF X(28295)

Barycentrics    1/((a - b) (a - c) (a^3 + 3 a^2 b - 5 a b^2 + 9 b^3 - 5 a^2 c + 6 a b c - 5 b^2 c - 5 a c^2 + 3 b c^2 + c^3) (a^3 - 5 a^2 b - 5 a b^2 + b^3 + 3 a^2 c + 6 a b c + 3 b^2 c - 5 a c^2 - 5 b c^2 + 9 c^3)) : :

X(28296) lies on these lines: {30, 511}, {3424, 4049}, {3429, 5466}

X(28296) = isogonal conjugate of X(28295)


X(28297) =  POINT PORRIMA(1,4)

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

X(28297) lies on these lines: {2, 4398}, {30, 511}, {190, 4395}, {192, 17392}, {320, 4409}, {594, 17254}, {1086, 17264}, {1266, 4422}, {1278, 4399}, {1698, 4364}, {3589, 3729}, {3616, 4363}, {3617, 4419}, {3623, 4454}, {3631, 17276}, {3635, 4796}, {3644, 17365}, {3663, 17359}, {3943, 4440}, {4346, 17269}, {4361, 6172}, {4373, 17265}, {4429, 24441}, {4461, 17255}, {4472, 19862}, {4478, 6646}, {4643, 4668}, {4664, 29622}, {4686, 17332}, {4718, 17390}, {4740, 17330}, {4764, 17362}, {4788, 17388}, {6173, 17243}, {6329, 17351}, {6707, 17116}, {7321, 29575}, {16834, 20583}, {17119, 20073}, {17281, 20582}, {17337, 25269}, {17369, 17399}

X(28297) = isogonal conjugate of X(28298)


X(28298) =  ISOGONAL CONJUGATE OF X(28297)

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

X(28298) lies on the circumcircle and these lines:

X(28298) = isogonal conjugate of X(28297)
X(28298) = circumcircle-antipode of X(28299)


X(28299) =  CIRCUMCIRCLE-ANTIPODE OF X(28298)

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

X(28299) lies on the circumcircle and these lines:

X(28299) = isogonal conjugate of X(28300)
X(28299) = circumcircle-antipode of X(28298)


X(28300) =  ISOGONAL CONJUGATE OF X(28299)

Barycentrics    1/((a - b) (a - c) (a^3 + 4 a^2 b - 5 a b^2 + 12 b^3 - 7 a^2 c + 8 a b c - 5 b^2 c - 7 a c^2 + 4 b c^2 + c^3) (a^3 - 7 a^2 b - 7 a b^2 + b^3 + 4 a^2 c + 8 a b c + 4 b^2 c - 5 a c^2 - 5 b c^2 + 12 c^3)) : :

X(28300) lies on these lines: {30, 511}

X(28300) = isogonal conjugate of X(28299)


X(28301) =  POINT PORRIMA(1,5)

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

X(28301) lies on these lines: {2, 1266}, {10, 24441}, {30, 511}, {239, 17487}, {551, 4363}, {903, 3912}, {1086, 4908}, {1125, 10022}, {1278, 3686}, {1644, 24407}, {1731, 3929}, {1738, 19875}, {2321, 3620}, {3008, 4370}, {3241, 4454}, {3618, 3729}, {3644, 17378}, {3663, 3763}, {3664, 4718}, {3679, 4419}, {3707, 20073}, {3828, 4364}, {3879, 4788}, {3943, 4887}, {3950, 17313}, {4058, 17255}, {4060, 6646}, {4072, 7232}, {4346, 4873}, {4398, 17342}, {4409, 17374}, {4416, 4764}, {4431, 17271}, {4440, 17310}, {4480, 4700}, {4643, 4669}, {4665, 4745}, {4686, 17330}, {4795, 17318}, {5750, 17320}, {6666, 17262}, {17355, 17382}, {17488, 29617}

X(28301) = isogonal conjugate of X(28302)


X(28302) =  ISOGONAL CONJUGATE OF X(28301)

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

X(28302) lies on the circumcircle and these lines:

X(28302) = isogonal conjugate of X(28301)
X(28302) = circumcircle-antipode of X(28303)


X(28303) =  CIRCUMCIRCLE-ANTIPODE OF X(28302)

Barycentrics    a^2 (a - b) (a - c) (a^3 + 5 a^2 b - 5 a b^2 + 15 b^3 - 9 a^2 c + 10 a b c - 5 b^2 c - 9 a c^2 + 5 b c^2 + c^3) (a^3 - 9 a^2 b - 9 a b^2 + b^3 + 5 a^2 c + 10 a b c + 5 b^2 c - 5 a c^2 - 5 b c^2 + 15 c^3) : :

X(28303) lies on the circumcircle and these lines:

X(28303) = isogonal conjugate of X(28304)
X(28303) = circumcircle-antipode of X(28302)


X(28304) =  ISOGONAL CONJUGATE OF X(28303)

Barycentrics    1/((a - b) (a - c) (a^3 + 5 a^2 b - 5 a b^2 + 15 b^3 - 9 a^2 c + 10 a b c - 5 b^2 c - 9 a c^2 + 5 b c^2 + c^3) (a^3 - 9 a^2 b - 9 a b^2 + b^3 + 5 a^2 c + 10 a b c + 5 b^2 c - 5 a c^2 - 5 b c^2 + 15 c^3)) : :

X(28304) lies on these lines: {30, 511}, {4401, 9708}

X(28304) = isogonal conjugate of X(28303)


X(28305) =  CIRCUMCIRCLE-ANTIPODE OF X(8700)

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

X(28305) lies on the circumcircle and these lines:

X(28305) = isogonal conjugate of X(28306)
X(28305) = circumcircle-antipode of X(8700)


X(28306) =  ISOGONAL CONJUGATE OF X(28305)

Barycentrics    1/((a - b) (a - c) (a^3 + 5 a^2 b + 5 a b^2 + b^3 - 2 a^2 c - 4 a b c - 2 b^2 c - 5 a c^2 - 5 b c^2 - 6 c^3) (a^3 - 2 a^2 b - 5 a b^2 - 6 b^3 + 5 a^2 c - 4 a b c - 5 b^2 c + 5 a c^2 - 2 b c^2 + c^3)) : :

X(28306) lies on these lines: {30, 511}

X(28306) = isogonal conjugate of X(28305)


X(28307) =  CIRCUMCIRCLE-ANTIPODE OF X(17223)

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

X(28307) lies on the circumcircle and these lines:

X(28307) = isogonal conjugate of X(28308)
X(28307) = circumcircle-antipode of X(17223)


X(28308) =  ISOGONAL CONJUGATE OF X(28307)

Barycentrics    1/((a - b) (a - c) (a^3 + 7 a^2 b + 7 a b^2 + b^3 - 3 a^2 c - 6 a b c - 3 b^2 c - 5 a c^2 - 5 b c^2 - 9 c^3) (a^3 - 3 a^2 b - 5 a b^2 - 9 b^3 + 7 a^2 c - 6 a b c - 5 b^2 c + 7 a c^2 - 3 b c^2 + c^3)) : :

X(28308) lies on these lines: {30, 511}

X(28308) = isogonal conjugate of X(28307)


X(28309) =  POINT PORRIMA(1,-4)

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

X(28309) lies on these lines: {1, 10022}, {2, 3943}, {8, 24441}, {30, 511}, {45, 4405}, {192, 4399}, {239, 4370}, {551, 4472}, {594, 17320}, {903, 6542}, {1086, 17310}, {1266, 4727}, {1278, 17378}, {2321, 17382}, {3241, 4363}, {3589, 3875}, {3631, 17274}, {3644, 17333}, {3679, 4364}, {3828, 25358}, {3932, 19875}, {4072, 17356}, {4360, 7227}, {4361, 18230}, {4422, 4908}, {4431, 17045}, {4452, 17309}, {4454, 20049}, {4478, 17246}, {4643, 4677}, {4659, 4795}, {4686, 17390}, {4708, 4745}, {4718, 17332}, {4740, 17392}, {4764, 17365}, {4788, 17334}, {4852, 6329}, {6707, 17319}, {7263, 17313}, {8028, 27921}, {17151, 17243}, {17301, 20582}, {17342, 17366}, {17487, 20016}

X(28309) = isogonal conjugate of X(28310)


X(28310) =  ISOGONAL CONJUGATE OF X(28309)

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

X(28310) lies on the circumcircle and these lines:

X(28310) = isogonal conjugate of X(28309)
X(28310) = circumcircle-antipode of X(28311)


X(28311) =  CIRCUMCIRCLE-ANTIPODE OF X(28310)

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

X(28311) lies on the circumcircle and these lines:

X(28311) = isogonal conjugate of X(28312)
X(28311) = circumcircle-antipode of X(28310)


X(28312) =  ISOGONAL CONJUGATE OF X(28311)

Barycentrics    1/((a - b) (a - c) (a^3 + 9 a^2 b + 9 a b^2 + b^3 - 4 a^2 c - 8 a b c - 4 b^2 c - 5 a c^2 - 5 b c^2 - 12 c^3) (a^3 - 4 a^2 b - 5 a b^2 - 12 b^3 + 9 a^2 c - 8 a b c - 5 b^2 c + 9 a c^2 - 4 b c^2 + c^3)) : :

X(28312) lies on these lines: {30, 511}

X(28312) = isogonal conjugate of X(28311)


X(28313) =  POINT PORRIMA(1,-5)

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

X(28313) lies on these lines: {2, 3950}, {30, 511}, {1125, 17318}, {1266, 17297}, {1278, 3664}, {2321, 3763}, {3008, 17160}, {3244, 4659}, {3618, 3875}, {3620, 3663}, {3625, 4419}, {3626, 17251}, {3633, 4454}, {3634, 4665}, {3635, 4363}, {3644, 17346}, {3672, 4058}, {3686, 4718}, {3729, 4856}, {3828, 4078}, {3879, 4764}, {3946, 17359}, {4000, 4072}, {4029, 17119}, {4060, 17246}, {4133, 4353}, {4364, 4691}, {4416, 4788}, {4452, 21255}, {4480, 20016}, {4643, 4701}, {4686, 17392}, {4740, 29574}, {4887, 6542}, {4896, 29605}, {4909, 17116}, {4980, 18698}, {6173, 17314}, {17117, 25072}, {17399, 29604}, {24199, 29575}

X(28313) = isogonal conjugate of X(28310)


X(28314) =  ISOGONAL CONJUGATE OF X(28313)

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

X(28314) lies on the circumcircle and these lines:

X(28314) = isogonal conjugate of X(28313)
X(28314) = circumcircle-antipode of X(28315)


X(28315) =  CIRCUMCIRCLE-ANTIPODE OF X(28314)

Barycentrics    a^2 (a - b) (a - c) (a^3 + 11 a^2 b + 11 a b^2 + b^3 - 5 a^2 c - 10 a b c - 5 b^2 c - 5 a c^2 - 5 b c^2 - 15 c^3) (a^3 - 5 a^2 b - 5 a b^2 - 15 b^3 + 11 a^2 c - 10 a b c - 5 b^2 c + 11 a c^2 - 5 b c^2 + c^3) : :

X(28315) lies on the circumcircle and these lines:

X(28315) = isogonal conjugate of X(28316)
X(28315) = circumcircle-antipode of X(28314)


X(28316) =  ISOGONAL CONJUGATE OF X(28315)

Barycentrics    1/((a - b) (a - c) (a^3 + 11 a^2 b + 11 a b^2 + b^3 - 5 a^2 c - 10 a b c - 5 b^2 c - 5 a c^2 - 5 b c^2 - 15 c^3) (a^3 - 5 a^2 b - 5 a b^2 - 15 b^3 + 11 a^2 c - 10 a b c - 5 b^2 c + 11 a c^2 - 5 b c^2 + c^3)) : :

X(28316) lies on these lines: {10, 14351}, {30, 511}

X(28316) = isogonal conjugate of X(28315)


X(28317) =  ISOGONAL CONJUGATE OF X(4715)

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

X(28317) lies on the circumcircle and these lines:

X(28317) = isogonal conjugate of X(4715)
X(28317) = circumcircle-antipode of X(28318)


X(28318) =  CIRCUMCIRCLE-ANTIPODE OF X(28317)

Barycentrics    a^2 (a - b) (a - c) (2 a^3 + a^2 b - 10 a b^2 + 3 b^3 + 2 a b c - 10 b^2 c + b c^2 + 2 c^3) (2 a^3 + 2 b^3 + a^2 c + 2 a b c + b^2 c - 10 a c^2 - 10 b c^2 + 3 c^3) : :

X(28318) lies on the circumcircle and these lines:

X(28318) = isogonal conjugate of X(28319)
X(28318) = circumcircle-antipode of X(28317)


X(28319) =  ISOGONAL CONJUGATE OF X(28318)

Barycentrics    1/((a - b) (a - c) (2 a^3 + a^2 b - 10 a b^2 + 3 b^3 + 2 a b c - 10 b^2 c + b c^2 + 2 c^3) (2 a^3 + 2 b^3 + a^2 c + 2 a b c + b^2 c - 10 a c^2 - 10 b c^2 + 3 c^3)) : :

X(28319) lies on these lines: {30, 511}

X(28319) = isogonal conjugate of X(28318)


X(28320) =  CIRCUMCIRCLE-ANTIPODE OF X(8696)

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

X(28320) lies on the circumcircle and these lines:

X(28320) = isogonal conjugate of X(28321)
X(28320) = circumcircle-antipode of X(28319)


X(28321) =  ISOGONAL CONJUGATE OF X(28320)

Barycentrics    1/((a - b) (a - c) (2 a^3 + 3 a^2 b - 10 a b^2 + 9 b^3 - 4 a^2 c + 6 a b c - 10 b^2 c - 4 a c^2 + 3 b c^2 + 2 c^3) (2 a^3 - 4 a^2 b - 4 a b^2 + 2 b^3 + 3 a^2 c + 6 a b c + 3 b^2 c - 10 a c^2 - 10 b c^2 + 9 c^3)) : :

X(28321) lies on these lines: {30, 511}

X(28321) = isogonal conjugate of X(28320)


X(28322) =  POINT PORRIMA(2,5)

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

X(28322) lies on these lines: {30, 511}, {190, 6687}, {3618, 17301}, {3620, 17229}, {3622, 4454}, {3624, 4363}, {3644, 4889}, {3729, 3763}, {3834, 4440}, {3912, 4409}, {4419, 4708}, {4643, 4678}, {4659, 17251}, {4681, 17392}, {4686, 17346}, {4718, 17389}, {4726, 17334}, {4796, 20057}, {6172, 17348}, {6173, 17262}, {17239, 17254}, {17258, 28633}, {17294, 17345}, {19876, 24441}

X(28322) = isogonal conjugate of X(28323)


X(28323) =  ISOGONAL CONJUGATE OF X(28322)

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

X(28323) lies on the circumcircle and these lines:

X(28323) = isogonal conjugate of X(28322)
X(28323) = circumcircle-antipode of X(28324)


X(28324) =  CIRCUMCIRCLE-ANTIPODE OF X(28323)

Barycentrics    a^2 (a - b) (a - c) (2 a^3 + 5 a^2 b - 10 a b^2 + 15 b^3 - 8 a^2 c + 10 a b c - 10 b^2 c - 8 a c^2 + 5 b c^2 + 2 c^3) (2 a^3 - 8 a^2 b - 8 a b^2 + 2 b^3 + 5 a^2 c + 10 a b c + 5 b^2 c - 10 a c^2 - 10 b c^2 + 15 c^3) : :

X(28324) lies on the circumcircle and these lines:

X(28324) = isogonal conjugate of X(28316)
X(28324) = circumcircle-antipode of X(28314)


X(28325) =  ISOGONAL CONJUGATE OF X(28324)

Barycentrics    1/((a - b) (a - c) (2 a^3 + 5 a^2 b - 10 a b^2 + 15 b^3 - 8 a^2 c + 10 a b c - 10 b^2 c - 8 a c^2 + 5 b c^2 + 2 c^3) (2 a^3 - 8 a^2 b - 8 a b^2 + 2 b^3 + 5 a^2 c + 10 a b c + 5 b^2 c - 10 a c^2 - 10 b c^2 + 15 c^3)) : :

X(28325) lies on these lines: {30, 511}

X(28325) = isogonal conjugate of X(28324)


X(28326) =  ISOGONAL CONJUGATE OF X(4725)

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

X(28326) lies on the circumcircle and these lines:

X(28326) = isogonal conjugate of X(4725)
X(28326) = circumcircle-antipode of X(28327)


X(28327) =  CIRCUMCIRCLE-ANTIPODE OF X(28326)

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

X(28327) lies on the circumcircle and these lines:

X(28327) = isogonal conjugate of X(28328)
X(28327) = circumcircle-antipode of X(28326)


X(28328) =  ISOGONAL CONJUGATE OF X(28327)

Barycentrics    1/((a - b) (a - c) (2 a^3 + 4 a^2 b + 4 a b^2 + 2 b^3 - a^2 c - 2 a b c - b^2 c - 10 a c^2 - 10 b c^2 - 3 c^3) (2 a^3 - a^2 b - 10 a b^2 - 3 b^3 + 4 a^2 c - 2 a b c - 10 b^2 c + 4 a c^2 - b c^2 + 2 c^3)) : :

X(28328) lies on these lines: {10, 4949}, {30, 511}

X(28328) = isogonal conjugate of X(28327)


X(28329) =  POINT PORRIMA(2,-3)

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

X(28329) lies on these lines: {2, 3723}, {8, 4708}, {30, 511}, {37, 29617}, {44, 20016}, {75, 4889}, {145, 4670}, {239, 4727}, {594, 4464}, {597, 2321}, {599, 3875}, {1992, 17351}, {2345, 4910}, {3175, 21873}, {3244, 4665}, {3621, 4643}, {3623, 4798}, {3625, 4364}, {3632, 4690}, {3633, 4363}, {3635, 4472}, {3663, 22165}, {3729, 15534}, {3739, 17388}, {3823, 4716}, {3834, 6542}, {3879, 4726}, {3946, 20582}, {4007, 17385}, {4021, 4478}, {4060, 17045}, {4360, 17239}, {4361, 20195}, {4377, 17144}, {4399, 4698}, {4405, 29571}, {4419, 20053}, {4460, 4657}, {4641, 20046}, {4644, 20014}, {4677, 17251}, {4681, 17362}, {4686, 17377}, {4688, 17389}, {4691, 25358}, {4718, 17363}, {4739, 17390}, {4748, 20052}, {4795, 20049}, {4796, 20050}, {4856, 20583}, {4898, 17259}, {5212, 12035}, {11160, 17276}, {15533, 17345}, {16834, 17359}, {17119, 29605}, {17151, 17376}, {17160, 17374}, {17237, 20055}, {17294, 17382}, {17301, 21356}, {17309, 17356}, {17314, 17348}, {20058, 27921}

X(28329) = isogonal conjugate of X(28330)


X(28330) =  ISOGONAL CONJUGATE OF X(28329)

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

X(28330) lies on the circumcircle and these lines:

X(28330) = isogonal conjugate of X(28329)
X(28330) = circumcircle-antipode of X(28331)


X(28331) =  CIRCUMCIRCLE-ANTIPODE OF X(28330)

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

X(28331) lies on the circumcircle and these lines:

X(28331) = isogonal conjugate of X(28332)
X(28331) = circumcircle-antipode of X(28330)


X(28332) =  ISOGONAL CONJUGATE OF X(28331)

Barycentrics    1/((a - b) (a - c) (2 a^3 + 8 a^2 b + 8 a b^2 + 2 b^3 - 3 a^2 c - 6 a b c - 3 b^2 c - 10 a c^2 - 10 b c^2 - 9 c^3) (2 a^3 - 3 a^2 b - 10 a b^2 - 9 b^3 + 8 a^2 c - 6 a b c - 10 b^2 c + 8 a c^2 - 3 b c^2 + 2 c^3)) : :

X(28332) lies on these lines: {30, 511}

X(28332) = isogonal conjugate of X(28331)


X(28333) =  POINT PORRIMA(3,2)

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

X(28333) lies on these lines: {2, 7232}, {30, 511}, {44, 7238}, {144, 17243}, {320, 4422}, {597, 17274}, {1086, 20072}, {1125, 4796}, {1698, 4472}, {3589, 17345}, {3616, 4364}, {3617, 4363}, {3623, 4419}, {3629, 16834}, {3630, 3729}, {3631, 17351}, {3664, 4755}, {3928, 21363}, {4361, 20059}, {4370, 17297}, {4409, 17160}, {4416, 4688}, {4440, 4969}, {4454, 20052}, {4480, 17374}, {4488, 17309}, {4659, 4816}, {4664, 17334}, {4665, 4668}, {4670, 19862}, {4740, 17362}, {4741, 17369}, {6172, 17313}, {6329, 17235}, {6646, 7277}, {7227, 17344}, {7263, 16833}, {8584, 17301}, {10022, 17251}, {17246, 29584}, {17258, 29580}, {17281, 22165}, {17329, 17398}, {17333, 17392}, {17336, 29582}, {17340, 17361}, {17376, 29600}

X(28333) = isogonal conjugate of X(28334)


X(28334) =  ISOGONAL CONJUGATE OF X(28333)

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

X(28334) lies on the circumcircle and these lines:

X(28334) = isogonal conjugate of X(28333)
X(28334) = circumcircle-antipode of X(28335)


X(28335) =  CIRCUMCIRCLE-ANTIPODE OF X(28334)

Barycentrics    a^2 (a - b) (a - c) (3 a^3 + 2 a^2 b - 15 a b^2 + 6 b^3 - a^2 c + 4 a b c - 15 b^2 c - a c^2 + 2 b c^2 + 3 c^3) (3 a^3 - a^2 b - a b^2 + 3 b^3 + 2 a^2 c + 4 a b c + 2 b^2 c - 15 a c^2 - 15 b c^2 + 6 c^3) : :

X(28335) lies on the circumcircle and these lines:

X(28335) = isogonal conjugate of X(28336)
X(28335) = circumcircle-antipode of X(28334)


X(28336) =  ISOGONAL CONJUGATE OF X(28335)

Barycentrics    1/((a - b) (a - c) (3 a^3 + 2 a^2 b - 15 a b^2 + 6 b^3 - a^2 c + 4 a b c - 15 b^2 c - a c^2 + 2 b c^2 + 3 c^3) (3 a^3 - a^2 b - a b^2 + 3 b^3 + 2 a^2 c + 4 a b c + 2 b^2 c - 15 a c^2 - 15 b c^2 + 6 c^3)) : :

X(28336) lies on these lines: {30, 511}

X(28336) = isogonal conjugate of X(28335)


X(28337) =  POINT PORRIMA(3,-2)

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

X(28337) lies on these lines: {1, 25358}, {2, 4445}, {8, 4472}, {30, 511}, {141, 16834}, {145, 4364}, {319, 17045}, {597, 17294}, {1086, 20016}, {1100, 4478}, {1213, 29580}, {3241, 17251}, {3244, 4690}, {3589, 17372}, {3621, 4363}, {3625, 4670}, {3629, 17299}, {3630, 3875}, {3631, 4852}, {3632, 4665}, {3633, 4643}, {3635, 4708}, {3686, 4755}, {3759, 29577}, {3782, 20046}, {3879, 4399}, {4395, 17374}, {4405, 4675}, {4419, 20014}, {4422, 4969}, {4460, 17255}, {4464, 17344}, {4470, 20052}, {4545, 4909}, {4644, 20053}, {4664, 17332}, {4668, 4798}, {4740, 17365}, {4746, 4758}, {4851, 16833}, {4856, 6329}, {4910, 17272}, {4916, 17259}, {5839, 17243}, {8584, 17281}, {17275, 29597}, {17301, 22165}, {17318, 20050}, {17330, 17389}, {17337, 17386}, {17348, 29600}, {17360, 17395}, {17366, 17373}, {17369, 20055}, {17392, 29617}, {20011, 25349}, {20012, 25350}, {20015, 25355}, {20049, 24441}

X(28337) = isogonal conjugate of X(28338)


X(28338) =  ISOGONAL CONJUGATE OF X(28337)

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

X(28338) lies on the circumcircle and these lines:

X(28338) = isogonal conjugate of X(28337)
X(28338) = circumcircle-antipode of X(28339)


X(28339) =  CIRCUMCIRCLE-ANTIPODE OF X(28338)

Barycentrics    a^2 (a - b) (a - c) (3 a^3 + 7 a^2 b + 7 a b^2 + 3 b^3 - 2 a^2 c - 4 a b c - 2 b^2 c - 15 a c^2 - 15 b c^2 - 6 c^3) (3 a^3 - 2 a^2 b - 15 a b^2 - 6 b^3 + 7 a^2 c - 4 a b c - 15 b^2 c + 7 a c^2 - 2 b c^2 + 3 c^3) : :

X(28339) lies on the circumcircle and these lines:

X(28339) = isogonal conjugate of X(28336)
X(28339) = circumcircle-antipode of X(28334)


X(28340) =  ISOGONAL CONJUGATE OF X(28339)

Barycentrics    1/((a - b) (a - c) (3 a^3 + 7 a^2 b + 7 a b^2 + 3 b^3 - 2 a^2 c - 4 a b c - 2 b^2 c - 15 a c^2 - 15 b c^2 - 6 c^3) (3 a^3 - 2 a^2 b - 15 a b^2 - 6 b^3 + 7 a^2 c - 4 a b c - 15 b^2 c + 7 a c^2 - 2 b c^2 + 3 c^3)) : :

X(28340) lies on these lines: {30, 511}

X(28340) = isogonal conjugate of X(28339)


X(28341) =  (name pending)

Barycentrics    2 a^22-9 a^20 (b^2+c^2)+4 a^18 (3 b^4+8 b^2 c^2+3 c^4)-(b^2-c^2)^8 (b^6+b^4 c^2+b^2 c^4+c^6)+a^16 (3 b^6-41 b^4 c^2-41 b^2 c^4+3 c^6)+a^14 (-20 b^8+28 b^6 c^2+74 b^4 c^4+28 b^2 c^6-20 c^8)+2 a^2 (b^2-c^2)^6 (2 b^8-b^6 c^2-b^2 c^6+2 c^8)+2 a^10 b^2 c^2 (11 b^8+3 b^6 c^2+26 b^4 c^4+3 b^2 c^6+11 c^8)-2 a^8 (b^2-c^2)^2 (b^10+3 b^8 c^2-6 b^6 c^4-6 b^4 c^6+3 b^2 c^8+c^10)-a^4 (b^2-c^2)^4 (5 b^10-17 b^8 c^2-b^6 c^4-b^4 c^6-17 b^2 c^8+5 c^10)+a^12 (14 b^10-24 b^8 c^2-71 b^6 c^4-71 b^4 c^6-24 b^2 c^8+14 c^10)+2 a^6 (b^2-c^2)^2 (b^12-10 b^10 c^2+b^8 c^4-6 b^6 c^6+b^4 c^8-10 b^2 c^10+c^12) : :
Barycentrics    R^2 S^4 + (-92 R^6-21 R^2 SB SC+99 R^4 SW+4 SB SC SW-35 R^2 SW^2+4 SW^3) S^2 + 132 R^6 SB SC-157 R^4 SB SC SW+63 R^2 SB SC SW^2-8 SB SC SW^3 : :

As a point on the Euler line, X(28341) has Shinagawa coefficients {92 R^6-99 R^4 SW-4 SW^3-R^2 (S^2-35 SW^2), -132 R^6+157 R^4 SW-4 S^2 SW+8 SW^3+21 R^2 (S^2-3 SW^2)}.

See Tran Quang Hung and Ercole Suppa, Hyacinthos 28654.

X(28341) lies on this line: {2,3}


X(28342) =  (name pending)

Barycentrics    2 a^22-11 a^20 (b^2+c^2)+23 a^18 (b^2+c^2)^2-(b^2-c^2)^8 (b^6+c^6)-a^16 (19 b^6+67 b^4 c^2+67 b^2 c^4+19 c^6)+a^14 (-6 b^8+28 b^6 c^2+62 b^4 c^4+28 b^2 c^6-6 c^8)+a^2 (b^2-c^2)^6 (5 b^8-3 b^6 c^2-5 b^4 c^4-3 b^2 c^6+5 c^8)-a^4 (b^2-c^2)^4 (9 b^10-11 b^8 c^2-7 b^6 c^4-7 b^4 c^6-11 b^2 c^8+9 c^10)+a^8 (b^2-c^2)^2 (12 b^10+13 b^8 c^2+29 b^6 c^4+29 b^4 c^6+13 b^2 c^8+12 c^10)+a^12 (28 b^10+18 b^8 c^2-13 b^6 c^4-13 b^4 c^6+18 b^2 c^8+28 c^10)+2 a^6 (b^2-c^2)^2 (2 b^12-5 b^10 c^2-6 b^6 c^6-5 b^2 c^10+2 c^12)-a^10 (28 b^12+7 b^10 c^2+13 b^8 c^4-24 b^6 c^6+13 b^4 c^8+7 b^2 c^10+28 c^12) : :
Barycentrics    (5 R^2-2 SW) S^4 + (-160 R^6-51 R^2 SB SC+164 R^4 SW+14 SB SC SW-55 R^2 SW^2+6 SW^3) S^2 + 192 R^6 SB SC - 212 R^4 SB SC SW+81 R^2 SB SC SW^2-10 SB SC SW^3 : :

As a point on the Euler line, X(28342) has Shinagawa coefficients {(5 R^2 - 2 SW) (32 R^4 - S^2 - 20 R^2 SW + 3 SW^2), -192 R^6 + 212 R^4 SW - 14 S^2 SW + 10 SW^3 + R^2 (51 S^2 - 81 SW^2)}.

See Tran Quang Hung and Ercole Suppa, Hyacinthos 28654.

X(28342) lies on this line: {2,3}


X(28343) =  MIDPOINT OF X(6) AND X(112)

Barycentrics    a^2 (a^4-b^4+b^2 c^2-c^4) (2 a^6-a^4 b^2-b^6-a^4 c^2+b^4 c^2+b^2 c^4-c^6) : :
X(28343) = X[127]-2*X[3589], X[141]-2*X[6720], X[1297]-3*X[5085], 5*X[3618]-X[13219], 3*X[5050]+X[13310], X[10749]-3*X[14561], 5*X[12017]-X[13115], X[12384]+3*X[25406], X[13200]+3*X[14853], X[13221]+3*X[16475], 3*X[16225]-X[19161], 2*X[19130]-X[19163]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28655.

X(28343) lies on these lines: {6,74}, {127,3589}, {132,1503}, {141,6720}, {518,11722}, {611,13312}, {613,13311}, {1297,5085}, {1384,14649}, {1428,3320}, {1691,13195}, {1974,13166}, {2330,6020}, {2492,6593}, {2794,5480}, {2799,5026}, {3618,13219}, {5039,14676}, {5050,13310}, {8744,18374}, {9019,10317}, {9142,21309}, {9157,17810}, {10749,14561}, {11610,14495}, {12017,13115}, {12145,19124}, {12384,25406}, {13200,14853}, {13221,16475}, {16225,19161}, {19130,19163}

X(28343) = midpoint of X(6) and X(112)
X(28343) = reflection of X(i) in X(j) for these {i,j}: {127,3589}, {141,6720}, {19163,19130}


X(28344) =  MIDPOINT OF X(7) AND X(934)

Barycentrics    (a+b-c) (a-b+c) (2 a^2-a b-b^2-a c+2 b c-c^2) (a^4 b-2 a^3 b^2+2 a b^4-b^5+a^4 c+2 a^3 b c-2 a b^3 c-b^4 c-2 a^3 c^2+2 b^3 c^2-2 a b c^3+2 b^2 c^3+2 a c^4-b c^4-c^5) : :
X(28344) = 2*X[142]-X[5514], X[972]-3*X[21151]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28655.

X(28344) lies on these lines: {7,104}, {142,5514}, {658,13257}, {971,1543}, {972,21151}, {1360,3323}, {3321,12831}, {4617,15252}, {6366,10427}

X(28344) = midpoint of X(7) and X(934)
X(28344) = reflection of X(5514) in X(142)


X(28345) =  MIDPOINT OF X(9) AND X(101)

Barycentrics    a (a^2-2 a b+b^2-2 a c+b c+c^2) (2 a^3-a^2 b-b^3-a^2 c+b^2 c+b c^2-c^3) : :
X(28295) = X[103]-3*X[21153], X[116]-2*X[6666], X[142]-2*X[6710], X[150]-5*X[18230]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28655.

X(28345) lies on these lines: {2,14154}, {9,48}, {103,21153}, {116,6666}, {118,516}, {142,6710}, {150,18230}, {518,11712}, {528,21090}, {954,11028}, {1001,2809}, {3022,15837}, {3887,6594}, {5375,16586}, {5526,15730}

X(28345) = midpoint of X(9) and X(101)
X(28345) = reflection of X(i) in X(j) for these {i,j}: {116,6666}, {142,6710}


X(28346) =  MIDPOINT OF X(10) AND X(101)

Barycentrics    (a^2+a b-b^2+a c-b c-c^2) (2 a^3-a^2 b-b^3-a^2 c+b^2 c+b c^2-c^3) : :
X(28346) = 3*X[2]+X[1282], X[103]-3*X[10164], X[116]-2*X[3634], X[150]-5*X[1698], X[152]+3*X[165], 3*X[551]-X[10695], 7*X[9780]+X[20096], 3*X[10175]-X[10739]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28655.

X(28346) lies on these lines: {2,1282}, {10,98}, {103,10164}, {116,3634}, {118,516}, {120,24685}, {150,1698}, {152,165}, {519,11712}, {544,3828}, {551,10695}, {1125,2809}, {1362,3911}, {2786,9508}, {2801,3035}, {2808,6684}, {2810,6686}, {3033,6685}, {3842,6690}, {4712,24582}, {6541,17927}, {9780,20096}, {10175,10739}, {11028,13405}, {13411,18413}, {14543,21914}

X(28346) = midpoint of X(10) and X(101)
X(28346) = reflection of X(i) in X(j) for these {i,j}: {116,3634}, {1125,6710}


X(28347) =  MIDPOINT OF X(11) AND (2720)

Barycentrics    (a^5-a^4 b-2 a^3 b^2+2 a^2 b^3+a b^4-b^5-a^4 c+5 a^3 b c-2 a^2 b^2 c-3 a b^3 c+b^4 c-2 a^3 c^2-2 a^2 b c^2+4 a b^2 c^2+2 a^2 c^3-3 a b c^3+a c^4+b c^4-c^5) (2 a^7-2 a^6 b-3 a^5 b^2+3 a^4 b^3+a b^6-b^7-2 a^6 c+8 a^5 b c-3 a^4 b^2 c-4 a^3 b^3 c+4 a^2 b^4 c-4 a b^5 c+b^6 c-3 a^5 c^2-3 a^4 b c^2+8 a^3 b^2 c^2-4 a^2 b^3 c^2-a b^4 c^2+3 b^5 c^2+3 a^4 c^3-4 a^3 b c^3-4 a^2 b^2 c^3+8 a b^3 c^3-3 b^4 c^3+4 a^2 b c^4-a b^2 c^4-3 b^3 c^4-4 a b c^5+3 b^2 c^5+a c^6+b c^6-c^7) : :
X(28347) = X[11]+X[2720], X[1737]+X[15524], X[2745]-3*X[21154]

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28655.

X(28347) lies on these lines: {11,2720}, {521,3035}, {522,10271}, {1737,15524}, {2745,21154}, {3660,6001}

X(28347) = midpoint of X(i) and X(j) for these {i,j}: {11,2720}, {1737,15524}






leftri  Collineation mappings involving Gemini triangle 81: X(28348) - X(28403)  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 81, as in centers X(28348)-X(28403). Then

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

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 19, 2018)

underbar




X(28348) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28348) lies on these lines:


X(28349) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28349) lies on these lines:


X(28350) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28350) lies on these lines:


X(28351) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28351) lies on these lines:


X(28352) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28352) lies on these lines:


X(28353) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28353) lies on these lines:


X(28354) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28354) lies on these lines:


X(28355) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28355) lies on these lines:


X(28356) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 81

Barycentrics    a (a^4 b + a^3 b^2 + a^4 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(28356) lies on these lines:


X(28357) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28357) lies on these lines:


X(28358) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28358) lies on these lines:


X(28359) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28359) lies on these lines:


X(28360) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28360) lies on these lines:


X(28361) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28361) lies on these lines:


X(28362) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28362) lies on these lines:


X(28363) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28363) lies on these lines:


X(28364) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28364) lies on these lines:


X(28365) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28365) lies on these lines:


X(28366) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 81

Barycentrics    a (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(28366) lies on these lines:


X(28367) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28367) lies on these lines:


X(28368) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28368) lies on these lines:


X(28369) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28369) lies on these lines:


X(28370) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28370) lies on these lines:


X(28371) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28371) lies on these lines:


X(28372) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 81

Barycentrics    a (b - c) (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(28372) lies on these lines:


X(28373) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28373) lies on these lines:


X(28374) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28374) lies on these lines:


X(28375) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28375) lies on these lines:


X(28376) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28376) lies on these lines:


X(28377) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(23277) lies on these lines:


X(28378) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28378) lies on these lines:


X(28379) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28379) lies on these lines:


X(28380) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28380) lies on these lines:


X(28381) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28381) lies on these lines:


X(28382) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28382) lies on these lines:


X(28383) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28383) lies on these lines:


X(28384) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28384) lies on these lines:


X(28385) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28385) lies on these lines:


X(28386) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28386) lies on these lines:


X(28387) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28387) lies on these lines:


X(28388) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28388) lies on these lines:


X(28389) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28389) lies on these lines:


X(28390) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28390) lies on these lines:


X(28391) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28391) lies on these lines:


X(28392) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28392) lies on these lines:


X(28393) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28393) lies on these lines:


X(28394) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28394) lies on these lines:


X(28395) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28395) lies on these lines:


X(28396) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28396) lies on these lines:


X(28397) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28397) lies on these lines:


X(28398) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28398) lies on these lines:


X(28399) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28399) lies on these lines:


X(28400) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28400) lies on these lines:


X(28401) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28401) lies on these lines:


X(28402) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28402) lies on these lines:


X(28403) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 81

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

X(28403) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 81: X(28404) - X(28441)  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 82, as in centers X(28404)-X(28441). Then

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

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 26, 2018)

underbar




X(28404) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 82

Barycentrics    (a^2 - b^2 - c^2) (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(28404) lies on these lines:


X(28405) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28405) lies on these lines:


X(28406) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28406) lies on these lines:


X(28407) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28407) lies on these lines:


X(28408) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28408) lies on these lines:


X(28409) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28409) lies on these lines:


X(28410) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28410) lies on these lines:


X(28411) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28411) lies on these lines:


X(28412) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28412) lies on these lines:


X(28413) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28413) lies on these lines:


X(28414) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28414) lies on these lines:


X(28415) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28415) lies on these lines:


X(28416) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28416) lies on these lines:


X(28417) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28417) lies on these lines:


X(28418) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 82

Barycentrics    (a^2 - b^2 - c^2) (a^6 b + a^4 b^3 - a^3 b^4 - a b^6 + a^6 c + a^4 b^2 c - a^2 b^4 c - b^6 c + a^4 b c^2 + a b^4 c^2 + a^4 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(28418) lies on these lines:


X(28419) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28419) lies on these lines:

X(28419) = isotomic conjugate of isogonal conjugate of X(23115)


X(28420) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28420) lies on these lines:


X(28421) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28421) lies on these lines:


X(28422) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28422) lies on these lines:


X(28423) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28423) lies on these lines:


X(28424) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28424) lies on these lines:


X(28425) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28425) lies on these lines:


X(28426) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28426) lies on these lines:


X(28427) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28427) lies on these lines:


X(28428) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28428) lies on these lines:


X(28429) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28429) lies on these lines:


X(28430) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28430) lies on these lines:


X(28431) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28431) lies on these lines:


X(28432) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28432) lies on these lines:


X(28433) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28433) lies on these lines:


X(28434) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28434) lies on these lines:


X(28435) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28435) lies on these lines:


X(28436) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28436) lies on these lines:


X(28437) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28437) lies on these lines:


X(28438) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28438) lies on these lines:


X(28439) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 82

Barycentrics    (a^2 - b^2 - c^2) (a^5 b - a b^5 + a^5 c - 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(28439) lies on these lines:


X(28440) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28440) lies on these lines:


X(28441) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 82

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

X(28441) lies on these lines:


X(28442) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(24), WHERE A'B'C' = GEMINI TRIANGLE 82

Barycentrics    (a^2 - b^2 - c^2) (a^14 - a^12 b^2 - a^10 b^4 + a^8 b^6 - a^6 b^8 + a^4 b^10 + a^2 b^12 - b^14 - a^12 c^2 - 2 a^10 b^2 c^2 - 3 a^8 b^4 c^2 + 4 a^6 b^6 c^2 + a^4 b^8 c^2 - 2 a^2 b^10 c^2 + 3 b^12 c^2 - a^10 c^4 - 3 a^8 b^2 c^4 + 2 a^6 b^4 c^4 - 2 a^4 b^6 c^4 + 7 a^2 b^8 c^4 - 3 b^10 c^4 + a^8 c^6 + 4 a^6 b^2 c^6 - 2 a^4 b^4 c^6 - 12 a^2 b^6 c^6 + b^8 c^6 - a^6 c^8 + a^4 b^2 c^8 + 7 a^2 b^4 c^8 + b^6 c^8 + a^4 c^10 - 2 a^2 b^2 c^10 - 3 b^4 c^10 + a^2 c^12 + 3 b^2 c^12 - c^14) : :

X(28442) lies on these lines:


X(28443) =  EULER LINE INTERCEPT OF X(36)X(4870)

Barycentrics    a (3 a^6-3 a^5 (b+c)+a^4 (-6 b^2+b c-6 c^2)-2 b c (b^2-c^2)^2+6 a^3 (b^3+c^3)+a^2 (3 b^4+b^3 c+6 b^2 c^2+b c^3+3 c^4)-3 a (b^5-b^4 c-b c^4+c^5)) : :
X(28443) = (6 r+7 R) X[3] + 2 R X[4]

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28681.

X(28443) lies on these lines: {1,22937}, {2,3}, {36,4870}, {79,5204}, {191,1385}, {399,16164}, {517,5426}, {551,22765}, {758,4930}, {993,3655}, {999,5427}, {1125,16159}, {1482,4428}, {2771,3576}, {3579,3922}, {3584,5172}, {3612,17637}, {3647,13465}, {3648,5303}, {3654,11849}, {3656,5248}, {3679,12331}, {3683,13624}, {5096,10168}, {5217,5441}, {5251,18524}, {5453,16948}, {7701,7987}, {10543,10573}, {11263,16150}, {12645,21677}, {15178,16126}, {16118,17605}, {16143,26202}, {18253,18526}, {25055,26286}

X(28443) = midpoint of X(21) and X(21161)
X(28443) = reflection of X(i) in X(j), for these {i, j}: {3,21161}, {5055,15671}, {21161,5428}


X(28444) =  EULER LINE INTERCEPT OF X(35)X(18518)

Barycentrics    a (-3 a^6+3 a^5 (b+c)+4 b c (b^2-c^2)^2+a^4 (6 b^2-8 b c+6 c^2)-6 a^3 (b^3+c^3)+a^2 (-3 b^4+4 b^3 c-6 b^2 c^2+4 b c^3-3 c^4)+3 a (b^5-b^4 c-b c^4+c^5)) : :
X(28444) = (3 r+R) X[3] + 2 R X[4]

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28681.

X(28444) lies on these lines: {2,3}, {35,18518}, {55,18519}, {84,24299}, {355,4421}, {498,18542}, {519,10679}, {551,5450}, {958,3654}, {999,11551}, {1001,3653}, {1319,7284}, {1385,1709}, {1470,3582}, {1482,11260}, {1727,2099}, {1836,18493}, {2077,19875}, {3058,10949}, {3085,18545}, {3189,12645}, {3241,12000}, {3295,22759}, {3652,12635}, {3655,4428}, {3656,10680}, {3679,11248}, {3829,11928}, {3873,10247}, {3928,24474}, {4302,18499}, {4640,12702}, {4653,18451}, {4861,8148}, {4870,22766}, {5010,18491}, {5204,9955}, {5217,18480}, {5432,18516}, {6001,10179}, {6284,18544}, {6767,12735}, {7171,13151}, {8069,11237}, {8071,11238}, {10039,18525}, {10056,10058}, {10269,25055}, {12686,24927}, {15171,18543}, {15338,18517}, {15446,26437}, {24467,24473}

X(28444) = midpoint of X(1012) and X(16370)
X(28444) = reflection of X(i) in X(j), for these {i, j}: {3,16370}, {16370,6914}


X(28445) =  EULER LINE INTERCEPT OF X(1385)X(1710)

Barycentrics    a (-3 a^9+9 a^5 b (b-c)^2 c+3 a^8 (b+c)+2 b (b-c)^4 c (b+c)^3+a^7 (6 b^2-7 b c+6 c^2)+a^6 (-9 b^3+b^2 c+b c^2-9 c^3)+a (b^2-c^2)^2 (3 b^4-5 b^3 c+6 b^2 c^2-5 b c^3+3 c^4)-a^2 (b-c)^2 (3 b^5+3 b^4 c+10 b^3 c^2+10 b^2 c^3+3 b c^4+3 c^5)+a^4 (9 b^5-9 b^4 c+8 b^3 c^2+8 b^2 c^3-9 b c^4+9 c^5)+a^3 (-6 b^6+3 b^5 c+12 b^4 c^2-14 b^3 c^3+12 b^2 c^4+3 b c^5-6 c^6)) : :
X(28445) = (9 r^2+20 r R+14 R^2-3 s^2) X[3] + 4 R (r+R) X[4]

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28681.

X(28445) lies on these lines: {2,3}, {1385,1710}, {2217,3655}


X(28446) =  EULER LINE INTERCEPT OF X(1385)X(1762)

Barycentrics    a (3 a^9-3 a^8 (b+c)-2 b (b-c)^4 c (b+c)^3+a^7 (-6 b^2+b c-6 c^2)+a^5 b c (3 b^2+16 b c+3 c^2)+a^6 (9 b^3+5 b^2 c+5 b c^2+9 c^3)-a (b^2-c^2)^2 (3 b^4-5 b^3 c+2 b^2 c^2-5 b c^3+3 c^4)+a^3 (b+c)^2 (6 b^4-21 b^3 c+22 b^2 c^2-21 b c^3+6 c^4)+a^2 (b-c)^2 (3 b^5+9 b^4 c+16 b^3 c^2+16 b^2 c^3+9 b c^4+3 c^5)-a^4 (9 b^5+3 b^4 c+8 b^3 c^2+8 b^2 c^3+3 b c^4+9 c^5)) : :
X(28446) = (9 r^2+32 r R+28 R^2-3 s^2) X[3] + 4 R (r+2 R) X[4]

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28681.

X(28446) lies on these lines: {2,3}, {1385, 1762}, {11903, 12132}


X(28447) =  EULER LINE INTERCEPT OF X(182)X(15162)

Barycentrics    a (4 a^4 b c-2 b c (b^2-c^2)^2-2 a^2 b c (b^2+c^2)-3 a^3 Sqrt[a^6-a^4 (b^2+c^2)+(b^2-c^2)^2 (b^2+c^2)-a^2 (b^4-3 b^2 c^2+c^4)]+3 a (b^2+c^2) Sqrt[a^6-a^4 (b^2+c^2)+(b^2-c^2)^2 (b^2+c^2)-a^2 (b^4-3 b^2 c^2+c^4)]) : :
X(28447) = (2R-3 OH) X[3] - 2 R X[4]

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28681.

X(28447) lies on these lines: {2,3}, {182,15162}, {1385,2100}, {2575,15041}, {14500,20127}, {14810,15163}


X(28448) =  EULER LINE INTERCEPT OF X(182)X(15163)

Barycentrics    a (4 a^4 b c-2 b c (b^2-c^2)^2-2 a^2 b c (b^2+c^2)+3 a^3 Sqrt[a^6-a^4 (b^2+c^2)+(b^2-c^2)^2 (b^2+c^2)-a^2 (b^4-3 b^2 c^2+c^4)]-3 a (b^2+c^2) Sqrt[a^6-a^4 (b^2+c^2)+(b^2-c^2)^2 (b^2+c^2)-a^2 (b^4-3 b^2 c^2+c^4)]) : :
X(28448) = (3 OH+2R) X[3] - 2R X[4]

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28681.

X(28448) lies on these lines: {2,3}, {182,15163}, {1385,2101}, {2574,15041}, {14499,20127}, {14810,15162}


X(28449) =  EULER LINE INTERCEPT OF X(1385)X(2941)

Barycentrics    a (-3 a^6-13 a^4 b c-3 a^5 (b+c)+2 b c (b^2-c^2)^2+3 a (b+c) (b^2+c^2)^2+a^2 (3 b^4+11 b^3 c+6 b^2 c^2+11 b c^3+3 c^4)) : :
X(28449) = (3 r^2+r R-3 s^2) X[3] + 2 r R X[4]

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28681.

X(28449) lies on these lines: {2,3}, {1385,2941}


X(28450) =  EULER LINE INTERCEPT OF X(1385)X(2960)

Barycentrics    a (-3 a^9+7 a^6 b c (b+c)+2 b (b-c)^4 c (b+c)^3+12 a^5 b c (b^2-b c+c^2)+a^7 (6 b^2-7 b c+6 c^2)+a^2 b (b-c)^2 c (3 b^3-b^2 c-b c^2+3 c^3)-2 a^4 b c (6 b^3-b^2 c-b c^2+6 c^3)+a (b^2-c^2)^2 (3 b^4-2 b^3 c+6 b^2 c^2-2 b c^3+3 c^4)-a^3 (6 b^6+3 b^5 c-6 b^4 c^2+14 b^3 c^3-6 b^2 c^4+3 b c^5+6 c^6)) : :
X(28450) = (3 r^2+7 r R+7 R^2-3 s^2) X[3] + 2 R (r+R) X[4]

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28681.

X(28450) lies on these lines: {2,3}, {1385,2960}


X(28451) =  EULER LINE INTERCEPT OF X(527)X(3653)

Barycentrics    a (-9 a^6+9 a^5 (b+c)+8 b c (b^2-c^2)^2+2 a^4 (9 b^2+b c+9 c^2)-18 a^3 (b^3+c^3)-a^2 (9 b^4+10 b^3 c+18 b^2 c^2+10 b c^3+9 c^4)+9 a (b^5-b^4 c-b c^4+c^5)) : :
X(28451) = (9 r+14 R) X[3] + 4 R X[4]

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28681.

X(28451) lies on these lines: {2,3}, {527,3653}, {1385,3929}, {3655,5325}


X(28452) =  EULER LINE INTERCEPT OF X(11)X(18407)

Barycentrics    2 a^7-2 a^6 b-3 a^5 b^2+3 a^4 b^3+a b^6-b^7-2 a^6 c+2 a^5 b c-a^4 b^2 c-4 a^3 b^3 c+2 a^2 b^4 c+2 a b^5 c+b^6 c-3 a^5 c^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+3 b^5 c^2+3 a^4 c^3-4 a^3 b c^3-2 a^2 b^2 c^3-4 a b^3 c^3-3 b^4 c^3+2 a^2 b c^4-a b^2 c^4-3 b^3 c^4+2 a b c^5+3 b^2 c^5+a c^6+b c^6-c^7 : :
X(28452) = (R-r) X[3] - (r+2 R) X[4]

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28681.

X(28452) lies on these lines: {2,3}, {11,18407}, {36,18406}, {56,18517}, {57,80}, {142,13151}, {355,529}, {386,13408}, {388,18518}, {495,18524}, {497,18499}, {515,5883}, {517,5891}, {519,24474}, {528,3656}, {542,4260}, {551,24299}, {553,18389}, {942,5434}, {952,3873}, {997,12699}, {1385,6253}, {1389,3241}, {1478,11502}, {1482,3189}, {1699,5840}, {1737,7354}, {2771,11246}, {3058,15950}, {3086,18544}, {3296,18526}, {3476,15934}, {3601,5443}, {3679,5709}, {3878,28194}, {4293,18519}, {4299,18761}, {4511,22791}, {5138,5476}, {5229,18542}, {5442,10483}, {5587,5841}, {5708,18391}, {5755,17330}, {5842,5886}, {6284,9955}, {6796,10197}, {7956,10738}, {7965,24466}, {9940,28208}, {9956,11827}, {10056,11501}, {10072,26475}, {10532,11239}, {10954,11237}, {11227,28160}, {11235,22753}, {11499,26332}, {11826,22793}, {12943,18516}, {14986,18543}, {15171,18493}

X(28452) = midpoint of X(4) and X(17579)
X(28452) = reflection of X(i) in X(j), for these {i, j}: {7491,11113}, {11113,5}


X(28453) =  EULER LINE INTERCEPT OF X(55)X(9897)

Barycentrics    a (3 a^6-3 a^5 (b+c)+a^4 (-6 b^2+5 b c-6 c^2)-4 b c (b^2-c^2)^2+6 a^3 (b^3+c^3)+a^2 (3 b^4-b^3 c+6 b^2 c^2-b c^3+3 c^4)-3 a (b^5-b^4 c-b c^4+c^5)) : :
X(28453) = (6 r+5 R) X[3] + 4 R X[4]

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28681.

X(28453)) lies on these lines: {1,13465}, {2,3}, {55,9897}, {191,1482}, {758,10247}, {993,3656}, {1001,18515}, {1385,7701}, {1621,12773}, {1749,2099}, {2771,5426}, {3647,8148}, {3653,5450}, {3655,5248}, {3679,11849}, {4265,11178}, {4421,5790}, {4428,22758}, {4995,10058}, {5204,16118}, {5901,14450}, {6690,10742}, {10543,12647}, {10902,28208}, {12409,25055}, {12702,22937}, {13089,26287}, {13624,16143}, {16132,26202}, {16159,18493}, {19875,26285}

X(28453) = reflection of X(i) in X(j), for these {i, j}: {5054,15672}, {10246,5426}


X(28454) =  EULER LINE INTERCEPT OF X(517)X(11202)

Barycentrics    a (-3 a^9+2 a^6 b c (b+c)-2 a^2 b^2 (b-c)^2 c^2 (b+c)+b (b-c)^4 c (b+c)^3+a^5 b c (3 b^2-8 b c+3 c^2)+a^7 (6 b^2-2 b c+6 c^2)+a^4 b c (-3 b^3+b^2 c+b c^2-3 c^3)+a (b^2-c^2)^2 (3 b^4-b^3 c+4 b^2 c^2-b c^3+3 c^4)+a^3 (-6 b^6+4 b^4 c^2-4 b^3 c^3+4 b^2 c^4-6 c^6)) : :
X(28454) = (3 r^2+11 r R+10 R^2-3 s^2) X[3] + R (r+2 R) X[4]

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28681.

X(28454) lies on these lines: {2,3}, {517,11202}, {1385,8141}, {3654,5285}


X(28455) =  EULER LINE INTERCEPT OF X(1385)X(9572)

Barycentrics    a (-6 a^9-22 a^5 b^2 c^2+a^6 b c (b+c)+2 a^4 b^2 c^2 (b+c)+2 b (b-c)^4 c (b+c)^3+a^7 (12 b^2-b c+12 c^2)-a^2 b (b-c)^2 c (3 b^3+7 b^2 c+7 b c^2+3 c^3)+2 a (b^2-c^2)^2 (3 b^4-b^3 c+4 b^2 c^2-b c^3+3 c^4)-a^3 (b+c)^2 (12 b^4-27 b^3 c+28 b^2 c^2-27 b c^3+12 c^4)) : :
X(28455) = (6 r^2+25 r R+26 R^2-6 s^2) X[3] + 2 R (r+2 R) X[4]

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28681.

X(28455) lies on these lines: {2,3}, {1385,9572}


X(28456) =  EULER LINE INTERCEPT OF X(1385)X(9573)

Barycentrics    a (6 a^9-7 a^6 b c (b+c)-2 b (b-c)^4 c (b+c)^3+a^7 (-12 b^2+7 b c-12 c^2)-2 a^5 b c (6 b^2-5 b c+6 c^2)-a^2 b (b-c)^2 c (3 b^3-b^2 c-b c^2+3 c^3)+2 a^4 b c (6 b^3-b^2 c-b c^2+6 c^3)-2 a (b^2-c^2)^2 (3 b^4-b^3 c+4 b^2 c^2-b c^3+3 c^4)+a^3 (b+c)^2 (12 b^4-21 b^3 c+28 b^2 c^2-21 b c^3+12 c^4)) : :
X(28456) = (6 r^2+19 r R+14 R^2-6 s^2) X[3] + 2 R (r+2 R) X[4]

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28681.

X(28456) lies on these lines: {2,3}, {1385,9573}, {10251,12645}


X(28457) =  EULER LINE INTERCEPT OF X(1385)X(10251)

Barycentrics    -4 a^10+3 a^7 b c (b+c)-(b^2-c^2)^4 (b^2+c^2)+a^8 (9 b^2-3 b c+9 c^2)-6 a^5 b c (b^3+c^3)-2 a^6 (b^4-3 b^3 c+5 b^2 c^2-3 b c^3+c^4)+2 a^2 (b^2-c^2)^2 (3 b^4+4 b^2 c^2+3 c^4)-a^4 (b+c)^2 (8 b^4-13 b^3 c+16 b^2 c^2-13 b c^3+8 c^4)+3 a^3 b c (b^5-b^4 c-b c^4+c^5) :
X(28457) = (5 r^2+17 r R+14 R^2-5 s^2) X[3] + ((r+2 R)^2-s^2) X[4]

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28681.

X(28457) lies on these lines: {2,3}, {1385,10251}, {3656,18453}, {8251,25055}


X(28458) =  EULER LINE INTERCEPT OF X(165)X(5841)

Barycentrics    -2 a^7+2 a^6 (b+c)-2 a^2 b (b-c)^2 c (b+c)-a (b-c)^4 (b+c)^2+(b-c)^4 (b+c)^3+4 a^3 b c (2 b^2-b c+2 c^2)+a^5 (3 b^2-10 b c+3 c^2)+a^4 (-3 b^3+b^2 c+b c^2-3 c^3) : :
X(28458) = (3 R-r) X[3] + r X[4]

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28681.

X(28458) lies on these lines: {2,3}, {165,5841}, {517,5434}, {519,5884}, {528,3655}, {529,3654}, {553,24474}, {1385,3058}, {1768,3359}, {2077,3584}, {2550,18519}, {2829,26446}, {3474,12702}, {3576,5840}, {3579,7354}, {3820,10742}, {4299,22759}, {4304,13151}, {4316,7688}, {4413,18516}, {4430,5844}, {4995,26285}, {5298,15908}, {5790,14647}, {6284,7743}, {7080,18545}, {10056,11248}, {10072,10269}, {10106,16004}, {10246,15170}, {10247,11038}, {10270,19875}, {10310,11237}, {10385,16202}, {10525,11238}, {10950,13145}, {20292,22791}, {26200,28198}

X(28458) = midpoint of X(i) and X(j), for these {i, j}: {376,17579}, {3058,11826}
X(28458) = reflection of X(i) in X(j), for these {i, j}: {3058,1385}, {11113,549}, {24474,553}


X(28459) =  EULER LINE INTERCEPT OF X(165)X(5840)

Barycentrics    -2 a^7+2 a^6 (b+c)-2 a^2 b (b-c)^2 c (b+c)-a (b-c)^4 (b+c)^2+(b-c)^4 (b+c)^3-4 a^3 b c (b^2+b c+c^2)+a^5 (3 b^2+2 b c+3 c^2)+a^4 (-3 b^3+b^2 c+b c^2-3 c^3) : :
X(28459) = (r+3 R) X[3] - r X[4]

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28681.

X(28459) lies on these lines: {2,3}, {165,5840}, {226,13151}, {515,10176}, {517,3058}, {528,3654}, {529,3655}, {553,10202}, {580,3017}, {582,1834}, {912,17781}, {952,3681}, {997,18481}, {1385,5434}, {1482,15170}, {1708,5722}, {1737,3579}, {2550,18499}, {2551,18518}, {3336,16113}, {3428,11238}, {3475,10246}, {3576,5841}, {3582,11012}, {3583,7688}, {3584,10902}, {3586,3587}, {3820,18524}, {3925,18407}, {4302,11502}, {4654,18443}, {5178,5690}, {5298,26286}, {5506,5691}, {5584,10525}, {5758,15933}, {5842,26446}, {6253,9956}, {7354,13624}, {7742,10953}, {8148,15172}, {10056,10267}, {10072,11249}, {10157,28160}, {10268,19875}, {10385,10679}, {10526,11237}, {12702,15171}, {18544,19843}

X(28459) = midpoint of X(i) and X(j), for these {i, j}: {376,11114}, {5434,11827}
X(28459) = reflection of X(i) in X(j), for these {i, j}: {1482,15170}, {5434,1385}, {11112,549}


X(28460) =  EULER LINE INTERCEPT OF X(79)X(4870)

Barycentrics    5 a^7-5 a^6 (b+c)+a (b-c)^4 (b+c)^2-(b-c)^4 (b+c)^3+a^5 (-9 b^2+b c-9 c^2)-a^2 (b-c)^2 (3 b^3+b^2 c+b c^2+3 c^3)+a^4 (9 b^3-b^2 c-b c^2+9 c^3)+a^3 (3 b^4+b^3 c+10 b^2 c^2+b c^3+3 c^4) : :
X(28460) = (8 r+9 R) X[3] - 2 r X[4]

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28681.

X(28460) lies on these lines: {2,3}, {79,4870}, {519,16139}, {758,3655}, {1385,16113}, {2193,3163}, {3579,5441}, {3612,18977}, {3647,18481}, {3652,4297}, {3653,16159}, {3683,26202}, {5427,10072}, {5655,16164}, {8148,10385}, {10543,12702}, {16140,21578}, {18253,18525}, {22937,28204}

X(28460) = midpoint of X(i) and X(j), for these {i, j}: {376,15677}, {3534,13743}, {3651,15678}
X(28460) = reflection of X(i) in X(j), for these {i, j}: {2,5428}, {381,15670}, {3651,8703}, {3830,6841}, {3845,10021}, {5499,12100}, {5655,16164}, {6175,549}, {6841,15673}, {13743,17525}, {15679,5499}


X(28461) =  EULER LINE INTERCEPT OF X(104)X(551)

Barycentrics    a*(3*a^6-3*(b+c)*a^5-(6*b^2-7*b*c+6*c^2)*a^4+6*(b^3+c^3)*a^3+(b^2+c^2)*(3*b^2-2*b*c+3*c^2)*a^2-3*(b^4-c^4)*(b-c)*a-5*(b^2-c^2)^2*b*c) : :
X(28461) = 2 (3 r+2 R) X[3] + 5 R X[4]

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28681.

X(28461) lies on these lines: {2,3}, {104,551}, {191,11531}, {758,16200}, {944,4428}, {999,16133}, {1385,16138}, {1482,19919}, {1621,3655}, {2077,3828}, {2975,3656}, {3241,22758}, {3584,10058}, {3822,10728}, {4861,11278}, {5303,9955}, {5441,10039}, {5450,25055}, {5603,11194}, {10308,16132}, {11281,16116}

X(28461) = midpoint of X(21161) and X(21669)
X(28461) = reflection of X(i) in X(j), for these {i, j}: {3651,21161}, {21161,21}}


X(28462) =  EULER LINE INTERCEPT OF X(1385)X(16309)

Barycentrics    -10 a^10+4 a^9 (b+c)-(b-c)^4 (b+c)^6-2 a (b-c)^4 (b+c)^3 (b^2-b c+c^2)+a^8 (21 b^2-8 b c+21 c^2)-2 a^7 (5 b^3-b^2 c-b c^2+5 c^3)-2 a^6 (b^4-4 b^3 c+22 b^2 c^2-4 b c^3+c^4)+4 a^2 (b^2-c^2)^2 (3 b^4-b^3 c+4 b^2 c^2-b c^3+3 c^4)+2 a^3 (b-c)^2 (b^5+3 b^4 c-2 b^3 c^2-2 b^2 c^3+3 b c^4+c^5)+2 a^5 (3 b^5-6 b^4 c+5 b^3 c^2+5 b^2 c^3-6 b c^4+3 c^5)+a^4 (-20 b^6+6 b^5 c+28 b^4 c^2-20 b^3 c^3+28 b^2 c^4+6 b c^5-20 c^6) : :
X(28462) = (6 r^2+21 r R+21 R^2-5 s^2) X[3] + (3 r R+6 R^2-s^2) X[4]

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28681.

X(28462) lies on these lines: {2,3}, {1385,16309}


X(28463) =  EULER LINE INTERCEPT OF X(1385)X(19919)

Barycentrics    a (-6 a^6+6 a^5 (b+c)+5 b c (b^2-c^2)^2+4 a^4 (3 b^2-b c+3 c^2)-12 a^3 (b^3+c^3)-a^2 (6 b^4+b^3 c+12 b^2 c^2+b c^3+6 c^4)+6 a (b^5-b^4 c-b c^4+c^5)) : :
X(28463) = (12 r+13 R) X[3] + 5 R X[4]

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28681.

X(28463) lies on these lines: {2,3}, {1385,19919}, {1397,10222}, {5426,16200}, {7987,16138}, {11278,22937}, {11531,16139}


X(28464) =  EULER LINE INTERCEPT OF X(1385)X(21375)

Barycentrics    a (-3 a^6+2 b c (b^2-c^2)^2+2 a^2 b c (b^2+c^2)+a^4 (3 b^2-4 b c+3 c^2)-3 a^3 (b^3+c^3)+3 a (b^5+b^3 c^2+b^2 c^3+c^5)) : :
X(28464) = (9 r^2+14 r R-3 s^2) X[3] + 4 r R X[4]

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28681.

X(28464) lies on these lines: {2,3}, {1385,21375}, {3052,12702}


X(28465) =  EULER LINE INTERCEPT OF X(57)X(5444)

Barycentrics    -4 a^7+4 a^6 (b+c)-(b-c)^4 (b+c)^3+a (b-c)^2 (b+c)^4+a^5 (9 b^2+2 b c+9 c^2)+2 a^2 (b-c)^2 (3 b^3+4 b^2 c+4 b c^2+3 c^3)-a^4 (9 b^3+b^2 c+b c^2+9 c^3)-2 a^3 (3 b^4+2 b^3 c+4 b^2 c^2+2 b c^3+3 c^4) : :
X(28465) = (5 r+7 R) X[3] + (r+2 R) X[4]

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28681.

X(28465) lies on these lines: {2,3}, {57,5444}, {119,17009}, {214,5745}, {519,24299}, {551,24474}, {758,10165}, {942,5298}, {997,18253}, {1385,21677}, {1737,10543}, {2771,11227}, {3601,5445}, {3649,22937}, {3655,5791}, {4260,10168}, {4995,24929}, {5427,5432}, {5708,16137}, {5709,25055}, {6174,12619}, {6699,16164}, {10246,24477}, {11231,21155}, {11281,16139}, {15174,18391}

X(28465) = midpoint of X(i) and X(j), for these {i, j}: {2,21161}, {3524,15671}


X(28466) =  EULER LINE INTERCEPT OF X(36)X(4654)

Barycentrics    a (3 a^6-3 a^5 (b+c)-2 b c (b^2-c^2)^2-2 a^4 (3 b^2+b c+3 c^2)+6 a^3 (b^3+c^3)+a^2 (3 b^4+4 b^3 c+6 b^2 c^2+4 b c^3+3 c^4)-3 a (b^5-b^4 c-b c^4+c^5)) : :
X(28466) = (3 r+5 R) X[3] + R X[4]

See Antreas Hatzipolakis and Angel Montesdeoca, Hyacinthos 28681.

X(28466) lies on these lines: {2,3}, {36,4654}, {55,3654}, {56,3653}, {63,13151}, {517,4428}, {519,10267}, {527,10269}, {551,11249}, {582,19765}, {846,7986}, {912,3576}, {958,28204}, {971,17502}, {993,5325}, {997,13624}, {1385,11194}, {1480,8616}, {1708,24929}, {1737,5217}, {3189,5690}, {3241,16202}, {3428,3656}, {3601,10399}, {3679,10902}, {3715,12738}, {3828,6796}, {3873,10246}, {3927,4511}, {3928,18443}, {4995,8069}, {5010,11502}, {5248,28194}, {5260,18518}, {5298,8071}, {5434,7742}, {7330,7987}, {10072,26357}, {10202,21165}, {11012,25055}, {11496,28198}, {11499,19875}, {14831,22076}, {15931,22758}

X(28466) = midpoint of X(3) and X(16418)
X(28466) = reflection of X(3560) in X(16418)






leftri  Points on circumcircle and line at infinity: X(28467) - X(28585)  rightri

Suppose that X = x : y : z is a point on the line at infinity. All the lines that meet in X are parallel, so that X can be regarded as a direction in the plane of the reference triangle ABC. Let X' be the isogonal conjugate of X, so that X' lies on the circumcircle. Let X'' be the circumcircle-antipode of X', and let X''' be its isogonal conjugate, on the line at infinity. As a direction, X''' is perpendicular to X. In this section, X is given by the form h (- 2 a^3 + b^3 + c^3) + j (a^2 (b + c) + k a ( b^2 + c^2) - ( j + k) (b c^2 + b^2 c) : : , where h, j, k are constants. (Clark Kimberling, November 27, 2018)

In the table below, Columns 1-3 show h, j, k.

Column 4. h (- 2 a^3 + b^3 + c^3) + j (a^2 (b + c) + k a ( b^2 + c^2) - ( j + k) (b c^2 + b^2 c) : : , on infinity line, referenced below as x : y : z

Column 5. (isogonal conjugate of x : y : z) = a^2/x + b^2/y + c^2/z : : on circumcircle, referenced below as u : v : w

Column 6. (antipode of u : v : w) = (a^2+b^2-c^2)(a^2-b^2+c^2)u + 2a^2 (a^2-b^2-c^2)v + 2a^2 (a^2-b^2-c^2)w : : on circumcircle, referenced below as u1 : v1 : w1

Column 7. (isogonal conjugate of u1 : v1 : w1) = a^2/u1 + b^2/v1 + c^2/w1

For each row, let X be the point in Column 4 and X' the point in Column 7. Let U be any point in the finite plane of ABC. Then the lines UX and UX' are perpendicular.

In the table below, certain points in Column 4 are here given names of the form Point Procyon(h,j,k).

h j k Column 4 Column 5Column 6 Column 7
1 0 0 752 753 28467 28468
0 1 0 740 741 6010 6002
0 0 2 726 727 284698 28470
1 1 0 516 103 101 514
1 0 1 17768 28471 28469 28473
0 1 1 536 739 28474 28475
1 1 1 2796 2712 2705 2780
1 -1 0 5847 28476 28477 28478
1 0 -1 5846 28479 28480 28481
0 1 -1 518 105 1292 3309
1 1 1 519 106 1293 3667
1 -1 1 17770 28482 28483 28477
1 1 -1 17766 28485 28486 28487
1 2 0 17764 28488 28489 28490
1 -2 0 17762 28491 28492 28493
2 1 0 28494 28495 28496 28497
2 -1 0 28498 28499 28500 28501
1 0 2 17767 28502 28500 28504
1 0 -2 17769 28505 28506 28507
2 0 1 28508 28509 28510 28511
2 0 -1 28512 28513 28514 28515
0 1 2 28516 28517 28518 28519
0 1 -2 537 2382 28520 28521
0 2 1 28522 28523 28524 28525
0 2 -1 519 106 1293 3667
1 1 2 28526 28527 28528 28529
1 2 1 28530 28531 28532 28533
2 1 1 28534 28535 28536 28537
1 1 -2 519 106 1293 3667
1 -2 1 524 111 1296 1499
-2 1 2 28538 28539 28540 28541
1 2 2 28542 28543 28544 28545
2 1 2 28546 28547 28548 28549
2 2 1 28550 28551 28552 28553
-1 2 2 28552 28543 28544 28545
2 -1 2 28558 28559 28560 28561
2 2 -1 28562 28563 28564 28565
2 1 -1 28566 28567 28568 28569
2 -1 1 28570 28571 28572 28573
1 2 -2 17765 28574 28575 28576
1 -2 2 17771 28577 28578 28579
0 1 -1 518 105 1292 3309
1 0 -1 5846 28479 28580 28581
1 -1 0 5847 28476 28477 28478
0 -1 3 28582 28583 28584 28585
underbar
A few more Points Procyon are given as follows. The appearance of (h; i,j,k) in the following list means that X(h) = Point Procyon(i,j,k):
(28472; -1,2,3), (28484; 0,3,1), (28503; -1,0,3), (28484; 0,3,1), (28581; 0,3,-1), (28554; 0,1,4), (28555; 0,1,3), (28556; 1,2,3), (28580; 1,3,0), (28581; 0,3,-1), (28583; 0,-1,3)




X(28467) =  CIRCUMCIRCLE-ANTIPODE OF X(753)

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

X(28467) lies on the circumcircle and these lines:

X(28467) = isogonal conjugate of X(28468)
X(28467) = circumcircle-antipode of X(753)


X(28468) =  ISOGONAL CONJUGATE OF X(28467)

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

X(28468) lies on these lines: {30, 511}, {321, 4462}, {649, 3904}, {1764, 4063}, {2051, 4049}, {2403, 17147}, {3666, 3669}, {3776, 4707}, {3835, 10015}, {3960, 6589}, {4120, 4391}, {4424, 4905}, {4801, 21116}, {4927, 7178}, {14349, 21130}

X(28468) = isogonal conjugate of X(28467)


X(28469) =  CIRCUMCIRCLE-ANTIPODE OF X(727)

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

X(28469) lies on the circumcircle and these lines:

X(28469) = isogonal conjugate of X(28470)
X(28469) = circumcircle-antipode of X(727)


X(28470) =  ISOGONAL CONJUGATE OF X(28469)

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

X(28470) lies on these lines: {2, 8643}, {3, 23867}, {4, 17922}, {8, 4498}, {10, 4401}, {30, 511}, {75, 30183}, {335, 30187}, {649, 21302}, {659, 4147}, {663, 3835}, {667, 17072}, {669, 28373}, {996, 23837}, {1027, 5691}, {1960, 21260}, {3669, 4504}, {3777, 4922}, {4040, 23605}, {4049, 14458}, {4057, 20316}, {4106, 4162}, {4129, 4794}, {4163, 11068}, {4367, 24720}, {4380, 4729}, {4879, 24719}, {8642, 25901}, {13246, 14837}, {14199, 20979}, {18197, 21300}, {21831, 22043}

X(28470) = isogonal conjugate of X(28469)


X(28471) =  ISOGONAL CONJUGATE OF X(17768)

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

X(28471) lies on the circumcircle and these lines:

X(28471) = isogonal conjugate of X(17768)


X(28472) =  POINT PROCYON(-1,2,3)

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

X(28472) lies on these lines: {1, 7227}, {8, 17246}, {30, 511}, {321, 17726}, {984, 4399}, {1698, 3932}, {2345, 3616}, {3617, 3672}, {3623, 4461}, {3790, 17366}, {3883, 4718}, {4310, 17309}, {4353, 17229}, {4684, 4727}, {6541, 17385}, {11038, 17314}, {17343, 20052}, {17388, 24349}


X(28473) =  ISOGONAL CONJUGATE OF X(28469)

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

X(28473) lies on these lines: {1, 7178}, {3, 4367}, {5, 21051}, {30, 511}, {40, 1019}, {663, 10015}, {885, 1389}, {946, 4129}, {1482, 4879}, {3904, 21302}, {4040, 21120}, {4162, 21185}, {4534, 5511}, {4784, 12702}, {4791, 4990}, {4806, 22791}, {4807, 11362}, {4895, 21118}, {5216, 12435}, {14432, 21052}, {15909, 23893}

X(28473) = isogonal conjugate of X(28469)


X(28474) =  CIRCUMCIRCLE-ANTIPODE OF X(739)

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

X(28474) lies on the circumcircle and these lines:

X(28474) = isogonal conjugate of X(28475)
X(28474) = circumcircle-antipode of X(739)


X(28475) =  ISOGONAL CONJUGATE OF X(28474)

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

X(28475) is the infinite point of the normal to hyperbola {{A,B,C,X(1),X(2)}} at X(2). (Randy Hutson, January 15, 2019)

X(28475) lies on these lines: {1, 4106}, {2, 30234}, {3, 21005}, {4, 6591}, {8, 4380}, {10, 4394}, {30, 511}, {104, 9081}, {405, 8642}, {667, 14431}, {905, 21301}, {3803, 4391}, {4162, 4170}, {4401, 20317}, {4761, 4790}, {4922, 24719}, {5752, 20983}, {6050, 21051}, {8142, 15599}

X(28475) = isogonal conjugate of X(28474)


X(28476) =  ISOGONAL CONJUGATE OF X(5847)

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

X(28476) lies on the circumcircle and these lines:

X(28476) = isogonal conjugate of X(17768)


X(28477) =  CIRCUMCIRCLE-ANTIPODE OF X(28476)

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

X(28477) lies on the circumcircle and these lines:

X(28477) = isogonal conjugate of X(28478)
X(28477) = circumcircle-antipode of X(28476)


X(28478) =  ISOGONAL CONJUGATE OF X(28477)

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

X(28478) lies on these lines: {3, 8637}, {30, 511}, {649, 6332}, {905, 3798}, {3569, 30094}, {3669, 4897}, {3835, 14837}, {4063, 11068}, {4106, 7178}, {4170, 21185}, {4382, 23755}, {4468, 4498}, {14321, 20317}

X(28478) = isogonal conjugate of X(28477)


X(28479) =  ISOGONAL CONJUGATE OF X(5846)

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

X(28479) lies on the circumcircle and these lines:

X(28479) = isogonal conjugate of X(5846)


X(28480) =  CIRCUMCIRCLE-ANTIPODE OF X(28479)

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

X(28480) lies on the circumcircle and these lines:

X(28480) = isogonal conjugate of X(28481)
X(28480) = circumcircle-antipode of X(28479)


X(28481) =  ISOGONAL CONJUGATE OF X(28480)

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

X(28481) lies on these lines: {30, 511}, {3569, 13251}, {3803, 6332}, {4106, 21185}, {4897, 4905}, {9135, 13250}, {10015, 21301}

X(28481) = isogonal conjugate of X(28480)


X(28482) =  ISOGONAL CONJUGATE OF X(17770)

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

X(28482) lies on the circumcircle and these lines:

X(28482) = isogonal conjugate of X(17770)


X(28483) =  CIRCUMCIRCLE-ANTIPODE OF X(28482)

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

X(28483) lies on the circumcircle and these lines:

X(28483) = isogonal conjugate of X(28477)
X(28483) = circumcircle-antipode of X(28482)


X(28484) =  POINT PROCYON(0,3,1)

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

X(28484) lies on these lines: {1, 4686}, {8, 3644}, {10, 4681}, {30, 511}, {37, 1698}, {43, 22034}, {44, 4716}, {75, 3616}, {141, 4133}, {192, 3617}, {984, 4668}, {1001, 17151}, {1125, 4739}, {1266, 4966}, {1278, 3623}, {1279, 4693}, {1386, 3875}, {1738, 3943}, {2321, 3844}, {2901, 3812}, {3175, 3740}, {3246, 3685}, {3666, 4365}, {3706, 17147}, {3723, 24342}, {3729, 4663}, {3739, 3993}, {3797, 29590}, {3821, 4527}, {3823, 6541}, {3826, 3950}, {3923, 4852}, {4026, 4431}, {4358, 4706}, {4361, 15254}, {4519, 4850}, {4655, 17372}, {4726, 24325}, {4764, 24349}, {4788, 20052}, {4891, 24165}, {4980, 27804}, {5564, 9791}, {5880, 17314}, {15481, 17262}, {17299, 24248}, {25354, 28633}

X(28484) = isogonal conjugate of X(28483)


X(28485) =  ISOGONAL CONJUGATE OF X(17766)

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

X(28485) lies on the circumcircle and these lines:

X(28485) = isogonal conjugate of X(17766)


X(28486) =  CIRCUMCIRCLE-ANTIPODE OF X(28485)

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

X(28486) lies on the circumcircle and these lines:

X(28486) = isogonal conjugate of X(28487)
X(28486) = circumcircle-antipode of X(28485)


X(28487) =  ISOGONAL CONJUGATE OF X(28486)

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

X(28487) lies on these lines: {4, 21108}, {30, 511}, {905, 13246}, {2530, 4142}, {3670, 4905}, {3777, 4458}, {4049, 14492}, {4462, 4696}, {20518, 23807}, {21132, 21301}, {21201, 23797}

X(28487) = isogonal conjugate of X(28486)


X(28488) =  ISOGONAL CONJUGATE OF X(17764)

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

X(28488) lies on the circumcircle and these lines:

X(28488) = isogonal conjugate of X(17764)


X(28489) =  CIRCUMCIRCLE-ANTIPODE OF X(28488)

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

X(28489) lies on the circumcircle and these lines:

X(28489) = isogonal conjugate of X(28490)
X(28489) = circumcircle-antipode of X(28488)


X(28490) =  ISOGONAL CONJUGATE OF X(28489)

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

X(28490) lies on these lines: {30, 511}, {4504, 21185}

X(28490) = isogonal conjugate of X(28489)


X(28491) =  ISOGONAL CONJUGATE OF X(17772)

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

X(28491) lies on the circumcircle and these lines:

X(28491) = isogonal conjugate of X(17772)


X(28492) =  CIRCUMCIRCLE-ANTIPODE OF X(28491)

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

X(28492) lies on the circumcircle and these lines:

X(28492) = isogonal conjugate of X(28493)
X(28492) = circumcircle-antipode of X(28491)


X(28493) =  ISOGONAL CONJUGATE OF X(28492)

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

X(28493) lies on these lines: {30, 511}, {4522, 4834}, {4790, 8045}, {4949, 20317}

X(28493) = isogonal conjugate of X(28492)


X(28494) =  POINT PROCYON(2,1,0)

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

X(28494) lies on these lines: {10, 15492}, {30, 511}, {238, 25351}, {896, 21282}, {902, 4892}, {1215, 4450}, {1279, 24692}, {3416, 4535}, {3622, 4307}, {3624, 15485}, {3722, 17491}, {3791, 20064}, {3823, 4759}, {4432, 4645}, {4434, 5057}, {4660, 4672}, {4974, 24715}, {9780, 25611}

X(28494) = isogonal conjugate of X(28495)


X(28495) =  ISOGONAL CONJUGATE OF X(28494)

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

X(28495) lies on the circumcircle and these lines:

X(28495) = isogonal conjugate of X(28494)


X(28496) =  CIRCUMCIRCLE-ANTIPODE OF X(28495)

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

X(28496) lies on the circumcircle and these lines:

X(28496) = isogonal conjugate of X(28497)
X(28496) = circumcircle-antipode of X(28495)


X(28497) =  ISOGONAL CONJUGATE OF X(28496)

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

X(28497) lies on these lines: {30, 511}

X(28497) = isogonal conjugate of X(28496)


X(28498) =  POINT PROCYON(2,-1,0)

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

X(28498) lies on these lines: {1, 17249}, {10, 16669}, {30, 511}, {238, 17266}, {1698, 16468}, {2308, 28595}, {3244, 17246}, {3416, 4672}, {3616, 17300}, {3617, 4307}, {3626, 7227}, {3791, 6327}, {3923, 4535}, {4349, 17245}, {4645, 4974}, {4722, 28599}, {6682, 17726}, {17469, 20290}

X(28498) = isogonal conjugate of X(28499)


X(28499) =  ISOGONAL CONJUGATE OF X(28498)

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

X(28499) lies on the circumcircle and these lines:

X(28499) = isogonal conjugate of X(28498)


X(28500) =  CIRCUMCIRCLE-ANTIPODE OF X(28499)

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

X(28500) lies on the circumcircle and these lines:

X(28500) = isogonal conjugate of X(28501)
X(28500) = circumcircle-antipode of X(28499)


X(28501) =  ISOGONAL CONJUGATE OF X(28500)

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

X(28501) lies on these lines: {30, 511}

X(28501) = isogonal conjugate of X(28500)


X(28502) =  ISOGONAL CONJUGATE OF X(17767)

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

X(28502) lies on the circumcircle and these lines:

X(28502) = isogonal conjugate of X(17767)


X(28503) =  POINT PROCYON(-1,0,3)

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

X(28503) lies on these lines: {1, 3943}, {2, 1390}, {8, 4389}, {10, 4395}, {30, 511}, {145, 5695}, {238, 4370}, {551, 6541}, {903, 4645}, {984, 17330}, {1279, 4908}, {2325, 3246}, {3006, 17070}, {3161, 8692}, {3416, 17274}, {3679, 7174}, {3712, 20045}, {3790, 17342}, {3844, 4353}, {4026, 17320}, {4030, 17147}, {4362, 4884}, {4387, 19993}, {4422, 4439}, {4966, 17310}, {6057, 7191}, {6542, 24841}, {16496, 17299}, {17313, 25557}, {17378, 24349}, {17395, 29659}, {27747, 29639}


X(28504) =  ISOGONAL CONJUGATE OF X(28500)

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

X(28504) lies on these lines: {30, 511}

X(28504) = isogonal conjugate of X(28500)


X(28505) =  ISOGONAL CONJUGATE OF X(17769)

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

X(28505) lies on the circumcircle and these lines:

X(28505) = isogonal conjugate of X(17769)


X(28506) =  CIRCUMCIRCLE-ANTIPODE OF X(28505)

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

X(28506) lies on the circumcircle and these lines:

X(28506) = isogonal conjugate of X(28507)
X(28506) = circumcircle-antipode of X(28505)


X(28507) =  ISOGONAL CONJUGATE OF X(28506)

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

X(28507) lies on these lines: {30, 511}

X(28507) = isogonal conjugate of X(28506)


X(28508) =  POINT PROCYON(2,0,1)

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

X(28508) lies on these lines: {10, 17331}, {30, 511}, {238, 24692}, {896, 21241}, {902, 17491}, {1125, 3662}, {1742, 22836}, {3244, 17364}, {3626, 4416}, {3629, 4743}, {3634, 17353}, {3636, 3664}, {3836, 4759}, {3923, 17286}, {4312, 16825}, {4473, 4645}, {4660, 24695}, {9965, 29844}, {15492, 17369}

X(28498) = isogonal conjugate of X(28509)


X(28509) =  ISOGONAL CONJUGATE OF X(28508)

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

X(28509) lies on the circumcircle and these lines:

X(28509) = isogonal conjugate of X(28508)


X(28510) =  CIRCUMCIRCLE-ANTIPODE OF X(28508)

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

X(28510) lies on the circumcircle and these lines:

X(28510) = isogonal conjugate of X(28511)
X(28510) = circumcircle-antipode of X(28509)


X(28511) =  ISOGONAL CONJUGATE OF X(28510)

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

X(28511) lies on these lines: {30, 511}

X(28511) = isogonal conjugate of X(28510)


X(28512) =  POINT PROCYON(2,0,-1)

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

X(28512) lies on these lines: {1, 17236}, {10, 16468}, {30, 511}, {1125, 3883}, {2308, 28599}, {3625, 4431}, {3634, 17337}, {3635, 3879}, {3686, 4691}, {3932, 4759}, {4085, 4991}, {4450, 4970}, {4697, 4914}, {14459, 20095}

X(28512) = isogonal conjugate of X(28513)


X(28513) =  ISOGONAL CONJUGATE OF X(28512)

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

X(28513) lies on the circumcircle and these lines:

X(28513) = isogonal conjugate of X(28512)


X(28514) =  CIRCUMCIRCLE-ANTIPODE OF X(28513)

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

X(28514) lies on the circumcircle and these lines:

X(28514) = isogonal conjugate of X(28515)
X(28514) = circumcircle-antipode of X(28513)


X(28515) =  ISOGONAL CONJUGATE OF X(28514)

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

X(28515) lies on these lines: {30, 511}

X(28515) = isogonal conjugate of X(28514)


X(28516) =  POINT PROCYON(0,1,2)

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

X(28516) lies on these lines: {1, 3644}, {10, 4686}, {30, 511}, {37, 19862}, {75, 1089}, {141, 4535}, {190, 4974}, {192, 3616}, {321, 6682}, {984, 1278}, {1086, 6541}, {1125, 4681}, {1215, 17147}, {1266, 3836}, {1738, 4439}, {1757, 17160}, {3097, 4479}, {3175, 24165}, {3623, 4788}, {3634, 4739}, {3663, 3773}, {3729, 4672}, {3752, 4135}, {3775, 4431}, {3797, 17266}, {3840, 22034}, {3932, 25351}, {3989, 4980}, {3993, 4718}, {3994, 17495}, {4398, 29674}, {4505, 24731}, {4668, 4764}, {4716, 4753}, {5557, 25474}, {5625, 17319}, {6057, 24169}, {16825, 17262}, {17755, 29590}, {17759, 17793}

X(28516) = isogonal conjugate of X(28517)


X(28517) =  ISOGONAL CONJUGATE OF X(28516)

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

X(28517) lies on the circumcircle and these lines:

X(28517) = isogonal conjugate of X(28516)


X(28518) =  CIRCUMCIRCLE-ANTIPODE OF X(28517)

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

X(28518) lies on the circumcircle and these lines:

X(28518) = isogonal conjugate of X(28518)
X(28518) = circumcircle-antipode of X(28517)


X(28519) =  ISOGONAL CONJUGATE OF X(28518)

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

X(28519) lies on these lines: {30, 511}

X(28519) = isogonal conjugate of X(28518)


X(28520) =  CIRCUMCIRCLE-ANTIPODE OF X(2382)

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

X(28520) lies on the circumcircle and these lines:

X(28520) = isogonal conjugate of X(28521)
X(28520) = circumcircle-antipode of X(2382)


X(28521) =  ISOGONAL CONJUGATE OF X(28520)

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

X(28521) lies on these lines: {3, 23866}, {30, 511}, {1960, 25380}, {3716, 6161}, {4444, 28843}, {4504, 4905}, {4730, 4830}, {4794, 25666}, {5400, 16576}, {9810, 24809}, {9811, 24810}

X(28521) = isogonal conjugate of X(28520)


X(28522) =  POINT PROCYON(0,2,1)

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

X(28522) lies on these lines: {1, 1278}, {8, 4788}, {10, 192}, {30, 511}, {37, 1574}, {43, 4135}, {75, 1125}, {141, 4527}, {190, 4716}, {238, 17160}, {312, 6686}, {321, 4970}, {551, 4740}, {984, 3626}, {1575, 20688}, {1698, 4704}, {1738, 6541}, {1739, 22220}, {1962, 4980}, {2321, 3821}, {2901, 5883}, {3008, 3797}, {3210, 3840}, {3244, 24349}, {3616, 4821}, {3624, 4772}, {3635, 4764}, {3636, 4686}, {3663, 4133}, {3670, 22167}, {3696, 4691}, {3739, 19878}, {3741, 4365}, {3775, 17246}, {3795, 4479}, {3828, 4664}, {3836, 3943}, {3842, 4681}, {3844, 4535}, {3875, 3923}, {3946, 24295}, {3989, 17163}, {4169, 19957}, {4442, 21241}, {4655, 17299}, {4672, 4852}, {4699, 19862}, {4706, 24003}, {4726, 15569}, {4759, 4974}, {4871, 17495}, {4967, 25354}, {5564, 24697}, {6700, 20171}, {8720, 17733}, {14459, 17484}, {16825, 17151}, {17319, 24342}, {17389, 27494}, {19789, 29642}, {19791, 20106}, {19925, 20430}, {20103, 20173}, {21330, 24167}, {21927, 25639}

X(28522) = isogonal conjugate of X(28523)


X(28523) =  ISOGONAL CONJUGATE OF X(28522)

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

X(28523) lies on the circumcircle and these lines:

X(28523) = isogonal conjugate of X(28512)


X(28524) =  CIRCUMCIRCLE-ANTIPODE OF X(28523)

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

X(28524) lies on the circumcircle and these lines:

X(28524) = isogonal conjugate of X(28525)
X(28524) = circumcircle-antipode of X(28523)


X(28525) =  ISOGONAL CONJUGATE OF X(28524)

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

X(28525) lies on these lines: {30, 511}, {4106, 4504}

X(28525) = isogonal conjugate of X(28524)


X(28526) =  POINT PROCYON(1,1,2)

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

X(28526) lies on these lines: {1, 7240}, {10, 2996}, {30, 511}, {45, 3634}, {69, 4133}, {72, 17635}, {190, 1738}, {238, 1266}, {345, 4138}, {988, 1125}, {990, 22836}, {1699, 9742}, {1721, 3811}, {1757, 4480}, {2321, 4655}, {2325, 3836}, {3008, 17738}, {3011, 4427}, {3120, 3977}, {3161, 7613}, {3175, 11246}, {3626, 4660}, {3636, 4353}, {3664, 3993}, {3678, 18252}, {3685, 4440}, {3696, 17334}, {3717, 24715}, {3751, 4780}, {3874, 12723}, {3875, 24695}, {3878, 12721}, {3881, 12722}, {3946, 4672}, {3980, 4656}, {4001, 4365}, {4011, 24177}, {4028, 5905}, {4052, 7612}, {4054, 4414}, {4078, 5880}, {4373, 16020}, {4398, 4676}, {4409, 4702}, {4684, 4693}, {4899, 24821}, {4967, 24697}, {5121, 17777}, {5695, 17276}, {5904, 12530}, {6541, 24692}, {6685, 24259}, {6686, 24260}, {6745, 21093}, {7263, 15254}, {8669, 12512}, {8781, 11599}, {9791, 17116}, {10444, 17733}, {10445, 17748}, {10916, 21629}, {17156, 20078}, {17304, 19862}, {19878, 24295}, {20103, 24283}, {20881, 23690}

X(28526) = isogonal conjugate of X(28527)


X(28527) =  ISOGONAL CONJUGATE OF X(28526)

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

X(28527) lies on the circumcircle and these lines:

X(28527) = isogonal conjugate of X(28526)


X(28528) =  CIRCUMCIRCLE-ANTIPODE OF X(28527)

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

X(28528) lies on the circumcircle and these lines:

X(28528) = isogonal conjugate of X(28529)
X(28528) = circumcircle-antipode of X(28527)


X(28529) =  ISOGONAL CONJUGATE OF X(28528)

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

X(28529) lies on these lines: {30, 511}, {3676, 4504}, {4107, 7658}

X(28529) = isogonal conjugate of X(28528)


X(28530) =  POINT PROCYON(1,2,1)

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

X(28530) lies on these lines: {30, 511}, {45, 1213}, {86, 3445}, {141, 5695}, {230, 1281}, {238, 4395}, {325, 5992}, {553, 4891}, {594, 24723}, {596, 15172}, {1001, 7263}, {1086, 3685}, {1266, 1279}, {1284, 4436}, {1376, 13097}, {1654, 4461}, {1738, 4422}, {2325, 3823}, {2475, 4918}, {2550, 17262}, {3058, 17155}, {3120, 3712}, {3434, 4884}, {3589, 3923}, {3624, 4657}, {3629, 24695}, {3631, 4655}, {3696, 17332}, {3704, 24851}, {3755, 17351}, {3883, 4686}, {3886, 17276}, {3932, 24715}, {3943, 4645}, {4026, 7227}, {4046, 4683}, {4054, 4689}, {4307, 17318}, {4312, 4851}, {4344, 15590}, {4356, 4670}, {4361, 5698}, {4418, 4854}, {4427, 4442}, {4663, 4780}, {4672, 6329}, {4676, 17366}, {4693, 4966}, {4702, 24231}, {4733, 24697}, {5263, 17246}, {5880, 17243}, {6651, 29607}, {7613, 17265}, {8421, 24438}, {17385, 25354}

X(28530) = isogonal conjugate of X(28531)


X(28531) =  ISOGONAL CONJUGATE OF X(28530)

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

X(28531) lies on the circumcircle and these lines:

X(28531) = isogonal conjugate of X(28530)


X(28532) =  CIRCUMCIRCLE-ANTIPODE OF X(28531)

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

X(28532) lies on the circumcircle and these lines:

X(28532) = isogonal conjugate of X(28533)
X(28532) = circumcircle-antipode of X(28531)


X(28533) =  ISOGONAL CONJUGATE OF X(28532)

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

X(28533) lies on these lines: {30, 511}

X(28533) = isogonal conjugate of X(28532)


X(28534) =  POINT PROCYON(2,1,1)

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

X(28534) lies on these lines: {2, 1155}, {7, 1319}, {9, 484}, {30, 511}, {36, 1001}, {40, 11236}, {44, 24715}, {46, 15297}, {65, 11114}, {142, 5122}, {144, 5176}, {320, 4702}, {390, 5048}, {405, 4338}, {497, 2094}, {551, 5126}, {553, 3660}, {901, 4945}, {903, 14190}, {908, 6174}, {960, 1770}, {962, 11260}, {1012, 5735}, {1086, 3246}, {1386, 17301}, {1709, 3928}, {1878, 1890}, {1889, 19723}, {2077, 11495}, {2078, 4428}, {2550, 5080}, {2951, 5538}, {3058, 10391}, {3218, 10707}, {3245, 3679}, {3416, 24280}, {3629, 4780}, {3654, 6923}, {3685, 17297}, {3689, 17484}, {3696, 17346}, {3742, 11246}, {3748, 17483}, {3811, 28645}, {3814, 3826}, {3829, 8727}, {3834, 4432}, {3844, 3923}, {4031, 17051}, {4295, 11111}, {4333, 5730}, {4421, 5537}, {4645, 17264}, {4660, 17351}, {4663, 24695}, {4689, 24725}, {4693, 17374}, {4759, 6687}, {4859, 8692}, {4863, 20078}, {4870, 8543}, {4930, 15681}, {4973, 7743}, {5078, 16403}, {5263, 17254}, {5440, 15228}, {5493, 12607}, {5535, 11372}, {5542, 25405}, {5570, 5572}, {5695, 17294}, {5759, 13528}, {5784, 17579}, {5805, 22835}, {5903, 11662}, {6068, 6735}, {7262, 21949}, {8545, 11237}, {9589, 12513}, {10431, 28610}, {11240, 12701}, {12514, 17528}, {15569, 17392}, {17564, 21616}, {17615, 17781}, {21578, 25558}, {22765, 28444}, {24344, 24710}

X(28534) = isogonal conjugate of X(28535)


X(28535) =  ISOGONAL CONJUGATE OF X(28534)

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

X(28535) lies on the circumcircle and these lines:

X(28535) = isogonal conjugate of X(28534)


X(28536) =  CIRCUMCIRCLE-ANTIPODE OF X(28535)

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

X(28536) lies on the circumcircle and these lines:

X(28536) = isogonal conjugate of X(28537)
X(28536) = circumcircle-antipode of X(28535)


X(28537) =  ISOGONAL CONJUGATE OF X(28536)

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

X(28537) lies on these lines: {3, 4378}, {30, 511}, {885, 14497}, {953, 9093}, {1482, 4775}, {1565, 3328}

X(28537) = isogonal conjugate of X(28536)


X(28538) =  POINT PROCYON(-2,1,1)

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

X(28538) lies on these lines: {1, 599}, {2, 1386}, {6, 3679}, {8, 1992}, {10, 597}, {30, 511}, {69, 3241}, {81, 4914}, {141, 551}, {145, 4741}, {355, 20423}, {896, 4141}, {902, 4933}, {1001, 29573}, {1125, 20582}, {1352, 3656}, {3242, 15533}, {3246, 3912}, {3589, 3828}, {3626, 20583}, {3654, 11179}, {3696, 29617}, {3751, 4677}, {3763, 16491}, {3823, 4974}, {3838, 4362}, {3883, 15569}, {4660, 4852}, {4669, 8584}, {4693, 4727}, {4702, 6542}, {5087, 17763}, {5263, 29615}, {5434, 24471}, {5790, 14848}, {5881, 11477}, {7982, 15069}, {7983, 11161}, {7984, 13169}, {8550, 11362}, {9588, 10541}, {9955, 25561}, {11725, 19662}, {16475, 19875}, {16666, 29659}, {20049, 20080}, {21358, 25055}

X(28538) = isogonal conjugate of X(28539)


X(28539) =  ISOGONAL CONJUGATE OF X(28538)

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

X(28539) lies on the circumcircle and these lines:

X(28539) = isogonal conjugate of X(28534)


X(28540) =  CIRCUMCIRCLE-ANTIPODE OF X(28539)

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

X(28540) lies on the circumcircle and these lines:

X(28540) = isogonal conjugate of X(28541)
X(28540) = circumcircle-antipode of X(28539)


X(28541) =  ISOGONAL CONJUGATE OF X(28540)

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

X(28541) lies on these lines: {30, 511}, {21145, 24719}

X(28541) = isogonal conjugate of X(28540)


X(28542) =  POINT PROCYON(1,2,2)

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

X(28542) lies on these lines: {2, 17593}, {30, 511}, {1266, 4432}, {2325, 25351}, {3679, 24723}, {3729, 4085}, {3773, 24248}, {3775, 17254}, {3821, 17359}, {3836, 17264}, {3923, 17301}, {3943, 24692}, {3993, 17392}, {4395, 4759}, {4398, 10436}, {4407, 4419}, {4429, 19875}, {4439, 24715}, {4440, 4693}, {4480, 4753}, {4527, 4655}, {4709, 17334}, {5257, 17340}

X(28542) = isogonal conjugate of X(28543)


X(28543) =  ISOGONAL CONJUGATE OF X(28542)

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

X(28543) lies on the circumcircle and these lines:

X(28543) = isogonal conjugate of X(28542)


X(28544) =  CIRCUMCIRCLE-ANTIPODE OF X(28543)

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

X(28544) lies on the circumcircle and these lines:

X(28544) = isogonal conjugate of X(28545)
X(28544) = circumcircle-antipode of X(28543)


X(28545) =  ISOGONAL CONJUGATE OF X(28544)

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

X(28545) lies on these lines: {30, 511}

X(28545) = isogonal conjugate of X(28544)


X(28546) =  POINT PROCYON(2,1,2)

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

X(28546) lies on these lines: {30, 511}, {86, 4862}, {1281, 7925}, {1654, 4488}, {3618, 4672}, {3620, 4655}, {3763, 3923}, {4310, 5625}, {4427, 4892}, {5550, 9791}

X(28546) = isogonal conjugate of X(28546)


X(28547) =  ISOGONAL CONJUGATE OF X(28546)

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

X(28547) lies on the circumcircle and these lines:

X(28547) = isogonal conjugate of X(28546)


X(28548) =  CIRCUMCIRCLE-ANTIPODE OF X(28547)

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

X(28548) lies on the circumcircle and these lines:

X(28548) = isogonal conjugate of X(28549)
X(28548) = circumcircle-antipode of X(28547)


X(28549) =  ISOGONAL CONJUGATE OF X(28548)

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

X(28549) lies on these lines: {30, 511}

X(28549) = isogonal conjugate of X(28548)


X(28550) =  POINT PROCYON(2,2,1)

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

X(28550) lies on these lines: {10, 17336}, {30, 511}, {1698, 3923}, {1738, 4759}, {3616, 24248}, {3617, 4660}, {3625, 17347}, {3635, 17365}, {3663, 17394}, {3685, 24692}, {3729, 4668}, {3821, 17400}, {4427, 21241}, {4691, 17332}, {8720, 12699}, {17266, 17738}

X(28550) = isogonal conjugate of X(28543)


X(28551) =  ISOGONAL CONJUGATE OF X(28550)

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

X(28551) lies on the circumcircle and these lines:

X(28551) = isogonal conjugate of X(28550)


X(28552) =  CIRCUMCIRCLE-ANTIPODE OF X(28551)

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

X(28552) lies on the circumcircle and these lines:

X(28552) = isogonal conjugate of X(28553)
X(28552) = circumcircle-antipode of X(28551)


X(28553) =  ISOGONAL CONJUGATE OF X(28552)

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

X(28553) lies on these lines: {30, 511}

X(28553) = isogonal conjugate of X(28552)


X(28554) =  POINT PROCYON(0,1,4)

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

X(28554) lies on these lines: {2, 3994}, {30, 511}, {37, 19883}, {75, 3992}, {984, 4695}, {1266, 4439}, {3729, 16491}, {3842, 4688}, {4535, 29594}, {4664, 24325}, {4686, 4732}, {4753, 17160}, {4937, 24003}


X(28555) =  POINT PROCYON(0,1,3)

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

X(28555) lies on these lines: {1, 4718}, {8, 4764}, {10, 4726}, {30, 511}, {37, 3624}, {75, 3701}, {192, 3622}, {982, 22034}, {984, 3987}, {1266, 3932}, {1278, 3696}, {1386, 3729}, {2999, 4942}, {3175, 3742}, {3210, 3967}, {3644, 20057}, {3663, 3844}, {3773, 17235}, {3790, 4398}, {3823, 4439}, {3834, 6541}, {3842, 4739}, {3848, 24165}, {3875, 4663}, {3943, 24231}, {3950, 25557}, {3952, 4706}, {3994, 16610}, {4003, 4671}, {4009, 17495}, {4078, 7263}, {4361, 15481}, {4387, 4906}, {4392, 4519}, {4662, 24068}, {4681, 15808}, {4688, 19876}, {4693, 4864}, {4716, 24821}, {4980, 27812}, {5220, 17151}, {15254, 17262}


X(28556) =  POINT PROCYON(1,2,3)

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

X(28556) lies on these lines: {30, 511}, {3631, 4527}, {3731, 17303}, {3826, 17262}, {3977, 17070}, {4133, 17345}, {4398, 5550}, {4440, 4966}, {4733, 17258}


X(28557) =  POINT PROCYON(1,3,2)

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

X(28557) lies on these lines: {1, 7222}, {10, 17262}, {30, 511}, {1125, 7263}, {1266, 3616}, {1278, 3883}, {1698, 1738}, {2321, 24248}, {3617, 3717}, {3623, 4344}, {3663, 5695}, {3729, 3755}, {3875, 24280}, {3923, 3946}, {3950, 5880}, {3977, 4442}, {4133, 4655}, {4312, 17314}, {4349, 17318}, {4356, 4363}, {4387, 24177}, {4402, 15601}, {4431, 24723}, {4440, 4684}, {4452, 7290}, {4657, 17067}, {4676, 4989}, {4693, 24231}, {4700, 4716}, {4887, 4966}, {4967, 9791}, {5698, 17151}, {15590, 15600}, {17246, 19868}


X(28558) =  POINT PROCYON(2,-1,2)

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

X(28558) lies on these lines: {2, 896}, {30, 511}, {44, 24692}, {86, 2163}, {89, 24710}, {320, 4432}, {597, 3821}, {599, 3923}, {903, 17960}, {1046, 17677}, {1125, 7238}, {1213, 15492}, {1281, 7840}, {1654, 7229}, {1992, 24248}, {3178, 3650}, {3834, 4759}, {4096, 17781}, {4416, 4732}, {4427, 4933}, {4434, 17484}, {4439, 4480}, {4753, 20072}, {4865, 20078}, {5625, 9791}, {5695, 15533}, {5988, 22329}, {6629, 12258}, {7413, 28609}, {8669, 28645}, {11160, 24280}, {19875, 24342}, {19883, 25354}, {20582, 24295}

X(28558) = isogonal conjugate of X(28559)


X(28559) =  ISOGONAL CONJUGATE OF X(28558)

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

X(28559) lies on the circumcircle and these lines:

X(28559) = isogonal conjugate of X(28558)


X(28560) =  CIRCUMCIRCLE-ANTIPODE OF X(28559)

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

X(28560) lies on the circumcircle and these lines:

X(28560) = isogonal conjugate of X(28561)
X(28560) = circumcircle-antipode of X(28559)


X(28561) =  ISOGONAL CONJUGATE OF X(28560)

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

X(28561) lies on these lines: {30, 511}, {4879, 21145}

X(28561) = isogonal conjugate of X(28560)


X(28562) =  POINT PROCYON(2,2,-1)

Barycentrics    -4 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(28562) lies on these lines: {1, 24692}, {2, 902}, {10, 598}, {30, 511}, {551, 3821}, {597, 4085}, {1125, 15810}, {3058, 20359}, {3241, 24248}, {3246, 25351}, {3474, 29844}, {3634, 14762}, {3663, 11057}, {3679, 3923}, {3729, 4677}, {3741, 4450}, {3755, 4991}, {3828, 24295}, {3915, 17679}, {3993, 21829}, {4052, 14458}, {4141, 4427}, {4669, 17346}, {4685, 25306}, {4709, 29617}, {4745, 14537}, {4956, 17763}, {5255, 17677}, {8669, 12699}, {9580, 29649}, {17310, 17738}

X(28562) = isogonal conjugate of X(28563)


X(28563) =  ISOGONAL CONJUGATE OF X(28562)

Barycentrics    a^2/(-4 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(28563) lies on the circumcircle and these lines:

X(28563) = isogonal conjugate of X(28562)


X(28564) =  CIRCUMCIRCLE-ANTIPODE OF X(28563)

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

X(28564) lies on the circumcircle and these lines:

X(28564) = isogonal conjugate of X(28565)
X(28564) = circumcircle-antipode of X(28563)


X(28565) =  ISOGONAL CONJUGATE OF X(28564)

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

X(28565) lies on these lines: {30, 511}, {262, 4049}, {3777, 21145}, {3835, 21132}, {4468, 17760}, {5466, 30094}

X(28565) = isogonal conjugate of X(28564)


X(28566) =  POINT PROCYON(2,1,-1)

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

X(28566) lies on these lines: {1, 7232}, {8, 17351}, {30, 511}, {44, 391}, {145, 17276}, {238, 1698}, {320, 3623}, {390, 4851}, {1279, 3616}, {1386, 4660}, {1757, 4668}, {2550, 17348}, {3058, 4891}, {3242, 17345}, {3246, 3836}, {3416, 17229}, {3666, 4450}, {3739, 3883}, {3744, 6327}, {3886, 17372}, {4307, 4670}, {4418, 4914}, {4434, 5087}, {4461, 20052}, {4514, 20101}, {4640, 4865}, {4641, 5014}, {5263, 17239}, {7290, 17356}

X(28566) = isogonal conjugate of X(28567)


X(28567) =  ISOGONAL CONJUGATE OF X(28562)

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

X(28567) lies on the circumcircle and these lines:

X(28567) = isogonal conjugate of X(28566)


X(28568) =  CIRCUMCIRCLE-ANTIPODE OF X(28567)

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

X(28568) lies on the circumcircle and these lines:

X(28568) = isogonal conjugate of X(28569)
X(28568) = circumcircle-antipode of X(28567)


X(28569) =  ISOGONAL CONJUGATE OF X(28568)

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

X(28569) lies on these lines: {30, 511}, {986, 4905}, {3803, 3904}, {4385, 4462}, {9956, 19971}

X(28569) = isogonal conjugate of X(28568)


X(28570) =  POINT PROCYON(2,-1,1)

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

X(28570) lies on these lines: {1, 17255}, {30, 511}, {44, 966}, {238, 3624}, {320, 1279}, {1001, 17376}, {1100, 24723}, {1386, 4655}, {3246, 15808}, {3416, 17351}, {3629, 3755}, {3685, 17374}, {3723, 9791}, {3744, 20064}, {3745, 4683}, {3836, 6687}, {3844, 4672}, {3883, 17365}, {3923, 17229}, {4307, 4643}, {4312, 4361}, {4349, 4364}, {4429, 16669}, {4641, 6327}, {4660, 4663}, {4676, 17231}, {4678, 20072}, {4682, 4703}, {4851, 5698}, {4852, 24248}, {4864, 20057}, {4974, 24692}, {5263, 17344}, {5695, 17372}, {5880, 17348}, {7232, 7290}, {9746, 9766}, {15601, 17265}, {16469, 17290}, {16475, 17382}, {17299, 24280}, {24342, 28633}

X(28570) = isogonal conjugate of X(28571)


X(28571) =  ISOGONAL CONJUGATE OF X(28570)

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

X(28571) lies on the circumcircle and these lines:

X(28571) = isogonal conjugate of X(28570)


X(28572) =  CIRCUMCIRCLE-ANTIPODE OF X(28571)

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

X(28572) lies on the circumcircle and these lines:

X(28572) = isogonal conjugate of X(28573)
X(28572) = circumcircle-antipode of X(28571)


X(28573) =  ISOGONAL CONJUGATE OF X(28572)

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

X(28573) lies on these lines: {30, 511}, {3669, 3931}

X(28573) = isogonal conjugate of X(28572)


X(28574) =  ISOGONAL CONJUGATE OF X(17765)

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

X(28571) lies on the circumcircle and these lines:

X(28571) = isogonal conjugate of X(28570)


X(28575) =  CIRCUMCIRCLE-ANTIPODE OF X(28574)

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

X(28575) lies on the circumcircle and these lines:

X(28575) = isogonal conjugate of X(28576)
X(28575) = circumcircle-antipode of X(28574)


X(28576) =  ISOGONAL CONJUGATE OF X(28575)

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

X(28576) lies on these lines: {30, 511}, {3953, 4905}, {21185, 28591}

X(28576) = isogonal conjugate of X(28575)


X(28577) =  ISOGONAL CONJUGATE OF X(17771)

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

X(28577) lies on the circumcircle and these lines:

X(28577) = isogonal conjugate of X(17771)


X(28578) =  CIRCUMCIRCLE-ANTIPODE OF X(28577)

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

X(28578) lies on the circumcircle and these lines:

X(28578) = isogonal conjugate of X(28579)
X(28578) = circumcircle-antipode of X(28577)


X(28579) =  ISOGONAL CONJUGATE OF X(28578)

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

X(28579) lies on these lines: {30, 511}

X(28579) = isogonal conjugate of X(28578)


X(28580) =  POINT PROCYON(1,3,0)

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

X(28580) lies on these lines: {1, 1266}, {2, 968}, {6, 4780}, {8, 17333}, {10, 45}, {30, 511}, {551, 4356}, {553, 7248}, {903, 24231}, {1086, 4702}, {1125, 17067}, {1731, 12514}, {1836, 4028}, {2177, 4054}, {2321, 4660}, {2550, 4078}, {3008, 4432}, {3011, 4442}, {3241, 4307}, {3246, 4395}, {3416, 4133}, {3679, 3717}, {3686, 4709}, {3696, 17330}, {3751, 24280}, {3755, 3923}, {3828, 17359}, {3869, 15076}, {3886, 17274}, {3912, 4693}, {3932, 4908}, {4061, 4703}, {4085, 17355}, {4429, 17342}, {4645, 17310}, {4672, 4743}, {4689, 27747}, {4779, 16020}, {5263, 17320}, {5880, 17313}, {15953, 24696}, {16484, 24199}, {17195, 18792}, {17271, 24723}, {17748, 18483}, {20470, 23386}, {24693, 29571}


X(28581) =  POINT PROCYON(0,3,-1)

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

X(28581) lies on these lines: {1, 3696}, {2, 4891}, {6, 3886}, {8, 37}, {10, 4698}, {11, 4819}, {30, 511}, {42, 3706}, {44, 3685}, {55, 17156}, {75, 145}, {141, 3755}, {192, 3621}, {238, 4702}, {239, 1279}, {244, 4706}, {312, 4849}, {321, 20011}, {390, 5839}, {756, 4113}, {960, 22271}, {968, 4042}, {984, 3632}, {1001, 17348}, {1086, 4684}, {1100, 5263}, {1125, 4732}, {1150, 4689}, {1278, 20014}, {1738, 3834}, {1757, 4693}, {1999, 3996}, {2550, 4851}, {2667, 10459}, {2886, 4028}, {3175, 3681}, {3187, 3744}, {3210, 21342}, {3241, 4688}, {3242, 3875}, {3243, 17151}, {3244, 4709}, {3246, 4974}, {3416, 17372}, {3617, 4687}, {3622, 4751}, {3623, 4699}, {3625, 3993}, {3626, 3842}, {3633, 4726}, {3644, 20054}, {3666, 3896}, {3679, 4755}, {3680, 10435}, {3689, 17763}, {3717, 3943}, {3723, 16830}, {3740, 4685}, {3751, 5695}, {3752, 10453}, {3783, 20530}, {3797, 20016}, {3821, 4743}, {3823, 3912}, {3844, 4085}, {3883, 17362}, {3913, 15624}, {3923, 4663}, {3930, 20593}, {3950, 24393}, {3999, 17495}, {4009, 21805}, {4022, 4642}, {4026, 17239}, {4043, 4696}, {4061, 5743}, {4356, 4364}, {4358, 19998}, {4359, 4883}, {4429, 17231}, {4519, 21870}, {4645, 17374}, {4646, 10449}, {4671, 20048}, {4673, 20018}, {4676, 16669}, {4678, 27268}, {4686, 20050}, {4704, 20052}, {4718, 20053}, {4720, 17015}, {4733, 28633}, {4740, 20049}, {4780, 17235}, {4812, 20035}, {4952, 20015}, {4981, 27804}, {5014, 19791}, {5223, 17262}, {5542, 7263}, {5836, 22316}, {5880, 17376}, {7174, 17318}, {8667, 9746}, {10950, 11997}, {12410, 18619}, {12645, 20430}, {15888, 21926}, {16610, 29824}, {16834, 27474}, {17045, 19868}, {17344, 24723}, {17345, 24248}, {18134, 21949}, {20013, 20171}, {20036, 20923}, {20040, 20891}, {20041, 20892}, {20047, 22016}, {20363, 21897}, {27483, 29580}


X(28582) =  POINT PROCYON(0,-1,3)

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

X(28582) lies on these lines: {1, 4681}, {8, 4686}, {10, 4739}, {30, 511}, {37, 2275}, {75, 3617}, {145, 3644}, {190, 1279}, {192, 3623}, {210, 17155}, {226, 4884}, {238, 24821}, {244, 4009}, {291, 20530}, {312, 21342}, {321, 20068}, {335, 17266}, {596, 5044}, {942, 24068}, {982, 3967}, {984, 1698}, {1086, 3717}, {1266, 4899}, {1278, 20052}, {1463, 4553}, {1575, 17794}, {3159, 5045}, {3175, 3873}, {3210, 4849}, {3242, 3729}, {3416, 17345}, {3589, 4353}, {3666, 17165}, {3685, 4864}, {3696, 4668}, {3740, 24165}, {3742, 3971}, {3751, 4852}, {3790, 17231}, {3834, 3932}, {3883, 17334}, {3891, 4641}, {3943, 4684}, {3952, 16610}, {3977, 17724}, {3994, 17449}, {3995, 4883}, {3999, 4358}, {4011, 4906}, {4052, 24386}, {4078, 25557}, {4310, 17279}, {4361, 5223}, {4363, 7174}, {4664, 24485}, {4683, 4914}, {4698, 19862}, {4706, 21805}, {4756, 7292}, {4862, 4901}, {4952, 17784}, {4981, 27812}, {5087, 21093}, {5220, 17348}, {5542, 17243}, {5695, 16496}, {7227, 19868}, {10453, 22034}, {12782, 25102}, {15481, 16825}, {16602, 27538}, {16814, 16823}, {17278, 27549}, {26718, 28655}, {27481, 29622}


X(28583) =  ISOGONAL CONJUGATE OF X(28582)

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

X(28583) lies on the circumcircle and these lines:

X(28583) = isogonal conjugate of X(28482)
X(28583) = circumcircle-antipode of X(28583)


X(28584) =  CIRCUMCIRCLE-ANTIPODE OF X(28583)

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

X(28584) lies on the circumcircle and these lines:

X(28584) = isogonal conjugate of X(28585)
X(28584) = circumcircle-antipode of X(28583)


X(28585) =  ISOGONAL CONJUGATE OF X(28584)

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

X(28585) lies on these lines: {30, 511}, {3803, 21302}

X(28585) = isogonal conjugate of X(28584)






leftri  Centers associated with the Gemini triangles 2-28: X(28586) - X(28661)  rightri

These centers were contributed by Randy Hutson, November 27, 2018. The Gemini triangles are introduced in the preamble just before X(24537).

underbar



X(28586) = CENTROID OF CROSS-TRIANGLE OF ABC AND GEMINI TRIANGLE 2

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

Let A'B'C' be the 2nd circumperp triangle. Let A" be the trilinear pole of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(81). A", B", C" are collinear on line X(239)X(514), the trilinear polar of X(86). A"B"C" is also the (degenerate) cross-triangle of ABC and Gemini triangle 2. X(28586) is the centroid of A"B"C".

X(28586) lies on these lines: {2, 16554}, {239, 514}, {2250, 3219}


X(28587) = CENTER OF THE {GEMINI 7, GEMINI 8}-CIRCUMCONIC

Barycentrics    (b - c)^2 (2 a^6 - 5 a^5 (b + c) + a^4 (3 b^2 + 13 b c + 3 c^2) + 2 a^3 (b + c) (b^2 - 7 b c + c^2) - a^2 (4 b^4 - 6 b^3 c - 8 b^2 c^2 - 6 b c^3 + 4 c^4) + a (b + c) (3 b^4 - 6 b^3 c + 4 b^2 c^2 - 6 b c^3 + 3 c^4) - (b - c)^2 (b^2 + c^2) (b^2 + b c + c^2)) : :

X(28587) lies on these lines: {1086, 28590}, {8257, 16549}


X(28588) = X(277)X(14837)∩X(1734)X(7658)

Barycentrics    (b - c) (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(28588) is the perspector of the Gemini triangle 7 and the tangential triangle, wrt the Gemini triangle 7, of the {Gemini 7, Gemini 8}-circumconic.

X(28588) lies on these lines: {277, 14837}, {1734, 7658}


X(28589) = X(100)X(2736)∩X(497)X(3676)

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

X(28589) is the perspector of the Gemini triangle 8 and the tangential triangle, wrt the Gemini triangle 7, of the {Gemini 7, Gemini 8}-circumconic.

X(28589) lies on these lines: {100, 2736}, {165, 11068}, {497, 3676}, {649, 15487}, {2254, 3667}, {2820, 4025}, {3309, 4897}


X(28590) = X(522)X(4000)∩X(905)X(918)

Barycentrics    (b - c) (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(28590) is the perspector of the Gemini triangle 7 and the tangential triangle, wrt the Gemini triangle 8, of the {Gemini 7, Gemini 8}-circumconic.

X(28590) lies on these lines: {522, 4000}, {905, 918}, {1086, 28587}, {2254, 3667}, {4063, 14837}


X(28591) = X(404)X(4401)∩X(1019)X(6006)

Barycentrics    (b - c) (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(28591) is the perspector of the Gemini triangle 8 and the tangential triangle, wrt the Gemini triangle 8, of the {Gemini 7, Gemini 8}-circumconic.

X(28591) lies on these lines: {404, 4401}, {1019, 6006}, {3667, 4017}, {7284, 23836}


X(28592) = X(37)X(714)∩X(292)X(4697)

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

X(28592) is the perspector of the Gemini triangle 15 and the tangential triangle, wrt the Gemini triangle 15, of the {Gemini 15, Gemini 16}-circumconic.

X(28592) lies on these lines: {2, 28596}, {37, 714}, {292, 4697}, {740, 21814}, {893, 4434}, {6682, 8620}, {16577, 16600}

X(28592) = complement of complement of X(28596)


X(28593) = X(10)X(37)∩X(714)X(3963)

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

X(28593) is the perspector of the Gemini triangle 16 and the tangential triangle, wrt the Gemini triangle 15, of the {Gemini 15, Gemini 16}-circumconic.

X(28593) lies on these lines: {2, 28597}, {10, 37}, {714, 3963}, {3728, 4033}, {3741, 17229}, {5301, 10791}, {27800, 28633}

X(28593) = complement of complement of X(28597)
X(28593) = trilinear product X(37)*X(27020)
X(28593) = barycentric product X(10)*X(27020)


X(28594) = X(9)X(595)∩X(10)X(37)

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

X(28594) is the perspector of the Gemini triangle 15 and the tangential triangle, wrt the Gemini triangle 16, of the {Gemini 15, Gemini 16}-circumconic.

X(28594) lies on these lines: {1, 1390}, {2, 28598}, {9, 595}, {10, 37}, {33, 7322}, {38, 16549}, {42, 4006}, {72, 3997}, {101, 5293}, {169, 3731}, {213, 3678}, {612, 17742}, {756, 3294}, {758, 2295}, {762, 2238}, {984, 3730}, {1018, 2292}, {1107, 24036}, {1334, 4456}, {1722, 16673}, {2176, 10176} et al)

X(28594) = complement of complement of X(28598)
X(28594) = trilinear product X(i)*X(j) for these {i,j}: {10, 5280}, {37, 3920}, {42, 17289}, {57, 4538}
X(28594) = barycentric product X(i)*X(j) for these {i,j}: {7, 4538}, {10, 3920}, {37, 17289}, {321, 5280}


X(28595) = X(2)X(4865)∩X(10)X(12)

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

X(28595) is the perspector of the Gemini triangle 16 and the tangential triangle, wrt the Gemini triangle 16, of the {Gemini 15, Gemini 16}-circumconic.

X(28595) lies on these lines: {2, 4865}, {10, 12}, {306, 4085}, {740, 3969}, {752, 5294}, {1213, 4071}, {2895, 4753}, {3006, 6682}, {3416, 3791}, {3703, 3821}, {3741, 3844}, {3773, 3914}, {3846, 24003}, {3932, 4425}, {4026, 10180}, {4365, 4535}, {4442, 6535}, {4645, 4697}, {4655, 20078}, {4672, 6327}, {4854, 6541} et al

X(28595) = complement of X(17469)
X(28595) = trilinear product X(i)*X(j) for these {i,j}: {10, 17598}, {37, 17291}
X(28595) = barycentric product X(i)*X(j) for these {i,j}: {10, 17291}, {321, 17598}


X(28596) = X(192)X(714)∩X(2350)X(17147)

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

X(28596) is the perspector of the Gemini triangle 17 and the tangential triangle, wrt the Gemini triangle 17, of the {Gemini 17, Gemini 18}-circumconic.

X(28596) lies on these lines: {2, 28592}, {192, 714}, {2350, 17147}

X(28596) = anticomplement of anticomplement of X(28592)


X(28597) = X(8)X(192)∩X(75)X(4553)

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

X(28597) is the perspector of the Gemini triangle 18 and the tangential triangle, wrt the Gemini triangle 17, of the {Gemini 17, Gemini 18}-circumconic.

X(28597) lies on these lines: {2, 28593}, {8, 192}, {75, 4553}, {350, 4651}, {519, 2309}, {1964, 20044}, {2276, 17135} et al

X(28597) = anticomplement of anticomplement of X(28593)


X(28598) = X(1)X(4568)∩X(8)X(192)

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

X(28598) is the perspector of the Gemini triangle 17 and the tangential triangle, wrt the Gemini triangle 18, of the {Gemini 17, Gemini 18}-circumconic.

X(28598) lies on these lines: {1, 4568}, {2, 28594}, {8, 192}, {239, 3294}, {536, 20911}, {2295, 9055} et al

X(28598) = anticomplement of anticomplement of X(28594)


X(28599) = X(2)X(4434)∩X(8)X(79)

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

X(28599) is the perspector of the Gemini triangle 18 and the tangential triangle, wrt the Gemini triangle 18, of the {Gemini 17, Gemini 18}-circumconic.

X(28599) lies on these lines: {2, 4434}, {8, 79}, {528, 3969}, {2887, 20045}, {3006, 5745}, {3416, 4863}, {3679, 6539}, {3703, 4427}, {3936, 4030}, {3952, 4388} et al

X(28599) = anticomplement of X(17469)


X(28600) = CENTROID OF VERTEX-TRIANGLE OF GEMINI TRIANGLES 19 AND 20

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

X(28600) lies on these lines: {1, 1575}, {2, 210}, {37, 291}, {42, 4682}, {43, 4038}, {86, 17792}, {513, 875}, {551, 14839}, {672, 15254}, {760, 5883}, {1001, 17754}, {1386, 24512}, {1390, 3315} et al

X(28600) = complement of X(3789)


X(28601) = CENTROID OF SIDE-TRIANGLE OF GEMINI TRIANGLES 19 AND 20

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

X(28601) lies on these lines: {10, 4777}, {513, 10176}, {522, 3828}, {900, 3035}


X(28602) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 19

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

The side-triangle of ABC and Gemini triangle 19 is degenerate, lying on line X(2786)X(9508), the trilinear polar of X(6542).

X(28602) lies on these lines: {2, 523}, {120, 2977}, {512, 10176}, {513, 3740}, {514, 3828}, {867, 5520}, {900, 1635}, {2605, 3961}, {2786, 9508} et al

X(28602) = tripolar centroid of X(6542)
X(28602) = trilinear product X(i)*X(j) for these {i,j}: {44, 2786}, {519, 9508}, {900, 1757}, {1635, 6542}, {1960, 20947}, {3762, 17735}, {4358, 5029}
X(28602) = barycentric product X(i)*X(j) for these {i,j}: {519, 2786}, {900, 6542}, {1635, 20947}, {1757, 3762}, {3264, 5029}, {4358, 9508}


X(28603) = CENTROID OF SIDE-TRIANGLE OF ABC AND GEMINI TRIANGLE 20

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

The side-triangle of ABC and Gemini triangle 20 is degenerate, lying on line X(4770)X(4777), the trilinear polar of X(4671).

X(28603) lies on these lines: {2, 14422}, {10, 900}, {514, 3837}, {891, 4728}, {2787, 4763}, {3679, 4800}, {4770, 4777} et al

X(28603) = tripolar centroid of X(4671)
X(28603) = trilinear product X(i)*X(j) for these {i,j}: {45, 4728}, {513, 4937}, {536, 4893}, {891, 3679}, {899, 4777}, {3230, 4791}, {3768, 4671}, {4775, 6381}
X(28603) = barycentric product X(i)*X(j) for these {i,j}: {514, 4937}, {536, 4777}, {891, 4671}, {899, 4791}, {3679, 4728}, {4893, 6381}


X(28604) = PERSPECTOR OF GEMINI TRIANGLE 19 AND CROSS-TRIANGLE OF ABC AND GEMINI TRIANGLE 19

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

X(28604) lies on these lines: {2, 37}, {8, 4649}, {10, 894}, {69, 4470}, {86, 594}, {144, 5880}, {190, 1213}, {193, 3416}, {239, 4967}, {274, 3963}, {291, 3728}, {319, 4670}, {1100, 5564}, {1125, 4431}, {1224, 4647}, {1698, 3729}, {1761, 3219}, {1766, 7384} et al

X(28604) = anticomplement of X(17322)


X(28605) = PERSPECTOR OF GEMINI TRIANGLE 20 AND CROSS-TRIANGLE OF ABC AND GEMINI TRIANGLE 20

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

X(28605) lies on these lines: {1, 4365}, {2, 37}, {7, 6358}, {8, 79}, {63, 4659}, {69, 17483}, {76, 6539}, {92, 144}, {100, 9103}, {145, 3902}, {190, 5278}, {274, 27789}, {306, 4431}, {310, 6382}, {314, 8025}, {329, 14213}, {594, 3782}, {726, 7226}, {894, 3187}, {984, 21020}, {1089, 9780}, {1211, 4665}, {1215, 3240}, {1227, 4440}, {1230, 3596}, {1255, 15668}, {1621, 5695}, {1698, 4066}, {1909, 20055}, {1999, 10447}, {2321, 5249} et al

X(28605) = isotomic conjugate of X(25417)
X(28605) = anticomplement of X(28606)


X(28606) = COMPLEMENT OF X(28605)

Barycentrics    a (b^2 + c^2 + a b + a c + b c) : :
Barycentrics    r cos A + s sin A : :
Trilinears    b^2 + c^2 + a b + a c + b c : :

Let A' be the trilinear pole of the tangent to the Apollonius circle where it meets the A-excircle, and define B' and C' cyclically. Triangle A'B'C' is homothetic to ABC at X(37), to the medial triangle at X(3666), and to the anticomplementary triangle at X(28606).

X(28606) lies on these lines: {1, 21}, {2, 37}, {6, 3219}, {7, 464}, {8, 3896}, {9, 5256}, {22, 55}, {33, 7466}, {42, 984}, {43, 756}, {45, 4383}, {57, 1255}, {88, 4606}, {89, 27789}, {92, 18662}, {100, 612}, {145, 14552}, {171, 4414}, {210, 3240}, {222, 1442}, {223, 8545}, {226, 17080}, {227, 5261}, {239, 5278}, {241, 21454}, {244, 2108}, {306, 4357}, {319, 20017}, {329, 7961}, {333, 3187}, {335, 2296}, {354, 4392}, {386, 3876}, {391, 20043}, {394, 2256}, {404, 975}, {405, 5262}, {518, 7226}, {559, 19373}, {581, 12528}, {614, 5284}, {647, 4467}, {750, 1961}, {894, 19684}, {902, 17716}, {908, 4656}, {940, 3218}, {941, 5739}, {942, 13726}, {943, 1062}, {964, 7283}, {976, 20769}, {980, 16826}, {982, 3720}, {988, 5253}, {990, 7411}, {991, 11220}, {1001, 7191}, {1082, 7051}, {1100, 4641}, {1104, 16865}, {1107, 4393}, {1125, 26747}, {1150, 1999}, {1155, 4682}, {1211, 4364} et al

X(28606) = isogonal conjugate of X(2214)
X(28606) = complement of X(28605)
X(28606) = crossdifference of every pair of points on line X(661)X(667)
X(28606) = {X(1),X(63)}-harmonic conjugate of X(81)
X(28606) = {X(37),X(3666)}-harmonic conjugate of X(2)


X(28607) = PERSPECTOR OF ABC AND UNARY COFACTOR TRIANGLE OF GEMINI TRIANGLE 20

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

X(28607) lies on these lines: {6, 36}, {31, 7113}, {32, 9456}, {45, 993}, {81, 89}, {739, 4588}, {2214, 16884}, {2298, 2320}, {2423, 21007} et al

X(28607) = isogonal conjugate of X(4671)
X(28607) = crossdifference of every pair of points on line X(4770)X(4777)
X(28607) = trilinear product X(i)*X(j) for these {i,j}: {6, 2163}, {31, 89}, {56, 2364}, {560, 20569}, {604, 2320}, {649, 4588}, {1919, 4597}, {4604, 9780}
X(28607) = barycentric product X(i)*X(j) for these {i,j}: {1, 2163}, {6, 89}, {36, 20569}, {56, 2320}, {57, 2364}, {513, 4588}, {649, 4604}, {4597, 9780}


X(28608) = CENTROID OF GEMINI TRIANGLE 21

Barycentrics    a^3 + 8 a^2 (b + c) + a (11 b^2 + 38 b c + 11 c^2) + 4 (b + c) (b^2 + 4 b c + c^2) : :

X(28608) lies on these lines: {2, 2321}, {10, 3296}, {6553, 19875}


X(28609) = CENTROID OF GEMINI TRIANGLE 22

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

X(28609) lies on these lines: {1, 529}, {2, 7}, {4, 519}, {10, 5714}, {12, 12526}, {30, 1490}, {40, 10786}, {72, 3679}, {78, 9579}, {165, 17768}, {200, 1836}, {306, 4873}, {321, 4007}, {346, 4035}, {381, 5715}, {388, 15829}, {405, 5563}, {442, 19875}, {497, 3243}, {516, 3158}, {518, 1699}, {524, 10888}, {528, 1750}, {535, 18446}, {551, 3487}, {758, 5587}, {946, 5811}, {950, 3241}, {954, 4428}, {960, 5290}, {962, 2136}, {1329, 3339}, {1376, 4312}, {1697, 11239}, {1698, 28645}, {1706, 4295}, {1707, 17719}, {1728, 3582}, {1743, 3772}, {1745, 3191}, {1754, 23693}, {1763, 16548}, {1864, 11238}, {2093, 17757} et al

X(28609) = complement of X(28610)
X(28609) = X(154)-of-2nd-extouch-triangle


X(28610) = ANTICOMPLEMENT OF X(28609)

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

X(28610) lies on these lines: {2, 7}, {8, 529}, {20, 519}, {21, 3304}, {27, 4921}, {30, 9799}, {38, 4344}, {46, 5815}, {72, 9858}, {84, 6766}, {88, 20014}, {165, 5850}, {200, 8544}, {210, 10861}, {222, 3160}, {312, 4488}, {333, 7320}, {376, 1071}, {518, 5918}, {528, 10430}, {529, 3474}, {551, 11036}, {758, 5731}, {938, 11113}, {962, 1709}, {1259, 13587}, {1478, 5775}, {1707, 4310}, {1776, 11415} et al

X(28610) = anticomplement of X(28609)
X(28610) = inner-Conway-to-Conway similarity image of X(2)
X(28610) = X(2)-of-A"B"C", as defined at X(18228)


X(28611) = PERSPECTOR OF GEMINI TRIANGLE 21 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 21 AND 22

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

X(28611) lies on these lines: {1, 3996}, {2, 3743}, {8, 3881}, {10, 38}, {46, 4384}, {75, 1089}, {274, 17731}, {321, 3634}, {333, 3336}, {1125, 24589} et al X(28611) = {X(2),X(28612)}-harmonic conjugate of X(4647)


X(28612) = {X(4647),X(28611)}-HARMONIC CONJUGATE OF X(2)

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

X(28612) lies on these lines: {1, 3896}, {2, 3743}, {8, 2891}, {10, 75}, {46, 5271}, {58, 3980}, {65, 9552}, {191, 5278}, {274, 4658}, {312, 3634}, {321, 1698}, {551, 4673}, {758, 9534}, {942, 3696}, {1089, 9780}, {1125, 19804}, {1150, 3336}, {1441, 3339}, {1724, 4418}, {1788, 6358} et al

X(28612) = anticomplement of X(27784)
X(28612) = {X(4647),X(28611)}-harmonic conjugate of X(2)


X(28613) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 21 AND 22

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

X(28613) lies on the line {1224, 4647}

X(28613) = isogonal conjugate of X(28614)


X(28614) = PERSPECTOR OF UNARY COFACTOR TRIANGLES OF GEMINI TRIANGLES 21 AND 22

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

X(28614) lies on these lines: {31, 18755}, {213, 1333}, {1203, 4272}, {28637, 28649}

X(28614) = isogonal conjugate of X(28613)
X(28614) = barycentric product X(1203)*X(1961)


X(28615) = PERSPECTOR OF ABC AND UNARY COFACTOR TRIANGLE OF GEMINI TRIANGLE 22

Barycentrics    a^3/(2 a + b + c) : :

X(28615) lies on these lines: {6, 595}, {9, 2214}, {31, 872}, {37, 81}, {44, 2298}, {213, 1333}, {292, 4629}, {604, 5035}, {739, 8701}, {1268, 14621}, {28625, 28649} et al

X(28615) = isogonal conjugate of X(4359)
X(28615) = crossdifference of every pair of points on line X(4977)X(4983)
X(28615) = trilinear product X(i)*X(j) for these {i,j}: {6, 1126}, {31, 1255}, {32, 1268}, {42, 1171}, {669, 4632}, {798, 4596}
X(28615) = barycentric product X(i)*X(j) for these {i,j}: {1, 1126}, {6, 1255}, {31, 1268}, {37, 1171}, {692, 4608}, {798, 4632}


X(28616) = X(7)X(10)∩X(75)X(329)

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

Let A21B21C21 and A22B22C22 be the Gemini triangles 21 and 22, resp. Let LA be the line through A21 parallel to BC, and define LB, LC cyclically. Let A'21 = LB∩LC, and define B'21, C'21 cyclically. Triangle A'21B'21C'21 is homothetic to ABC at X(5936). Let MA be the line through A22 parallel to BC, and define MB, MC cyclically. Let A'22 = MB∩MC, and define B'22, C'22 cyclically. Triangle A'22B'22C'22 is homothetic to ABC at X(7). Triangles A'21B'21C'21 and A'22B'22C'22 are homothetic at X(28616).

X(28616) lies on these lines: {7, 10}, {75, 329}, {144, 2270}, {150, 4738}, {962, 4647}, {2999, 3672}


X(28617) = CENTROID OF GEMINI TRIANGLE 23

Barycentrics    15 a^3 + 46 a^2 (b + c) + a (37 b^2 + 82 b c + 37 c^2) + 2 (b + c) (3 b^2 + 14 b c + 3 c^2) : :

X(28617) lies on these lines: {2, 1449}, {3624, 4252}

X(28617) = complement of X(30707)


X(28618) = PERSPECTOR OF GEMINI TRIANGLE 23 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 23 AND 24

Barycentrics    (a^2 + 3 b^2 + 3 c^2 + 5 a b + 5 a c + 6 b c)/(b + c) : :

X(28618) lies on these lines: {1, 14007}, {2, 4658}, {21, 36}, {58, 5550}, {81, 19862}, {86, 3624}, {272, 28626}, {314, 6533}, {474, 5132}, {551, 14005}, {1010, 25055}, {4803, 20057}, {5235, 19878}, {5439, 18417} et al

X(28618) = {X(2),X(28620)}-harmonic conjugate of X(28619)


X(28619) = PERSPECTOR OF GEMINI TRIANGLE 24 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 23 AND 24

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

X(28619) lies on these lines: {1, 75}, {2, 4658}, {10, 5333}, {21, 551}, {58, 3616}, {72, 28639}, {81, 1125}, {333, 3624}, {519, 14005}, {1014, 3671}, {1255, 3159}, {1408, 15950}, {1412, 3485}, {1437, 11281}, {1444, 11551}, {1621, 4278} et al

X(28619) = {X(2),X(28620)}-harmonic conjugate of X(28618)


X(28620) = {X(28618),X(28619)}-HARMONIC CONJUGATE OF X(2)

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

X(28620) lies on these lines: {1, 4720}, {2, 4658}, {21, 25055}, {58, 86}, {81, 3624}, {333, 19862}, {386, 15668}, {519, 14007}, {551, 1010}, {1001, 4278}, {3487, 28641}

X(28620) = {X(28618),X(28619)}-harmonic conjugate of X(2)


X(28621) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 23 AND 24

Barycentrics    1/((b + c) (2 a^2 + a b + a c + b c) (b^2 + c^2 + a b + a c + b c)) : :

X(28621) lies on the line {4658, 10449}

X(28621) = isogonal conjugate of X(28622)
X(28621) = trilinear pole of line X(4840)X(28623)


X(28622) = PERSPECTOR OF UNARY COFACTOR TRIANGLES OF GEMINI TRIANGLES 23 AND 24

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

X(28622) lies on these lines: {6, 2200}, {37, 3678}, {42, 2176}, {1918, 18755}, {3293, 4050}, {28643, 28649}

X(28622) = isogonal conjugate of X(28621)
X(28622) = crossdifference of every pair of points on line X(4840)X(28623)
X(28622) = barycentric product X(10)*X(386)*X(17379)
X(28622) = barycentric product X(42)*X(5224)*X(17379)


X(28623) = X(30)X(511)∩X(3261)X(4025)

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

X(28623) is the infinite point of the perspectrix of Gemini triangles 23 and 24.

X(28623) lies on these lines: {30, 511}, {338, 23820}, {656, 4391}, {905, 8062}, {1459, 7253}, {1577, 23800}, {1734, 4086}, {1769, 4811}, {2254, 2517}, {3239, 6586}, {3261, 4025}, {3700, 21348}, {4017, 7650}, {4036, 17072}, {4057, 23405}, {4064, 20294}, {4467, 20906}, {4840, 4932}, {4985, 21189}, {6133, 9508}, {15419, 17215}, {20517, 21179}, {22084, 23978}

X(28623) = isogonal conjugate of X(28624)
X(28623) = crossdifference of every pair of points on line X(6)X(2200)


X(28624) = ISOGONAL CONJUGATE OF X(28623)

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

X(28624) lies on the circumcircle and these lines: {99, 1331}, {104, 1468}, {105, 2282}, {107, 8750}, {675, 1246}

X(28624) = isogonal conjugate of X(28623)
X(28624) = trilinear pole of line X(6)X(2200)
X(28624) = Ψ(X(2), X(71))
X(28624) = Ψ(X(4), X(42))
X(28624) = Ψ(X(6), X(2200))
X(28624) = Λ(X(3261), X(4025))


X(28625) = PERSPECTOR OF ABC AND UNARY COFACTOR TRIANGLE OF GEMINI TRIANGLE 24

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

X(28625) lies on these lines: {2, 319}, {6, 35}, {37, 3678}, {44, 941}, {80, 1989}, {111, 8652}, {1126, 2174}, {1169, 4273}, {1171, 1333}, {1400, 2594}, {1825, 1880}, {2350, 5153}, {28615, 28649} et al

X(28625) = isogonal conjugate of X(5333)
X(28625) = crossdifference of every pair of points on line X(4716)X(4802)
X(28625) = trilinear product X(42)*X(25417)
X(28625) = barycentric product X(37)*X(25417)


X(28626) = ISOTOMIC CONJUGATE OF X(9780)

Barycentrics    1/(a + 3 b + 3 c) : :

Let A23B23C23 be the Gemini triangle 23. Let LA be the line through A23 parallel to BC, and define LB and LC cyclically. Let A'23 = LB∩LC, and define B'23, C'23 cyclically. Triangle A'23B'23C'23 is homothetic to ABC at X(28626).

X(28626) lies on these lines: {1, 5936}, {2, 1449}, {7, 1125}, {8, 1268}, {27, 5333}, {75, 3616}, {86, 5550}, {272, 28618}, {273, 3160}, {335, 4473}, {673, 15668}, {1014, 4423} et al

X(28626) = isotomic conjugate of X(9780)


X(28627) = X(2)X(610)∩X(7)X(1125)

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

Let A'23B'23C'23 be as at X(28626). Let A24B24C24 be the Gemini triangle 24. Let MA be the line through A24 parallel to BC, and define MB and MC cyclically. Let A'24 = MB∩MC, and define B'24 and C'24 cyclically. Triangles A'23B'23C'23 and A'24B'24C'24 are homothetic at X(28627).

X(28627) lies on these lines: {2, 610}, {7, 1125}, {9, 5736}, {21, 10436}, {63, 86}, {307, 6857}, {614, 2309}, {3945, 5273}, {5232, 20106}, {5738, 5745}, {5750, 14021} et al

X(28627) = barycentric product X(3945)*X(5271)


X(28628) = PERSPECTOR OF GEMINI TRIANGLE 23 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 21 AND 23

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

X(28628) lies on these lines: {1, 442}, {2, 65}, {3, 142}, {4, 3838}, {5, 6261}, {7, 15823}, {8, 17718}, {10, 3940}, {21, 1836}, {40, 6690}, {46, 3624}, {56, 5249}, {57, 4999}, {63, 3649}, {78, 3925}, {224, 4666}, {226, 958}, {329, 5302}, {354, 10527}, {355, 3822}, {377, 497}, {405, 12047}, {474, 11507}, {498, 3753}, {499, 5439}, {517, 10198}, {518, 3487}, {529, 5290}, {551, 11235}, {758, 5791}, {936, 3826}, {944, 6984}, {956, 13407}, {988, 1086}, {993, 11263}, {997, 8728}, {1056, 11260} et al

X(28628) = {X(2),X(28629)}-harmonic conjugate of X(3812)


X(28629) = {X(3812),X(28628)}-HARMONIC CONJUGATE OF X(2)

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

X(28629) lies on these lines: {1, 142}, {2, 65}, {4, 12520}, {7, 958}, {8, 3475}, {9, 3671}, {10, 3487}, {20, 5880}, {21, 3474}, {40, 631}, {46, 6857}, {55, 404}, {56, 9776}, {144, 5302}, {226, 2551}, {329, 3649}, {355, 3824}, {377, 3486}, {388, 5249}, {405, 4295}, {442, 18391}, {452, 1836}, {498, 26725}, {516, 5436}, {517, 6989}, {518, 11036}, {527, 5234}, {938, 2886}, {942, 24477}, {944, 6901}, {946, 6865}, {948, 1042}, {962, 1001}, {1104, 4307}, {1118, 11109}, {1191, 16020}, {1329, 5226}, {1376, 5703}, {1385, 6885} et al

X(28629) = {X(3812),X(28628)}-harmonic conjugate of X(2)


X(28630) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 21 AND 23

Barycentrics    1/(a^4 (b + c) + a^3 (b^2 + b c + c^2) + a^2 b c (b + c) - a b c (b^2 + 3 b c + c^2) - b c (b + c) (b^2 + b c + c^2)) : :

The perspectrix of Gemini triangles 21 and 23 passes through X(4369).

X(28630) lies on these lines: {894, 2292}, {1211, 1909}, {3666, 17103} et al

X(28630) = isogonal conjugate of X(28631)


X(28631) = PERSPECTOR OF UNARY COFACTOR TRIANGLES OF GEMINI TRIANGLES 21 AND 23

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

X(28631) lies on these lines: {21, 37}, {55, 869}, {213, 4281}, {312, 4426}, {960, 1914}

X(28631) = isogonal conjugate of X(28630)


X(28632) = X(1)X(5936)∩X(3945)X(18217)

Barycentrics    15 a^5 + 61 a^4 (b + c) + 2 a^3 (49 b^2 + 78 b c + 49 c^2) + 10 a^2 (b + c) (7 b^2 + 10 b c + 7 c^2) + a (3 b + c) (b + 3 c) (5 b^2 + 6 b c + 5 c^2) - (b - c)^2 (b + c) (3 b + c) (b + 3 c) : :

Let A'21B'21C'21 be as at X(28616) and let A'23B'23C'23 be as at X(28626). Triangles A'21B'21C'21 and A'23B'23C'23 are homothetic at X(28632).

X(28632) lies on these lines: {1, 5936}, {3945, 18217}


X(28633) = PERSPECTOR OF GEMINI TRIANGLE 21 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 21 AND 24

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

X(28633) lies on these lines: {2, 3723}, {10, 141}, {75, 4708}, {86, 4725}, {193, 4670}, {239, 1268}, {519, 6707}, {536, 1213}, {594, 4698}, {966, 5936}, {1125, 4399}, {27800, 28593} et al

X(28633) = complement of X(3723)
X(28633) = {X(2),X(28635)}-harmonic conjugate of X(28634)


X(28634) = PERSPECTOR OF GEMINI TRIANGLE 24 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 21 AND 24

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

X(28634) lies on these lines: {1, 4399}, {2, 3723}, {6, 4967}, {7, 4690}, {8, 3739}, {9, 4665}, {10, 3946}, {45, 4431}, {69, 4688}, {75, 1654}, {141, 3679}, {142, 3626}, {145, 28639}, {193, 4795}, {319, 4675}, {320, 4772}, {519, 15668}, {524, 4034}, {536, 966}, {1100, 4798}, {1213, 3875}, {1449, 4472}, {1698, 4405}, {1743, 7227} et al

X(28634) = {X(2),X(28635)}-harmonic conjugate of X(28633)


X(28635) = {X(28633),X(28634)}-HARMONIC CONJUGATE OF X(2)

Barycentrics    a^2 - 3 b^2 - 3 c^2 - 4 a b - 4 a c - 10 b c : :

X(28635) lies on these lines: {2, 3723}, {6, 5936}, {8, 15668}, {10, 4000}, {75, 4748}, {145, 28641}, {966, 3729}, {1654, 7222} et al

X(28635) = {X(28633),X(28634)}-harmonic conjugate of X(2)


X(28636) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 21 AND 24

Barycentrics    1/(a^3 (b + c) + a^2 (6 b^2 + 11 b c + 6 c^2) + 6 a (b + c)^3 + (2 b + c) (b + 2 c) (b^2 + b c + c^2)) : :

X(28636) lies on these lines: (pending)

X(28636) = isogonal conjugate of X(28637)


X(28637) = PERSPECTOR OF UNARY COFACTOR TRIANGLES OF GEMINI TRIANGLES 21 AND 24

Barycentrics    a^2 (a^3 (b + c) + a^2 (6 b^2 + 11 b c + 6 c^2) + 6 a (b + c)^3 + (2 b + c) (b + 2 c) (b^2 + b c + c^2)) : :

X(28637) lies on these lines: {1171, 1333}, {2276, 4272}, {28614, 28649}

X(28637) = isogonal conjugate of X(28636)


X(28638) = X(7)X(10)∩X(1212)X(3666)

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

Let A'21B'21C'21 be as at X(28616) and let A'24B'24C'24 be as at X(28627). Triangles A'21B'21C'21 and A'24B'24C'24 are homothetic at X(28638).

X(28638) lies on these lines: {7, 10}, {1212, 3666}, {3295, 4460}, {3672, 18249}


X(28639) = PERSPECTOR OF GEMINI TRIANGLE 22 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 22 AND 23

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

X(28639) lies on these lines: {1, 3696}, {2, 319}, {6, 4698}, {9, 4755}, {10, 4478}, {37, 86}, {44, 4687}, {69, 4708}, {72, 28619}, {75, 3723}, {141, 1125}, {142, 214}, {144, 4795}, {145, 28634}, {524, 5257}, {536, 10436}, {597, 6666}, {1001, 3941}, {1213, 3879}, {1255, 3175}, {1279, 3616} et al

X(28639) = complement of X(17275)
X(28639) = {X(2),X(28641)}-harmonic conjugate of X(28640)


X(28640) = PERSPECTOR OF GEMINI TRIANGLE 23 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 22 AND 23

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

X(28640) lies on these lines: {1, 4399}, {2, 319}, {3, 142}, {37, 4798}, {86, 4643}, {141, 3624}, {551, 4361}, {1444, 5333} et al

X(28640) = {X(2),X(28641)}-harmonic conjugate of X(28639)


X(28641) = {X(28639),X(28640)}-HARMONIC CONJUGATE OF X(2)

Barycentrics    5 a^2 + b^2 + c^2 + 8 a b + 8 a c + 6 b c : :

X(28641) lies on these lines: {1, 4371}, {2, 319}, {8, 6707}, {86, 1778}, {141, 5550}, {142, 25055}, {145, 28635}, {346, 4798}, {1125, 4349}, {2345, 16826}, {3487, 28620} et al

X(28641) = {X(28639),X(28640)}-harmonic conjugate of X(2)


X(28642) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 22 AND 23

Barycentrics    1/(a^3 (b + c) + a^2 (2 b^2 + 3 b c + 2 c^2) + 2 a b c (b + c) + b c (b^2 + b c + c^2)) : :

X(28642) lies on the line {1962, 3757}

X(28642) = isogonal conjugate of X(28643)


X(28643) = PERSPECTOR OF UNARY COFACTOR TRIANGLES OF GEMINI TRIANGLES 22 AND 23

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

X(28643) lies on these lines: {6, 5364}, {37, 81}, {172, 2194}, {1911, 3725}, {28622, 28649} et al

X(28643) = isogonal conjugate of X(28642)


X(28644) = X(7)X(1125)∩X(405)X(934)

Barycentrics    3 a^4 + 2 a^3 (b + c) - 24 a^2 b c - 2 a (b + c) (b^2 + 6 b c + c^2) - (b - c)^2 (3 b^2 + 2 b c + 3 c^2) : :

Let A'22B'22C'22 be as at X(28616) and let A'23B'23C'23 be as at X(28626). Triangles A'22B'22C'22 and A'23B'23C'23 are homothetic at X(28644).

X(28644) lies on these lines: {7, 1125}, {405, 934}, {1212, 5308}, {3160, 5129}


X(28645) = PERSPECTOR OF GEMINI TRIANGLE 22 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 22 AND 24

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

X(28645) lies on these lines: {2, 28646}, {8, 1836}, {10, 11544}, {65, 17484}, {72, 79}, {210, 14450}, {226, 18253}, {329, 3812}, {518, 12699}, {527, 1125}, {529, 3244}, {758, 18480}, {946, 5852}, {960, 5905}, {1698, 28609}, {1898, 5057} et al

X(28645) = complement of X(28646)
X(28645) = {X(2),X(28647)}-harmonic conjugate of X(28646)


X(28646) = PERSPECTOR OF GEMINI TRIANGLE 24 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 22 AND 24

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

X(28646) lies on these lines: {1, 3650}, {2, 28645}, {9, 5586}, {10, 527}, {40, 5852}, {63, 3649}, {65, 20078}, {144, 3812}, {518, 6361}, {519, 15685}, {529, 3632}, {758, 18481}, {960, 9965}, {3419, 16118}, {3616, 3683}, {3624, 3928}, {3636, 11194}, {3648, 3868}, {3901, 5441}, {3913, 5850}, {3951, 11246} et al

X(28646) = complement of X(28647)
X(28646) = anticomplement of X(28645)
X(28646) = {X(2),X(28647)}-harmonic conjugate of X(28645)


X(28647) = {X(28645),X(28646)}-HARMONIC CONJUGATE OF X(2)

Barycentrics    5 a^4 + 10 a^3 (b + c) - 2 a^2 (b - c)^2 - 2 a (b + c) (5 b^2 - 4 b c + 5 c^2) - 3 (b^2 - c^2)^2 : :

X(28647) lies on these lines: {1, 527}, {2, 28645}, {8, 15679}, {144, 5302}, {329, 5221}, {529, 20050}, {960, 20059}, {962, 5852}, {1788, 17484} et al

X(28647) = anticomplement of X(28646)
X(28647) = {X(28645),X(28646)}-harmonic conjugate of X(2)


X(28648) = TRILINEAR POLE OF PERSPECTRIX OF GEMINI TRIANGLES 22 AND 24

Barycentrics    1/(7 a^4 (b + c) + a^3 (19 b^2 + 39 b c + 19 c^2) + 3 a^2 (b + c) (6 b^2 + 11 b c + 6 c^2) + a (8 b^4 + 29 b^3 c + 43 b^2 c^2 + 29 b c^3 + 8 c^4) + (b + c) (2 b + c) (b + 2 c) (b^2 + b c + c^2)) : :

X(28648) lies on these lines: (pending)

X(28648) = isogonal conjugate of X(28649)


X(28649) = PERSPECTOR OF UNARY COFACTOR TRIANGLES OF GEMINI TRIANGLES 22 AND 24

Barycentrics    a^2 (7 a^4 (b + c) + a^3 (19 b^2 + 39 b c + 19 c^2) + 3 a^2 (b + c) (6 b^2 + 11 b c + 6 c^2) + a (8 b^4 + 29 b^3 c + 43 b^2 c^2 + 29 b c^3 + 8 c^4) + (b + c) (2 b + c) (b + 2 c) (b^2 + b c + c^2)) : :

X(28649) lies on these lines: {28614, 28637}, {28615, 28625}, {28622, 28643}

X(28649) = isogonal conjugate of X(28648)


X(28650) = HOMOTHETIC CENTER OF ABC AND VERTEX-TRIANGLE OF GEMINI TRIANGLES 25 AND 26

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

X(28650) lies on these lines: {1, 3723}, {2, 3723}, {7, 10588}, {75, 3634}, {86, 1698}, {335, 4751}, {903, 19876}, {1268, 3875} et al

X(28650) = {X(2),X(28652)}-harmonic conjugate of X(28651)


X(28651) = HOMOTHETIC CENTER OF MEDIAL TRIANGLE AND VERTEX-TRIANGLE OF GEMINI TRIANGLES 25 AND 26

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

X(28651) lies on these lines: {2, 3723}, {10, 3873}, {4359, 6376}

X(28651) = complement of X(27789)
X(28651) = {X(2),X(28652)}-harmonic conjugate of X(28650)


X(28652) = HOMOTHETIC CENTER OF ANTICOMPLEMENTARY TRIANGLE AND VERTEX-TRIANGLE OF GEMINI TRIANGLES 25 AND 26

Barycentrics    4 a^3 + 17 a^2 (b + c) + a (21 b^2 + 46 b c + 21 c^2) + (b + c) (8 b^2 + 19 b c + 8 c^2) : :

X(28652) lies on these lines: {2, 3723}, {1268, 3995}

X(28652) = {X(28650),X(28651)}-harmonic conjugate of X(2)


X(28653) = PERSPECTOR OF GEMINI TRIANGLE 25 AND CROSS-TRIANGLE OF ABC AND GEMINI TRIANGLE 25

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

X(28653) lies on these lines: {1, 5564}, {2, 37}, {7, 10588}, {10, 86}, {69, 3844}, {190, 5257}, {274, 313}, {320, 1698}, {594, 6707}, {757, 5247}, {894, 1213}, {966, 3758}, {1125, 4360}, {1444, 5260}, {1654, 4670}, {1826, 14013} et al

X(28653) = {X(86),X(1268)}-harmonic conjugate of X(10)


X(28654) = TRILINEAR PRODUCT OF VERTICES OF GEMINI TRIANGLE 26

Barycentrics    b^2 c^2 (b + c)^2 : :

X(28654) lies on these lines: {2, 1240}, {10, 20966}, {42, 4710}, {63, 4494}, {76, 6539}, {199, 835}, {312, 3969}, {313, 321}, {346, 7017}, {429, 3695}, {561, 8024}, {668, 2895}, {1089, 6535}, {1228, 27801}, {1269, 4980}, {1500, 3948}, {1920, 3266}, {2295, 3765} et al

X(28654) = isotomic conjugate of X(593)
X(28654) = barycentric square of X(321)
X(28654) = trilinear product X(i)*X(j) for these {i,j}: {2, 1089}, {8, 6358}, {10, 321}, {12, 312}, {37, 313}, {42, 27801}, {63, 7141}, {75, 594}, {76, 756}, {92, 3695}, {190, 4036}, {201, 7017}, {264, 3949}, {274, 6535}, {310, 762}, {318, 26942}, {341, 6354}, {561, 1500}, {668, 4024}, {872, 1502}, {1240, 21810}, {1928, 7109}, {1969, 3690}, {1978, 4705}, {2171, 3596}, {3969, 6757}, {4079, 6386}, {4359, 6538}, {4647, 6539}, {14624, 18697}
X(28654) = barycentric product X(i)*X(j) for these {i,j}: {10, 313}, {12, 3596}, {37, 27801}, {69, 7141}, {75, 1089}, {76, 594}, {264, 3695}, {310, 6535}, {312, 6358}, {321, 321}, {561, 756}, {668, 4036}, {762, 6385}, {872, 1928}, {1228, 14624}, {1230, 6539}, {1240, 20653}, {1269, 6538}, {1500, 1502}, {1969, 3949}, {1978, 4024}, {3690, 18022}, {4705, 6386}, {7017, 26942}


X(28655) = CENTROID OF GEMINI TRIANGLE 27

Barycentrics    a^3 - 6 a^2 (b + c) - a (3 b^2 - 34 b c + 3 c^2) + 4 (b + c) (b^2 - 4 b c + c^2) : :

X(28655) lies on these lines: {2, 2415}, {145, 4654}, {519, 6552}, {551, 8688}, {1279, 3699} et al


X(28656) = PERSPECTOR OF GEMINI TRIANGLE 27 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 27 AND 28

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

X(28656) lies on these lines: {2, 28657}, {5, 10}


X(28657) = PERSPECTOR OF GEMINI TRIANGLE 28 AND CROSS-TRIANGLE OF GEMINI TRIANGLES 27 AND 28

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

X(28657) lies on these lines: {2, 28656}, {3452, 12610}


X(28658) = PERSPECTOR OF ABC AND UNARY COFACTOR TRIANGLE OF GEMINI TRIANGLE 28

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

X(28658) lies on these lines: {2, 44}, {6, 36}, {37, 758}, {42, 2245}, {79, 1989}, {111, 4588}, {172, 5549}, {941, 1100}, {1218, 20569}, {1400, 1464} et al

X(28658) = isogonal conjugate of X(5235)
X(28658) = crossdifference of every pair of points on line X(4693)X(4775)
X(28658) = trilinear product X(i)*X(j) for these {i,j}: {37, 2163}, {42, 89}, {65, 2364}, {512, 4604}, {661, 4588}, {798, 4597}, {1400, 2320}, {1918, 20569}, {4017, 5549}
X(28658) = barycentric product X(i)*X(j) for these {i,j}: {10, 2163}, {37, 89}, {65, 2320}, {213, 20569}, {226, 2364}, {512, 4597}, {523, 4588}, {661, 4604}, {5549, 7178}


X(28659) = TRILINEAR PRODUCT OF VERTICES OF GEMINI TRIANGLE 27

Barycentrics    b^3 c^3 (a - b - c) : :
Barycentrics    b^2 c^2/(1 - cos A) : :
Barycentrics    csc^2 A csc^2(A/2) : :

X(28659) lies on these lines: {8, 4087}, {75, 1237}, {76, 321}, {78, 7257}, {192, 1921}, {304, 1978}, {312, 28660}, {318, 3718}, {349, 1502}, {668, 3869}, {700, 7242}, {1909, 4485}, {1969, 18022}, {2171, 17786}, {2292, 6376} et al

X(28659) = isotomic conjugate of X(604)
X(28659) = polar conjugate of X(1395)
X(28659) = barycentric square of isotomic conjugate of X(266)
X(28659) = trilinear product X(i)*X(j) for these {i,j}: {2, 3596}, {8, 76}, {9, 561}, {21, 27801}, {55, 1502}, {75, 312}, {78, 1969}, {85, 341}, {92, 3718}, {264, 345}, {281, 305}, {304, 318}, {313, 333}, {314, 321}, {331, 1265}, {349, 1043}, {668, 4391}
X(28659) = barycentric product X(i)*X(j) for these {i,j}: {8, 561}, {9, 1502}, {55, 1928}, {75, 3596}, {76, 312}, {78, 18022}, {264, 3718}, {304, 7017}, {305, 318}, {313, 314}, {333, 27801}, {341, 6063}, {345, 1969}


X(28660) = TRILINEAR PRODUCT OF VERTICES OF GEMINI TRIANGLE 28

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

X(28660) lies on these lines: {2, 39}, {8, 314}, {10, 4476}, {29, 332}, {58, 5209}, {75, 2292}, {81, 3765}, {85, 6385}, {86, 313}, {92, 304}, {99, 1311}, {257, 1921}, {312, 28659}, {325, 3142}, {333, 3691}, {350, 1193}, {668, 17751}, {670, 1121}, {978, 3760}, {1043, 7257}, {1231, 1952}, {1237, 20947}, {1334, 17787} et al

X(28660) = isotomic conjugate of X(1400)
X(28660) = trilinear product X(i)*X(j) for these {i,j}: {2, 314}, {8, 274), {9, 310}, {11, 4601}, {21, 76}, {29, 304}, {55, 6385}, {60, 27801}, {75, 333}, {81, 3596}, {85, 1043}, {86, 312}, {92, 332}, {99, 4391}, {109, 799}, {313, 2185}, {650, 670}, {668, 4560}
X(28660) = barycentric product X(i)*X(j) for these {i,j}: {8, 310}, {9, 6385}, {10, 18021}, {21, 561}, {29, 305}, {75, 314}, {76, 333}, {86, 3596}, {261, 313}, {264, 332}, {274, 312}, {650, 4602}, {664, 670}, {668, 18155}, {799, 4391}, {1043, 6063}, {1978, 4560}, {2185, 27801}, {4601, 4858}


X(28661) = X(1)X(8055)∩X(8)X(3452)

Barycentrics    (a - b - c) (3 a^3 - 3 a^2 (b + c) - a (7 b^2 - 10 b c + 7 c^2) - (b + c) (b^2 - 6 b c + c^2)) : :

Let A27B27C27 and A28B28C28 be the Gemini triangles 27 and 28, resp. Let LA be the line through A27 parallel to BC, and define LB, LC cyclically. Let A'27 = LB∩LC, and define B'27, C'27 cyclically. Triangle A'27B'27C'27 is homothetic to ABC at X(6557). Let MA be the line through A28 parallel to BC, and define MB and MC cyclically. Let A'28 = MB∩MC, and define B'28 and C'28 cyclically. Triangle A'28B'28C'28 is homothetic to ABC at X(8). Triangles A'27B'27C'27 and A'28B'28C'28 are homothetic at X(28661).

X(28661) lies on these lines: {1, 8055}, {8, 3452}, {37, 2275}, {56, 4488}, {341, 4345}, {1120, 3622} et al


X(28662) =  X(6)X(110)∩X(126)X(3589)

Barycentrics    a^2*(2*a^6 - b^6 - c^6 - 3*a^4*(b^2 + c^2) - 6*a^2*(b^4 - 3*b^2*c^2 + c^4)) : :
X(28662) = X[126]-2*X[3589], X[141]-2*X[6719], 2*X[182]-X[14688], X[1296]-3*X[5085], 5*X[3618]-X[14360], 3*X[5050]+X[11258], X[10748]-3*X[14561], X[14654]+3*X[14853], X[14666]+X[20423]

See Francisco Javier García Capitán AdGeom 5028.

X(28662) lies on these lines: {6,110}, {126,3589}, {141,6719}, {182,14688}, {187,1084}, {511,14650}, {518,11721}, {524,5914}, {543,597}, {1296,5085}, {1428,3325}, {1503,5512}, {1576,21309}, {2330,6019}, {2492,2780}, {3618,14360}, {5027,6088}, {5050,11258}, {5166,9019}, {5480,23699},{6094,11166}, {6096,21448}, {10748,14561}, {14654,14853}, {14666,20423}

X(28662) = midpoint of X(i) and X(j) for these {i,j}: {6,111},{14666,20423}
X(28662) = reflection of X(i) in X(j) for these {i,j}: {126,3589},{141,6719},{14688,182}
X(28662) = radical trace of circles {{X(6),X(13),X(16)}} and {{X(6),X(14),X(15)}}






leftri  Collineation mappings involving Gemini triangle 83: X(28663) - X(28693)  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 83, as in centers X(28663)-X(28693). Then

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

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 27, 2018)

underbar




X(28663) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28663) lies on these lines:


X(28664) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28664) lies on these lines:


X(28665) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28665) lies on these lines:


X(28666) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28666) lies on these lines:


X(28667) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28667) lies on these lines:


X(28668) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28668) lies on these lines:


X(28669) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28669) lies on these lines:


X(28670) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28670) lies on these lines:


X(28671) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28671) lies on these lines:


X(28672) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28672) lies on these lines:


X(28673) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28673) lies on these lines:


X(28674) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28674) lies on these lines:


X(28675) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28675) lies on these lines:


X(28676) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28676) lies on these lines:


X(28677) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28677) lies on these lines:


X(28678) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28678) lies on these lines:


X(28679) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28679) lies on these lines:


X(28680) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28680) lies on these lines:


X(28681) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28681) lies on these lines:


X(28682) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28682) lies on these lines:


X(28683) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28683) lies on these lines:


X(28684) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28684) lies on these lines:


X(28685) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 83

Barycentrics    (b^2 + c^2) (-5 a^8 - 7 a^6 b^2 - a^4 b^4 + 7 a^2 b^6 + 6 b^8 - 7 a^6 c^2 + 13 a^4 b^2 c^2 + 3 a^2 b^4 c^2 - 5 b^6 c^2 - a^4 c^4 + 3 a^2 b^2 c^4 - 2 b^4 c^4 + 7 a^2 c^6 - 5 b^2 c^6 + 6 c^8) : :

X(28685) lies on these lines:


X(28686) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28686) lies on these lines:


X(28687) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28687) lies on these lines:


X(28688) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28688) lies on these lines:


X(28689) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28689) lies on these lines:


X(28690) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28690) lies on these lines:


X(28691) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28691) lies on these lines:


X(28692) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = GEMINI TRIANGLE 83

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

X(28692) lies on these lines:


X(28693) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(24), WHERE A'B'C' = GEMINI TRIANGLE 83

Barycentrics    (b^2 + c^2) (-a^14 + 2 a^10 b^4 + a^8 b^6 - a^6 b^8 - 2 a^4 b^10 + b^14 + 5 a^10 b^2 c^2 + 4 a^8 b^4 c^2 - 4 a^6 b^6 c^2 - 2 a^4 b^8 c^2 - a^2 b^10 c^2 - 2 b^12 c^2 + 2 a^10 c^4 + 4 a^8 b^2 c^4 + 4 a^6 b^4 c^4 - 2 a^2 b^8 c^4 + a^8 c^6 - 4 a^6 b^2 c^6 + 6 a^2 b^6 c^6 + b^8 c^6 - a^6 c^8 - 2 a^4 b^2 c^8 - 2 a^2 b^4 c^8 + b^6 c^8 - 2 a^4 c^10 - a^2 b^2 c^10 - 2 b^2 c^12 + c^14) : :

X(28693) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 84: X(28694) - X(28733)  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 84, as in centers X(28694)-X(28733). Then

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

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 27, 2018)

underbar




X(28694) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28694) lies on these lines:


X(28695) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28695) lies on these lines:


X(28696) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28696) lies on these lines:


X(28697) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28697) lies on these lines:


X(28698) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28698) lies on these lines:


X(28699) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28699) lies on these lines:


X(28700) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28700) lies on these lines:


X(28701) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28701) lies on these lines:


X(28702) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28702) lies on these lines:


X(28703) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28703) lies on these lines:


X(28704) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28704) lies on these lines:


X(28705) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28705) lies on these lines:


X(28706) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28706) lies on these lines:


X(28707) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28707) lies on these lines:


X(28708) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28708) lies on these lines:


X(28709) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28709) lies on these lines:


X(28710) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28710) lies on these lines:


X(28711) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28711) lies on these lines:


X(28712) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28712) lies on these lines:


X(28713) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28713) lies on these lines:


X(28714) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28714) lies on these lines:


X(28715) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28715) lies on these lines:


X(28716) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28716) lies on these lines:


X(28717) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28717) lies on these lines:


X(28718) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28718) lies on these lines:


X(28719) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28719) lies on these lines:


X(28720) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28720) lies on these lines:


X(28721) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 84

Barycentrics    (a^2 - b^2 - c^2) (7 a^8 + 4 a^6 b^2 - 6 a^4 b^4 - 4 a^2 b^6 - b^8 + 4 a^6 c^2 + 4 a^2 b^4 c^2 - 8 b^6 c^2 - 6 a^4 c^4 + 4 a^2 b^2 c^4 + 18 b^4 c^4 - 4 a^2 c^6 - 8 b^2 c^6 - c^8) : :

X(28721) lies on these lines:


X(28722) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28722) lies on these lines:


X(28723) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28723) lies on these lines:


X(28724) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28724) lies on these lines:


X(28725) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28725) lies on these lines:


X(28726) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28726) lies on these lines:


X(28727) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28727) lies on these lines:


X(28728) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28728) lies on these lines:


X(28729) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28729) lies on these lines:


X(28730) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28730) lies on these lines:


X(28731) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 84

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

X(28731) lies on these lines:


X(28732) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(24), WHERE A'B'C' = GEMINI TRIANGLE 84

Barycentrics    (a^2 - b^2 - c^2) (a^14 - 3 a^10 b^4 + 3 a^6 b^8 - a^2 b^12 - 6 a^10 b^2 c^2 + 2 a^8 b^4 c^2 + 6 a^2 b^10 c^2 - 2 b^12 c^2 - 3 a^10 c^4 + 2 a^8 b^2 c^4 + 2 a^6 b^4 c^4 - 7 a^2 b^8 c^4 + 6 b^10 c^4 + 4 a^2 b^6 c^6 - 4 b^8 c^6 + 3 a^6 c^8 - 7 a^2 b^4 c^8 - 4 b^6 c^8 + 6 a^2 b^2 c^10 + 6 b^4 c^10 - a^2 c^12 - 2 b^2 c^12) : :

X(28732) lies on these lines:


X(28733) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(26), WHERE A'B'C' = GEMINI TRIANGLE 84

Barycentrics    (a^2 - b^2 - c^2) (a^14 - 3 a^10 b^4 + 3 a^6 b^8 - a^2 b^12 - 8 a^10 b^2 c^2 + 2 a^6 b^6 c^2 + 2 a^4 b^8 c^2 + 6 a^2 b^10 c^2 - 2 b^12 c^2 - 3 a^10 c^4 + 2 a^6 b^4 c^4 - 2 a^4 b^6 c^4 - 3 a^2 b^8 c^4 + 6 b^10 c^4 + 2 a^6 b^2 c^6 - 2 a^4 b^4 c^6 - 4 a^2 b^6 c^6 - 4 b^8 c^6 + 3 a^6 c^8 + 2 a^4 b^2 c^8 - 3 a^2 b^4 c^8 - 4 b^6 c^8 + 6 a^2 b^2 c^10 + 6 b^4 c^10 - a^2 c^12 - 2 b^2 c^12) : :

X(28733) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 85: X(28734) - X(28780)  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 85, as in centers X(28734)-X(28780). Then

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

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 29, 2018)

underbar




X(28734) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28734) lies on these lines:


X(28735) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28735) lies on these lines:


X(28736) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28736) lies on these lines:


X(28737) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28737) lies on these lines:


X(28738) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28738) lies on these lines:


X(28739) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28739) lies on these lines:


X(28740) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28740) lies on these lines:


X(28741) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28741) lies on these lines:


X(28742) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28742) lies on these lines:


X(28743) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 85

Barycentrics    (a - b) (a - c) (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(28743) lies on these lines:


X(28744) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28744) lies on these lines:


X(28745) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28745) lies on these lines:


X(28746) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28746) lies on these lines:


X(28747) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28747) lies on these lines:


X(28748) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28748) lies on these lines:


X(28749) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28749) lies on these lines:


X(28750) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28750) lies on these lines:


X(28751) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 85

Barycentrics    a^5 b - a^4 b^2 - a^2 b^4 + a b^5 + a^5 c - 2 a^4 b c + 2 a^3 b^2 c - 2 a b^4 c + b^5 c - a^4 c^2 + 2 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(28751) lies on these lines:


X(28752) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28752) lies on these lines:


X(28753) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28753) lies on these lines:


X(28754) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28754) lies on these lines:


X(28755) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28755) lies on these lines:


X(28756) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28756) lies on these lines:


X(28757) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28757) lies on these lines:


X(28758) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 85

Barycentrics    (b - c) (a^4 b - a^2 b^3 + a^4 c - a^3 b c + 2 a^2 b^2 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(28758) lies on these lines:


X(28759) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28759) lies on these lines:


X(28760) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28760) lies on these lines:


X(28761) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28761) lies on these lines:


X(28762) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28762) lies on these lines:


X(28763) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28763) lies on these lines:


X(28764) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28764) lies on these lines:


X(28765) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28765) lies on these lines:


X(28766) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28766) lies on these lines:


X(28767) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28767) lies on these lines:


X(28768) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28768) lies on these lines:


X(28769) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28769) lies on these lines:


X(28770) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28770) lies on these lines:


X(28771) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28771) lies on these lines:


X(28772) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28772) lies on these lines:


X(28773) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28773) lies on these lines:


X(28774) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28774) lies on these lines:


X(28775) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28775) lies on these lines:


X(28776) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28776) lies on these lines:


X(28777) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28777) lies on these lines:


X(28778) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28778) lies on these lines:


X(28779) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28779) lies on these lines:


X(28780) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 85

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

X(28780) lies on these lines:


X(28781) =  ISOGONAL CONJUGATE OF X(14362)

Barycentrics    a^2 (3 a^4-(b^2-c^2)^2-2 a^2 (b^2+c^2)) (a^12-6 a^10 (b^2-c^2)-2 a^2 (b^2-c^2)^4 (3 b^2+5 c^2)+a^8 (15 b^4+14 b^2 c^2-29 c^4)+(b^2-c^2)^4 (b^4+10 b^2 c^2+5 c^4)-4 a^6 (5 b^6+5 b^4 c^2-b^2 c^4-9 c^6)+a^4 (15 b^8-20 b^6 c^2+50 b^4 c^4-36 b^2 c^6-9 c^8)) (a^12+6 a^10 (b^2-c^2)-2 a^2 (b^2-c^2)^4 (5 b^2+3 c^2)+(b^2-c^2)^4 (5 b^4+10 b^2 c^2+c^4)+a^8 (-29 b^4+14 b^2 c^2+15 c^4)+4 a^6 (9 b^6+b^4 c^2-5 b^2 c^4-5 c^6)+a^4 (-9 b^8-36 b^6 c^2+50 b^4 c^4-20 b^2 c^6+15 c^8)) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28686.

X(28781) lies on these lines: {3,2130}, {154,1033}, {2060,14365}

X(28781) = isogonal conjugate of X(14362)

X(28782) =  X(3)X(2131)∩X(1436)X(7037)

Barycentrics    -a^2 (a^2-b^2-c^2) (a^8-4 a^6 (b^2-c^2)+(b^2-c^2)^4-4 a^2 (b^2-c^2) (b^2+c^2)^2+2 a^4 (3 b^4+2 b^2 c^2-5 c^4)) (a^8+4 a^6 (b^2-c^2)+(b^2-c^2)^4+4 a^2 (b^2-c^2) (b^2+c^2)^2+a^4 (-10 b^4+4 b^2 c^2+6 c^4)) (a^16-8 a^14 (b^2+c^2)-56 a^10 (b^2-c^2)^2 (b^2+c^2)-8 a^2 (b^2-c^2)^6 (b^2+c^2)+(b^2-c^2)^6 (b^4+14 b^2 c^2+c^4)+4 a^12 (7 b^4-10 b^2 c^2+7 c^4)+2 a^8 (b^2-c^2)^2 (35 b^4+114 b^2 c^2+35 c^4)-8 a^6 (b^2-c^2)^2 (7 b^6+25 b^4 c^2+25 b^2 c^4+7 c^6)+4 a^4 (b^2-c^2)^2 (7 b^8+50 b^4 c^4+7 c^8)) : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28686.

X(228782) lies on these lines: {3,2131}, {1436,7037}


X(28783) =  ISOGONAL CONJUGATE OF X(14361)

Barycentrics    a^2 (a^2-b^2-c^2) (a^8-4 a^6 (b^2-c^2)+(b^2-c^2)^4-4 a^2 (b^2-c^2) (b^2+c^2)^2+2 a^4 (3 b^4+2 b^2 c^2-5 c^4)) (a^8+4 a^6 (b^2-c^2)+(b^2-c^2)^4+4 a^2 (b^2-c^2) (b^2+c^2)^2+a^4 (-10 b^4+4 b^2 c^2+6 c^4)) : :
Barycentrics    S^4 + (96 R^4-SB SC-32 R^2 SW+2 SW^2) S^2 + 256 R^6 SB+256 R^6 SC-64 R^4 SB SC-128 R^4 SB SW-128 R^4 SC SW+24 R^2 SB SC SW+16 R^2 SB SW^2+16 R^2 SC SW^2-2 SB SC SW^2 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28686.

X(28783) lies on these lines: {3,1033}, {6,14092}, {154,577}, {198,1035}, {1032,3964}, {1092,15905}, {1105,20792}, {1598,13855}, {15341,16391}

X(28783) = isogonal conjugate of X(14361)

X(28784) =  X(3)X(3341)∩X(6)X(2188)

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

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28686.

X(28784) lies on these lines: {3,3341}, {6,2188}, {25,1436}, {56,64}


X(28785) =  ISOGONAL CONJUGATE OF X(14365)

Barycentrics    a^2 (a^4+b^4+2 b^2 c^2-3 c^4-2 a^2 (b^2-c^2)) (a^4-3 b^4+2 b^2 c^2+c^4+2 a^2 (b^2-c^2)) (5 a^12+(b^2-c^2)^6-10 a^10 (b^2+c^2)+36 a^6 (b^2-c^2)^2 (b^2+c^2)+a^8 (-9 b^4+34 b^2 c^2-9 c^4)-a^4 (b^2-c^2)^2 (29 b^4+54 b^2 c^2+29 c^4)+2 a^2 (b^2-c^2)^2 (3 b^6+13 b^4 c^2+13 b^2 c^4+3 c^6)) : :
Barycentrics    (16 R^2+SB+SC-4 SW) S^4 + (1536 R^6+64 R^4 SB+64 R^4 SC-16 R^2 SB SC-896 R^4 SW-16 R^2 SB SW-16 R^2 SC SW+4 SB SC SW+160 R^2 SW^2-8 SW^3) S^2 -1024 R^6 SB SC+640 R^4 SB SC SW-128 R^2 SB SC SW^2+8 SB SC SW^3 : :

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28686.

X(28785) lies on these lines: {3,2130}, {6,14092}, {25,64}, {56,7037}, {1073,9786}, {1301,1498}, {1436,2155}, {1620,11589}, {2060,14362}, {15394,17928}

X(28785) = isogonal conjugate of X(14365)
X(28785) = X(3)-Ceva conjugate of X(64)

X(28786) =  X(3)X(307)∩X(4)X(916)

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

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28686.

X(28786) lies on these lines: {3,307}, {4,916}, {6,226}, {64,516}, {66,674}, {71,440}, {73,6356}, {74,1305}, {272,1175}, {349,2893}, {912,1243}, {1246,15467}, {2218,7083}, {2772,11744}, {6817,8814}, {8804,21091}


X(28787) =  X(3)X(6511)∩X(4)X(912)

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

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28686.

X(28787) lies on these lines: {3,6511}, {4,912}, {6,169}, {54,10202}, {64,517}, {65,23604}, {66,518}, {69,20235}, {71,18674}, {72,21015}, {74,13397}, {81,1175}, {520,3657}, {1177,2836}, {1245,2650}, {2771,11744}, {3874,9028}, {3962,10693}, {8673,10099}, {9940,14528} {8804,21091}

X(28787) = orthocenter of extraversion triangle of X(65)


X(28788) =  X(4)X(5906)∩X(6)X(1210)

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

See Antreas Hatzipolakis and Ercole Suppa, Hyacinthos 28686.

X(28788) lies on these lines: {4,5906}, {6,1210}, {64,515}, {66,8679}, {73,18641}, {2779,11744}






leftri  Collineation mappings involving Gemini triangle 86: X(28789) - X(28837)  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 86, as in centers X(28789)-X(28837). Then

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

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, November 30, 2018)

underbar




X(28789) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28789) lies on these lines:


X(28790) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28790) lies on these lines:


X(28791) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28791) lies on these lines:


X(28792) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28792) lies on these lines:


X(28793) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28793) lies on these lines:


X(28794) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 86

Barycentrics    (a - b - c) (a^5 + a^4 b - a b^4 - b^5 + 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 + 10 a b^2 c^2 - 4 a b c^3 - a c^4 + b c^4 - c^5) : :

X(28794) lies on these lines:


X(28795) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28795) lies on these lines:


X(28796) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28796) lies on these lines:


X(28797) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28797) lies on these lines:


X(28798) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28798) lies on these lines:


X(28799) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28799) lies on these lines:


X(28800) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28800) lies on these lines:


X(28801) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28801) lies on these lines:


X(28802) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28802) lies on these lines:


X(28803) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28803) lies on these lines:


X(28804) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28804) lies on these lines:


X(28805) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28805) lies on these lines:


X(28806) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28806) lies on these lines:


X(28807) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28807) lies on these lines:


X(28808) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28808) lies on these lines:


X(28809) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28809) lies on these lines:


X(28810) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28810) lies on these lines:


X(28811) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28811) lies on these lines:


X(28812) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28812) lies on these lines:


X(28813) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28813) lies on these lines:


X(28814) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28814) lies on these lines:


X(28815) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28815) lies on these lines:


X(28816) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28816) lies on these lines:


X(28817) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28817) lies on these lines:


X(28818) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28818) lies on these lines:


X(28819) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28819) lies on these lines:


X(28820) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28820) lies on these lines:


X(28821) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28821) lies on these lines:


X(28822) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28822) lies on these lines:


X(28823) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28823) lies on these lines:


X(28824) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28824) lies on these lines:


X(28825) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28825) lies on these lines:


X(28826) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28826) lies on these lines:


X(28827) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28827) lies on these lines:


X(28828) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28828) lies on these lines:


X(28829) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28829) lies on these lines:


X(28830) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28830) lies on these lines:


X(28831) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28831) lies on these lines:


X(28832) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28832) lies on these lines:


X(28833) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 86

Barycentrics    (a - b - c) (b - c) (a^4 + a b^3 + 2 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(28833) lies on these lines:


X(28834) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 86

Barycentrics    (a - b - c) (b - c) (a^4 - a^3 b - a^2 b^2 + a b^3 - a^3 c + 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(28834) lies on these lines:


X(28835) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28835) lies on these lines:


X(28836) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28836) lies on these lines:


X(28837) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 86

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

X(28837) lies on these lines:






leftri  Points on circumcircle and line at infinity: X(28838) - X(28915)  rightri

Suppose that X = x : y : z is a point on the line at infinity. All the lines that meet in X are parallel, so that X can be regarded as a direction in the plane of the reference triangle ABC. Let X' be the isogonal conjugate of X, so that X' lies on the circumcircle. Let X'' be the circumcircle-antipode of X', and let X''' be its isogonal conjugate, on the line at infinity. As a direction, X''' is perpendicular to X. In this section, X is given by the form (b - c) (h a^2 + j (b^2 + c^2) + k b c + (h - j + k)(a b + a c) : : , where h, j, k are constants. (Clark Kimberling, November 26, 2018)

In the table below, Columns 1-3 show h, j, k.

Column 4. (b - c) (h a^2 + j (b^2 + c^2) + k b c + (h - j + k)(a b + a c) : : : : , on infinity line, referenced below as x : y : z

Column 5. (isogonal conjugate of x : y : z) = a^2/x + b^2/y + c^2/z : : on circumcircle, referenced below as u : v : w

Column 6. (antipode of u : v : w) = (a^2+b^2-c^2)(a^2-b^2+c^2)u + 2a^2 (a^2-b^2-c^2)v + 2a^2 (a^2-b^2-c^2)w : : on circumcircle, referenced below as u1 : v1 : w1

Column 7. (isogonal conjugate of u1 : v1 : w1) = a^2/u1 + b^2/v1 + c^2/w1

For each row, let X be the point in Column 4 and X' the point in Column 7. Let U be any point in the finite plane of ABC. Then the lines UX and UX' are perpendicular.

In the table below, the points in Column 4 are here given names of the form Point Propus(h,j,k).

h j k Column 4 Column 5Column 6 Column 7
1 0 0 513 100 104 5617
0 1 0 918 919 28838ew 28839
0 0 1 514 101 103 516
1 1 0 514 101 103 516
1 0 1 28840 28841 28842 28843
0 1 1 824 825 28844 28845
1 -1 0 28846 28847 28848 28849
1 0 -1 812 813 12032 28850
0 1 -1 28851 28852 28853 28854
1 1 1 514 101 103 516
-1 1 1 2786 2702 2700 2784
1 -1 1 28855 28856 28857 28858
1 1 -1 514 101 103 516
2 1 1 28859 28860 28861 28862
1 2 1 28863 28864 28865 28866
1 1 2 514 101 103 516
-2 1 1 28867 28868 28869 28870
1 -2 1 28871 28872 28873 28874
1 1 -2 514 101 103 516
-1 2 1 27484 28875 28876 28877
1 -1 2 28878 28879 28880 28881
2 1 -1 28882 28883 28884 28885
2 -1 1 28886 28887 28888 28889
1 2 -1 28890 28891 28892 28893
-1 1 2 522 109 102 515
1 2 2 28894 28895 28896 28897
2 1 2 4977 8701 28173 28174/td>
-1 2 2 514 101 103 516
-1 2 2 28898 28899 28900 28901
2 -1 2 28902 28903 28904 28905
2 2 -1 514 101 103 516
-2 2 1 28906 28907 28908 28909
1 -2 2 28910 28911 28912 28913
2 1 -2 6084 6078 28914 28915
underbar




X(28838) =  CIRCUMCIRCLE-ANTIPODE OF X(919)

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

X(28838) lies on the circumcircle and these lines:

X(28838) = isogonal conjugate of X(28839)
X(28838) = circumcircle-antipode of X(919)


X(28839) =  ISOGONAL CONJUGATE OF X(28838)

Barycentrics    1/(a^5 - a^4 b - a b^4 + b^5 - a^4 c + 2 a^2 b^2 c - b^4 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 a c^4 + 2 b c^4 - 2 c^5) (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 + 2 a^2 b c^2 - a b^2 c^2 - b^3 c^2 + b^2 c^3 - a c^4 - b c^4 + c^5) : :

X(28839) lies on these lines:

X(28839) = isogonal conjugate of X(28838)


X(28840) =  POINT PROPUS(1,0,1)

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

X(28840) lies on these lines: {2, 661}, {30, 511}, {650, 4932}, {693, 4813}, {798, 1019}, {850, 4842}, {1577, 4960}, {3679, 4761}, {4017, 24417}, {4077, 4654}, {4120, 4789}, {4367, 4455}, {4379, 4776}, {4467, 4988}, {4486, 21146}, {4608, 4838}, {4724, 4817}, {4753, 4784}, {4763, 4893}, {4794, 28843}, {4801, 23794}, {4822, 17166}, {4841, 4897}, {4978, 20954}, {4979, 17494}, {5592, 13745}, {14838, 14991}, {15936, 23730}, {20509, 23755}, {20908, 20949}, {24687, 24721}, {24718, 24720}

X(28841) = isogonal conjugate of X(28842)


X(28841) =  ISOGONAL CONJUGATE OF X(28840)

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

X(28841) lies on the circumcircle and these lines:

X(28841) = isogonal conjugate of X(28840)


X(28842) =  CIRCUMCIRCLE-ANTIPODE OF X(29941)

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

X(28842) lies on the circumcircle and these lines:

X(28842) = isogonal conjugate of X(28843)
X(28842) = circumcircle-antipode of X(28841)


X(28843) =  ISOGONAL CONJUGATE OF X(28842)

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

X(28843) lies on these lines:

X(28843) = isogonal conjugate of X(28842)


X(28844) =  CIRCUMCIRCLE-ANTIPODE OF X(825)

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

X(28844) lies on the circumcircle and these lines:

X(28844) = isogonal conjugate of X(28845)
X(28844) = circumcircle-antipode of X(825)


X(28845) =  ISOGONAL CONJUGATE OF X(28844)

Barycentrics    1/((a^5 - a^3 b^2 - a^2 b^3 + b^5 + a^3 c^2 + b^3 c^2 - 2 c^5) (a^5 + a^3 b^2 - 2 b^5 - a^3 c^2 - a^2 c^3 + b^2 c^3 + c^5)) : :

X(28845) lies on these lines: {1, 4056}, {3, 23849}, {4, 1973}, {10, 910}, {30, 511}, {116, 5144}, {1631, 20305}, {1836, 24268}, {2239, 6999}, {2887, 4112}, {3773, 9857}, {3836, 24294}, {4085, 10791}, {4136, 4769}, {4349, 6610}, {5074, 11712}, {5587, 9746}, {5698, 24247}, {5731, 10186}, {5880, 24249}, {17798, 26012}, {24266, 24703}, {24291, 24723}

X(28845) = isogonal conjugate of X(28844)


X(28846) =  POINT PROPUS(1-1,0)

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

X(28846) lies on these lines: {30, 511}, {63, 649}, {226, 3676}, {650, 3798}, {661, 4025}, {894, 28960}, {905, 3709}, {1019, 2484}, {1577, 15413}, {1635, 4786}, {2509, 14838}, {3064, 5307}, {3239, 4369}, {4106, 21104}, {4120, 4379}, {4129, 21188}, {4374, 4391}, {4375, 24333}, {4401, 21003}, {4453, 4776}, {4481, 23829}, {4486, 24720}, {4521, 5745}, {4728, 21183}, {4750, 4893}, {4813, 16892}, {4885, 14321}, {4905, 24462}, {4949, 23813}, {5905, 20295}, {6590, 7192}, {7180, 25098}, {7265, 22044}, {7658, 25666}, {14349, 23785}, {14837, 21195}, {20078, 26853}, {25353, 25381}

X(28846) = isogonal conjugate of X(28847)


X(28847) =  ISOGONAL CONJUGATE OF X(28846)

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

X(28847) lies on the circumcircle and these lines:

X(28847) = isogonal conjugate of X(28846)


X(28848) =  CIRCUMCIRCLE-ANTIPODE OF X(28847)

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

X(28848) lies on the circumcircle and these lines:

X(28848) = isogonal conjugate of X(28849)
X(28848) = circumcircle-antipode of X(28847)


X(28849) =  ISOGONAL CONJUGATE OF X(28848)

Barycentrics    1/((a^5 - a^4 b - a b^4 + b^5 - 2 a^4 c + 4 a^2 b^2 c - 2 b^4 c - 2 a^2 b c^2 - 2 a b^2 c^2 + 3 a c^4 + 3 b c^4 - 2 c^5) (a^5 - 2 a^4 b + 3 a b^4 - 2 b^5 - a^4 c - 2 a^2 b^2 c + 3 b^4 c + 4 a^2 b c^2 - 2 a b^2 c^2 - a c^4 - 2 b c^4 + c^5)) : :

X(28849) lies on these lines: {1, 348}, {6, 21629}, {8, 10025}, {10, 220}, {30, 511}, {40, 3208}, {46, 7131}, {69, 1721}, {165, 29573}, {193, 9801}, {307, 4336}, {329, 28124}, {946, 16825}, {1125, 21258}, {1699, 16833}, {1742, 3879}, {1944, 3332}, {3241, 11200}, {3755, 9620}, {3911, 5091}, {3912, 9441}, {4028, 7580}, {4295, 16091}, {4851, 11495}, {5657, 9746}, {5745, 24264}, {6776, 12717}, {8558, 12514}, {10164, 29600}, {10431, 17156}

X(28849) = isogonal conjugate of X(28842)


X(28850) =  ISOGONAL CONJUGATE OF X(12032)

Barycentrics    1/((-a^3 b^2 + 2 a^2 b^3 - a b^4 + a^4 c - b^4 c - a^3 c^2 + 2 b^3 c^2 - a^2 c^3 - b^2 c^3 + a c^4) (a^4 b - a^3 b^2 - a^2 b^3 + a b^4 - a^3 c^2 - b^3 c^2 + 2 a^2 c^3 + 2 b^2 c^3 - a c^4 - b c^4)) : :

X(28850) lies on these lines: {1, 85}, {2, 10186}, {3, 8301}, {4, 1840}, {5, 20531}, {8, 3177}, {10, 1146}, {30, 511}, {40, 21384}, {55, 24268}, {75, 1742}, {80, 14947}, {105, 9317}, {118, 21090}, {145, 20089}, {165, 16833}, {190, 9355}, {239, 9441}, {659, 19903}, {991, 24293}, {1001, 24249}, {1121, 3679}, {1125, 6706}, {1215, 24255}, {1282, 3732}, {1376, 24266}, {1441, 2293}, {1699, 29573}, {1721, 3875}, {1754, 3791}, {1959, 20556}, {2170, 13576}, {2223, 16609}, {2310, 4552}, {2321, 21629}, {2550, 24247}, {2901, 22035}, {2951, 17151}, {3059, 21084}, {3576, 9746}, {3773, 12618}, {3817, 29600}, {3826, 10012}, {3886, 12652}, {3971, 5927}, {4019, 21278}, {4032, 21746}, {4073, 25252}, {4096, 15064}, {4361, 11495}, {4362, 7580}, {4432, 24294}, {4672, 10791}, {4974, 13329}, {5263, 24291}, {5400, 24003}, {5572, 13563}, {5587, 24808}, {5723, 24980}, {6996, 18788}, {8053, 21231}, {8226, 29653}, {8299, 21232}, {8727, 29671}, {10167, 24165}, {10883, 29643}, {11220, 17155}, {13257, 21093}, {13632, 26446}, {16112, 17262}, {17784, 28124}, {19541, 29649}, {21320, 23774}

X(28850) = isogonal conjugate of X(12032)


X(28851) =  POINT PROPUS(0,1-1)

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

X(28851) lies on these lines: {30, 511}, {661, 3776}, {693, 4120}, {2403, 25237}, {3261, 3762}, {3676, 25666}, {3835, 4927}, {3960, 6586}, {3970, 4079}, {4049, 17758}, {4369, 4468}, {4453, 4893}, {4462, 23755}, {4500, 4931}, {4522, 21146}, {4707, 21130}, {4773, 4897}, {4776, 6545}, {4818, 4824}, {4928, 21183}, {10015, 21195}, {16552, 21390}, {16601, 21348}, {20880, 20906}, {21222, 21225}

X(28851) = isogonal conjugate of X(28847)


X(28852) =  ISOGONAL CONJUGATE OF X(28851)

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

X(28852) lies on the circumcircle and these lines:

X(28852) = isogonal conjugate of X(28851)


X(28853) =  CIRCUMCIRCLE-ANTIPODE OF X(28852)

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

X(28853) lies on the circumcircle and these lines:

X(28853) = isogonal conjugate of X(28854)
X(28853) = circumcircle-antipode of X(28852)


X(28854) =  ISOGONAL CONJUGATE OF X(28853)

Barycentrics    1/((a^5 - 2 a^4 b + a^3 b^2 + a^2 b^3 - 2 a b^4 + b^5 - 2 a^4 c + 4 a^2 b^2 c - 2 b^4 c + a^3 c^2 - 2 a^2 b c^2 - 2 a b^2 c^2 + b^3 c^2 - 2 a^2 c^3 - 2 b^2 c^3 + 4 a c^4 + 4 b c^4 - 2 c^5) (a^5 - 2 a^4 b + a^3 b^2 - 2 a^2 b^3 + 4 a b^4 - 2 b^5 - 2 a^4 c - 2 a^2 b^2 c + 4 b^4 c + a^3 c^2 + 4 a^2 b c^2 - 2 a b^2 c^2 - 2 b^3 c^2 + a^2 c^3 + b^2 c^3 - 2 a c^4 - 2 b c^4 + c^5)) : :

X(28854) lies on these lines: {1, 17078}, {2, 9441}, {30, 511}, {1721, 17274}, {1742, 17378}, {3058, 20358}, {4343, 15936}, {11495, 17313}, {12699, 16825}, {18788, 29573}

X(28854) = isogonal conjugate of X(28853)


X(28855) =  POINT PROPUS(1,-1,1)

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

X(28855) lies on these lines: {30, 511}, {661, 4453}, {1639, 4369}, {3294, 16820}, {3709, 3960}, {3762, 4374}, {3835, 21183}, {4468, 4932}, {4776, 21204}, {4791, 18160}, {4984, 17494}, {21116, 21297}, {22037, 22044}

X(28855) = isogonal conjugate of X(28856)


X(28856) =  ISOGONAL CONJUGATE OF X(28855)

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

X(28856) lies on the circumcircle and these lines:

X(28856) = isogonal conjugate of X(28851)


X(28857) =  CIRCUMCIRCLE-ANTIPODE OF X(28856)

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

X(28857) lies on the circumcircle and these lines:

X(28857) = isogonal conjugate of X(28858)
X(28857) = circumcircle-antipode of X(28856)


X(28858) =  ISOGONAL CONJUGATE OF X(28853)

Barycentrics    1/((a^5 - 2 a^4 b + a^3 b^2 + a^2 b^3 - 2 a b^4 + b^5 - 3 a^4 c + 6 a^2 b^2 c - 3 b^4 c - 3 a^2 b c^2 - 3 a b^2 c^2 - a^2 c^3 - b^2 c^3 + 5 a c^4 + 5 b c^4 - 2 c^5) (a^5 - 3 a^4 b - a^2 b^3 + 5 a b^4 - 2 b^5 - 2 a^4 c - 3 a^2 b^2 c + 5 b^4 c + a^3 c^2 + 6 a^2 b c^2 - 3 a b^2 c^2 - b^3 c^2 + a^2 c^3 - 2 a c^4 - 3 b c^4 + c^5)) : :

X(28858) lies on these lines: {30, 511}, {1699, 4384}, {5603, 24331}, {9441, 10164}, {9778, 17316}, {25352, 26446}

X(28858) = isogonal conjugate of X(28857)


X(28859) =  POINT PROPUS(2,1,1)

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

X(28859) lies on these lines: {30, 511}, {693, 23731}, {1019, 16755}, {3004, 4932}, {3776, 4817}, {4380, 4988}, {4406, 20908}, {4500, 20295}, {4784, 4818}, {4790, 21196}, {14349, 21123}, {25381, 28602}

X(28859) = isogonal conjugate of X(28856)


X(28860) =  ISOGONAL CONJUGATE OF X(28859)

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

X(28860) lies on the circumcircle and these lines:

X(28860) = isogonal conjugate of X(28859)


X(28861) =  CIRCUMCIRCLE-ANTIPODE OF X(28856)

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

X(28861) lies on the circumcircle and these lines:

X(28861) = isogonal conjugate of X(28862)
X(28861) = circumcircle-antipode of X(28860)


X(28862) =  ISOGONAL CONJUGATE OF X(28861)

Barycentrics    1/((a^5 - a^3 b^2 - a^2 b^3 + b^5 + 2 a^4 c - 4 a^2 b^2 c + 2 b^4 c + 3 a^3 c^2 + 2 a^2 b c^2 + 2 a b^2 c^2 + 3 b^3 c^2 - 2 a^2 c^3 - 2 b^2 c^3 - 2 a c^4 - 2 b c^4 - 2 c^5) (a^5 + 2 a^4 b + 3 a^3 b^2 - 2 a^2 b^3 - 2 a b^4 - 2 b^5 + 2 a^2 b^2 c - 2 b^4 c - a^3 c^2 - 4 a^2 b c^2 + 2 a b^2 c^2 - 2 b^3 c^2 - a^2 c^3 + 3 b^2 c^3 + 2 b c^4 + c^5)) : :

X(28862) lies on these lines: {30, 511}, {40, 7380}, {9778, 10186}

X(28862) = isogonal conjugate of X(28861)


X(28863) =  POINT PROPUS(1,2,1)

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

X(28863) lies on these lines: {30, 511}, {1577, 20916}, {1639, 3004}, {2530, 21349}, {3762, 4509}, {3776, 6590}, {3835, 4944}, {3904, 4529}, {4369, 4453}, {4408, 4791}, {4467, 4984}, {4789, 6545}, {4838, 26824}, {4931, 21297}, {4958, 20295}

X(28863) = isogonal conjugate of X(28864)


X(28864) =  ISOGONAL CONJUGATE OF X(28863)

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

X(28864) lies on the circumcircle and these lines:

X(28864) = isogonal conjugate of X(28863)


X(28865) =  CIRCUMCIRCLE-ANTIPODE OF X(28864)

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

X(28865) lies on the circumcircle and these lines:

X(28865) = isogonal conjugate of X(28866)
X(28865) = circumcircle-antipode of X(28864)


X(28866) =  ISOGONAL CONJUGATE OF X(28865)

Barycentrics    1/((2 a^5 - a^4 b - a^3 b^2 - a^2 b^3 - a b^4 + 2 b^5 + 3 a^3 c^2 + 3 b^3 c^2 - 2 a^2 c^3 - 2 b^2 c^3 + a c^4 + b c^4 - 4 c^5) (2 a^5 + 3 a^3 b^2 - 2 a^2 b^3 + a b^4 - 4 b^5 - a^4 c + b^4 c - a^3 c^2 - 2 b^3 c^2 - a^2 c^3 + 3 b^2 c^3 - a c^4 + 2 c^5)) : :

X(28866) lies on these lines: {30, 511}, {1699, 6996}, {6999, 9778}, {10171, 19512}

X(28866) = isogonal conjugate of X(28865)


X(28867) =  POINT PROPUS(-2,1,1)

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

X(28867) lies on these lines: {30, 511}, {661, 27486}, {1577, 4406}, {1638, 3835}, {3700, 4932}, {3776, 20295}, {3798, 25666}, {4467, 4813}, {4481, 4502}, {4500, 7192}, {4522, 4784}, {4750, 4776}, {4763, 4786}, {4885, 4949}, {4940, 21212}, {4979, 25259}, {23800, 24417}

X(28867) = isogonal conjugate of X(28869)


X(28868) =  ISOGONAL CONJUGATE OF X(28867)

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

X(28868) lies on the circumcircle and these lines:

X(28868) = isogonal conjugate of X(28867)


X(28869) =  CIRCUMCIRCLE-ANTIPODE OF X(28868)

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

X(28869) lies on the circumcircle and these lines:

X(28869) = isogonal conjugate of X(28870)
X(28869) = circumcircle-antipode of X(28868)


X(28870) =  ISOGONAL CONJUGATE OF X(28869)

Barycentrics    1/((a^5 - a^3 b^2 - a^2 b^3 + b^5 - 2 a^4 c + 4 a^2 b^2 c - 2 b^4 c - a^3 c^2 - 2 a^2 b c^2 - 2 a b^2 c^2 - b^3 c^2 + 2 a^2 c^3 + 2 b^2 c^3 + 2 a c^4 + 2 b c^4 - 2 c^5) (a^5 - 2 a^4 b - a^3 b^2 + 2 a^2 b^3 + 2 a b^4 - 2 b^5 - 2 a^2 b^2 c + 2 b^4 c - a^3 c^2 + 4 a^2 b c^2 - 2 a b^2 c^2 + 2 b^3 c^2 - a^2 c^3 - b^2 c^3 - 2 b c^4 + c^5)) : :

X(28870) lies on these lines: {1, 17095}, {10, 6603}, {30, 511}, {145, 11200}, {1742, 17377}, {3241, 10186}, {5886, 16825}, {6542, 9441}, {7988, 16833}

X(28870) = isogonal conjugate of X(28869)


X(28871) =  POINT PROPUS(1,-2,1)

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

X(28871) lies on these lines: {30, 511}, {1638, 25666}, {4828, 18160}

X(28871) = isogonal conjugate of X(28872)


X(28872) =  ISOGONAL CONJUGATE OF X(28871)

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

X(28872) lies on the circumcircle and these lines:

X(28872) = isogonal conjugate of X(28871)


X(28873) =  CIRCUMCIRCLE-ANTIPODE OF X(28872)

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

X(28873) lies on the circumcircle and these lines:

X(28873) = isogonal conjugate of X(28874)
X(28873) = circumcircle-antipode of X(28872)


X(28874) =  ISOGONAL CONJUGATE OF X(28873)

Barycentrics    1/((2 a^5 - 3 a^4 b + a^3 b^2 + a^2 b^3 - 3 a b^4 + 2 b^5 - 4 a^4 c + 8 a^2 b^2 c - 4 b^4 c + a^3 c^2 - 4 a^2 b c^2 - 4 a b^2 c^2 + b^3 c^2 - 2 a^2 c^3 - 2 b^2 c^3 + 7 a c^4 + 7 b c^4 - 4 c^5) (2 a^5 - 4 a^4 b + a^3 b^2 - 2 a^2 b^3 + 7 a b^4 - 4 b^5 - 3 a^4 c - 4 a^2 b^2 c + 7 b^4 c + a^3 c^2 + 8 a^2 b c^2 - 4 a b^2 c^2 - 2 b^3 c^2 + a^2 c^3 + b^2 c^3 - 3 a c^4 - 4 b c^4 + 2 c^5)) : :

X(28874) lies on these lines: {30, 511}, {3623, 11200}, {9441, 17266}

X(28874) = isogonal conjugate of X(28873)


X(28875) =  ISOGONAL CONJUGATE OF X(27484)

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

X(28875) lies on the circumcircle and these lines:

X(28875) = isogonal conjugate of X(27484)


X(28876) =  CIRCUMCIRCLE-ANTIPODE OF X(28875)

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

X(28876) lies on the circumcircle and these lines:

X(28876) = isogonal conjugate of X(28877)
X(28876) = circumcircle-antipode of X(28875)


X(28877) =  ISOGONAL CONJUGATE OF X(28876)

Barycentrics    1/((2 a^5 - a^4 b - a^3 b^2 - a^2 b^3 - a b^4 + 2 b^5 - 2 a^4 c + 4 a^2 b^2 c - 2 b^4 c + a^3 c^2 - 2 a^2 b c^2 - 2 a b^2 c^2 + b^3 c^2 + 3 a c^4 + 3 b c^4 - 4 c^5) (2 a^5 - 2 a^4 b + a^3 b^2 + 3 a b^4 - 4 b^5 - a^4 c - 2 a^2 b^2 c + 3 b^4 c - a^3 c^2 + 4 a^2 b c^2 - 2 a b^2 c^2 - a^2 c^3 + b^2 c^3 - a c^4 - 2 b c^4 + 2 c^5)) : :

X(28877) lies on these lines: {30, 511}, {4781, 29616}, {5222, 7384}, {9746, 24808}

X(28877) = isogonal conjugate of X(28876)


X(28878) =  POINT PROPUS(1,-1,2)

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

X(28878) lies on these lines: {30, 511}, {661, 3676}, {2490, 7653}, {3669, 3709}, {4369, 4521}, {4374, 4462}, {4468, 7192}, {4765, 4897}, {4776, 21183}, {4801, 23819}, {4817, 8689}, {4932, 11068}, {17066, 20317}

X(28878) = isogonal conjugate of X(28879)


X(28879) =  ISOGONAL CONJUGATE OF X(28878)

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

X(28879) lies on the circumcircle and these lines:

X(28879) = isogonal conjugate of X(28878)


X(28880) =  CIRCUMCIRCLE-ANTIPODE OF X(28879)

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

X(28880) lies on the circumcircle and these lines:

X(28880) = isogonal conjugate of X(28881)
X(28880) = circumcircle-antipode of X(28879)


X(28881) =  ISOGONAL CONJUGATE OF X(28880)

Barycentrics    1/((a^5 - 3 a^4 b + 2 a^3 b^2 + 2 a^2 b^3 - 3 a b^4 + b^5 - 4 a^4 c + 8 a^2 b^2 c - 4 b^4 c - 4 a^2 b c^2 - 4 a b^2 c^2 - 2 a^2 c^3 - 2 b^2 c^3 + 7 a c^4 + 7 b c^4 - 2 c^5) (a^5 - 4 a^4 b - 2 a^2 b^3 + 7 a b^4 - 2 b^5 - 3 a^4 c - 4 a^2 b^2 c + 7 b^4 c + 2 a^3 c^2 + 8 a^2 b c^2 - 4 a b^2 c^2 - 2 b^3 c^2 + 2 a^2 c^3 - 3 a c^4 - 4 b c^4 + c^5)) : :

X(28881) lies on these lines: {30, 511}, {40, 1334}, {946, 17050}, {962, 20244}, {4349, 6767}, {4356, 15934}, {11200, 16200}

X(28881) = isogonal conjugate of X(28880)


X(28882) =  POINT PROPUS(2,1,-1)

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

X(28882) lies on these lines: {30, 511}, {649, 3776}, {985, 4817}, {1577, 18076}, {1639, 3835}, {4106, 4944}, {4369, 21183}, {4380, 4984}, {4382, 4500}, {4394, 21212}, {4522, 24719}, {4776, 6546}, {4932, 21104}, {4958, 25259}, {11068, 25666}, {14825, 17192}, {21385, 21389}

X(28882) = isogonal conjugate of X(28883)


X(28883) =  ISOGONAL CONJUGATE OF X(28882)

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

X(28883) lies on the circumcircle and these lines:

X(28883) = isogonal conjugate of X(28882)


X(28884) =  CIRCUMCIRCLE-ANTIPODE OF X(28883)

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

X(28884) lies on the circumcircle and these lines:

X(28884) = isogonal conjugate of X(28886)
X(28884) = circumcircle-antipode of X(28884)


X(28885) =  ISOGONAL CONJUGATE OF X(28884)

Barycentrics    1/((a^5 - 2 a^4 b + a^3 b^2 + a^2 b^3 - 2 a b^4 + b^5 + 3 a^3 c^2 + 3 b^3 c^2 - 4 a^2 c^3 - 4 b^2 c^3 + 2 a c^4 + 2 b c^4 - 2 c^5) (a^5 + 3 a^3 b^2 - 4 a^2 b^3 + 2 a b^4 - 2 b^5 - 2 a^4 c + 2 b^4 c + a^3 c^2 - 4 b^3 c^2 + a^2 c^3 + 3 b^2 c^3 - 2 a c^4 + c^5)) : :

X(28885) lies on these lines: {30, 511}, {31, 5222}, {1699, 2887}, {6327, 29616}, {6679, 10164}, {11246, 20358}

X(28885) = isogonal conjugate of X(28884)


X(28886) =  POINT PROPUS(2,-1,1)

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

X(28886) lies on these lines: {30, 511}, {3762, 4406}, {3776, 4813}, {4444, 4776}

X(28886) = isogonal conjugate of X(28887)


X(28887) =  ISOGONAL CONJUGATE OF X(28886)

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

X(28887) lies on the circumcircle and these lines:

X(28887) = isogonal conjugate of X(28886)


X(28888) =  CIRCUMCIRCLE-ANTIPODE OF X(28887)

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

X(28888) lies on the circumcircle and these lines:

X(28888) = isogonal conjugate of X(28889)
X(28888) = circumcircle-antipode of X(28887)


X(28889) =  ISOGONAL CONJUGATE OF X(28888)

Barycentrics    1/((a^5 - 2 a^4 b + a^3 b^2 + a^2 b^3 - 2 a b^4 + b^5 - 4 a^4 c + 8 a^2 b^2 c - 4 b^4 c - a^3 c^2 - 4 a^2 b c^2 - 4 a b^2 c^2 - b^3 c^2 + 6 a c^4 + 6 b c^4 - 2 c^5) (a^5 - 4 a^4 b - a^3 b^2 + 6 a b^4 - 2 b^5 - 2 a^4 c - 4 a^2 b^2 c + 6 b^4 c + a^3 c^2 + 8 a^2 b c^2 - 4 a b^2 c^2 + a^2 c^3 - b^2 c^3 - 2 a c^4 - 4 b c^4 + c^5)) : :

X(28889) lies on these lines: {30, 511}, {750, 5308}, {7988, 16832}, {18788, 29602}

X(28889) = isogonal conjugate of X(28888)


X(28890) =  POINT PROPUS(1,2,-1)

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

X(28890) lies on these lines: {2, 21115}, {30, 511}, {764, 21349}, {1638, 10196}, {1639, 21204}, {3762, 18150}, {3776, 4468}, {4444, 4518}, {4453, 4763}, {4458, 26275}, {4707, 24125}, {4789, 21116}, {4928, 6545}, {25380, 28602}

X(28890) = isogonal conjugate of X(28891)


X(28891) =  ISOGONAL CONJUGATE OF X(28890)

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

X(28891) lies on the circumcircle and these lines:

X(28891) = isogonal conjugate of X(28890)


X(28892) =  CIRCUMCIRCLE-ANTIPODE OF X(28891)

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

X(28892) lies on the circumcircle and these lines:

X(28892) = isogonal conjugate of X(28893)
X(28892) = circumcircle-antipode of X(28891)


X(28893) =  ISOGONAL CONJUGATE OF X(28892)

Barycentrics    1/((2 a^5 - 3 a^4 b + a^3 b^2 + a^2 b^3 - 3 a b^4 + 2 b^5 - 2 a^4 c + 4 a^2 b^2 c - 2 b^4 c + 3 a^3 c^2 - 2 a^2 b c^2 - 2 a b^2 c^2 + 3 b^3 c^2 - 4 a^2 c^3 - 4 b^2 c^3 + 5 a c^4 + 5 b c^4 - 4 c^5) (2 a^5 - 2 a^4 b + 3 a^3 b^2 - 4 a^2 b^3 + 5 a b^4 - 4 b^5 - 3 a^4 c - 2 a^2 b^2 c + 5 b^4 c + a^3 c^2 + 4 a^2 b c^2 - 2 a b^2 c^2 - 4 b^3 c^2 + a^2 c^3 + 3 b^2 c^3 - 3 a c^4 - 2 b c^4 + 2 c^5)) : :

X(28893) lies on these lines: {30, 511}, {40, 24808}, {8227, 21554}

X(28893) = isogonal conjugate of X(28892)


X(28894) =  POINT PROPUS(1,2,2)

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

X(28894) lies on these lines: {30, 511}, {650, 16757}, {693, 20950}, {1491, 21349}, {1577, 4408}, {3004, 4885}, {3700, 4940}, {4024, 4106}, {4380, 17161}, {4382, 4838}, {4391, 18158}, {4394, 21196}, {4411, 20908}, {4467, 4790}, {4468, 4841}, {4500, 23813}, {4776, 4944}, {4820, 20295}

X(28894) = isogonal conjugate of X(28895)


X(28895) =  ISOGONAL CONJUGATE OF X(28894)

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

X(28895) lies on the circumcircle and these lines:

X(28895) = isogonal conjugate of X(28894)


X(28896) =  CIRCUMCIRCLE-ANTIPODE OF X(28895)

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

X(28896) lies on the circumcircle and these lines:

X(28896) = isogonal conjugate of X(28897)
X(28896) = circumcircle-antipode of X(28895)


X(28897) =  ISOGONAL CONJUGATE OF X(28896)

Barycentrics    1/((2 a^5 - 2 a^3 b^2 - 2 a^2 b^3 + 2 b^5 + a^4 c - 2 a^2 b^2 c + b^4 c + 3 a^3 c^2 + a^2 b c^2 + a b^2 c^2 + 3 b^3 c^2 - a^2 c^3 - b^2 c^3 - a c^4 - b c^4 - 4 c^5) (2 a^5 + a^4 b + 3 a^3 b^2 - a^2 b^3 - a b^4 - 4 b^5 + a^2 b^2 c - b^4 c - 2 a^3 c^2 - 2 a^2 b c^2 + a b^2 c^2 - b^3 c^2 - 2 a^2 c^3 + 3 b^2 c^3 + b c^4 + 2 c^5)) : :

X(28897) lies on these lines: {30, 511}, {381, 9746}, {3755, 18907}

X(28897) = isogonal conjugate of X(28896)


X(28898) =  POINT PROPUS(-1,2,2)

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

X(28898) lies on these lines: {2, 4944}, {30, 511}, {37, 905}, {75, 4391}, {192, 17496}, {650, 4467}, {693, 4820}, {984, 1734}, {1577, 4411}, {1638, 3700}, {2400, 27475}, {3004, 4940}, {3239, 17069}, {3739, 21192}, {3776, 23813}, {4106, 16892}, {4364, 23809}, {4379, 4931}, {4408, 20908}, {4468, 4976}, {4828, 24002}, {4897, 6590}, {7178, 30181}, {8061, 27485}, {18155, 21438}, {20950, 23794}, {27484, 28132}

X(28898) = isogonal conjugate of X(28899)


X(28899) =  ISOGONAL CONJUGATE OF X(28894)

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

X(28899) lies on the circumcircle and these lines:

X(28899) = isogonal conjugate of X(28898)


X(28900) =  CIRCUMCIRCLE-ANTIPODE OF X(28899)

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

X(28900) lies on the circumcircle and these lines:

X(28900) = isogonal conjugate of X(28901)
X(28900) = circumcircle-antipode of X(28900)


X(28901) =  ISOGONAL CONJUGATE OF X(28900)

Barycentrics    1/((2 a^5 - 2 a^3 b^2 - 2 a^2 b^3 + 2 b^5 - a^4 c + 2 a^2 b^2 c - b^4 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 + a c^4 + b c^4 - 4 c^5) (2 a^5 - a^4 b + a^3 b^2 + a^2 b^3 + a b^4 - 4 b^5 - a^2 b^2 c + b^4 c - 2 a^3 c^2 + 2 a^2 b c^2 - a b^2 c^2 + b^3 c^2 - 2 a^2 c^3 + b^2 c^3 - b c^4 + 2 c^5)) : :

X(28901) lies on these lines: {30, 511}, {944, 11200}, {3655, 10186}, {5587, 16788}, {5790, 9746}

X(28901) = isogonal conjugate of X(28900)


X(28902) =  POINT PROPUS(2,-1,2)

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

X(28902) lies on these lines: {30, 511}, {661, 1638}, {1019, 22108}, {2527, 4932}, {4406, 4462}, {4521, 7653}, {4813, 21104}, {4897, 27486}, {21131, 23755}

X(28902) = isogonal conjugate of X(28903)


X(28903) =  ISOGONAL CONJUGATE OF X(28902)

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

X(28903) lies on the circumcircle and these lines:

X(28903) = isogonal conjugate of X(28902)


X(28904) =  CIRCUMCIRCLE-ANTIPODE OF X(28903)

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

X(28904) lies on the circumcircle and these lines:

X(28904) = isogonal conjugate of X(28905)
X(28904) = circumcircle-antipode of X(28903)


X(28905) =  ISOGONAL CONJUGATE OF X(28904)

Barycentrics    1/((a^5 - 3 a^4 b + 2 a^3 b^2 + 2 a^2 b^3 - 3 a b^4 + b^5 - 5 a^4 c + 10 a^2 b^2 c - 5 b^4 c - a^3 c^2 - 5 a^2 b c^2 - 5 a b^2 c^2 - b^3 c^2 - a^2 c^3 - b^2 c^3 + 8 a c^4 + 8 b c^4 - 2 c^5) (a^5 - 5 a^4 b - a^3 b^2 - a^2 b^3 + 8 a b^4 - 2 b^5 - 3 a^4 c - 5 a^2 b^2 c + 8 b^4 c + 2 a^3 c^2 + 10 a^2 b c^2 - 5 a b^2 c^2 - b^3 c^2 + 2 a^2 c^3 - b^2 c^3 - 3 a c^4 - 5 b c^4 + c^5)) : :

X(28905) lies on these lines: {30, 511}, {1482, 11200}

X(28905) = isogonal conjugate of X(28904)


X(28906) =  POINT PROPUS(-2,2,1)

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

X(28906) lies on these lines: {30, 511}, {693, 4958}, {1639, 4897}, {3835, 4453}, {4369, 4944}, {4374, 4791}, {4786, 10196}, {4932, 25259}

X(28906) = isogonal conjugate of X(28907)


X(28907) =  ISOGONAL CONJUGATE OF X(28906)

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

X(28907) lies on the circumcircle and these lines:

X(28907) = isogonal conjugate of X(28906)


X(28908) =  CIRCUMCIRCLE-ANTIPODE OF X(28907)

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

X(28908) lies on the circumcircle and these lines:

X(28908) = isogonal conjugate of X(28909)
X(28908) = circumcircle-antipode of X(28907)


X(28909) =  ISOGONAL CONJUGATE OF X(28908)

Barycentrics    1/((2 a^5 - a^4 b - a^3 b^2 - a^2 b^3 - a b^4 + 2 b^5 - 3 a^4 c + 6 a^2 b^2 c - 3 b^4 c - 3 a^2 b c^2 - 3 a b^2 c^2 + a^2 c^3 + b^2 c^3 + 4 a c^4 + 4 b c^4 - 4 c^5) (2 a^5 - 3 a^4 b + a^2 b^3 + 4 a b^4 - 4 b^5 - a^4 c - 3 a^2 b^2 c + 4 b^4 c - a^3 c^2 + 6 a^2 b c^2 - 3 a b^2 c^2 + b^3 c^2 - a^2 c^3 - a c^4 - 3 b c^4 + 2 c^5)) : :

X(28909) lies on these lines: {30, 511}, {165, 17310}, {239, 1699}, {3008, 10171}, {3912, 10164}, {6542, 9778}

X(28909) = isogonal conjugate of X(28908)


X(28910) =  POINT PROPUS(1,-2,2)

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

X(28910) lies on these lines: {30, 511}, {1001, 4378}, {3762, 4411}, {4391, 4828}, {4776, 6548}, {4940, 21104}, {6173, 23598}, {24616, 27484}

X(28910) = isogonal conjugate of X(28911)


X(28911) =  ISOGONAL CONJUGATE OF X(28906)

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

X(28911) lies on the circumcircle and these lines:

X(28911) = isogonal conjugate of X(28910)


X(28912) =  CIRCUMCIRCLE-ANTIPODE OF X(28911)

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

X(28912) lies on the circumcircle and these lines:

X(28912) = isogonal conjugate of X(28913)
X(28912) = circumcircle-antipode of X(28911)


X(28913) =  ISOGONAL CONJUGATE OF X(28912)

Barycentrics    1/((2 a^5 - 4 a^4 b + 2 a^3 b^2 + 2 a^2 b^3 - 4 a b^4 + 2 b^5 - 5 a^4 c + 10 a^2 b^2 c - 5 b^4 c + a^3 c^2 - 5 a^2 b c^2 - 5 a b^2 c^2 + b^3 c^2 - 3 a^2 c^3 - 3 b^2 c^3 + 9 a c^4 + 9 b c^4 - 4 c^5) (2 a^5 - 5 a^4 b + a^3 b^2 - 3 a^2 b^3 + 9 a b^4 - 4 b^5 - 4 a^4 c - 5 a^2 b^2 c + 9 b^4 c + 2 a^3 c^2 + 10 a^2 b c^2 - 5 a b^2 c^2 - 3 b^3 c^2 + 2 a^2 c^3 + b^2 c^3 - 4 a c^4 - 5 b c^4 + 2 c^5)) : :

X(28913) lies on these lines: {30, 511}

X(28913) = isogonal conjugate of X(28912)


X(28914) =  CIRCUMCIRCLE-ANTIPODE OF X(6078)

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

X(28914) lies on the circumcircle and these lines:

X(28914) = isogonal conjugate of X(28915)
X(28914) = circumcircle-antipode of X(6078)


X(28915) =  ISOGONAL CONJUGATE OF X(28914)

Barycentrics    1/((a^5 - 3 a^4 b + 2 a^3 b^2 + 2 a^2 b^3 - 3 a b^4 + b^5 - a^4 c + 2 a^2 b^2 c - b^4 c + 3 a^3 c^2 - a^2 b c^2 - a b^2 c^2 + 3 b^3 c^2 - 5 a^2 c^3 - 5 b^2 c^3 + 4 a c^4 + 4 b c^4 - 2 c^5) (a^5 - a^4 b + 3 a^3 b^2 - 5 a^2 b^3 + 4 a b^4 - 2 b^5 - 3 a^4 c - a^2 b^2 c + 4 b^4 c + 2 a^3 c^2 + 2 a^2 b c^2 - a b^2 c^2 - 5 b^3 c^2 + 2 a^2 c^3 + 3 b^2 c^3 - 3 a c^4 - b c^4 + c^5)) : :

X(28915) lies on these lines: {1, 1358}, {3, 105}, {4, 10743}, {5, 120}, {10, 3039}, {20, 20097}, {30, 511}, {40, 5540}, {140, 6714}, {381, 10712}, {382, 10729}, {549, 9746}, {644, 14661}, {659, 19915}, {970, 3034}, {1001, 20328}, {1351, 10760}, {1385, 11716}, {1482, 10699}, {1565, 14942}, {4307, 15934}, {5901, 11730}, {8751, 20740}, {10246, 11200}, {10738, 10773}

X(28915) = isogonal conjugate of X(28914)






leftri  Collineation mappings involving Gemini triangle 87: X(28916) - X(28960)  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 87, as in centers X(28916)-X(28960). Then

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

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, December 2, 2018)

underbar




X(28916) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28916) lies on these lines:


X(28917) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28917) lies on these lines:


X(28918) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28918) lies on these lines:


X(28919) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 87

Barycentrics    (a - b - c) (2 a^7 + 2 a^6 b - 3 a^5 b^2 - 3 a^4 b^3 + 2 a^3 b^4 + 2 a^2 b^5 - a b^6 - b^7 + 2 a^6 c + 4 a^5 b c - a^4 b^2 c - 4 a^3 b^3 c - 2 a^2 b^4 c + b^6 c - 3 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 - 3 a^4 c^3 - 4 a^3 b c^3 - b^4 c^3 + 2 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(28919) lies on these lines:


X(28920) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28920) lies on these lines:


X(28921) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28921) lies on these lines:


X(28922) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28922) lies on these lines:


X(28923) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28923) lies on these lines:


X(28924) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28924) lies on these lines:


X(28925) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28925) lies on these lines:


X(28926) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28926) lies on these lines:


X(28927) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28927) lies on these lines:


X(28928) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28928) lies on these lines:


X(28929) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28929) lies on these lines:


X(28930) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28930) lies on these lines:


X(28931) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28931) lies on these lines:


X(28932) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28932) lies on these lines:


X(28933) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28933) lies on these lines:


X(28934) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28934) lies on these lines:


X(28935) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28935) lies on these lines:


X(28936) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28936) lies on these lines:


X(28937) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28937) lies on these lines:


X(28938) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28938) lies on these lines:


X(28939) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28939) lies on these lines:


X(28940) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28940) lies on these lines:


X(28941) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 87

Barycentrics    (a - b - c) (5 a^7 + 5 a^6 b - 3 a^5 b^2 - 3 a^4 b^3 - a^3 b^4 - a^2 b^5 - a b^6 - b^7 + 5 a^6 c + 10 a^5 b c - a^4 b^2 c - 4 a^3 b^3 c - 5 a^2 b^4 c - 6 a b^5 c + b^6 c - 3 a^5 c^2 - a^4 b c^2 + 2 a^3 b^2 c^2 + 6 a^2 b^3 c^2 + a b^4 c^2 - 5 b^5 c^2 - 3 a^4 c^3 - 4 a^3 b c^3 + 6 a^2 b^2 c^3 + 12 a b^3 c^3 + 5 b^4 c^3 - a^3 c^4 - 5 a^2 b c^4 + a b^2 c^4 + 5 b^3 c^4 - a^2 c^5 - 6 a b c^5 - 5 b^2 c^5 - a c^6 + b c^6 - c^7) : :

X(28941) lies on these lines:


X(28942) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28942) lies on these lines:


X(28943) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28943) lies on these lines:


X(28944) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28944) lies on these lines:


X(28945) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28945) lies on these lines:


X(28946) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28946) lies on these lines:


X(28947) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 87

Barycentrics    (a - b - c) (7 a^7 + 7 a^6 b - 3 a^5 b^2 - 3 a^4 b^3 - 3 a^3 b^4 - 3 a^2 b^5 - a b^6 - b^7 + 7 a^6 c + 14 a^5 b c - a^4 b^2 c - 4 a^3 b^3 c - 7 a^2 b^4 c - 10 a b^5 c + b^6 c - 3 a^5 c^2 - a^4 b c^2 + 2 a^3 b^2 c^2 + 10 a^2 b^3 c^2 + a b^4 c^2 - 9 b^5 c^2 - 3 a^4 c^3 - 4 a^3 b c^3 + 10 a^2 b^2 c^3 + 20 a b^3 c^3 + 9 b^4 c^3 - 3 a^3 c^4 - 7 a^2 b c^4 + a b^2 c^4 + 9 b^3 c^4 - 3 a^2 c^5 - 10 a b c^5 - 9 b^2 c^5 - a c^6 + b c^6 - c^7) : :

X(28947) lies on these lines:


X(28948) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28948) lies on these lines:


X(28949) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28949) lies on these lines:


X(28950) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28950) lies on these lines:


X(28951) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28951) lies on these lines:


X(28952) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28952) lies on these lines:


X(28953) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(89), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28953) lies on these lines:


X(28954) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28954) lies on these lines:


X(28955) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28955) lies on these lines:


X(28956) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 87

Barycentrics    (a - b - c) (a^5 - 2 a^3 b^2 + a b^4 + 3 a^3 b c - a^2 b^2 c - 5 a b^3 c - b^4 c - 2 a^3 c^2 - a^2 b c^2 + 10 a b^2 c^2 + b^3 c^2 - 5 a b c^3 + b^2 c^3 + a c^4 - b c^4) : :

X(28956) lies on these lines:


X(28957) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28957) lies on these lines:


X(28958) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28958) lies on these lines:


X(28959) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28959) lies on these lines:


X(28960) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 87

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

X(28960) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 88: X(28961) - X(29007)  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 88, as in centers X(28961)-X(29007). Then

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

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, December 1, 2018)

underbar




X(28961) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 88

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

X(28961) lies on these lines:


X(28962) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 88

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

X(28962) lies on these lines:


X(28963) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 88

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

X(28963) lies on these lines:


X(28964) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    2 a^7 - 2 a^6 b - 3 a^5 b^2 + 3 a^4 b^3 + 2 a^3 b^4 - 2 a^2 b^5 - a b^6 + b^7 - 2 a^6 c + 4 a^5 b c + a^4 b^2 c - 4 a^3 b^3 c + 2 a^2 b^4 c - b^6 c - 3 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 + 3 a^4 c^3 - 4 a^3 b c^3 + b^4 c^3 + 2 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(28964) lies on these lines:


X(28965) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 88

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

X(28965) lies on these lines:


X(28966) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 88

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

X(28966) lies on these lines:


X(28967) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 88

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

X(28967) lies on these lines:


X(28968) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 88

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

X(28968) lies on these lines:


X(28969) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    2 a^4 - 2 a^3 b - a^2 b^2 + b^4 - 2 a^3 c + 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(28969) lies on these lines:


X(28970) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 88

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

X(28970) lies on these lines:


X(28971) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 88

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

X(28971) lies on these lines:


X(28972) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 88

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

X(28972) lies on these lines:


X(28973) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 88

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

X(28973) lies on these lines:


X(28974) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 88

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

X(28974) lies on these lines:


X(28975) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 88

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

X(28975) lies on these lines:


X(28976) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 88

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

X(28976) lies on these lines:


X(28977) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 88

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

X(28977) lies on these lines:


X(28978) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 88

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

X(28978) lies on these lines:


X(28979) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 88

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

X(28979) lies on these lines:


X(28980) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    a^5 - 2 a^4 b + 2 a^2 b^3 - a b^4 - 2 a^4 c + 3 a^3 b c + a^2 b^2 c - a b^3 c - b^4 c + a^2 b c^2 + 2 a b^2 c^2 + b^3 c^2 + 2 a^2 c^3 - a b c^3 + b^2 c^3 - a c^4 - b c^4 : :

X(28980) lies on these lines:


X(28981) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    5 a^4 - 8 a^3 b + 2 a^2 b^2 + b^4 - 8 a^3 c + 16 a^2 b c - 4 a b^2 c - 4 b^3 c + 2 a^2 c^2 - 4 a b c^2 + 6 b^2 c^2 - 4 b c^3 + c^4 : :

X(28981) lies on these lines:


X(28982) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    a (a^4 - 2 a^2 b^2 + b^4 - a^2 b c + 4 a b^2 c - 3 b^3 c - 2 a^2 c^2 + 4 a b c^2 - 3 b c^3 + c^4) : :

X(28982) lies on these lines:


X(28983) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    (b - c) (a^5 - a^3 b^2 + 2 a^3 b c - a b^3 c + b^4 c - a^3 c^2 + a b^2 c^2 - a b c^3 + b c^4) : :

X(28983) lies on these lines:


X(28984) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    a (b - c) (a^4 - 2 a^3 b + 2 a b^3 - b^4 - 2 a^3 c + 2 a^2 b c - 2 b^2 c^2 + 2 a c^3 - c^4) : :

X(28984) lies on these lines:


X(28985) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    a^6 - a^5 b - a^3 b^3 + a^2 b^4 - a^5 c + 3 a^4 b c - a^3 b^2 c - a b^4 c - a^3 b c^2 + 2 a^2 b^2 c^2 + b^4 c^2 - a^3 c^3 - 2 b^3 c^3 + a^2 c^4 - a b c^4 + b^2 c^4 : :

X(28985) lies on these lines:


X(28986) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    2 a^3 b - 3 a^2 b^2 + b^4 + 2 a^3 c - 4 a^2 b c + a b^2 c + b^3 c - 3 a^2 c^2 + a b c^2 - 4 b^2 c^2 + b c^3 + c^4 : :

X(28986) lies on these lines:


X(28987) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    5 a^7 - 5 a^6 b - 3 a^5 b^2 + 3 a^4 b^3 - a^3 b^4 + a^2 b^5 - a b^6 + b^7 - 5 a^6 c + 10 a^5 b c + a^4 b^2 c - 4 a^3 b^3 c + 5 a^2 b^4 c - 6 a b^5 c - b^6 c - 3 a^5 c^2 + a^4 b c^2 + 2 a^3 b^2 c^2 - 6 a^2 b^3 c^2 + a b^4 c^2 + 5 b^5 c^2 + 3 a^4 c^3 - 4 a^3 b c^3 - 6 a^2 b^2 c^3 + 12 a b^3 c^3 - 5 b^4 c^3 - a^3 c^4 + 5 a^2 b c^4 + a b^2 c^4 - 5 b^3 c^4 + a^2 c^5 - 6 a b c^5 + 5 b^2 c^5 - a c^6 - b c^6 + c^7 : :

X(28987) lies on these lines:


X(28988) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    a^7 - a^6 b - a^3 b^4 + a^2 b^5 - a^6 c + 3 a^5 b c - 2 a^3 b^3 c + a^2 b^4 c - a b^5 c - 2 a^2 b^3 c^2 + 2 b^5 c^2 - 2 a^3 b c^3 - 2 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 b^3 c^4 + a^2 c^5 - a b c^5 + 2 b^2 c^5 : :

X(28988) lies on these lines:


X(28989) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    a^9 - a^8 b + 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^8 c + 2 a^7 b c - a^6 b^2 c + 2 a^5 b^3 c + a^4 b^4 c - 2 a^3 b^5 c + a^2 b^6 c - 2 a b^7 c + a^7 c^2 - a^6 b c^2 - 4 a^5 b^2 c^2 + 2 a^4 b^3 c^2 + 3 a^3 b^4 c^2 - 3 a^2 b^5 c^2 + 2 b^7 c^2 - a^6 c^3 + 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 + 2 a b^5 c^3 - 2 b^6 c^3 - a^5 c^4 + a^4 b c^4 + 3 a^3 b^2 c^4 + a^2 b^3 c^4 + a^4 c^5 - 2 a^3 b c^5 - 3 a^2 b^2 c^5 + 2 a b^3 c^5 - a^3 c^6 + a^2 b c^6 - 2 b^3 c^6 + a^2 c^7 - 2 a b c^7 + 2 b^2 c^7 : :

X(28989) lies on these lines:


X(28990) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    a^9 + 2 a^8 b - 2 a^7 b^2 - 4 a^6 b^3 + 2 a^5 b^4 + 2 a^4 b^5 - 2 a^3 b^6 + a b^8 + 2 a^8 c - a^7 b c - a^6 b^2 c + 3 a^5 b^3 c + a^4 b^4 c + a^3 b^5 c - 3 a^2 b^6 c - 3 a b^7 c + b^8 c - 2 a^7 c^2 - a^6 b c^2 + 4 a^5 b^2 c^2 + a^4 b^3 c^2 - a^2 b^5 c^2 - 2 a b^6 c^2 + b^7 c^2 - 4 a^6 c^3 + 3 a^5 b c^3 + a^4 b^2 c^3 - 6 a^3 b^3 c^3 + 4 a^2 b^4 c^3 + 3 a b^5 c^3 - b^6 c^3 + 2 a^5 c^4 + a^4 b c^4 + 4 a^2 b^3 c^4 + 2 a b^4 c^4 - b^5 c^4 + 2 a^4 c^5 + a^3 b c^5 - a^2 b^2 c^5 + 3 a b^3 c^5 - b^4 c^5 - 2 a^3 c^6 - 3 a^2 b c^6 - 2 a b^2 c^6 - b^3 c^6 - 3 a b c^7 + b^2 c^7 + a c^8 + b c^8 : :

X(28990) lies on these lines:


X(28991) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    a^10 - 2 a^6 b^4 + a^2 b^8 + 3 a^8 b c - a^7 b^2 c - 3 a^6 b^3 c + 5 a^5 b^4 c + a^4 b^5 c - 3 a^3 b^6 c - a^2 b^7 c - a b^8 c - a^7 b c^2 + a^5 b^3 c^2 + 2 a^4 b^4 c^2 + a^3 b^5 c^2 - 4 a^2 b^6 c^2 - a b^7 c^2 + 2 b^8 c^2 - 3 a^6 b c^3 + a^5 b^2 c^3 + 2 a^4 b^3 c^3 - 2 a^3 b^4 c^3 + a^2 b^5 c^3 + a b^6 c^3 - 2 a^6 c^4 + 5 a^5 b c^4 + 2 a^4 b^2 c^4 - 2 a^3 b^3 c^4 + 6 a^2 b^4 c^4 + a b^5 c^4 - 2 b^6 c^4 + a^4 b c^5 + a^3 b^2 c^5 + a^2 b^3 c^5 + a b^4 c^5 - 3 a^3 b c^6 - 4 a^2 b^2 c^6 + a b^3 c^6 - 2 b^4 c^6 - a^2 b c^7 - a b^2 c^7 + a^2 c^8 - a b c^8 + 2 b^2 c^8 : :

X(28991) lies on these lines:


X(28992) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    a^10 - 3 a^9 b + 6 a^7 b^3 - 2 a^6 b^4 - 4 a^5 b^5 + 2 a^3 b^7 + a^2 b^8 - a b^9 - 3 a^9 c + 3 a^8 b c + 2 a^7 b^2 c - 4 a^6 b^3 c + 4 a^5 b^4 c - 2 a^4 b^5 c - 2 a^3 b^6 c + 4 a^2 b^7 c - a b^8 c - b^9 c + 2 a^7 b c^2 - 2 a^6 b^2 c^2 - 6 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 + 2 a b^7 c^2 + 2 b^8 c^2 + 6 a^7 c^3 - 4 a^6 b c^3 - 6 a^5 b^2 c^3 + 8 a^4 b^3 c^3 - 2 a^3 b^4 c^3 - 4 a^2 b^5 c^3 + 2 a b^6 c^3 - 2 a^6 c^4 + 4 a^5 b c^4 + 2 a^4 b^2 c^4 - 2 a^3 b^3 c^4 + 2 a^2 b^4 c^4 - 2 a b^5 c^4 - 2 b^6 c^4 - 4 a^5 c^5 - 2 a^4 b c^5 + 2 a^3 b^2 c^5 - 4 a^2 b^3 c^5 - 2 a b^4 c^5 + 2 b^5 c^5 - 2 a^3 b c^6 - 2 a^2 b^2 c^6 + 2 a b^3 c^6 - 2 b^4 c^6 + 2 a^3 c^7 + 4 a^2 b c^7 + 2 a b^2 c^7 + a^2 c^8 - a b c^8 + 2 b^2 c^8 - a c^9 - b c^9 : :

X(28992) lies on these lines:


X(28993) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    7 a^7 - 7 a^6 b - 3 a^5 b^2 + 3 a^4 b^3 - 3 a^3 b^4 + 3 a^2 b^5 - a b^6 + b^7 - 7 a^6 c + 14 a^5 b c + a^4 b^2 c - 4 a^3 b^3 c + 7 a^2 b^4 c - 10 a b^5 c - b^6 c - 3 a^5 c^2 + a^4 b c^2 + 2 a^3 b^2 c^2 - 10 a^2 b^3 c^2 + a b^4 c^2 + 9 b^5 c^2 + 3 a^4 c^3 - 4 a^3 b c^3 - 10 a^2 b^2 c^3 + 20 a b^3 c^3 - 9 b^4 c^3 - 3 a^3 c^4 + 7 a^2 b c^4 + a b^2 c^4 - 9 b^3 c^4 + 3 a^2 c^5 - 10 a b c^5 + 9 b^2 c^5 - a c^6 - b c^6 + c^7 : :

X(28993) lies on these lines:


X(28994) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    3 a^7 - 3 a^6 b - 3 a^5 b^2 + 3 a^4 b^3 + a^3 b^4 - a^2 b^5 - a b^6 + b^7 - 3 a^6 c + 4 a^5 b c + a^4 b^2 c + 3 a^2 b^4 c - 4 a b^5 c - b^6 c - 3 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 + 3 a^4 c^3 - 2 a^2 b^2 c^3 + 8 a b^3 c^3 - b^4 c^3 + a^3 c^4 + 3 a^2 b c^4 + a b^2 c^4 - b^3 c^4 - a^2 c^5 - 4 a b c^5 + b^2 c^5 - a c^6 - b c^6 + c^7 : :

X(28994) lies on these lines:


X(28995) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    a^7 - a^6 b - a^5 b^2 + a^4 b^3 - 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 - 2 a b^5 c - a^5 c^2 + a^4 b c^2 + a^3 b^2 c^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 - 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 - 2 a b c^5 + b^2 c^5 : :

X(28995) lies on these lines:


X(28996) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    (a + b - c) (a - b + c) (a^4 - a^2 b^2 - 4 a^2 b c + 5 a b^2 c - b^3 c - a^2 c^2 + 5 a b c^2 - 2 b^2 c^2 - b c^3) : :

X(28996) lies on these lines:


X(28997) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    (a + b - c) (a - b + c) (a^4 - a^2 b^2 - 2 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 - b c^3) : :

X(28997) lies on these lines:


X(28998) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    (a - b) (a - c) (a^5 - a^3 b^2 + a^3 b c + a^2 b^2 c + 2 b^4 c - a^3 c^2 + a^2 b c^2 + 2 a b^2 c^2 - 2 b^3 c^2 - 2 b^2 c^3 + 2 b c^4) : :

X(28998) lies on these lines:


X(28999) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    (a - b) (a - c) (a^2 - a b - a c + 2 b c) (a^2 + b^2 - 2 b c + c^2) : :

X(28999) lies on these lines:


X(29000) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    (a - b) (a - c) (a^5 - a^4 b + a^2 b^3 - a b^4 - a^4 c + 2 a^3 b c - a^2 b^2 c + a b^3 c + b^4 c - a^2 b c^2 - b^3 c^2 + a^2 c^3 + a b c^3 - b^2 c^3 - a c^4 + b c^4) : :

X(29000) lies on these lines:


X(29001) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + a^4 c - 5 a^3 b c + 6 a^2 b^2 c - a b^3 c - b^4 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 - a b c^3 + b^2 c^3 + a c^4 - b c^4 : :

X(29001) lies on these lines:


X(29002) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    (b - c) (-4 a^4 + 7 a^3 b - 3 a^2 b^2 - a b^3 + b^4 + 7 a^3 c - 14 a^2 b c + 7 a b^2 c - 2 b^3 c - 3 a^2 c^2 + 7 a b c^2 - a c^3 - 2 b c^3 + c^4) : :

X(29002) lies on these lines:


X(29003) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    b^2 c^2 (b - c) (-a + b + c) (2 a^3 - a^2 b + b^3 - a^2 c + 2 a b c - b^2 c - b c^2 + c^3) : :

X(29003) lies on these lines:


X(29004) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    (b - c) (a^5 - a^3 b^2 - a^3 b c + 3 a^2 b^2 c - a^3 c^2 + 3 a^2 b c^2 - 5 a b^2 c^2 + b^3 c^2 + b^2 c^3) : :

X(29004) lies on these lines:


X(29005) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    b c (b - c) (3 a^3 - 3 a^2 b - a b^2 + b^3 - 3 a^2 c + 6 a b c - b^2 c - a c^2 - b c^2 + c^3) : :

X(29005) lies on these lines:


X(29006) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    a (a - b) (a - c) (a^2 b^2 - b^4 - 3 a^2 b c + a b^2 c + 2 b^3 c + a^2 c^2 + a b c^2 - 4 b^2 c^2 + 2 b c^3 - c^4) : :

X(29006) lies on these lines:


X(29007) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 88

Barycentrics    a (a + b - c) (a - b + c) (a^2 - 2 a b + b^2 - 2 a c + 3 b c + c^2) : :

X(29007) lies on these lines:


X(29008) =  X(1)X(18341)∩X(11)X(11700)

Barycentrics    2 a^10-3 a^9 b-5 a^8 b^2+9 a^7 b^3+2 a^6 b^4-9 a^5 b^5+4 a^4 b^6+3 a^3 b^7-4 a^2 b^8+b^10-3 a^9 c+14 a^8 b c-7 a^7 b^2 c-22 a^6 b^3 c+24 a^5 b^4 c+a^4 b^5 c-15 a^3 b^6 c+8 a^2 b^7 c+a b^8 c-b^9 c-5 a^8 c^2-7 a^7 b c^2+34 a^6 b^2 c^2-14 a^5 b^3 c^2-27 a^4 b^4 c^2+22 a^3 b^5 c^2+a^2 b^6 c^2-a b^7 c^2-3 b^8 c^2+9 a^7 c^3-22 a^6 b c^3-14 a^5 b^2 c^3+44 a^4 b^3 c^3-10 a^3 b^4 c^3-8 a^2 b^5 c^3-3 a b^6 c^3+4 b^7 c^3+2 a^6 c^4+24 a^5 b c^4-27 a^4 b^2 c^4-10 a^3 b^3 c^4+6 a^2 b^4 c^4+3 a b^5 c^4+2 b^6 c^4-9 a^5 c^5+a^4 b c^5+22 a^3 b^2 c^5-8 a^2 b^3 c^5+3 a b^4 c^5-6 b^5 c^5+4 a^4 c^6-15 a^3 b c^6+a^2 b^2 c^6-3 a b^3 c^6+2 b^4 c^6+3 a^3 c^7+8 a^2 b c^7-a b^2 c^7+4 b^3 c^7-4 a^2 c^8+a b c^8-3 b^2 c^8-b c^9+c^10 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28692.

X(29008) lies on these lines: {1,18341}, {11,11700}, {109,16173}, {117,11715}, {942,1387}, {1125,3738}, {2802,6718}, {2817,6713}, {3576,10771}, {5450,11798}






leftri  Points on circumcircle and line at infinity: X(29009) - X(29157)  rightri

Suppose that X = x : y : z is a point on the line at infinity. All the lines that meet in X are parallel, so that X can be regarded as a direction in the plane of the reference triangle ABC. Let X' be the isogonal conjugate of X, so that X' lies on the circumcircle. Let X'' be the circumcircle-antipode of X', and let X''' be its isogonal conjugate, on the line at infinity. As a direction, X''' is perpendicular to X. (Clark Kimberling, December 3, 2018)

In this section, X is given by the form

(b - c) (h a^3 + i (b^3 + c^3) + j a^2 (b + c) + (i - h) (b c^2 + b^2 c) + k a b c) : : , where h, i, j, k are constants.

In the table below, Columns 1-4 show h, i, j, k.

Column 5. (b - c) (h a^3 + i (b^3 + c^3) + j a^2 (b + c) + (i - h) (b c^2 + b^2 c) + k a b c) : : , on infinity line, referenced below as x : y : z

Column 6. (isogonal conjugate of x : y : z) = a^2/x + b^2/y + c^2/z : : on circumcircle, referenced below as u : v : w

Column 7. (antipode of u : v : w) = (a^2+b^2-c^2)(a^2-b^2+c^2)u + 2a^2 (a^2-b^2-c^2)v + 2a^2 (a^2-b^2-c^2)w : : on circumcircle, referenced below as u1 : v1 : w1

Column 8. (isogonal conjugate of u1 : v1 : w1) = a^2/u1 + b^2/v1 + c^2/w1

For each row, let X be the point in Column 5 and X' the point in Column 8. Let U be any point in the finite plane of ABC. Then the lines UX and UX' are perpendicular.

In the table below, the points in Column 5 are here given names of the form Point Polaris(h,i,j,k).

h i j k Column 5 Column 6Column 7 Column 8
1 0 0 0 814 815 29009 29010
0 2 0 0 826 827 29011 29012
0 0 1 0 512 99 98 511
0 0 0 1 514 101 103 516
1 1 0 0 514 101 103 516
1 0 1 0 29013 29014 29015 29016
1 0 0 1 2787 2703 2699 2783
0 2 1 0 523 110 74 30
0 2 0 2 29017 29018 29019 29020
0 0 1 1 513 100 104 517
0 1 1 1 29021 29022 29023 29024
1 0 2 2 6002 6010 741 740
1 1 0 1 514 101 103 516
1 1 1 0 29025 29026 29027 29028
1 1 1 1 29029 29030 29031 29032
-1 1 0 1 522 109 102 515
2 0 0 -11 29033 29034 29035 29036
1 1 0 -1 514 101 103 516
0 -1 1 0 525 112 1297 1503
0 -1 0 1 29037 29038 29039 29040
0 0 -1 1 4083 932 15323 15310
0 -1 1 1 23875 29041 29042 29043
0 1 -1 1 23876 290441 29045 29046
0 1 1 -1 29047 29048 29049 29050
-1 0 1 1 29051 29052 29053 29054
1 0 -1 1 3907 29055 29056 29057
1 0 1 -1 29058 29059 29060 29061
1 -1 0 0 29062 29063 29064 29065
1 -0 -1 0 29066 29067 29068 29069
1 0 0 -1 29070 29071 29072 29073
-1 1 1 0 29074 29075 29076 29077
1 -1 1 0 29078 29079 29080 29081
1 1 -1 0 29082 29083 29084 29085
-1 1 1 1 29086 29087 29088 29089
1 -1 1 1 29090 29091 29092 29093
1 1 -1 1 29094 29095 29096 29097
1 1 1 -1 29098 29099 29100 29101
1 1 -1 -1 29102 29103 29104 29105
1 -1 1 -1 29106 29107 29108 29109
1 -1 -1 1 29110 29111 29112 29113
2 1 1 1 29114 29115 . .
1 2 1 1 29116 29117 . .
1 1 2 1 29118 29119 . .
1 1 1 2 29120 29121 . .
2 2 1 1 29122 29123 . .
2 1 2 1 29124 29125 . .
2 1 1 2 29126 29127 . .
1 2 2 1 29128 29129 . .
1 2 1 2 29130 29131 . .
1 1 2 2 29132 29133 . .
1 2 2 2 29134 29135 . .
2 1 2 2 29136 29137 . .
2 2 1 2 29138 29139 . .
2 2 2 1 29140 29141 . .
0 1 1 2 29142 29143 . .
0 1 2 1 29144 29145 . .
0 2 1 1 29146 29147 . .
1 0 1 2 29148 29149 . .
1 0 2 1 29150 29151 . .
2 0 1 1 29152 29153 . .
1 1 0 2 514 101 103 516
1 2 0 1 29154 29155 . .
2 1 0 1 29156 29157 . .
1 1 2 0 29158 29159 . .
1 2 1 0 29160 29161 . .
2 1 1 0 29162 29163 . .
0 2 2 1 29164 29164 . .
0 2 1 2 29166 29167 . .
0 1 2 2 29167 29169 . .
1 0 2 2 29170 29171 . .
1 2 0 2 29172 29173 . .
1 2 2 0 29174 29175 . .
2 0 1 2 29176 29177 . .
2 0 2 1 29178 29179 . .
0 1 3 0 3800 907 29180 29181
2 0 -1 0 29182 29183 . .
2 2 0 1 514 101 103 516
2 2 1 0 29184 29185 . .
-1 0 1 2 29186 29187 . .
-1/td> 0 2 1 29188 291689 . .
-1/td> 1 0 2 29190 291691 . .
-1/td> 1 2 0 29192 291693 . .
-1 2 0 1 29194 29195 . .
-1 2 1 0 29196 29197 . .
0 -1 1 2 918 919 28838 29199
0/td> -1 2 1 29200 29201 . .
0 2 -1 1 29202 29203 . .
0 2 1 -1 29204 29205 . .
0 1 -1 2 3910 8687 29206 29207
0 1 2 -1 29208 29209 29210 29211
1 -1 0 2 29212 29213 29214 29215
1 -1 2 0 29216 29217 29218 29219
1/td> 2 -1 0 29220 29221 29222 20223
1/td> 2 0 -1 29224 29225 . .
0/td> 0 -1 3 29226 29227 29228 29229
2 -1 0 1 29230 29231 . .
2 -1 1 0 29232 29233 29234 29235
2 0 -1 1 29236 29237 . .
2 0 1 -1 29239 29239 . .
2 1 -1 0 29240 29241 29242 29243
2 1 0 -1 29244 29245 . .
-1 0 2 2 29246 29247 . .
-1 2 0 2 29248 29249 . .
-1 2 2 0 29250 29251 . .
0 -1 2 2 29252 29253 29254 20255
0 2 -1 2 29256 29257 29258 20259
0 2 2 -1 29260 29261 29262 20263
2 -1 0 2 29264 29265 . .
2 -1 2 0 29266 29267 . .
2 0 -1 2 29268 29269 . .
2 0 2 -1 29270 29271 . .
2 2 -1 0 29272 29273 . .
2 2 0 -1 514 101 103 516
2 0 -1 -1 29274 29275 . .
2 -1 0 -1 29276 29277 . .
2 -1 -1 0 29278 29279 . .
0 2 -1 -1 29280 29281 29282 20283
0 -1 2 -1 29284 29285 29286 20287
0 -1 -1 2 29288 29289 29290 20291
-1 2 0 -1 29292 29293 . .
-1 2 -1 0 29294 29295 29296 29297
-1 0 2 -1/td> 29298 29299 29300 29301
-1 0 -1 2 29302 29303 . .
-1 -1 2 0 29304 29305 29306 20307
-1 -1 0 2 514 101 103 516
0 0 1 2 6372 8708 29308 29309
0 0 2 1 6005 6013 29310 29311
0 1 0 2 29312 29313 29314 29315
0 1 2 0 7927 7953 29316 29317
0 2 0 1 29318 29319 29320 29321
0 2 1 0 7950 7954 29322 29323
1 0 0 2 29324 29325 29326 29327
1 0 2 0 29328 29329 29330 29331
1 2 0 0 29332 29333 29334 29335
2 1 0 0 29336 29337 29338 29339
2 0 1 0 29340 29341 29342 29343
2 0 0 1 29344 29345 29346 29347
0 0 -1 2 891 898 29348 29349
0 0 2 -1 29350 20351 29352 29353
0 -1 0 2 29354 29355 29356 29357
0 -1 2 0 690 691 842 542
0 2 0 -1 29358 29359 29360 29361
0 2 -1 0 3906 11636 14388 11645
-1 0 0 2 29362 29363 29364 29365
-1 0 2 0 29366 29367 29368 29369
-1 2 0 0 29370 29371 29372 29373
2 0 0 -1 29374 29375 29376 29377
underbar




X(29009) =  CIRCUMCIRCLE-ANTIPODE OF X(815)

Barycentrics    a^2 (a^3 b^3 - a b^5 + a^5 c - a^3 b^2 c + a^2 b^3 c - b^5 c + a b^3 c^2 - 2 a^3 c^3 - a b^2 c^3 + b^3 c^3 + a c^5) (a^5 b - 2 a^3 b^3 + a b^5 - 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(29009) lies on the circumcircle and these lines:

X(29009) = isogonal conjugate of X(29010)
X(29009) = circumcircle-antipode of X(815)


X(29010) =  ISOGONAL CONJUGATE OF X(29009)

Barycentrics    1/((a^3 b^3 - a b^5 + a^5 c - a^3 b^2 c + a^2 b^3 c - b^5 c + a b^3 c^2 - 2 a^3 c^3 - a b^2 c^3 + b^3 c^3 + a c^5) (a^5 b - 2 a^3 b^3 + a b^5 - 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(29010) lies on these lines: {3, 75}, {4, 192}, {5, 37}, {10, 24269}, {20, 1278}, {30, 511}, {48, 24332}, {92, 20760}, {100, 20887}, {140, 3739}, {312, 19540}, {321, 4192}, {335, 24833}, {355, 984}, {376, 4740}, {381, 4664}, {382, 3644}, {495, 3931}, {546, 4681}, {547, 4755}, {548, 4726}, {549, 4688}, {550, 4686}, {631, 4699}, {942, 4032}, {944, 24349}, {946, 3993}, {1009, 26665}, {1214, 20256}, {1284, 23690}, {1385, 24325}, {1656, 4687}, {1657, 4764}, {1733, 2223}, {1943, 22161}, {1944, 17976}, {2901, 15488}, {3090, 27268}, {3091, 4704}, {3146, 4788}, {3149, 20171}, {3522, 4821}, {3523, 4772}, {3526, 4751}, {3530, 4739}, {3627, 4718}, {3628, 4698}, {3696, 5690}, {3797, 6996}, {3842, 9956}, {4021, 5719}, {4043, 19648}, {4087, 16085}, {4358, 19546}, {4431, 5295}, {4451, 5015}, {4671, 19647}, {4709, 11362}, {4812, 19548}, {5536, 13244}, {5770, 27472}, {5901, 15569}, {6796, 10104}, {11997, 15171}, {12588, 24248}, {15908, 21927}, {16056, 17862}, {16850, 24547}, {17479, 20242}, {17592, 17718}, {18750, 22149}, {19513, 20891}, {19514, 20892}, {19541, 20173}, {19542, 19791}, {19549, 20923}, {19646, 22016}, {20879, 22060}, {21072, 21243}, {21168, 27484}

X(29010) = isogonal conjugate of X(29009)


X(29011) =  CIRCUMCIRCLE-ANTIPODE OF X(827)

Barycentrics    a^2 (a^6 - a^4 b^2 - a^2 b^4 + b^6 + a^4 c^2 + b^4 c^2 - 2 c^6) (a^6 + a^4 b^2 - 2 b^6 - a^4 c^2 - a^2 c^4 + b^2 c^4 + c^6) : :

X(29011) lies on the circumcircle and these lines:

X(29011) = isogonal conjugate of X(29012)
X(29011) = circumcircle-antipode of X(827)


X(29012) =  ISOGONAL CONJUGATE OF X(29011)

Barycentrics    1/((a^6 - a^4 b^2 - a^2 b^4 + b^6 + a^4 c^2 + b^4 c^2 - 2 c^6) (a^6 + a^4 b^2 - 2 b^6 - a^4 c^2 - a^2 c^4 + b^2 c^4 + c^6)) : :

X(29012) lies on these lines: {2, 6030}, {3, 2916}, {4, 83}, {5, 5092}, {6, 382}, {20, 1352}, {22, 11550}, {23, 125}, {26, 20299}, {30, 511}, {40, 12783}, {55, 12944}, {56, 12954}, {66, 3357}, {67, 11559}, {69, 3529}, {74, 1287}, {98, 8784}, {99, 5207}, {110, 5189}, {113, 7574}, {114, 5999}, {115, 1691}, {140, 17712}, {141, 550}, {143, 18128}, {147, 9866}, {159, 12085}, {184, 7391}, {206, 18569}, {316, 12215}, {323, 24981}, {376, 11178}, {381, 5085}, {383, 6774}, {389, 7553}, {401, 15595}, {427, 13394}, {428, 5943}, {468, 6723}, {485, 8993}, {486, 13984}, {546, 3589}, {548, 18358}, {549, 25561}, {575, 3627}, {576, 3146}, {597, 15687}, {599, 15681}, {611, 12943}, {613, 12953}, {620, 5031}, {626, 4048}, {858, 1495}, {944, 7977}, {946, 12264}, {1080, 6771}, {1209, 13564}, {1350, 1657}, {1351, 5073}, {1370, 9306}, {1386, 22793}, {1428, 3583}, {1469, 10483}, {1478, 10064}, {1479, 10080}, {1506, 5116}, {1513, 6036}, {1539, 6593}, {1568, 14157}, {1587, 19092}, {1588, 19091}, {1658, 6697}, {1843, 6240}, {1853, 9909}, {1899, 7500}, {2070, 15061}, {2076, 6781}, {2080, 10991}, {2330, 3585}, {3060, 11225}, {3094, 7756}, {3153, 15462}, {3448, 15107}, {3543, 11179}, {3575, 12144}, {3581, 16003}, {3796, 5064}, {3819, 7667}, {3830, 5050}, {3843, 12017}, {3853, 13470}, {4299, 12589}, {4302, 12588}, {5026, 5103}, {5039, 7737}, {5059, 5921}, {5093, 15684}, {5097, 8550}, {5111, 5477}, {5133, 22352}, {5159, 15448}, {5169, 15080}, {5171, 6308}, {5181, 12367}, {5251, 9840}, {5448, 14862}, {5449, 15579}, {5562, 16659}, {5596, 5878}, {5611, 23001}, {5615, 23010}, {5621, 5899}, {5642, 10989}, {5643, 7693}, {5651, 16063}, {5654, 6759}, {5691, 9903}, {5870, 6274}, {5871, 6275}, {5892, 13490}, {5907, 16655}, {6146, 13598}, {6284, 13078}, {6296, 9735}, {6297, 9736}, {6313, 9738}, {6317, 9739}, {6329, 15807}, {6699, 7575}, {6756, 9729}, {7354, 18983}, {7387, 14852}, {7401, 13347}, {7418, 14811}, {7464, 16163}, {7519, 18911}, {7540, 9730}, {7684, 20415}, {7685, 20416}, {7687, 11799}, {7710, 9765}, {7728, 19140}, {7750, 14994}, {7753, 13331}, {8597, 18800}, {8598, 19662}, {8703, 21167}, {8721, 9737}, {9698, 12055}, {9825, 17704}, {9833, 13346}, {9834, 12476}, {9835, 12477}, {9838, 12994}, {9839, 12995}, {9863, 9990}, {9967, 18563}, {9969, 11819}, {9970, 10721}, {9973, 18565}, {10113, 20301}, {10116, 10263}, {10128, 10219}, {10154, 23332}, {10182, 18281}, {10193, 18324}, {10282, 23335}, {10295, 12140}, {10296, 13202}, {10313, 13236}, {10328, 21248}, {10575, 19161}, {10722, 12177}, {10733, 11579}, {11180, 15683}, {11245, 21849}, {11381, 12225}, {11416, 25321}, {11442, 20062}, {11500, 12339}, {11574, 12605}, {11745, 15012}, {12007, 22330}, {12083, 18474}, {12084, 15577}, {12105, 20397}, {12107, 20191}, {12110, 12206}, {12111, 15084}, {12112, 15063}, {12113, 12795}, {12114, 12924}, {12115, 13112}, {12116, 13113}, {12121, 12584}, {12134, 15644}, {12156, 14912}, {12220, 15086}, {12250, 20079}, {12290, 15103}, {12294, 18560}, {12295, 18325}, {12359, 14864}, {12362, 16621}, {12902, 16010}, {12974, 21736}, {13414, 14807}, {13415, 14808}, {14070, 23329}, {14458, 22712}, {14683, 23061}, {15030, 16658}, {15069, 17800}, {15116, 25564}, {15573, 15588}, {15688, 21358}, {15761, 20300}, {15800, 19150}, {16625, 18914}, {18374, 18403}, {18383, 23300}, {18390, 18534}, {18438, 18562}, {18859, 19596}, {19145, 23251}, {19146, 23261}, {19149, 22802}, {19160, 28343}, {20304, 25338}, {20850, 26958}

X(29012) = isogonal conjugate of X(29011)


X(29013) =  POINT POLARIS(1,0,1,0)

Barycentrics    (b - c) (-a^3 - a^2 b - a^2 c + b^2 c + b c^2) : :

X(29013) lies on these lines: {30, 511}, {649, 1577}, {650, 4129}, {663, 4170}, {667, 4010}, {693, 1019}, {905, 4106}, {1734, 21301}, {2504, 3798}, {2530, 24719}, {2533, 4834}, {3716, 4401}, {3762, 4498}, {3766, 17899}, {3835, 14838}, {3960, 23724}, {4063, 4380}, {4367, 4810}, {4369, 4823}, {4382, 4978}, {4462, 21385}, {4486, 8630}, {4560, 14349}, {9508, 21260}, {17925, 23800}, {21348, 22043}

X(29013) = isogonal conjugate of X(29014)


X(29014) =  ISOGONAL CONJUGATE OF X(29013)

Barycentrics    a^2/((b - c) (a^3 + a^2 b + a^2 c - b^2 c - b c^2)) : :

X(29014) lies on the circumcircle and these lines:

X(29014) = isogonal conjugate of X(29013)


X(29015) =  CIRCUMCIRCLE-ANTIPODE OF X(29014)

Barycentrics    a^2 (-a^4 b^2 + a^3 b^3 + a^2 b^4 - a b^5 + a^5 c - a^3 b^2 c + a^2 b^3 c - b^5 c + a b^3 c^2 + b^4 c^2 - 2 a^3 c^3 - a b^2 c^3 + b^3 c^3 - b^2 c^4 + a c^5) (a^5 b - 2 a^3 b^3 + a b^5 - a^4 c^2 - a^3 b 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 + b^3 c^3 + a^2 c^4 + b^2 c^4 - a c^5 - b c^5) : :

X(29015) lies on the circumcircle and these lines:

X(29015) = isogonal conjugate of X(29016)
X(29015) = circumcircle-antipode of X(29014)


X(29016) =  ISOGONAL CONJUGATE OF X(29015)

Barycentrics    1/((-a^4 b^2 + a^3 b^3 + a^2 b^4 - a b^5 + a^5 c - a^3 b^2 c + a^2 b^3 c - b^5 c + a b^3 c^2 + b^4 c^2 - 2 a^3 c^3 - a b^2 c^3 + b^3 c^3 - b^2 c^4 + a c^5) (a^5 b - 2 a^3 b^3 + a b^5 - a^4 c^2 - a^3 b 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 + b^3 c^3 + a^2 c^4 + b^2 c^4 - a c^5 - b c^5)) : :

X(29016) lies on these lines: {1, 1441}, {3, 4361}, {4, 2901}, {5, 17243}, {8, 25252}, {10, 5721}, {30, 511}, {40, 5767}, {75, 991}, {92, 3190}, {101, 242}, {145, 3332}, {239, 13329}, {427, 21072}, {596, 12675}, {990, 3875}, {1125, 17043}, {1736, 4552}, {1754, 3187}, {1818, 4858}, {1876, 4605}, {2223, 7235}, {2321, 12618}, {3159, 5777}, {3175, 5927}, {3971, 15064}, {4133, 21629}, {4300, 4647}, {4358, 5400}, {4360, 13727}, {4716, 9441}, {4851, 5805}, {5295, 15852}, {5732, 17151}, {5759, 5839}, {5779, 17262}, {6358, 14547}, {9940, 24176}, {14872, 24068}, {16833, 21153}, {17792, 24269}, {22001, 26893}, {24257, 24309}

X(29016) = isogonal conjugate of X(29015)


X(29017) =  POINT POLARIS(0,1,0,1)

Barycentrics    (b - c) (b^3 + a b c + b^2 c + b c^2 + c^3) : :

X(29017) lies on these lines: {30, 511}, {693, 3801}, {1491, 21124}, {3777, 16892}, {4024, 21118}, {4088, 4490}, {4122, 4391}, {4142, 4874}, {4522, 21051}, {7650, 15416}

X(29017) = isogonal conjugate of X(29018)


X(29018) =  ISOGONAL CONJUGATE OF X(29017)

Barycentrics    a^2/((b - c) (b^3 + a b c + b^2 c + b c^2 + c^3)) : :

X(29018) lies on the circumcircle and these lines:

X(29018) = isogonal conjugate of X(29017)


X(29019) =  CIRCUMCIRCLE-ANTIPODE OF X(29018)

Barycentrics    a^2 (a^6 - a^4 b^2 - a^2 b^4 + b^6 + a^4 b c - a^3 b^2 c - a^2 b^3 c + a b^4 c + a^4 c^2 + b^4 c^2 + a^2 b c^3 + a b^2 c^3 - 2 a b c^4 - 2 c^6) (a^6 + a^4 b^2 - 2 b^6 + a^4 b c + a^2 b^3 c - 2 a b^4 c - a^4 c^2 - a^3 b c^2 + a b^3 c^2 - a^2 b c^3 - a^2 c^4 + a b c^4 + b^2 c^4 + c^6) : :

X(29019) lies on the circumcircle and these lines:

X(29019) = isogonal conjugate of X(29020)
X(29019) = circumcircle-antipode of X(29018)


X(29020) =  ISOGONAL CONJUGATE OF X(29019)

Barycentrics    1/((a^6 - a^4 b^2 - a^2 b^4 + b^6 + a^4 b c - a^3 b^2 c - a^2 b^3 c + a b^4 c + a^4 c^2 + b^4 c^2 + a^2 b c^3 + a b^2 c^3 - 2 a b c^4 - 2 c^6) (a^6 + a^4 b^2 - 2 b^6 + a^4 b c + a^2 b^3 c - 2 a b^4 c - a^4 c^2 - a^3 b c^2 + a b^3 c^2 - a^2 b c^3 - a^2 c^4 + a b c^4 + b^2 c^4 + c^6)) : :

X(29020) lies on these lines: {4, 17918}, {30, 511}, {4389, 18481}, {5267, 24251}, {5710, 12943}, {17369, 18480}

X(29020) = isogonal conjugate of X(29019)


X(29021) =  POINT POLARIS(0,1,1,1)

Barycentrics    (b - c) (a^2 b + b^3 + a^2 c + a b c + b^2 c + b c^2 + c^3) : :

X(29021) lies on these lines: {30, 511}, {1734, 21124}, {3776, 23789}, {4129, 4522}, {4369, 20517}, {4490, 4808}, {4905, 16892}, {6590, 21185}

X(29021) = isogonal conjugate of X(29022)


X(29022) =  ISOGONAL CONJUGATE OF X(29021)

Barycentrics    a^2/((b - c) (a^2 b + b^3 + a^2 c + a b c + b^2 c + b c^2 + c^3)) : :

X(29022) lies on the circumcircle and these lines:

X(29022) = isogonal conjugate of X(29021)


X(29023) =  CIRCUMCIRCLE-ANTIPODE OF X(29022)

Barycentrics    a^2 (a^6 - a^4 b^2 - a^2 b^4 + 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 + 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 - 2 c^6) (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 - a^4 c^2 - a^3 b c^2 + a b^3 c^2 - b^4 c^2 - a^2 b c^3 - a^2 c^4 + a b c^4 + 2 b^2 c^4 + c^6) : :

X(29023) lies on the circumcircle and these lines:

X(29023) = isogonal conjugate of X(29024)
X(29023) = circumcircle-antipode of X(29022)


X(29024) =  ISOGONAL CONJUGATE OF X(29023)

Barycentrics    1/((a^6 - a^4 b^2 - a^2 b^4 + 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 + 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 - 2 c^6) (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 - a^4 c^2 - a^3 b c^2 + a b^3 c^2 - b^4 c^2 - a^2 b c^3 - a^2 c^4 + a b c^4 + 2 b^2 c^4 + c^6)) : :

X(29024) lies on these lines: {20, 24309}, {30, 511}, {1766, 5691}, {3663, 7354}, {4297, 12610}, {4353, 18990}, {5086, 16566}, {10444, 10464}, {10483, 24248}

X(29024) = isogonal conjugate of X(29023)


X(29025) =  POINT POLARIS(1,1,1,0)

Barycentrics    (b - c) (a^3 + a^2 b + b^3 + a^2 c + c^3) : :

X(29025) lies on these lines: {30, 511}, {649, 3801}, {659, 8636}, {4142, 4782}, {4707, 4834}, {4992, 6332}

X(29025) = isogonal conjugate of X(29026)


X(29026) =  ISOGONAL CONJUGATE OF X(29025)

Barycentrics    a^2/((b - c) (a^3 + a^2 b + b^3 + a^2 c + c^3)) : :

X(29026) lies on the circumcircle and these lines:

X(29026) = isogonal conjugate of X(29025)


X(29027) =  CIRCUMCIRCLE-ANTIPODE OF X(29026)

Barycentrics    a^2 (a^6 - a^5 b - a^4 b^2 + 2 a^3 b^3 - a^2 b^4 - a b^5 + b^6 + 2 a^4 c^2 + a^3 b c^2 + a b^3 c^2 + 2 b^4 c^2 - a^3 c^3 - a^2 b c^3 - a b^2 c^3 - b^3 c^3 - a^2 c^4 - b^2 c^4 + a c^5 + b c^5 - 2 c^6) (a^6 + 2 a^4 b^2 - a^3 b^3 - a^2 b^4 + a b^5 - 2 b^6 - a^5 c + a^3 b^2 c - a^2 b^3 c + b^5 c - a^4 c^2 - a b^3 c^2 - b^4 c^2 + 2 a^3 c^3 + a b^2 c^3 - b^3 c^3 - a^2 c^4 + 2 b^2 c^4 - a c^5 + c^6) : :

X(29027) lies on the circumcircle and these lines:

X(29027) = isogonal conjugate of X(29028)
X(29027) = circumcircle-antipode of X(29026)


X(29028) =  ISOGONAL CONJUGATE OF X(29027)

Barycentrics    1/((a^6 - a^5 b - a^4 b^2 + 2 a^3 b^3 - a^2 b^4 - a b^5 + b^6 + 2 a^4 c^2 + a^3 b c^2 + a b^3 c^2 + 2 b^4 c^2 - a^3 c^3 - a^2 b c^3 - a b^2 c^3 - b^3 c^3 - a^2 c^4 - b^2 c^4 + a c^5 + b c^5 - 2 c^6) (a^6 + 2 a^4 b^2 - a^3 b^3 - a^2 b^4 + a b^5 - 2 b^6 - a^5 c + a^3 b^2 c - a^2 b^3 c + b^5 c - a^4 c^2 - a b^3 c^2 - b^4 c^2 + 2 a^3 c^3 + a b^2 c^3 - b^3 c^3 - a^2 c^4 + 2 b^2 c^4 - a c^5 + c^6)) : :

X(29028) lies on these lines: {3, 16706}, {4, 17280}, {5, 17357}, {30, 511}

X(29028) = isogonal conjugate of X(29027)


X(29029) =  POINT POLARIS(1,1,1,1)

Barycentrics    (b - c) (a^3 + a^2 b + b^3 + a^2 c + a b c + c^3) : :

X(29029) lies on these lines: {30, 511}, {1019, 3801}, {4369, 17048}, {4707, 4784}, {7192, 20247}

X(29029) = isogonal conjugate of X(29030)


X(29030) =  ISOGONAL CONJUGATE OF X(29029)

Barycentrics    a^2/((b - c) (a^3 + a^2 b + b^3 + a^2 c + a b c + c^3)) : :

X(29030) lies on the circumcircle and these lines:

X(29030) = isogonal conjugate of X(29029)


X(29031) =  CIRCUMCIRCLE-ANTIPODE OF X(29030)

Barycentrics    a^2 (a^6 - a^5 b - a^4 b^2 + 2 a^3 b^3 - a^2 b^4 - a b^5 + 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 + a b^3 c^2 + 2 b^4 c^2 - a^3 c^3 - b^3 c^3 - a^2 c^4 - 2 a b c^4 - b^2 c^4 + a c^5 + b c^5 - 2 c^6) (a^6 + 2 a^4 b^2 - a^3 b^3 - a^2 b^4 + a b^5 - 2 b^6 - a^5 c + a^4 b c + a^3 b^2 c - 2 a b^4 c + b^5 c - a^4 c^2 - a^3 b c^2 - b^4 c^2 + 2 a^3 c^3 - a^2 b c^3 + a b^2 c^3 - b^3 c^3 - a^2 c^4 + a b c^4 + 2 b^2 c^4 - a c^5 + c^6) : :

X(29031) lies on the circumcircle and these lines:

X(29031) = isogonal conjugate of X(29032)
X(29031) = circumcircle-antipode of X(29030)


X(29032) =  ISOGONAL CONJUGATE OF X(29031)

Barycentrics    1/((a^6 - a^5 b - a^4 b^2 + 2 a^3 b^3 - a^2 b^4 - a b^5 + 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 + a b^3 c^2 + 2 b^4 c^2 - a^3 c^3 - b^3 c^3 - a^2 c^4 - 2 a b c^4 - b^2 c^4 + a c^5 + b c^5 - 2 c^6) (a^6 + 2 a^4 b^2 - a^3 b^3 - a^2 b^4 + a b^5 - 2 b^6 - a^5 c + a^4 b c + a^3 b^2 c - 2 a b^4 c + b^5 c - a^4 c^2 - a^3 b c^2 - b^4 c^2 + 2 a^3 c^3 - a^2 b c^3 + a b^2 c^3 - b^3 c^3 - a^2 c^4 + a b c^4 + 2 b^2 c^4 - a c^5 + c^6)) : :

X(29032) lies on these lines: {30, 511}, {382, 5695}, {1621, 4220}, {1770, 10544}, {3430, 12699}, {3454, 22793}, {3579, 7683}, {3744, 4854}, {3821, 14810}, {17382, 17502}

X(29032) = isogonal conjugate of X(29031)


X(29033) =  POINT POLARIS(-1,2,0,0)

Barycentrics    (b - c) (-2 a^3 + a b c + 2 b^2 c + 2 b c^2) : :

X(29033) lies on these lines: {1, 4382}, {30, 511}, {659, 4791}, {667, 4823}, {1577, 4401}, {4010, 4794}, {4380, 4761}, {4474, 21385}, {4775, 4810}

X(29033) = isogonal conjugate of X(29034)


X(29034) =  ISOGONAL CONJUGATE OF X(29033)

Barycentrics    a^2/((b - c) (-2 a^3 + a b c + 2 b^2 c + 2 b c^2)) : :

X(29034) lies on the circumcircle and these lines:

X(29034) = isogonal conjugate of X(29033)


X(29035) =  CIRCUMCIRCLE-ANTIPODE OF X(29034)

Barycentrics    a^2 (2 a^2 b^3 - 2 a b^4 + 2 a^4 c - a^3 b c - a^2 b^2 c + 2 a b^3 c - 2 b^4 c - 2 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 - a b c^3 + 2 a c^4) (2 a^4 b - 2 a^3 b^2 - 2 a^2 b^3 + 2 a b^4 - a^3 b c + 2 a^2 b^2 c - a b^3 c - a^2 b c^2 - a b^2 c^2 + 2 a^2 c^3 + 2 a b c^3 + 2 b^2 c^3 - 2 a c^4 - 2 b c^4) : :

X(29035) lies on the circumcircle and these lines:

X(29035) = isogonal conjugate of X(29036)
X(29035) = circumcircle-antipode of X(29034)


X(29036) =  ISOGONAL CONJUGATE OF X(29035)

Barycentrics    1/((2 a^2 b^3 - 2 a b^4 + 2 a^4 c - a^3 b c - a^2 b^2 c + 2 a b^3 c - 2 b^4 c - 2 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 - a b c^3 + 2 a c^4) (2 a^4 b - 2 a^3 b^2 - 2 a^2 b^3 + 2 a b^4 - a^3 b c + 2 a^2 b^2 c - a b^3 c - a^2 b c^2 - a b^2 c^2 + 2 a^2 c^3 + 2 a b c^3 + 2 b^2 c^3 - 2 a c^4 - 2 b c^4)) : :

X(29036) lies on these lines: {30, 511}, {2223, 24209}, {3757, 10434}

X(29036) = isogonal conjugate of X(29035)


X(29037) =  POINT POLARIS(0,-1,0,1)

Barycentrics    (b - c) (-a^3 + b^3 - a b c + 2 b^2 c + 2 b c^2 + c^3) : :

X(29037) lies on these lines: {1, 7265}, {10, 21192}, {30, 511}, {663, 25259}, {905, 4522}, {1577, 4458}, {4024, 17166}, {4025, 17072}, {4036, 21187}, {4041, 4467}, {4086, 17899}, {4088, 4560}, {4122, 4367}, {4142, 4391}, {4705, 21196}, {4791, 20517}, {16892, 21301}, {21212, 21260}

X(29037) = isogonal conjugate of X(29038)


X(29038) =  ISOGONAL CONJUGATE OF X(29037)

Barycentrics    a^2/((b - c) (-a^3 + b^3 - a b c + 2 b^2 c + 2 b c^2 + c^3)) : :

X(29038) lies on the circumcircle and these lines:

X(29038) = isogonal conjugate of X(29037)


X(29039) =  CIRCUMCIRCLE-ANTIPODE OF X(29038)

Barycentrics    a^2 (a^6 + a^5 b - a^4 b^2 - 2 a^3 b^3 - a^2 b^4 + a b^5 + 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 b^3 c^2 + b^4 c^2 + a^3 c^3 + b^3 c^3 + 2 a b c^4 - a c^5 - b c^5 - 2 c^6) (a^6 + a^4 b^2 + a^3 b^3 - a b^5 - 2 b^6 + a^5 c - a^4 b c - a^3 b^2 c + 2 a b^4 c - b^5 c - a^4 c^2 + a^3 b c^2 - 2 a^3 c^3 + a^2 b c^3 - a b^2 c^3 + b^3 c^3 - a^2 c^4 - a b c^4 + b^2 c^4 + a c^5 + c^6) : :

X(29039) lies on the circumcircle and these lines:

X(29039) = isogonal conjugate of X(29040)
X(29039) = circumcircle-antipode of X(29038)


X(29040) =  ISOGONAL CONJUGATE OF X(29039)

Barycentrics    1/((a^6 + a^5 b - a^4 b^2 - 2 a^3 b^3 - a^2 b^4 + a b^5 + 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 b^3 c^2 + b^4 c^2 + a^3 c^3 + b^3 c^3 + 2 a b c^4 - a c^5 - b c^5 - 2 c^6) (a^6 + a^4 b^2 + a^3 b^3 - a b^5 - 2 b^6 + a^5 c - a^4 b c - a^3 b^2 c + 2 a b^4 c - b^5 c - a^4 c^2 + a^3 b c^2 - 2 a^3 c^3 + a^2 b c^3 - a b^2 c^3 + b^3 c^3 - a^2 c^4 - a b c^4 + b^2 c^4 + a c^5 + c^6)) : :

X(29040) lies on these lines: {4, 24257}, {10, 8424}, {30, 511}, {1010, 2360}, {1352, 24728}, {1761, 2321}, {1901, 20970}, {2939, 8931}, {2959, 9862}, {3579, 3773}, {3818, 3821}, {3946, 18483}, {4085, 18480}, {4431, 11683}, {4655, 18440}, {5092, 24295}, {20872, 21091}

X(29040) = isogonal conjugate of X(29039)


X(29041) =  ISOGONAL CONJUGATE OF X(23875)

Barycentrics    a^2/((b - c) (a^2 b - b^3 + a^2 c + a b c - b^2 c - b c^2 - c^3)) : :

X(29041) lies on the circumcircle and these lines:

X(29041) = isogonal conjugate of X(23875)


X(29042) =  CIRCUMCIRCLE-ANTIPODE OF X(29041)

Barycentrics    a^2 (a^6 - a^4 b^2 - a^2 b^4 + b^6 - a^4 b c + a^3 b^2 c + a^2 b^3 c - a b^4 c - a^2 b c^3 - a b^2 c^3 + a^2 c^4 + 2 a b c^4 + b^2 c^4 - 2 c^6) (a^6 + a^2 b^4 - 2 b^6 - a^4 b c - a^2 b^3 c + 2 a b^4 c - a^4 c^2 + a^3 b c^2 - a b^3 c^2 + b^4 c^2 + a^2 b c^3 - a^2 c^4 - a b c^4 + c^6) : :

X(29042) lies on the circumcircle and these lines:

X(29042) = isogonal conjugate of X(29043)
X(29042) = circumcircle-antipode of X(29041)


X(29043) =  ISOGONAL CONJUGATE OF X(29042)

Barycentrics    1/((a^6 - a^4 b^2 - a^2 b^4 + b^6 - a^4 b c + a^3 b^2 c + a^2 b^3 c - a b^4 c - a^2 b c^3 - a b^2 c^3 + a^2 c^4 + 2 a b c^4 + b^2 c^4 - 2 c^6) (a^6 + a^2 b^4 - 2 b^6 - a^4 b c - a^2 b^3 c + 2 a b^4 c - a^4 c^2 + a^3 b c^2 - a b^3 c^2 + b^4 c^2 + a^2 b c^3 - a^2 c^4 - a b c^4 + c^6)) : :

X(29043) lies on these lines: {30, 511}, {1352, 24309}, {1618, 3814}, {3220, 21293}, {5248, 26130}, {13329, 28256}, {24253, 25639}

X(29043) = isogonal conjugate of X(29042)


X(29044) =  ISOGONAL CONJUGATE OF X(23876)

Barycentrics    a^2/((b - c) (a^2 b - b^3 + a^2 c - a b c - b^2 c - b c^2 - c^3)) : :

X(29044) lies on the circumcircle and these lines:

X(29044) = isogonal conjugate of X(23876)


X(29045) =  CIRCUMCIRCLE-ANTIPODE OF X(29044)

Barycentrics    a^2 (a^6 - a^4 b^2 - a^2 b^4 + b^6 + a^4 b c - a^3 b^2 c - a^2 b^3 c + a b^4 c + a^2 b c^3 + a b^2 c^3 + a^2 c^4 - 2 a b c^4 + b^2 c^4 - 2 c^6) (a^6 + a^2 b^4 - 2 b^6 + a^4 b c + a^2 b^3 c - 2 a b^4 c - a^4 c^2 - a^3 b c^2 + a b^3 c^2 + b^4 c^2 - a^2 b c^3 - a^2 c^4 + a b c^4 + c^6) : :

X(29045) lies on the circumcircle and these lines:

X(29045) = isogonal conjugate of X(29047)
X(29045) = circumcircle-antipode of X(29044)


X(29046) =  ISOGONAL CONJUGATE OF X(29045)

Barycentrics    1/((a^6 - a^4 b^2 - a^2 b^4 + b^6 + a^4 b c - a^3 b^2 c - a^2 b^3 c + a b^4 c + a^2 b c^3 + a b^2 c^3 + a^2 c^4 - 2 a b c^4 + b^2 c^4 - 2 c^6) (a^6 + a^2 b^4 - 2 b^6 + a^4 b c + a^2 b^3 c - 2 a b^4 c - a^4 c^2 - a^3 b c^2 + a b^3 c^2 + b^4 c^2 - a^2 b c^3 - a^2 c^4 + a b c^4 + c^6)) : :

X(29046) lies on these lines: {10, 2182}, {30, 511}, {226, 1456}, {993, 3220}, {1478, 4307}, {2239, 13329}, {3814, 5150}, {4645, 24630}, {13478, 29635}

X(29046) = isogonal conjugate of X(29045)


X(29047) =  POINT POLARIS(0,1,1,-1)

Barycentrics    (b - c) (a^2 b + b^3 + a^2 c - a b c + b^2 c + b c^2 + c^3) : :

X(29047) lies on these lines: {30, 511}, {1491, 4808}, {1734, 16892}, {3806, 28374}, {4088, 14349}, {4170, 25259}, {4823, 23770}

X(29047) = isogonal conjugate of X(29048)


X(29048) =  ISOGONAL CONJUGATE OF X(29047)

Barycentrics    a^2/((b - c) (a^2 b + b^3 + a^2 c - a b c + b^2 c + b c^2 + c^3)) : :

X(29048) lies on the circumcircle and these lines:

X(29048) = isogonal conjugate of X(29047)


X(29049) =  CIRCUMCIRCLE-ANTIPODE OF X(29048)

Barycentrics    a^2 (a^6 - a^4 b^2 - a^2 b^4 + 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 + 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 - 2 c^6) (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 - a^4 c^2 + a^3 b c^2 - a b^3 c^2 - b^4 c^2 + a^2 b c^3 - a^2 c^4 - a b c^4 + 2 b^2 c^4 + c^6) : :

X(29049) lies on the circumcircle and these lines:

X(29049) = isogonal conjugate of X(29050)
X(29049) = circumcircle-antipode of X(29048)


X(29050) =  ISOGONAL CONJUGATE OF X(29049)

Barycentrics    1/((a^6 - a^4 b^2 - a^2 b^4 + 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 + 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 - 2 c^6) (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 - a^4 c^2 + a^3 b c^2 - a b^3 c^2 - b^4 c^2 + a^2 b c^3 - a^2 c^4 - a b c^4 + 2 b^2 c^4 + c^6)) : :

X(29050) lies on these lines: {4, 24309}, {30, 511}, {1766, 16545}, {3663, 6284}, {4353, 15171}

X(29050) = isogonal conjugate of X(29049)


X(29051) =  POINT POLARIS(-1,0,1,1)

Barycentrics    (b - c) (-a^3 + a^2 b + a^2 c + a b c + b^2 c + b c^2) : :

X(29051) lies on these lines: {1, 4978}, {30, 511}, {650, 17072}, {659, 2533}, {661, 21301}, {663, 693}, {667, 4369}, {905, 24720}, {1027, 1220}, {1577, 3716}, {1734, 4913}, {2254, 4560}, {3960, 23789}, {4041, 17494}, {4063, 4761}, {4142, 7178}, {4367, 21146}, {4378, 4504}, {4379, 8643}, {4391, 4724}, {4449, 4801}, {4462, 4474}, {4794, 4823}, {4822, 20295}, {4833, 14288}, {4895, 26824}, {5592, 8045}, {14838, 25380}, {16695, 23866}, {21222, 23738}, {21260, 25666}, {24560, 24561}, {25901, 26017}

X(29051) = isogonal conjugate of X(29052)


X(29052) =  ISOGONAL CONJUGATE OF X(29051)

Barycentrics    a^2/((b - c) (-a^3 + a^2 b + a^2 c + a b c + b^2 c + b c^2)) : :

X(29052) lies on the circumcircle and these lines:

X(29052) = isogonal conjugate of X(29051)


X(29053) =  CIRCUMCIRCLE-ANTIPODE OF X(29052)

Barycentrics    a^2 (a^4 b^2 + a^3 b^3 - a^2 b^4 - a b^5 + a^5 c + a^4 b c - a^3 b^2 c + 2 a^2 b^3 c - 2 a b^4 c - b^5 c - a^3 b c^2 + 2 a b^3 c^2 - b^4 c^2 - 2 a^3 c^3 - a^2 b c^3 - a b^2 c^3 + b^3 c^3 + a b c^4 + b^2 c^4 + a c^5) (a^5 b - 2 a^3 b^3 + a b^5 + 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 b^3 c^2 + b^4 c^2 + a^3 c^3 + 2 a^2 b c^3 + 2 a b^2 c^3 + b^3 c^3 - a^2 c^4 - 2 a b c^4 - b^2 c^4 - a c^5 - b c^5) : :

X(29053) lies on the circumcircle and these lines:

X(29053) = isogonal conjugate of X(29054)
X(29053) = circumcircle-antipode of X(29052)


X(29054) =  ISOGONAL CONJUGATE OF X(29053)

Barycentrics    1/((a^4 b^2 + a^3 b^3 - a^2 b^4 - a b^5 + a^5 c + a^4 b c - a^3 b^2 c + 2 a^2 b^3 c - 2 a b^4 c - b^5 c - a^3 b c^2 + 2 a b^3 c^2 - b^4 c^2 - 2 a^3 c^3 - a^2 b c^3 - a b^2 c^3 + b^3 c^3 + a b c^4 + b^2 c^4 + a c^5) (a^5 b - 2 a^3 b^3 + a b^5 + 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 b^3 c^2 + b^4 c^2 + a^3 c^3 + 2 a^2 b c^3 + 2 a b^2 c^3 + b^3 c^3 - a^2 c^4 - 2 a b c^4 - b^2 c^4 - a c^5 - b c^5)) : :

X(29054) lies on these lines: {1, 4032}, {3, 24325}, {4, 984}, {5, 3842}, {20, 24349}, {30, 511}, {37, 946}, {40, 75}, {65, 13569}, {192, 962}, {335, 6999}, {388, 24248}, {611, 24268}, {958, 3923}, {1215, 4192}, {1278, 20070}, {3475, 17592}, {3663, 3931}, {3696, 11362}, {3729, 12717}, {3739, 6684}, {3821, 25466}, {3993, 4301}, {4295, 7201}, {4660, 5794}, {4687, 8227}, {4732, 5690}, {4847, 22001}, {6211, 6996}, {6796, 15624}, {9944, 12675}, {10624, 11997}, {12005, 13476}, {12689, 14872}, {12699, 20430}, {13464, 15569}, {16609, 23690}, {19546, 24003}

X(29054) = isogonal conjugate of X(29053)


X(29055) =  ISOGONAL CONJUGATE OF X(3907)

Barycentrics    a^2/((a - b - c) (b - c) (a^2 + b c)) : :

X(29055) lies on the circumcircle and these lines:

X(29055) = isogonal conjugate of X(3906)


X(29056) =  CIRCUMCIRCLE-ANTIPODE OF X(29055)

Barycentrics    a^2 (a^4 b^2 + a^3 b^3 - a^2 b^4 - a b^5 + a^5 c - 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 - 2 a^3 c^3 + a^2 b c^3 - a b^2 c^3 + b^3 c^3 - a b c^4 + b^2 c^4 + a c^5) (a^5 b - 2 a^3 b^3 + a b^5 - 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 b^3 c^2 + b^4 c^2 + a^3 c^3 + b^3 c^3 - a^2 c^4 + 2 a b c^4 - b^2 c^4 - a c^5 - b c^5) : :

X(29056) lies on the circumcircle and these lines:

X(29056) = isogonal conjugate of X(29057)
X(29056) = circumcircle-antipode of X(29055)


X(29057) =  ISOGONAL CONJUGATE OF X(29056)

Barycentrics    1/((a^4 b^2 + a^3 b^3 - a^2 b^4 - a b^5 + a^5 c - 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 - 2 a^3 c^3 + a^2 b c^3 - a b^2 c^3 + b^3 c^3 - a b c^4 + b^2 c^4 + a c^5) (a^5 b - 2 a^3 b^3 + a b^5 - 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 b^3 c^2 + b^4 c^2 + a^3 c^3 + b^3 c^3 - a^2 c^4 + 2 a b c^4 - b^2 c^4 - a c^5 - b c^5)) : :

X(29057) lies on these lines: {2, 10853}, {3, 3923}, {4, 240}, {5, 3821}, {6, 24257}, {20, 24280}, {30, 511}, {40, 3729}, {75, 6210}, {84, 309}, {104, 24852}, {140, 24295}, {182, 4672}, {190, 6211}, {312, 20368}, {355, 4660}, {846, 7413}, {946, 3663}, {990, 3736}, {1010, 8235}, {1045, 1047}, {1158, 1761}, {1281, 5999}, {1284, 4459}, {1350, 5695}, {1352, 4655}, {1423, 4008}, {1513, 5988}, {1699, 17591}, {1733, 1756}, {1901, 2092}, {2292, 15971}, {3120, 8229}, {3576, 4234}, {3724, 13265}, {3980, 19544}, {4011, 16434}, {4192, 24259}, {4220, 4418}, {4353, 13464}, {5450, 15952}, {5587, 17677}, {5699, 14538}, {5700, 14539}, {5777, 18252}, {6245, 21629}, {6684, 17355}, {6776, 24695}, {6796, 24309}, {6996, 17738}, {6998, 8245}, {7379, 9791}, {7427, 24402}, {7574, 19400}, {7609, 17277}, {7992, 7996}, {8095, 12726}, {8096, 12727}, {8227, 17304}, {9799, 9801}, {9942, 9944}, {9948, 9950}, {9959, 15973}, {9960, 9962}, {10175, 16052}, {10454, 15071}, {12528, 12530}, {12547, 12549}, {12608, 12610}, {12616, 12618}, {12650, 12652}, {12664, 12689}, {12669, 12718}, {12672, 12721}, {12673, 12719}, {12674, 12720}, {12675, 12722}, {12681, 12724}, {12685, 12728}, {12688, 17635}, {17649, 17651}, {19514, 25079}, {19540, 24260}, {19541, 24283}

X(29057) = isogonal conjugate of X(29056)


X(29058) =  POINT POLARIS(2,-1,0,0)

Barycentrics    (b - c) (-2 a^3 + b^3 + 3 b^2 c + 3 b c^2 + c^3) : :

X(29058) lies on these lines: {30, 511}, {667, 17989}, {1577, 4809}, {1960, 3700}, {4770, 4976}

X(29058) = isogonal conjugate of X(29059)


X(29059) =  ISOGONAL CONJUGATE OF X(29058)

Barycentrics    a^2/((b - c) (2 a^3 - b^3 - 3 b^2 c - 3 b c^2 - c^3)) : :

X(29059) lies on the circumcircle and these lines:

X(29059) = isogonal conjugate of X(29058)


X(29060) =  CIRCUMCIRCLE-ANTIPODE OF X(29059)

Barycentrics    a^2 (a^6 + 2 a^5 b - a^4 b^2 - 4 a^3 b^3 - a^2 b^4 + 2 a b^5 + b^6 + a^4 c^2 - 2 a^3 b c^2 - 2 a b^3 c^2 + b^4 c^2 + 2 a^3 c^3 + 2 a^2 b c^3 + 2 a b^2 c^3 + 2 b^3 c^3 - 2 a c^5 - 2 b c^5 - 2 c^6) (a^6 + a^4 b^2 + 2 a^3 b^3 - 2 a b^5 - 2 b^6 + 2 a^5 c - 2 a^3 b^2 c + 2 a^2 b^3 c - 2 b^5 c - a^4 c^2 + 2 a b^3 c^2 - 4 a^3 c^3 - 2 a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 + b^2 c^4 + 2 a c^5 + c^6) : :

X(29060) lies on the circumcircle and these lines:

X(29060) = isogonal conjugate of X(29061)
X(29060) = circumcircle-antipode of X(29059)


X(29061) =  ISOGONAL CONJUGATE OF X(29060)

Barycentrics    1/((a^6 + 2 a^5 b - a^4 b^2 - 4 a^3 b^3 - a^2 b^4 + 2 a b^5 + b^6 + a^4 c^2 - 2 a^3 b c^2 - 2 a b^3 c^2 + b^4 c^2 + 2 a^3 c^3 + 2 a^2 b c^3 + 2 a b^2 c^3 + 2 b^3 c^3 - 2 a c^5 - 2 b c^5 - 2 c^6) (a^6 + a^4 b^2 + 2 a^3 b^3 - 2 a b^5 - 2 b^6 + 2 a^5 c - 2 a^3 b^2 c + 2 a^2 b^3 c - 2 b^5 c - a^4 c^2 + 2 a b^3 c^2 - 4 a^3 c^3 - 2 a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 + b^2 c^4 + 2 a c^5 + c^6)) : :

X(29061) lies on these lines: {30, 511}, {17726, 24210}

X(29061) = isogonal conjugate of X(29060)


X(29062) =  POINT POLARIS(1,-1,0,0)

Barycentrics    (b - c) (-a^3 + b^3 + 2 b^2 c + 2 b c^2 + c^3) : :

X(29062) lies on these lines: {30, 511}, {663, 7265}, {667, 4122}, {1577, 20517}, {1734, 4467}, {4040, 25259}, {4142, 4791}, {4458, 4823}, {4522, 14838}, {17072, 21192}

X(29062) = isogonal conjugate of X(29063)


X(29063) =  ISOGONAL CONJUGATE OF X(29062)

Barycentrics    a^2/((b - c) (-a^3 + b^3 + 2 b^2 c + 2 b c^2 + c^3)) : :

X(29063) lies on the circumcircle and these lines:

X(29063) = isogonal conjugate of X(29062)


X(29064) =  CIRCUMCIRCLE-ANTIPODE OF X(29063)

Barycentrics    a^2 (a^6 + a^5 b - a^4 b^2 - 2 a^3 b^3 - a^2 b^4 + a b^5 + b^6 + a^4 c^2 - a^3 b 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 + b^3 c^3 - a c^5 - b c^5 - 2 c^6) (a^6 + a^4 b^2 + a^3 b^3 - a b^5 - 2 b^6 + a^5 c - a^3 b^2 c + a^2 b^3 c - b^5 c - a^4 c^2 + a b^3 c^2 - 2 a^3 c^3 - a b^2 c^3 + b^3 c^3 - a^2 c^4 + b^2 c^4 + a c^5 + c^6) : :

X(29064) lies on the circumcircle and these lines:

X(29064) = isogonal conjugate of X(29065)
X(29064) = circumcircle-antipode of X(29063)


X(29065) =  ISOGONAL CONJUGATE OF X(29060)

Barycentrics    1/((a^6 + a^5 b - a^4 b^2 - 2 a^3 b^3 - a^2 b^4 + a b^5 + b^6 + a^4 c^2 - a^3 b 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 + b^3 c^3 - a c^5 - b c^5 - 2 c^6) (a^6 + a^4 b^2 + a^3 b^3 - a b^5 - 2 b^6 + a^5 c - a^3 b^2 c + a^2 b^3 c - b^5 c - a^4 c^2 + a b^3 c^2 - 2 a^3 c^3 - a b^2 c^3 + b^3 c^3 - a^2 c^4 + b^2 c^4 + a c^5 + c^6)) : :

X(29065) lies on these lines: {3, 7087}, {22, 21072}, {30, 511}, {950, 4021}, {2172, 2908}, {10454, 10468}

X(29065) = isogonal conjugate of X(29064)


X(29066) =  POINT POLARIS(1,0,-1,0)

Barycentrics    (b - c) (-a^3 + a^2 b + a^2 c + b^2 c + b c^2) : :

X(29066) lies on these lines: {1, 693}, {8, 17494}, {10, 650}, {30, 511}, {145, 26824}, {386, 25667}, {649, 4761}, {659, 4774}, {663, 1577}, {667, 2533}, {936, 27417}, {950, 11934}, {996, 1027}, {1125, 4885}, {1734, 4560}, {1960, 4874}, {2517, 3737}, {2901, 4024}, {3616, 26985}, {3617, 26777}, {3669, 23789}, {3716, 4791}, {3762, 4474}, {3825, 15283}, {3924, 27712}, {3960, 24720}, {4010, 4775}, {4040, 4391}, {4107, 17031}, {4378, 4922}, {4385, 21611}, {4411, 24325}, {4449, 4978}, {4804, 4895}, {4905, 17496}, {5248, 8641}, {6702, 10006}, {7178, 20517}, {8142, 12512}, {8583, 26695}, {9780, 27115}, {10479, 24900}, {11019, 30235}, {12609, 23806}, {14349, 21301}, {14838, 17072}, {15280, 24387}, {17425, 28143}, {19853, 26049}, {19860, 26546}, {20083, 25684}, {23791, 28374}, {24987, 26641}, {26363, 28834}

X(29066) = isogonal conjugate of X(29067)


X(29067) =  ISOGONAL CONJUGATE OF X(29066)

Barycentrics    a^2/((b - c) (-a^3 + a^2 b + a^2 c + b^2 c + b c^2)) : :

X(29067) lies on the circumcircle and these lines:

X(29067) = isogonal conjugate of X(29066)


X(29068) =  CIRCUMCIRCLE-ANTIPODE OF X(29067)

Barycentrics    a^2 (a^3 b^2 - a b^4 + a^4 c - a^3 b c - a^2 b^2 c + 2 a b^3 c - b^4 c - a^3 c^2 + 2 a^2 b c^2 - a b^2 c^2 - a^2 c^3 - a b c^3 + b^2 c^3 + a c^4) (a^4 b - a^3 b^2 - a^2 b^3 + a b^4 - a^3 b c + 2 a^2 b^2 c - a b^3 c + a^3 c^2 - a^2 b c^2 - a b^2 c^2 + b^3 c^2 + 2 a b c^3 - a c^4 - b c^4) : :

X(29068) lies on the circumcircle and these lines:

X(29068) = isogonal conjugate of X(29069)
X(29068) = circumcircle-antipode of X(29067)


X(29069) =  ISOGONAL CONJUGATE OF X(29067)

Barycentrics    1/((a^3 b^2 - a b^4 + a^4 c - a^3 b c - a^2 b^2 c + 2 a b^3 c - b^4 c - a^3 c^2 + 2 a^2 b c^2 - a b^2 c^2 - a^2 c^3 - a b c^3 + b^2 c^3 + a c^4) (a^4 b - a^3 b^2 - a^2 b^3 + a b^4 - a^3 b c + 2 a^2 b^2 c - a b^3 c + a^3 c^2 - a^2 b c^2 - a b^2 c^2 + b^3 c^2 + 2 a b c^3 - a c^4 - b c^4)) : :

X(29069) lies on these lines: {3, 4363}, {4, 4419}, {5, 4364}, {10, 24336}, {20, 4454}, {30, 511}, {37, 24220}, {40, 4659}, {63, 321}, {75, 573}, {101, 1944}, {140, 4472}, {150, 17950}, {190, 6996}, {192, 10446}, {226, 1465}, {241, 24237}, {343, 21072}, {355, 4643}, {381, 24441}, {497, 7961}, {549, 10022}, {572, 894}, {631, 4470}, {944, 4644}, {946, 24424}, {990, 5757}, {993, 3923}, {1125, 25371}, {1385, 4670}, {1423, 17861}, {1478, 4424}, {1482, 17318}, {1730, 17862}, {1746, 3219}, {1756, 23690}, {1763, 20223}, {2183, 4858}, {2223, 4459}, {2901, 10441}, {3210, 9535}, {3262, 3882}, {3628, 25358}, {3655, 4795}, {3664, 4032}, {3821, 3822}, {3825, 25369}, {3868, 10454}, {3878, 24705}, {3927, 5786}, {4359, 21363}, {4389, 7377}, {4422, 19512}, {4440, 6999}, {4465, 19546}, {4665, 5690}, {4667, 5882}, {4708, 9956}, {4713, 19540}, {4748, 5818}, {5267, 24700}, {5745, 17355}, {5790, 17251}, {5816, 17257}, {5886, 7611}, {5905, 17147}, {6358, 22097}, {6796, 24315}, {7406, 20073}, {8609, 17197}, {9590, 24347}, {10478, 28606}, {11329, 26659}, {11500, 24328}, {16609, 24209}, {17134, 28968}, {17738, 24630}, {18252, 22325}, {19541, 24352}, {20245, 21078}, {20258, 25078}, {20348, 25252}, {20927, 29497}, {22003, 25083}, {24309, 24326}, {24319, 25639}, {24334, 25440}, {25065, 30097}, {25353, 25368}

X(29069) = isogonal conjugate of X(29068)


X(29070) =  POINT POLARIS(1,0,0,-1)

Barycentrics    (b - c) (-a^3 + a b c + b^2 c + b c^2) : :

X(29070) lies on these lines: {30, 511}, {650, 21260}, {659, 1577}, {663, 4382}, {667, 693}, {764, 17496}, {905, 19947}, {1019, 21146}, {2530, 4560}, {2533, 4063}, {3803, 7662}, {3837, 14838}, {4010, 4040}, {4170, 4810}, {4367, 4978}, {4378, 4801}, {4380, 4834}, {4401, 4823}, {4455, 18077}, {4705, 17494}, {4730, 21302}, {4885, 6050}, {4983, 20295}, {7234, 29771}, {14349, 24719}, {17072, 23791}, {17166, 26824}, {25901, 26546}

X(29070) = isogonal conjugate of X(29071)


X(29071) =  ISOGONAL CONJUGATE OF X(29070)

Barycentrics    a^2/((b - c) (-a^3 + a b c + b^2 c + b c^2)) : :

X(29071) lies on the circumcircle and these lines:

X(29071) = isogonal conjugate of X(29070)


X(29072) =  CIRCUMCIRCLE-ANTIPODE OF X(29071)

Barycentrics    a^2 (a^3 b^3 - a b^5 + a^5 c + a^4 b c - a^3 b^2 c + 2 a^2 b^3 c - 2 a b^4 c - b^5 c - a^3 b c^2 + 2 a b^3 c^2 - 2 a^3 c^3 - a^2 b c^3 - a b^2 c^3 + b^3 c^3 + a b c^4 + a c^5) (a^5 b - 2 a^3 b^3 + a b^5 + a^4 b c - 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 + 2 a^2 b c^3 + 2 a b^2 c^3 + b^3 c^3 - 2 a b c^4 - a c^5 - b c^5) : :

X(29072) lies on the circumcircle and these lines:

X(29072) = isogonal conjugate of X(29073)
X(29072) = circumcircle-antipode of X(29071)


X(29073) =  ISOGONAL CONJUGATE OF X(29072)

Barycentrics    1/((a^3 b^3 - a b^5 + a^5 c + a^4 b c - a^3 b^2 c + 2 a^2 b^3 c - 2 a b^4 c - b^5 c - a^3 b c^2 + 2 a b^3 c^2 - 2 a^3 c^3 - a^2 b c^3 - a b^2 c^3 + b^3 c^3 + a b c^4 + a c^5) (a^5 b - 2 a^3 b^3 + a b^5 + a^4 b c - 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 + 2 a^2 b c^3 + 2 a b^2 c^3 + b^3 c^3 - 2 a b c^4 - a c^5 - b c^5)) : :

X(29073) lies on these lines: {3, 16684}, {30, 511}, {2223, 23690}, {4660, 24269}

X(29073) = isogonal conjugate of X(29072)


X(29074) =  POINT POLARIS(-1,1,1,0)

Barycentrics    (b - c) (-a^3 + a^2 b + b^3 + a^2 c + 2 b^2 c + 2 b c^2 + c^3) : :

X(29074) lies on these lines: {30, 511}, {663, 4122}, {21831, 23282}

X(29074) = isogonal conjugate of X(29075)


X(29075) =  ISOGONAL CONJUGATE OF X(29074)

Barycentrics    a^2/((b - c) (-a^3 + a^2 b + b^3 + a^2 c + 2 b^2 c + 2 b c^2 + c^3)) : :

X(29075) lies on the circumcircle and these lines:

X(29075) = isogonal conjugate of X(29074)


X(29076) =  CIRCUMCIRCLE-ANTIPODE OF X(29075)

Barycentrics    a^2 (a^6 + a^5 b - a^4 b^2 - 2 a^3 b^3 - a^2 b^4 + a b^5 + b^6 + 2 a^4 c^2 - a^3 b c^2 - a b^3 c^2 + 2 b^4 c^2 + a^3 c^3 + a^2 b c^3 + a b^2 c^3 + b^3 c^3 - a^2 c^4 - b^2 c^4 - a c^5 - b c^5 - 2 c^6) (a^6 + 2 a^4 b^2 + a^3 b^3 - a^2 b^4 - a b^5 - 2 b^6 + a^5 c - a^3 b^2 c + a^2 b^3 c - b^5 c - a^4 c^2 + a b^3 c^2 - b^4 c^2 - 2 a^3 c^3 - a b^2 c^3 + b^3 c^3 - a^2 c^4 + 2 b^2 c^4 + a c^5 + c^6) : :

X(29076) lies on the circumcircle and these lines:

X(29076) = isogonal conjugate of X(29077)
X(29076) = circumcircle-antipode of X(29075)


X(29077) =  ISOGONAL CONJUGATE OF X(29076)

Barycentrics    1/((a^6 + a^5 b - a^4 b^2 - 2 a^3 b^3 - a^2 b^4 + a b^5 + b^6 + 2 a^4 c^2 - a^3 b c^2 - a b^3 c^2 + 2 b^4 c^2 + a^3 c^3 + a^2 b c^3 + a b^2 c^3 + b^3 c^3 - a^2 c^4 - b^2 c^4 - a c^5 - b c^5 - 2 c^6) (a^6 + 2 a^4 b^2 + a^3 b^3 - a^2 b^4 - a b^5 - 2 b^6 + a^5 c - a^3 b^2 c + a^2 b^3 c - b^5 c - a^4 c^2 + a b^3 c^2 - b^4 c^2 - 2 a^3 c^3 - a b^2 c^3 + b^3 c^3 - a^2 c^4 + 2 b^2 c^4 + a c^5 + c^6)) : :

X(29077) lies on these lines: {3, 17289}, {4, 17302}, {5, 17384}, {30, 511}

X(29077) = isogonal conjugate of X(29076)


X(29078) =  POINT POLARIS(1,-1,1,0)

Barycentrics    (b - c) (-a^3 - a^2 b + b^3 - a^2 c + 2 b^2 c + 2 b c^2 + c^3) : :

X(29078) lies on these lines: {30, 511}, {649, 4122}, {650, 18004}, {659, 25259}, {667, 7265}, {1491, 4467}, {3700, 4874}, {3837, 4025}, {4408, 20518}, {4522, 9508}, {4820, 7662}, {16892, 24719}, {21187, 21200}, {21192, 21260}

X(29078) = isogonal conjugate of X(29079)


X(29079) =  ISOGONAL CONJUGATE OF X(29078)

Barycentrics    a^2/((b - c) (-a^3 - a^2 b + b^3 - a^2 c + 2 b^2 c + 2 b c^2 + c^3)) : :

X(29079) lies on the circumcircle and these lines:

X(29079) = isogonal conjugate of X(29078)


X(29080) =  CIRCUMCIRCLE-ANTIPODE OF X(29079)

Barycentrics    a^2 (a^6 + a^5 b - a^4 b^2 - 2 a^3 b^3 - a^2 b^4 + a b^5 + b^6 - 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^2 c^4 + b^2 c^4 - a c^5 - b c^5 - 2 c^6) (a^6 + a^3 b^3 + a^2 b^4 - a b^5 - 2 b^6 + a^5 c - a^3 b^2 c + a^2 b^3 c - b^5 c - a^4 c^2 + a b^3 c^2 + b^4 c^2 - 2 a^3 c^3 - a b^2 c^3 + b^3 c^3 - a^2 c^4 + a c^5 + c^6) : :

X(29080) lies on the circumcircle and these lines:

X(29080) = isogonal conjugate of X(29081)
X(29080) = circumcircle-antipode of X(29079)


X(29081) =  ISOGONAL CONJUGATE OF X(29080)

Barycentrics    1/((a^6 + a^5 b - a^4 b^2 - 2 a^3 b^3 - a^2 b^4 + a b^5 + b^6 - 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^2 c^4 + b^2 c^4 - a c^5 - b c^5 - 2 c^6) (a^6 + a^3 b^3 + a^2 b^4 - a b^5 - 2 b^6 + a^5 c - a^3 b^2 c + a^2 b^3 c - b^5 c - a^4 c^2 + a b^3 c^2 + b^4 c^2 - 2 a^3 c^3 - a b^2 c^3 + b^3 c^3 - a^2 c^4 + a c^5 + c^6)) : :

X(29081) lies on these lines: {3, 3661}, {4, 4393}, {5, 17023}, {20, 20055}, {30, 511}, {140, 29604}, {355, 29659}

X(29081) = isogonal conjugate of X(29080)


X(29082) =  POINT POLARIS(1,1,-1,0)

Barycentrics    (b - c) (a^3 - a^2 b + b^3 - a^2 c + c^3) : :

X(29082) lies on these lines: {30, 511}, {663, 3801}, {667, 4707}, {1960, 20517}, {3777, 3904}, {3837, 6332}, {4142, 5592}, {4367, 8636}, {4449, 23747}, {4809, 8643}, {4874, 7178}

X(29082) = isogonal conjugate of X(29083)


X(29083) =  ISOGONAL CONJUGATE OF X(29082)

Barycentrics    a^2/((b - c) (a^3 - a^2 b + b^3 - a^2 c + c^3)) : :

X(29083) lies on the circumcircle and these lines:

X(29083) = isogonal conjugate of X(29082)


X(29084) =  CIRCUMCIRCLE-ANTIPODE OF X(29083)

Barycentrics    a^2 (a^6 - a^5 b - a^4 b^2 + 2 a^3 b^3 - a^2 b^4 - a b^5 + b^6 + 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^2 c^4 + b^2 c^4 + a c^5 + b c^5 - 2 c^6) (a^6 - a^3 b^3 + a^2 b^4 + a b^5 - 2 b^6 - a^5 c + a^3 b^2 c - a^2 b^3 c + b^5 c - a^4 c^2 - a b^3 c^2 + b^4 c^2 + 2 a^3 c^3 + a b^2 c^3 - b^3 c^3 - a^2 c^4 - a c^5 + c^6) : :

X(29084) lies on the circumcircle and these lines:

X(29084) = isogonal conjugate of X(29085)
X(29084) = circumcircle-antipode of X(29083)


X(29085) =  ISOGONAL CONJUGATE OF X(29084)

Barycentrics    1/((a^6 - a^5 b - a^4 b^2 + 2 a^3 b^3 - a^2 b^4 - a b^5 + b^6 + 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^2 c^4 + b^2 c^4 + a c^5 + b c^5 - 2 c^6) (a^6 - a^3 b^3 + a^2 b^4 + a b^5 - 2 b^6 - a^5 c + a^3 b^2 c - a^2 b^3 c + b^5 c - a^4 c^2 - a b^3 c^2 + b^4 c^2 + 2 a^3 c^3 + a b^2 c^3 - b^3 c^3 - a^2 c^4 - a c^5 + c^6)) : :

X(29085) lies on these lines: {3, 3662}, {4, 17350}, {5, 17353}, {30, 511}, {3073, 5398}, {3927, 5015}, {5266, 6147}, {6210, 26921}

X(29085) = isogonal conjugate of X(29084)


X(29086) =  POINT POLARIS(-1,1,1,1)

Barycentrics    (b - c) (-a^3 + a^2 b + b^3 + a^2 c + a b c + 2 b^2 c + 2 b c^2 + c^3) : :

X(29086) lies on these lines: {30, 511}, {1960, 8045}, {4024, 8632}, {4040, 4122}, {4808, 17494}, {17161, 21303}, {21196, 21261}

X(29086) = isogonal conjugate of X(29087)


X(29087) =  ISOGONAL CONJUGATE OF X(29086)

Barycentrics    a^2/((b - c) (-a^3 + a^2 b + b^3 + a^2 c + a b c + 2 b^2 c + 2 b c^2 + c^3)) : :

X(29087) lies on the circumcircle and these lines:

X(29087) = isogonal conjugate of X(29086)


X(29088) =  CIRCUMCIRCLE-ANTIPODE OF X(29087)

Barycentrics    a^2 (a^6 + a^5 b - a^4 b^2 - 2 a^3 b^3 - a^2 b^4 + a b^5 + 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 - a b^3 c^2 + 2 b^4 c^2 + a^3 c^3 + 2 a^2 b c^3 + 2 a b^2 c^3 + b^3 c^3 - a^2 c^4 - 2 a b c^4 - b^2 c^4 - a c^5 - b c^5 - 2 c^6) (a^6 + 2 a^4 b^2 + a^3 b^3 - a^2 b^4 - a b^5 - 2 b^6 + a^5 c + a^4 b c - a^3 b^2 c + 2 a^2 b^3 c - 2 a b^4 c - b^5 c - a^4 c^2 - a^3 b c^2 + 2 a b^3 c^2 - b^4 c^2 - 2 a^3 c^3 - a^2 b c^3 - a b^2 c^3 + b^3 c^3 - a^2 c^4 + a b c^4 + 2 b^2 c^4 + a c^5 + c^6) : :

X(29088) lies on the circumcircle and these lines:

X(29088) = isogonal conjugate of X(29089)
X(29088) = circumcircle-antipode of X(29087)


X(29089) =  ISOGONAL CONJUGATE OF X(29088)

Barycentrics    1/((a^6 + a^5 b - a^4 b^2 - 2 a^3 b^3 - a^2 b^4 + a b^5 + 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 - a b^3 c^2 + 2 b^4 c^2 + a^3 c^3 + 2 a^2 b c^3 + 2 a b^2 c^3 + b^3 c^3 - a^2 c^4 - 2 a b c^4 - b^2 c^4 - a c^5 - b c^5 - 2 c^6) (a^6 + 2 a^4 b^2 + a^3 b^3 - a^2 b^4 - a b^5 - 2 b^6 + a^5 c + a^4 b c - a^3 b^2 c + 2 a^2 b^3 c - 2 a b^4 c - b^5 c - a^4 c^2 - a^3 b c^2 + 2 a b^3 c^2 - b^4 c^2 - 2 a^3 c^3 - a^2 b c^3 - a b^2 c^3 + b^3 c^3 - a^2 c^4 + a b c^4 + 2 b^2 c^4 + a c^5 + c^6)) : :

X(29089) lies on these lines: {30, 511}, {572, 18481}

X(29089) = isogonal conjugate of X(29088)


X(29090) =  POINT POLARIS(1,-1,1,1)

Barycentrics    (b - c) (-a^3 - a^2 b + b^3 - a^2 c - a b c + 2 b^2 c + 2 b c^2 + c^3) : :

X(29090) lies on these lines: {30, 511}, {667, 25259}, {1019, 4122}, {4025, 21260}, {4367, 7265}, {4467, 4705}, {14838, 18004}, {21051, 21192}

X(29090) = isogonal conjugate of X(29091)


X(29091) =  ISOGONAL CONJUGATE OF X(29090)

Barycentrics    a^2/((b - c) (-a^3 - a^2 b + b^3 - a^2 c - a b c + 2 b^2 c + 2 b c^2 + c^3)) : :

X(29091) lies on the circumcircle and these lines:

X(29091) = isogonal conjugate of X(29090)


X(29092) =  CIRCUMCIRCLE-ANTIPODE OF X(29091)

Barycentrics    a^2 (a^6 + a^5 b - a^4 b^2 - 2 a^3 b^3 - a^2 b^4 + a b^5 + b^6 - a^4 b c + 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 + b^3 c^3 + a^2 c^4 + 2 a b c^4 + b^2 c^4 - a c^5 - b c^5 - 2 c^6) (a^6 + a^3 b^3 + a^2 b^4 - a b^5 - 2 b^6 + a^5 c - a^4 b c - a^3 b^2 c + 2 a b^4 c - b^5 c - a^4 c^2 + a^3 b c^2 + b^4 c^2 - 2 a^3 c^3 + a^2 b c^3 - a b^2 c^3 + b^3 c^3 - a^2 c^4 - a b c^4 + a c^5 + c^6) : :

X(29092) lies on the circumcircle and these lines:

X(29092) = isogonal conjugate of X(29093)
X(29092) = circumcircle-antipode of X(29091)


X(29093) =  ISOGONAL CONJUGATE OF X(29092)

Barycentrics    1/((a^6 + a^5 b - a^4 b^2 - 2 a^3 b^3 - a^2 b^4 + a b^5 + b^6 - a^4 b c + 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 + b^3 c^3 + a^2 c^4 + 2 a b c^4 + b^2 c^4 - a c^5 - b c^5 - 2 c^6) (a^6 + a^3 b^3 + a^2 b^4 - a b^5 - 2 b^6 + a^5 c - a^4 b c - a^3 b^2 c + 2 a b^4 c - b^5 c - a^4 c^2 + a^3 b c^2 + b^4 c^2 - 2 a^3 c^3 + a^2 b c^3 - a b^2 c^3 + b^3 c^3 - a^2 c^4 - a b c^4 + a c^5 + c^6)) : :

X(29093) lies on these lines: {30, 511}, {3818, 24257}, {3821, 18553}, {4852, 22793}, {20190, 24295}

X(29093) = isogonal conjugate of X(29092)


X(29094) =  POINT POLARIS(1,1,-1,1)

Barycentrics    (b - c) (a^3 - a^2 b + b^3 - a^2 c + a b c + c^3) : :

X(29094) lies on these lines: {1, 3801}, {8, 4808}, {30, 511}, {905, 7626}, {1960, 4142}, {2530, 3904}, {2605, 21121}, {4367, 4707}, {6332, 21260}

X(29094) = isogonal conjugate of X(29095)


X(29095) =  ISOGONAL CONJUGATE OF X(29094)

Barycentrics    a^2/((b - c) (a^3 - a^2 b + b^3 - a^2 c + a b c + c^3)) : :

X(29095) lies on the circumcircle and these lines:

X(29095) = isogonal conjugate of X(29094)


X(29096) =  CIRCUMCIRCLE-ANTIPODE OF X(29095)

Barycentrics    a^2 (a^6 - a^5 b - a^4 b^2 + 2 a^3 b^3 - a^2 b^4 - a b^5 + b^6 + a^4 b c - 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 - b^3 c^3 + a^2 c^4 - 2 a b c^4 + b^2 c^4 + a c^5 + b c^5 - 2 c^6) (a^6 - a^3 b^3 + a^2 b^4 + a b^5 - 2 b^6 - a^5 c + a^4 b c + a^3 b^2 c - 2 a b^4 c + b^5 c - a^4 c^2 - a^3 b c^2 + b^4 c^2 + 2 a^3 c^3 - a^2 b c^3 + a b^2 c^3 - b^3 c^3 - a^2 c^4 + a b c^4 - a c^5 + c^6) : :

X(29096) lies on the circumcircle and these lines:

X(29096) = isogonal conjugate of X(29097)
X(29096) = circumcircle-antipode of X(29095)


X(29097) =  ISOGONAL CONJUGATE OF X(29096)

Barycentrics    1/((a^6 - a^5 b - a^4 b^2 + 2 a^3 b^3 - a^2 b^4 - a b^5 + b^6 + a^4 b c - 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 - b^3 c^3 + a^2 c^4 - 2 a b c^4 + b^2 c^4 + a c^5 + b c^5 - 2 c^6) (a^6 - a^3 b^3 + a^2 b^4 + a b^5 - 2 b^6 - a^5 c + a^4 b c + a^3 b^2 c - 2 a b^4 c + b^5 c - a^4 c^2 - a^3 b c^2 + b^4 c^2 + 2 a^3 c^3 - a^2 b c^3 + a b^2 c^3 - b^3 c^3 - a^2 c^4 + a b c^4 - a c^5 + c^6)) : :

X(29097) lies on these lines: {21, 17202}, {30, 511}, {58, 12699}, {79, 983}, {191, 6210}, {573, 16139}, {1330, 6361}, {1742, 16132}, {3098, 4655}, {3454, 3579}, {3664, 16137}, {3818, 3923}, {3821, 5092}, {4672, 19130}, {5016, 11684}, {5695, 18440}, {6693, 9955}, {7683, 22793}, {8258, 18483}, {13624, 17235}, {17276, 18481}, {17351, 18480}

X(29097) = isogonal conjugate of X(29096)


X(29098) =  POINT POLARIS(1,1,1,-1)

Barycentrics    (b - c) (a^3 + a^2 b + b^3 + a^2 c - a b c + c^3) : :

X(29098) lies on these lines: {30, 511}, {3801, 4063}, {4782, 20517}, {4810, 7265}

X(29098) = isogonal conjugate of X(29099)


X(29099) =  ISOGONAL CONJUGATE OF X(29098)

Barycentrics    a^2/((b - c) (a^3 + a^2 b + b^3 + a^2 c - a b c + c^3)) : :

X(29099) lies on the circumcircle and these lines:

X(29099) = isogonal conjugate of X(29098)


X(29100) =  CIRCUMCIRCLE-ANTIPODE OF X(29099)

Barycentrics    a^2 (a^6 - a^5 b - a^4 b^2 + 2 a^3 b^3 - a^2 b^4 - a b^5 + 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 + a b^3 c^2 + 2 b^4 c^2 - a^3 c^3 - 2 a^2 b c^3 - 2 a b^2 c^3 - b^3 c^3 - a^2 c^4 + 2 a b c^4 - b^2 c^4 + a c^5 + b c^5 - 2 c^6) (a^6 + 2 a^4 b^2 - a^3 b^3 - a^2 b^4 + a b^5 - 2 b^6 - a^5 c - a^4 b c + a^3 b^2 c - 2 a^2 b^3 c + 2 a b^4 c + b^5 c - a^4 c^2 + a^3 b c^2 - 2 a b^3 c^2 - b^4 c^2 + 2 a^3 c^3 + a^2 b c^3 + a b^2 c^3 - b^3 c^3 - a^2 c^4 - a b c^4 + 2 b^2 c^4 - a c^5 + c^6) : :

X(29100) lies on the circumcircle and these lines:

X(29100) = isogonal conjugate of X(29101)
X(29100) = circumcircle-antipode of X(29099)


X(29101) =  ISOGONAL CONJUGATE OF X(29100)

Barycentrics    1/((a^6 - a^5 b - a^4 b^2 + 2 a^3 b^3 - a^2 b^4 - a b^5 + 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 + a b^3 c^2 + 2 b^4 c^2 - a^3 c^3 - 2 a^2 b c^3 - 2 a b^2 c^3 - b^3 c^3 - a^2 c^4 + 2 a b c^4 - b^2 c^4 + a c^5 + b c^5 - 2 c^6) (a^6 + 2 a^4 b^2 - a^3 b^3 - a^2 b^4 + a b^5 - 2 b^6 - a^5 c - a^4 b c + a^3 b^2 c - 2 a^2 b^3 c + 2 a b^4 c + b^5 c - a^4 c^2 + a^3 b c^2 - 2 a b^3 c^2 - b^4 c^2 + 2 a^3 c^3 + a^2 b c^3 + a b^2 c^3 - b^3 c^3 - a^2 c^4 - a b c^4 + 2 b^2 c^4 - a c^5 + c^6)) : :

X(29101) lies on these lines: {30, 511}

X(29101) = isogonal conjugate of X(29100)


X(29102) =  POINT POLARIS(1,1,-1,-1)

Barycentrics    (b - c) (a^3 - a^2 b + b^3 - a^2 c - a b c + c^3) : :

X(29102) lies on these lines: {30, 511}, {659, 4707}, {764, 3904}, {1960, 4458}, {3801, 4040}, {4448, 21145}, {4453, 14419}, {4808, 21302}, {6332, 23815}, {6545, 14432}, {14413, 21115}

X(29102) = isogonal conjugate of X(29103)


X(29103) =  ISOGONAL CONJUGATE OF X(29102)

Barycentrics    a^2/((b - c) (a^3 - a^2 b + b^3 - a^2 c - a b c + c^3)) : :

X(29103) lies on the circumcircle and these lines:

X(29103) = isogonal conjugate of X(29102)


X(29104) =  CIRCUMCIRCLE-ANTIPODE OF X(29103)

Barycentrics    a^2 (a^6 - a^5 b - a^4 b^2 + 2 a^3 b^3 - a^2 b^4 - a b^5 + b^6 - a^4 b c + 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 - 2 a^2 b c^3 - 2 a b^2 c^3 - b^3 c^3 + a^2 c^4 + 2 a b c^4 + b^2 c^4 + a c^5 + b c^5 - 2 c^6) (a^6 - a^3 b^3 + a^2 b^4 + a b^5 - 2 b^6 - a^5 c - a^4 b c + a^3 b^2 c - 2 a^2 b^3 c + 2 a b^4 c + b^5 c - a^4 c^2 + a^3 b c^2 - 2 a b^3 c^2 + b^4 c^2 + 2 a^3 c^3 + a^2 b c^3 + a b^2 c^3 - b^3 c^3 - a^2 c^4 - a b c^4 - a c^5 + c^6) : :

X(29104) lies on the circumcircle and these lines:

X(29104) = isogonal conjugate of X(29105)
X(29104) = circumcircle-antipode of X(29103)


X(29105) =  ISOGONAL CONJUGATE OF X(29104)

Barycentrics    1/((a^6 - a^5 b - a^4 b^2 + 2 a^3 b^3 - a^2 b^4 - a b^5 + b^6 - a^4 b c + 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 - 2 a^2 b c^3 - 2 a b^2 c^3 - b^3 c^3 + a^2 c^4 + 2 a b c^4 + b^2 c^4 + a c^5 + b c^5 - 2 c^6) (a^6 - a^3 b^3 + a^2 b^4 + a b^5 - 2 b^6 - a^5 c - a^4 b c + a^3 b^2 c - 2 a^2 b^3 c + 2 a b^4 c + b^5 c - a^4 c^2 + a^3 b c^2 - 2 a b^3 c^2 + b^4 c^2 + 2 a^3 c^3 + a^2 b c^3 + a b^2 c^3 - b^3 c^3 - a^2 c^4 - a b c^4 - a c^5 + c^6)) : :

X(29105) lies on these lines: {30, 511}, {580, 12699}, {1770, 4014}

X(29105) = isogonal conjugate of X(29104)


X(29106) =  POINT POLARIS(1,-1,1,-1)

Barycentrics    (b - c) (-a^3 - a^2 b + b^3 - a^2 c + a b c + 2 b^2 c + 2 b c^2 + c^3) : :

X(29106) lies on these lines: {30, 511}, {659, 7265}, {2530, 4467}, {3837, 21192}, {4025, 23815}, {4063, 4122}, {4142, 21261}

X(29106) = isogonal conjugate of X(29107)


X(29107) =  ISOGONAL CONJUGATE OF X(29106)

Barycentrics    a^2/((b - c) (-a^3 - a^2 b + b^3 - a^2 c + a b c + 2 b^2 c + 2 b c^2 + c^3)) : :

X(29107) lies on the circumcircle and these lines:

X(29107) = isogonal conjugate of X(29106)


X(29108) =  CIRCUMCIRCLE-ANTIPODE OF X(29109)

Barycentrics    a^2 (a^6 + a^5 b - a^4 b^2 - 2 a^3 b^3 - a^2 b^4 + a b^5 + b^6 + a^4 b c - 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 + 2 a^2 b c^3 + 2 a b^2 c^3 + b^3 c^3 + a^2 c^4 - 2 a b c^4 + b^2 c^4 - a c^5 - b c^5 - 2 c^6) (a^6 + a^3 b^3 + a^2 b^4 - a b^5 - 2 b^6 + a^5 c + a^4 b c - a^3 b^2 c + 2 a^2 b^3 c - 2 a b^4 c - b^5 c - a^4 c^2 - a^3 b c^2 + 2 a b^3 c^2 + b^4 c^2 - 2 a^3 c^3 - a^2 b c^3 - a b^2 c^3 + b^3 c^3 - a^2 c^4 + a b c^4 + a c^5 + c^6) : :

X(29108) lies on the circumcircle and these lines:

X(29108) = isogonal conjugate of X(29109)
X(29108) = circumcircle-antipode of X(29107)


X(29109) =  ISOGONAL CONJUGATE OF X(29108)

Barycentrics    1/((a^6 + a^5 b - a^4 b^2 - 2 a^3 b^3 - a^2 b^4 + a b^5 + b^6 + a^4 b c - 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 + 2 a^2 b c^3 + 2 a b^2 c^3 + b^3 c^3 + a^2 c^4 - 2 a b c^4 + b^2 c^4 - a c^5 - b c^5 - 2 c^6) (a^6 + a^3 b^3 + a^2 b^4 - a b^5 - 2 b^6 + a^5 c + a^4 b c - a^3 b^2 c + 2 a^2 b^3 c - 2 a b^4 c - b^5 c - a^4 c^2 - a^3 b c^2 + 2 a b^3 c^2 + b^4 c^2 - 2 a^3 c^3 - a^2 b c^3 - a b^2 c^3 + b^3 c^3 - a^2 c^4 + a b c^4 + a c^5 + c^6)) : :

X(29109) lies on these lines: {30, 511}, {355, 572}, {944, 17321}, {1385, 25498}

X(29109) = isogonal conjugate of X(29108)


X(29110) =  POINT POLARIS(1,-1,-1,1)

Barycentrics    (b - c) (-a^3 + a^2 b + b^3 + a^2 c - a b c + 2 b^2 c + 2 b c^2 + c^3) : :

X(29110) lies on these lines: {1, 4122}, {30, 511}, {38, 4041}, {1577, 4692}, {4467, 4730}, {4560, 4808}, {4707, 4774}, {4770, 21196}, {4775, 25259}, {4879, 7265}, {14838, 28602}

X(29110) = isogonal conjugate of X(29111)


X(29111) =  ISOGONAL CONJUGATE OF X(29110)

Barycentrics    a^2/((b - c) (-a^3 + a^2 b + b^3 + a^2 c - a b c + 2 b^2 c + 2 b c^2 + c^3)) : :

X(29111) lies on the circumcircle and these lines:

X(29111) = isogonal conjugate of X(29110)


X(29112) =  CIRCUMCIRCLE-ANTIPODE OF X(29111)

Barycentrics    a^2 (a^6 + a^5 b - a^4 b^2 - 2 a^3 b^3 - a^2 b^4 + a b^5 + 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 - a b^3 c^2 + 2 b^4 c^2 + a^3 c^3 + b^3 c^3 - a^2 c^4 + 2 a b c^4 - b^2 c^4 - a c^5 - b c^5 - 2 c^6) (a^6 + 2 a^4 b^2 + a^3 b^3 - a^2 b^4 - a b^5 - 2 b^6 + a^5 c - a^4 b c - a^3 b^2 c + 2 a b^4 c - b^5 c - a^4 c^2 + a^3 b c^2 - b^4 c^2 - 2 a^3 c^3 + a^2 b c^3 - a b^2 c^3 + b^3 c^3 - a^2 c^4 - a b c^4 + 2 b^2 c^4 + a c^5 + c^6) : :

X(29112) lies on the circumcircle and these lines:

X(29112) = isogonal conjugate of X(29113)
X(29112) = circumcircle-antipode of X(29111)


X(29113) =  ISOGONAL CONJUGATE OF X(29112)

Barycentrics    1/((a^6 + a^5 b - a^4 b^2 - 2 a^3 b^3 - a^2 b^4 + a b^5 + 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 - a b^3 c^2 + 2 b^4 c^2 + a^3 c^3 + b^3 c^3 - a^2 c^4 + 2 a b c^4 - b^2 c^4 - a c^5 - b c^5 - 2 c^6) (a^6 + 2 a^4 b^2 + a^3 b^3 - a^2 b^4 - a b^5 - 2 b^6 + a^5 c - a^4 b c - a^3 b^2 c + 2 a b^4 c - b^5 c - a^4 c^2 + a^3 b c^2 - b^4 c^2 - 2 a^3 c^3 + a^2 b c^3 - a b^2 c^3 + b^3 c^3 - a^2 c^4 - a b c^4 + 2 b^2 c^4 + a c^5 + c^6)) : :

X(29113) lies on these lines: {30, 511}, {1657, 5695}, {3818, 24728}

X(29113) = isogonal conjugate of X(29112)


X(29114) =  POINT POLARIS(2,1,1,1)

Barycentrics    (b - c) (2 a^3 + a^2 b + b^3 + a^2 c + a b c - b^2 c - b c^2 + c^3) : :

X(29114) lies on these lines: {30, 511}, {3803, 21201}

X(29114) = isogonal conjugate of X(29115)


X(29115) =  ISOGONAL CONJUGATE OF X(29114)

Barycentrics    a^2/((b - c) (2 a^3 + a^2 b + b^3 + a^2 c + a b c - b^2 c - b c^2 + c^3)) : :

X(29115) lies on the circumcircle and these lines:

X(29115) = isogonal conjugate of X(29114)


X(29116) =  POINT POLARIS(1,2,1,1)

Barycentrics    (b - c) (a^3 + a^2 b + 2 b^3 + a^2 c + a b c + b^2 c + b c^2 + 2 c^3) : :

X(29116) lies on these lines: {30, 511}, {3801, 4369}, {4913, 21124}

X(29116) = isogonal conjugate of X(29117)


X(29117) =  ISOGONAL CONJUGATE OF X(29116)

Barycentrics    a^2/((b - c) (a^3 + a^2 b + 2 b^3 + a^2 c + a b c + b^2 c + b c^2 + 2 c^3)) : :

X(29117) lies on the circumcircle and these lines:

X(29117) = isogonal conjugate of X(29116)


X(29118) =  POINT POLARIS(1,1,2,1)

Barycentrics    (b - c) (a^3 + 2 a^2 b + b^3 + 2 a^2 c + a b c + c^3) : :

X(29118) lies on these lines: {30, 511}, {649, 4142}, {1019, 4458}, {3801, 4784}, {4010, 8045}

X(29118) = isogonal conjugate of X(29119)


X(29119) =  ISOGONAL CONJUGATE OF X(29118)

Barycentrics    a^2/((b - c) (a^3 + 2 a^2 b + b^3 + 2 a^2 c + a b c + c^3)) : :

X(29119) lies on the circumcircle and these lines:

X(29119) = isogonal conjugate of X(29118)


X(29120) =  POINT POLARIS(1,1,1,2)

Barycentrics    (b - c) (a^3 + a^2 b + b^3 + a^2 c + 2 a b c + c^3) : :

X(29120) lies on these lines: {30, 511}, {4806, 6332}

X(29120) = isogonal conjugate of X(29121)


X(29121) =  ISOGONAL CONJUGATE OF X(29120)

Barycentrics    a^2/((b - c) (a^3 + a^2 b + b^3 + a^2 c + 2 a b c + c^3)) : :

X(29121) lies on the circumcircle and these lines:

X(29121) = isogonal conjugate of X(29120)


X(29122) =  POINT POLARIS(2,2,1,1)

Barycentrics    (b - c) (2 a^3 + a^2 b + 2 b^3 + a^2 c + a b c + 2 c^3) : :

X(29122) lies on these lines: {30, 511}

X(29122) = isogonal conjugate of X(29123)


X(29123) =  ISOGONAL CONJUGATE OF X(29122)

Barycentrics    a^2/((b - c) (2 a^3 + a^2 b + 2 b^3 + a^2 c + a b c + 2 c^3)) : :

X(29123) lies on the circumcircle and these lines:

X(29123) = isogonal conjugate of X(29122)


X(29124) =  POINT POLARIS(2,1,2,1)

Barycentrics    (b - c) (2 a^3 + 2 a^2 b + b^3 + 2 a^2 c + a b c - b^2 c - b c^2 + c^3) : :

X(29124) lies on these lines: {30, 511}

X(29124) = isogonal conjugate of X(29125)


X(29125) =  ISOGONAL CONJUGATE OF X(29124)

Barycentrics    a^2/((b - c) (2 a^3 + 2 a^2 b + b^3 + 2 a^2 c + a b c - b^2 c - b c^2 + c^3)) : :

X(29125) lies on the circumcircle and these lines:

X(29125) = isogonal conjugate of X(29124)


X(29126) =  POINT POLARIS(2,1,1,2)

Barycentrics    (b - c) (2 a^3 + a^2 b + b^3 + a^2 c + 2 a b c - b^2 c - b c^2 + c^3) : :

X(29126) lies on these lines: {30, 511}, {57, 1019}, {649, 10015}, {654, 4063}, {667, 26275}, {999, 4367}, {3452, 4129}, {3820, 21051}, {3904, 20295}, {4378, 23770}, {4707, 4897}, {4773, 21130}, {20508, 29487}

X(29126) = isogonal conjugate of X(29127)


X(29127) =  ISOGONAL CONJUGATE OF X(29126)

Barycentrics    a^2/((b - c) (2 a^3 + a^2 b + b^3 + a^2 c + 2 a b c - b^2 c - b c^2 + c^3)) : :

X(29127) lies on the circumcircle and these lines:

X(29127) = isogonal conjugate of X(29126)


X(29128) =  POINT POLARIS(1,2,2,1)

Barycentrics    (b - c) (a^3 + 2 a^2 b + 2 b^3 + 2 a^2 c + a b c + b^2 c + b c^2 + 2 c^3) : :

X(29128) lies on these lines: {30, 511}, {523, 8774}, {2850, 28863}, {5856, 6170}, {8063, 29188}, {9517, 28229}

X(29128) = isogonal conjugate of X(29129)


X(29129) =  ISOGONAL CONJUGATE OF X(29128)

Barycentrics    a^2/((b - c) (a^3 + 2 a^2 b + 2 b^3 + 2 a^2 c + a b c + b^2 c + b c^2 + 2 c^3)) : :

X(29129) lies on the circumcircle and these lines:

X(29129) = isogonal conjugate of X(29128)


X(29130) =  POINT POLARIS(1,2,1,2)

Barycentrics    (b - c) (a^3 + a^2 b + 2 b^3 + a^2 c + 2 a b c + b^2 c + b c^2 + 2 c^3) : :

X(29130) lies on these lines: {30, 511}

X(29130) = isogonal conjugate of X(29131)


X(29131) =  ISOGONAL CONJUGATE OF X(29130)

Barycentrics    a^2/((b - c) (a^3 + a^2 b + 2 b^3 + a^2 c + 2 a b c + b^2 c + b c^2 + 2 c^3)) : :

X(29131) lies on the circumcircle and these lines:

X(29131) = isogonal conjugate of X(29130)


X(29132) =  POINT POLARIS(1,1,2,2)

Barycentrics    (b - c) (a^3 + 2 a^2 b + b^3 + 2 a^2 c + 2 a b c + c^3) : :

X(29132) lies on these lines: {30, 511}, {1019, 20517}

X(29132) = isogonal conjugate of X(29133)


X(29133) =  ISOGONAL CONJUGATE OF X(29132)

Barycentrics    a^2/((b - c) (a^3 + 2 a^2 b + b^3 + 2 a^2 c + 2 a b c + c^3)) : :

X(29133) lies on the circumcircle and these lines:

X(29133) = isogonal conjugate of X(29132)


X(29134) =  POINT POLARIS(1,2,2,2)

Barycentrics    (b - c) (a^3 + 2 a^2 b + 2 b^3 + 2 a^2 c + 2 a b c + b^2 c + b c^2 + 2 c^3) : :

X(29134) lies on these lines: {30, 511}, {786, 29025}, {1938, 2853}, {2845, 9013}, {4844, 8677}, {8063, 9373}, {9240, 28859}

X(29134) = isogonal conjugate of X(29135)


X(29135) =  ISOGONAL CONJUGATE OF X(29134)

Barycentrics    a^2/((b - c) (a^3 + 2 a^2 b + 2 b^3 + 2 a^2 c + 2 a b c + b^2 c + b c^2 + 2 c^3)) : :

X(29135) lies on the circumcircle and these lines:

X(29135) = isogonal conjugate of X(29134)


X(29136) =  POINT POLARIS(2,1,2,2)

Barycentrics    (b - c) (2 a^3 + 2 a^2 b + b^3 + 2 a^2 c + 2 a b c - b^2 c - b c^2 + c^3) : :

X(29136) lies on these lines: {30, 511}, {1019, 3337}

X(29136) = isogonal conjugate of X(29137)


X(29137) =  ISOGONAL CONJUGATE OF X(29136)

Barycentrics    a^2/((b - c) (2 a^3 + 2 a^2 b + b^3 + 2 a^2 c + 2 a b c - b^2 c - b c^2 + c^3)) : :

X(29137) lies on the circumcircle and these lines:

X(29137) = isogonal conjugate of X(29136)


X(29138) =  POINT POLARIS(2,2,1,2)

Barycentrics    (b - c) (2 a^3 + a^2 b + 2 b^3 + a^2 c + 2 a b c + 2 c^3) : :

X(29138) lies on these lines: {30, 511}

X(29138) = isogonal conjugate of X(29139)


X(29139) =  ISOGONAL CONJUGATE OF X(29138)

Barycentrics    a^2/((b - c) (2 a^3 + a^2 b + 2 b^3 + a^2 c + 2 a b c + 2 c^3)) : :

X(29139) lies on the circumcircle and these lines:

X(29139) = isogonal conjugate of X(29138)


X(29140) =  POINT POLARIS(2,2,2,1)

Barycentrics    (b - c) (2 a^3 + 2 a^2 b + 2 b^3 + 2 a^2 c + a b c + 2 c^3) : :

X(29140) lies on these lines: {30, 511}

X(29140) = isogonal conjugate of X(29141)


X(29141) =  ISOGONAL CONJUGATE OF X(29140)

Barycentrics    a^2/((b - c) (2 a^3 + 2 a^2 b + 2 b^3 + 2 a^2 c + a b c + 2 c^3)) : :

X(29141) lies on the circumcircle and these lines:

X(29141) = isogonal conjugate of X(29140)


X(29142) =  POINT POLARIS(0,1,1,2)

Barycentrics    (b - c) (a^2 b + b^3 + a^2 c + 2 a b c + b^2 c + b c^2 + c^3) : :

X(29142) lies on these lines: {30, 511}, {2254, 21124}, {2530, 3004}, {2533, 10015}, {3716, 8045}, {3801, 21146}, {4142, 4369}, {4978, 23770}, {7662, 21185}, {21108, 21118}, {21125, 23753}

X(29142) = isogonal conjugate of X(29143)


X(29143) =  ISOGONAL CONJUGATE OF X(29142)

Barycentrics    a^2/((b - c) (a^2 b + b^3 + a^2 c + 2 a b c + b^2 c + b c^2 + c^3)) : :

X(29143) lies on the circumcircle and these lines:

X(29143) = isogonal conjugate of X(29142)


X(29144) =  POINT POLARIS(0,1,2,1)

Barycentrics    (b - c) (2 a^2 b + b^3 + 2 a^2 c + a b c + b^2 c + b c^2 + c^3) : :

X(29144) lies on these lines: {30, 511}, {4120, 4951}, {4522, 4806}, {21146, 29835}, {28199, 29246}

X(29144) = isogonal conjugate of X(29145)


X(29145) =  ISOGONAL CONJUGATE OF X(29144)

Barycentrics    a^2/((b - c) (2 a^2 b + b^3 + 2 a^2 c + a b c + b^2 c + b c^2 + c^3)) : :

X(29145) lies on the circumcircle and these lines:

X(29145) = isogonal conjugate of X(29144)


X(29146) =  POINT POLARIS(0,2,1,1)

Barycentrics    (b - c) (a^2 b + 2 b^3 + a^2 c + a b c + 2 b^2 c + 2 b c^2 + 2 c^3) : :

X(29146) lies on these lines: {30, 511}

X(29146) = isogonal conjugate of X(29147)


X(29147) =  ISOGONAL CONJUGATE OF X(29146)

Barycentrics    a^2/((b - c) (a^2 b + 2 b^3 + a^2 c + a b c + 2 b^2 c + 2 b c^2 + 2 c^3)) : :

X(29147) lies on the circumcircle and these lines:

X(29147) = isogonal conjugate of X(29146)


X(29148) =  POINT POLARIS(1,0,1,2)

Barycentrics    (b - c) (-a^3 - a^2 b - a^2 c - 2 a b c + b^2 c + b c^2) : :

X(29148) lies on these lines: {30, 511}, {63, 4063}, {226, 3669}, {649, 3762}, {667, 993}, {764, 24719}, {905, 4129}, {1019, 4391}, {1478, 4905}, {1577, 4379}, {3822, 21260}, {3835, 3960}, {4010, 4378}, {4170, 4449}, {4367, 4800}, {4369, 4791}, {4380, 21385}, {4474, 4761}, {4775, 4922}, {4897, 10015}, {5307, 17924}, {5745, 20317}, {14349, 17496}, {20295, 21222}

X(29148) = isogonal conjugate of X(29149)


X(29149) =  ISOGONAL CONJUGATE OF X(29148)

Barycentrics    a^2/((b - c) (-a^3 - a^2 b - a^2 c - 2 a b c + b^2 c + b c^2)) : :

X(29149) lies on the circumcircle and these lines:

X(29149) = isogonal conjugate of X(29148)


X(29150) =  POINT POLARIS(1,0,2,1)

Barycentrics    (b - c) (-a^3 - 2 a^2 b - 2 a^2 c - a b c + b^2 c + b c^2) : :

X(29150) lies on these lines: {21, 667}, {30, 511}, {79, 4905}, {191, 4063}, {442, 21260}, {764, 14450}, {1019, 4010}, {1577, 4784}, {2475, 21301}, {2530, 20295}, {3572, 21836}, {3647, 4782}, {3649, 3669}, {3960, 4992}, {4106, 23815}, {4129, 9508}, {4162, 10543}, {4170, 4367}, {4391, 4834}, {4462, 11684}, {4560, 4983}, {4806, 14838}, {4810, 4978}, {6161, 15680}, {11263, 19947}, {18014, 21201}, {18253, 20317}

X(29150) = isogonal conjugate of X(29151)


X(29151) =  ISOGONAL CONJUGATE OF X(29150)

Barycentrics    a^2/((b - c) (-a^3 - 2 a^2 b - 2 a^2 c - a b c + b^2 c + b c^2)) : :

X(29151) lies on the circumcircle and these lines:

X(29151) = isogonal conjugate of X(29150)


X(29152) =  POINT POLARIS(2,0,1,1)

Barycentrics    (b - c) (-2 a^3 - a^2 b - a^2 c - a b c + 2 b^2 c + 2 b c^2) : :

X(29152) lies on these lines: {30, 511}, {4391, 4782}, {4449, 4810}, {4800, 8643}, {17496, 24719}

X(29152) = isogonal conjugate of X(29153)


X(29153) =  ISOGONAL CONJUGATE OF X(29152)

Barycentrics    a^2/((b - c) (-2 a^3 - a^2 b - a^2 c - a b c + 2 b^2 c + 2 b c^2)) : :

X(29153) lies on the circumcircle and these lines:

X(29153) = isogonal conjugate of X(29152)


X(29154) =  POINT POLARIS(1,2,0,1)

Barycentrics    (b - c) (a^3 + 2 b^3 + a b c + b^2 c + b c^2 + 2 c^3) : :

X(29154) lies on these lines: {30, 511}

X(29154) = isogonal conjugate of X(29155)


X(29155) =  ISOGONAL CONJUGATE OF X(29154)

Barycentrics    a^2/((b - c) (a^3 + 2 b^3 + a b c + b^2 c + b c^2 + 2 c^3)) : :

X(29155) lies on the circumcircle and these lines:

X(29155) = isogonal conjugate of X(29154)


X(29156) =  POINT POLARIS(2,1,0,1)

Barycentrics    (b - c) (2 a^3 + b^3 + a b c - b^2 c - b c^2 + c^3) : :

X(29156) lies on these lines: {30, 511}, {3904, 24719}, {4782, 10015}

X(29156) = isogonal conjugate of X(29157)


X(29157) =  ISOGONAL CONJUGATE OF X(29154)

Barycentrics    a^2/((b - c) (2 a^3 + b^3 + a b c - b^2 c - b c^2 + c^3)) : :

X(29157) lies on the circumcircle and these lines:

X(29157) = isogonal conjugate of X(29156)


X(29158) =  POINT POLARIS(1,1,2,0)

Barycentrics    (b - c) (a^3 + 2 a^2 b + b^3 + 2 a^2 c + c^3) : :

X(29158) lies on these lines: {30, 511}, {649, 20517}, {3801, 4834}, {4401, 8636}, {4905, 23687}

X(29158) = isogonal conjugate of X(29159)


X(29159) =  ISOGONAL CONJUGATE OF X(29158)

Barycentrics    a^2/((b - c) (a^3 + 2 a^2 b + b^3 + 2 a^2 c + c^3)) : :

X(29159) lies on the circumcircle and these lines:

X(29159) = isogonal conjugate of X(29158)


X(29160) =  POINT POLARIS(1,2,1,0)

Barycentrics    (b - c) (a^3 + a^2 b + 2 b^3 + a^2 c + b^2 c + b c^2 + 2 c^3) : :

X(29160) lies on these lines: {30, 511}

X(29160) = isogonal conjugate of X(29161)


X(29161) =  ISOGONAL CONJUGATE OF X(29160)

Barycentrics    a^2/((b - c) (a^3 + a^2 b + 2 b^3 + a^2 c + b^2 c + b c^2 + 2 c^3)) : :

X(29161) lies on the circumcircle and these lines:

X(29161) = isogonal conjugate of X(29160)


X(29162) =  POINT POLARIS(2,1,1,0)

Barycentrics    (b - c) (2 a^3 + a^2 b + b^3 + a^2 c - b^2 c - b c^2 + c^3) : :

X(29162) lies on these lines: {30, 511}, {649, 2504}, {652, 4498}, {659, 21789}, {667, 676}, {2487, 21188}, {2977, 21051}, {3004, 4560}, {3669, 6591}, {3733, 17925}, {3803, 21185}, {4010, 4990}, {4063, 10015}, {4106, 6332}, {4367, 23770}, {4394, 14837}, {4462, 20298}, {4976, 21124}, {4979, 23755}, {8638, 21005}, {11068, 20317}

X(29162) = isogonal conjugate of X(29163)


X(29163) =  ISOGONAL CONJUGATE OF X(29162)

Barycentrics    a^2/((b - c) (2 a^3 + a^2 b + b^3 + a^2 c - b^2 c - b c^2 + c^3)) : :

X(29163) lies on the circumcircle and these lines:

X(29163) = isogonal conjugate of X(29162)


X(29164) =  POINT POLARIS(0,2,2,1)

Barycentrics    (b - c) (2 a^2 b + 2 b^3 + 2 a^2 c + a b c + 2 b^2 c + 2 b c^2 + 2 c^3) : :

X(29164) lies on these lines: {30, 511}

X(29164) = isogonal conjugate of X(29165)


X(29165) =  ISOGONAL CONJUGATE OF X(29164)

Barycentrics    a^2/((b - c) (2 a^2 b + 2 b^3 + 2 a^2 c + a b c + 2 b^2 c + 2 b c^2 + 2 c^3)) : :

X(29165) lies on the circumcircle and these lines:

X(29165) = isogonal conjugate of X(29164)


X(29166) =  POINT POLARIS(0,2,1,2)

Barycentrics    (b - c) (a^2 b + 2 b^3 + a^2 c + 2 a b c + 2 b^2 c + 2 b c^2 + 2 c^3) : :

X(29166) lies on these lines: {30, 511}

X(29166) = isogonal conjugate of X(29167)


X(29167) =  ISOGONAL CONJUGATE OF X(29166)

Barycentrics    a^2/((b - c) (a^2 b + 2 b^3 + a^2 c + 2 a b c + 2 b^2 c + 2 b c^2 + 2 c^3)) : :

X(29167) lies on the circumcircle and these lines:

X(29167) = isogonal conjugate of X(29166)


X(29168) =  POINT POLARIS(0,1,2,2)

Barycentrics    (b - c) (2 a^2 b + b^3 + 2 a^2 c + 2 a b c + b^2 c + b c^2 + c^3) : :

X(29168) lies on these lines: {30, 511}, {4802, 5849}, {6364, 8774}, {29138, 29184}

X(29168) = isogonal conjugate of X(29169)


X(29169) =  ISOGONAL CONJUGATE OF X(29168)

Barycentrics    a^2/((b - c) (2 a^2 b + b^3 + 2 a^2 c + 2 a b c + b^2 c + b c^2 + c^3)) : :

X(29169) lies on the circumcircle and these lines:

X(29169) = isogonal conjugate of X(29168)


X(29170) =  POINT POLARIS(1,0,2,2)

Barycentrics    (b - c) (-a^3 - 2 a^2 b - 2 a^2 c - 2 a b c + b^2 c + b c^2) : :

X(29170) lies on these lines: {30, 511}, {905, 4806}, {1019, 4874}, {3669, 4992}, {3762, 4834}, {3777, 20295}, {4170, 4378}, {4391, 4784}, {4801, 4810}

X(29170) = isogonal conjugate of X(29171)


X(29171) =  ISOGONAL CONJUGATE OF X(29170)

Barycentrics    a^2/((b - c) (-a^3 - 2 a^2 b - 2 a^2 c - 2 a b c + b^2 c + b c^2)) : :

X(29171) lies on the circumcircle and these lines:

X(29171) = isogonal conjugate of X(29170)


X(29172) =  POINT POLARIS(1,2,0,2)

Barycentrics    (b - c) (a^3 + 2 b^3 + 2 a b c + b^2 c + b c^2 + 2 c^3) : :

X(29172) lies on these lines: {30, 511}, {4379, 21145}

X(29172) = isogonal conjugate of X(29173)


X(29173) =  ISOGONAL CONJUGATE OF X(29170)

Barycentrics    a^2/((b - c) (a^3 + 2 b^3 + 2 a b c + b^2 c + b c^2 + 2 c^3)) : :

X(29173) lies on the circumcircle and these lines:

X(29173) = isogonal conjugate of X(29172)


X(29174) =  POINT POLARIS(1,2,2,0)

Barycentrics    (b - c) (a^3 + 2 a^2 b + 2 b^3 + 2 a^2 c + b^2 c + b c^2 + 2 c^3) : :

X(29174) lies on these lines: {30, 511}, {6364, 8774}, {8063, 9391}

X(29174) = isogonal conjugate of X(29175)


X(29175) =  ISOGONAL CONJUGATE OF X(29174)

Barycentrics    a^2/((b - c) (a^3 + 2 a^2 b + 2 b^3 + 2 a^2 c + b^2 c + b c^2 + 2 c^3)) : :

X(29175) lies on the circumcircle and these lines:

X(29175) = isogonal conjugate of X(29176)


X(29176) =  POINT POLARIS(2,0,1,2)

Barycentrics    (b - c) (-2 a^3 - a^2 b - a^2 c - 2 a b c + 2 b^2 c + 2 b c^2) : :

X(29176) lies on these lines: {30, 511}, {4170, 4922}, {4474, 4834}

X(29176) = isogonal conjugate of X(29177)


X(29177) =  ISOGONAL CONJUGATE OF X(29176)

Barycentrics    a^2/((b - c) (-2 a^3 - a^2 b - a^2 c - 2 a b c + 2 b^2 c + 2 b c^2)) : :

X(29177) lies on the circumcircle and these lines:

X(29177) = isogonal conjugate of X(29176)


X(29178) =  POINT POLARIS(2,0,2,1)

Barycentrics    (b - c) (-2 a^3 - 2 a^2 b - 2 a^2 c - a b c + 2 b^2 c + 2 b c^2) : :

X(29178) lies on these lines: {30, 511}, {649, 4791}, {667, 4800}, {1019, 4379}, {3762, 4380}, {3835, 16751}, {3960, 4106}, {4378, 4810}, {4401, 4448}

X(29178) = isogonal conjugate of X(29179)


X(29179) =  ISOGONAL CONJUGATE OF X(29178)

Barycentrics    a^2/((b - c) (-2 a^3 - 2 a^2 b - 2 a^2 c - a b c + 2 b^2 c + 2 b c^2)) : :

X(29179) lies on the circumcircle and these lines:

X(29179) = isogonal conjugate of X(29178)


X(29180) =  CIRCUMCIRCLE-ANTIPODE OF X(907)

Barycentrics    a^2 (a^6 - a^4 b^2 - a^2 b^4 + b^6 + 4 a^4 c^2 + 4 b^4 c^2 - 3 a^2 c^4 - 3 b^2 c^4 - 2 c^6) (a^6 + 4 a^4 b^2 - 3 a^2 b^4 - 2 b^6 - a^4 c^2 - 3 b^4 c^2 - a^2 c^4 + 4 b^2 c^4 + c^6) : :

X(29180) lies on the circumcircle and these lines:

X(29180) = isogonal conjugate of X(29181)
X(29180) = circumcircle-antipode of X(907)


X(29181) =  ISOGONAL CONJUGATE OF X(29180)

Barycentrics    2*a^6 + 3*a^4*b^2 - 4*a^2*b^4 - b^6 + 3*a^4*c^2 + b^4*c^2 - 4*a^2*c^4 + b^2*c^4 - c^6 : :

X(29181) lies on these lines: {2, 21167}, {3, 3589}, {4, 141}, {5, 3098}, {6, 20}, {7, 5716}, {9, 13442}, {22, 13394}, {23, 11064}, {30, 511}, {51, 7667}, {67, 10733}, {69, 3146}, {74, 25328}, {140, 14810}, {146, 2930}, {159, 2883}, {182, 550}, {193, 5059}, {206, 13346}, {230, 2076}, {316, 6393}, {323, 20063}, {343, 7391}, {376, 597}, {381, 20582}, {382, 1352}, {388, 10387}, {390, 28369}, {394, 7500}, {411, 4265}, {428, 3917}, {546, 24206}, {548, 5092}, {575, 12103}, {576, 12007}, {599, 3543}, {611, 4302}, {613, 4299}, {858, 15059}, {950, 24471}, {962, 3242}, {1001, 9840}, {1151, 13910}, {1152, 13972}, {1213, 7379}, {1351, 1657}, {1370, 13567}, {1386, 4297}, {1428, 15326}, {1469, 6284}, {1513, 5103}, {1593, 3867}, {1692, 6781}, {1843, 1885}, {1890, 5784}, {1992, 15683}, {1993, 20062}, {2330, 15338}, {2475, 26543}, {2549, 14532}, {2550, 5793}, {3056, 7354}, {3091, 3763}, {3094, 7745}, {3313, 3575}, {3416, 5691}, {3522, 3618}, {3529, 3629}, {3534, 5050}, {3580, 5189}, {3619, 3832}, {3620, 17578}, {3627, 3818}, {3630, 15069}, {3793, 10991}, {3826, 15973}, {3839, 21358}, {3844, 19925}, {3853, 18358}, {3932, 18788}, {4220, 6703}, {4846, 8546}, {5017, 5254}, {5039, 15048}, {5052, 7756}, {5073, 18440}, {5093, 11179}, {5096, 6909}, {5102, 8584}, {5157, 11424}, {5181, 13202}, {5188, 24256}, {5476, 8703}, {5562, 16621}, {5596, 17845}, {5621, 25320}, {5651, 10301}, {5654, 7387}, {5743, 26118}, {5893, 15585}, {5894, 15583}, {5895, 9924}, {5907, 16656}, {5943, 10691}, {6247, 17834}, {6403, 18560}, {6593, 16163}, {6643, 15873}, {6688, 7734}, {6696, 23300}, {6698, 7687}, {6707, 6998}, {6756, 15644}, {6995, 17811}, {7385, 17245}, {7386, 17810}, {7390, 15668}, {7396, 26958}, {7407, 17327}, {7417, 11053}, {7422, 24975}, {7464, 19457}, {7492, 14389}, {7519, 15066}, {7540, 13340}, {7553, 10625}, {7605, 15246}, {7710, 9766}, {7750, 18906}, {8353, 22486}, {8356, 22676}, {9589, 16496}, {9756, 13468}, {9820, 17714}, {9825, 13348}, {9909, 10192}, {9969, 12362}, {9970, 12121}, {10124, 25565}, {10295, 15472}, {10721, 14982}, {10723, 11646}, {10752, 25329}, {11178, 15687}, {11245, 21969}, {11412, 16655}, {11414, 12233}, {11459, 16654}, {11579, 20127}, {11694, 25566}, {12017, 15696}, {12084, 15578}, {12203, 12212}, {12241, 19161}, {12244, 16010}, {12305, 21736}, {12588, 12943}, {12589, 12953}, {13403, 21851}, {14689, 28343}, {14718, 17949}, {14790, 14852}, {14807, 15163}, {14808, 15162}, {14848, 15689}, {14893, 25561}, {15520, 19710}, {15533, 15640}, {15582, 22660}, {15680, 15988}, {15682, 22165}, {20300, 23335}, {21659, 26926}, {23049, 23328}

X(29181) = isogonal conjugate of X(29180)


X(29182) =  POINT POLARIS(2,0,-1,0)

Barycentrics    (b - c) (-2 a^3 + a^2 b + a^2 c + 2 b^2 c + 2 b c^2) : :

X(29182) lies on these lines: {30, 511}, {1577, 1960}, {4063, 4774}, {4922, 4978}

X(29182) = isogonal conjugate of X(29183)


X(29183) =  ISOGONAL CONJUGATE OF X(29182)

Barycentrics    a^2/((b - c) (-2 a^3 + a^2 b + a^2 c + 2 b^2 c + 2 b c^2)) : :

X(29183) lies on the circumcircle and these lines:

X(29183) = isogonal conjugate of X(29182)


X(29184) =  POINT POLARIS(2,2,1,0)

Barycentrics    (b - c) (2 a^3 + a^2 b + 2 b^3 + a^2 c + 2 c^3) : :

X(29184) lies on these lines: {30, 511}

X(29184) = isogonal conjugate of X(29185)


X(29185) =  ISOGONAL CONJUGATE OF X(29184)

Barycentrics    a^2/((b - c) (2 a^3 + a^2 b + 2 b^3 + a^2 c + 2 c^3)) : :

X(29185) lies on the circumcircle and these lines:

X(29185) = isogonal conjugate of X(29184)


X(29186) =  POINT POLARIS(-1,0,1,2)

Barycentrics    (b - c) (-a^3 + a^2 b + a^2 c + 2 a b c + b^2 c + b c^2) : :

X(29186) lies on these lines: {1, 4801}, {30, 511}, {663, 4978}, {667, 21146}, {693, 4040}, {905, 23789}, {1577, 4724}, {1734, 17494}, {3716, 4823}, {4170, 4382}, {4369, 4401}, {4498, 4761}, {4560, 4905}, {4818, 19992}, {4983, 24719}, {14838, 24720}, {19594, 23724}

X(29186) = isogonal conjugate of X(29187)


X(29187) =  ISOGONAL CONJUGATE OF X(29186)

Barycentrics    a^2/((b - c) (-a^3 + a^2 b + a^2 c + 2 a b c + b^2 c + b c^2)) : :

X(29187) lies on the circumcircle and these lines:

X(29187) = isogonal conjugate of X(29186)


X(29188) =  POINT POLARIS(-1,0,2,1)

Barycentrics    (b - c) (-a^3 + 2 a^2 b + 2 a^2 c + a b c + b^2 c + b c^2) : :

X(29188) lies on these lines: {1, 21146}, {30, 511}, {659, 4761}, {663, 3720}, {693, 4775}, {1577, 4800}, {1960, 4369}, {2533, 4040}, {4651, 4705}, {4730, 17494}, {4794, 4874}, {19947, 24720}

X(29188) = isogonal conjugate of X(29189)


X(29189) =  ISOGONAL CONJUGATE OF X(29188)

Barycentrics    a^2/((b - c) (-a^3 + 2 a^2 b + 2 a^2 c + a b c + b^2 c + b c^2)) : :

X(29189) lies on the circumcircle and these lines:

X(29189) = isogonal conjugate of X(29188)


X(29190) =  POINT POLARIS(-1,1,0,2)

Barycentrics    (b - c) (-a^3 + b^3 + 2 a b c + 2 b^2 c + 2 b c^2 + c^3) : :

X(29190) lies on these lines: {30, 511}, {693, 20517}, {4025, 23789}, {4142, 4823}, {4401, 8045}, {4467, 4905}, {4724, 7265}, {21192, 24720}

X(29190) = isogonal conjugate of X(29191)


X(29191) =  ISOGONAL CONJUGATE OF X(29190)

Barycentrics    a^2/((b - c) (-a^3 + b^3 + 2 a b c + 2 b^2 c + 2 b c^2 + c^3)) : :

X(29191) lies on the circumcircle and these lines:

X(29191) = isogonal conjugate of X(29190)


X(29192) =  POINT POLARIS(-1,1,2,0)

Barycentrics    (b - c) (-a^3 + 2 a^2 b + b^3 + 2 a^2 c + 2 b^2 c + 2 b c^2 + c^3) : :

X(29192) lies on these lines: {30, 511}, {2533, 20517}, {4122, 4775}

X(29192) = isogonal conjugate of X(29193)


X(29193) =  ISOGONAL CONJUGATE OF X(29192)

Barycentrics    a^2/((b - c) (-a^3 + 2 a^2 b + b^3 + 2 a^2 c + 2 b^2 c + 2 b c^2 + c^3)) : :

X(29193) lies on the circumcircle and these lines:

X(29193) = isogonal conjugate of X(29192)


X(29194) =  POINT POLARIS(-1,2,0,1)

Barycentrics    (b - c) (-a^3 + 2 b^3 + a b c + 3 b^2 c + 3 b c^2 + 2 c^3) : :

X(29194) lies on these lines: {30, 511}

X(29194) = isogonal conjugate of X(29195)


X(29195) =  ISOGONAL CONJUGATE OF X(29192)

Barycentrics    a^2/((b - c) (-a^3 + 2 b^3 + a b c + 3 b^2 c + 3 b c^2 + 2 c^3)) : :

X(29195) lies on the circumcircle and these lines:

X(29195) = isogonal conjugate of X(29194)


X(29196) =  POINT POLARIS(-1,2,1,0)

Barycentrics    (b - c) (-a^3 + a^2 b + 2 b^3 + a^2 c + 3 b^2 c + 3 b c^2 + 2 c^3) : :

X(29196) lies on these lines: {30, 511}

X(29196) = isogonal conjugate of X(29197)


X(29197) =  ISOGONAL CONJUGATE OF X(29192)

Barycentrics    a^2/((b - c) (-a^3 + a^2 b + 2 b^3 + a^2 c + 3 b^2 c + 3 b c^2 + 2 c^3)) : :

X(29197) lies on the circumcircle and these lines:

X(29197) = isogonal conjugate of X(29196)


X(29198) =  POINT POLARIS(0,0,1,3)

Barycentrics    a (b - c) (a b + a c + 3 b c) : :

X(29198) lies on these lines: {30, 511}, {351, 28374}, {661, 3777}, {667, 23394}, {764, 14349}, {1019, 4782}, {2254, 4490}, {2533, 4462}, {3250, 23751}, {4010, 4801}, {4040, 4378}, {4129, 23815}, {4367, 4724}, {4391, 21146}, {4498, 4784}, {4705, 4905}, {4790, 21832}, {4824, 22320}, {4834, 21385}, {7192, 16737}, {20949, 23807}, {20980, 21791}, {21051, 24720}, {21260, 23789}

X(29198) = isogonal conjugate of X(29433)


X(29199) =  ISOGONAL CONJUGATE OF X(29198)

Barycentrics    a/((b - c) (a b + a c + 3 b c)) : :

X(29199) lies on these lines:

X(29199) = isogonal conjugate of X(29198)


X(29200) =  POINT POLARIS(0,-1,2,1)

Barycentrics    (b - c) (-2 a^2 b + b^3 - 2 a^2 c - a b c + b^2 c + b c^2 + c^3) : :

X(29200) lies on these lines: {30, 511}, {2533, 25259}, {3776, 4992}, {17072, 18004}

X(29200) = isogonal conjugate of X(29201)


X(29201) =  ISOGONAL CONJUGATE OF X(29192)

Barycentrics    a^2/((b - c) (-2 a^2 b + b^3 - 2 a^2 c - a b c + b^2 c + b c^2 + c^3)) : :

X(29201) lies on the circumcircle and these lines:

X(29201) = isogonal conjugate of X(29200)


X(29202) =  POINT POLARIS(0,2,-1,1)

Barycentrics    (b - c) (-a^2 b + 2 b^3 - a^2 c + a b c + 2 b^2 c + 2 b c^2 + 2 c^3) : :

X(29202) lies on these lines: {30, 511}, {2528, 28374}, {4951, 21052}

X(29202) = isogonal conjugate of X(29203)


X(29203) =  ISOGONAL CONJUGATE OF X(29192)

Barycentrics    a^2/((b - c) (-a^2 b + 2 b^3 - a^2 c + a b c + 2 b^2 c + 2 b c^2 + 2 c^3)) : :

X(29203) lies on the circumcircle and these lines:

X(29203) = isogonal conjugate of X(29202)


X(29204) =  POINT POLARIS(0,2,1,-1)

Barycentrics    (b - c) (a^2 b + 2 b^3 + a^2 c - a b c + 2 b^2 c + 2 b c^2 + 2 c^3) : :

X(29204) lies on these lines: {30, 511}, {4728, 4951}

X(29204) = isogonal conjugate of X(29205)


X(29205) =  ISOGONAL CONJUGATE OF X(29192)

Barycentrics    a^2/((b - c) (a^2 b + 2 b^3 + a^2 c - a b c + 2 b^2 c + 2 b c^2 + 2 c^3)) : :

X(29205) lies on the circumcircle and these lines:

X(29205) = isogonal conjugate of X(29204)


X(29206) =  CIRCUMCIRCLE-ANTIPODE OF X(3910)

Barycentrics    a^2 (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 + 2 a^2 b c^3 + 2 a b^2 c^3 + a^2 c^4 - 4 a b c^4 + b^2 c^4 - 2 c^6) (a^6 + a^2 b^4 - 2 b^6 + 2 a^4 b c + 2 a^2 b^3 c - 4 a b^4 c - a^4 c^2 - 2 a^3 b c^2 + 2 a b^3 c^2 + b^4 c^2 - 2 a^2 b c^3 - a^2 c^4 + 2 a b c^4 + c^6) : :

X(29206) lies on the circumcircle and these lines:

X(29206) = isogonal conjugate of X(29207)
X(29206) = circumcircle-antipode of X(3910)


X(29207) =  ISOGONAL CONJUGATE OF X(29198)

Barycentrics    1/((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 + 2 a^2 b c^3 + 2 a b^2 c^3 + a^2 c^4 - 4 a b c^4 + b^2 c^4 - 2 c^6) (a^6 + a^2 b^4 - 2 b^6 + 2 a^4 b c + 2 a^2 b^3 c - 4 a b^4 c - a^4 c^2 - 2 a^3 b c^2 + 2 a b^3 c^2 + b^4 c^2 - 2 a^2 b c^3 - a^2 c^4 + 2 a b c^4 + c^6)) : :

X(29207) lies on these lines: {1, 10401}, {3, 16872}, {4, 608}, {11, 5061}, {30, 511}, {31, 19542}, {40, 5814}, {572, 4026}, {946, 4349}, {1329, 24265}, {1386, 12610}, {1565, 5018}, {1746, 3925}, {1766, 3416}, {1891, 12711}, {2050, 26098}, {2886, 13478}, {3703, 21375}, {3836, 19512}, {4300, 13442}, {4356, 5882}, {4388, 23512}, {4645, 6996}, {4999, 24251}, {5810, 11500}, {5928, 8270}, {6284, 10454}, {6327, 19645}, {16435, 26034}

X(29207) = isogonal conjugate of X(29206)


X(29208) =  POINT POLARIS(0,1,2,-1)

Barycentrics    (b - c) (2 a^2 b + b^3 + 2 a^2 c - a b c + b^2 c + b c^2 + c^3) : :

X(29208) lies on these lines: {30, 511}, {4522, 4992}, {4808, 14349}

X(29208) = isogonal conjugate of X(29209)


X(29209) =  ISOGONAL CONJUGATE OF X(29208)

Barycentrics    a^2/((b - c) (2 a^2 b + b^3 + 2 a^2 c - a b c + b^2 c + b c^2 + c^3)) : :

X(29209) lies on the circumcircle and these lines:

X(29209) = isogonal conjugate of X(29208)


X(29210) =  CIRCUMCIRCLE-ANTIPODE OF X(29109)

Barycentrics    a^2 (a^6 - a^4 b^2 - a^2 b^4 + b^6 - a^4 b c + a^3 b^2 c + a^2 b^3 c - a b^4 c + 3 a^4 c^2 + 3 b^4 c^2 - a^2 b c^3 - a b^2 c^3 - 2 a^2 c^4 + 2 a b c^4 - 2 b^2 c^4 - 2 c^6) (a^6 + 3 a^4 b^2 - 2 a^2 b^4 - 2 b^6 - a^4 b c - a^2 b^3 c + 2 a b^4 c - a^4 c^2 + a^3 b c^2 - a b^3 c^2 - 2 b^4 c^2 + a^2 b c^3 - a^2 c^4 - a b c^4 + 3 b^2 c^4 + c^6) : :

X(29210) lies on the circumcircle and these lines:

X(29210) = isogonal conjugate of X(29211)
X(29210) = circumcircle-antipode of X(29209)


X(29211) =  ISOGONAL CONJUGATE OF X(29210)

Barycentrics    1/((a^6 - a^4 b^2 - a^2 b^4 + b^6 - a^4 b c + a^3 b^2 c + a^2 b^3 c - a b^4 c + 3 a^4 c^2 + 3 b^4 c^2 - a^2 b c^3 - a b^2 c^3 - 2 a^2 c^4 + 2 a b c^4 - 2 b^2 c^4 - 2 c^6) (a^6 + 3 a^4 b^2 - 2 a^2 b^4 - 2 b^6 - a^4 b c - a^2 b^3 c + 2 a b^4 c - a^4 c^2 + a^3 b c^2 - a b^3 c^2 - 2 b^4 c^2 + a^2 b c^3 - a^2 c^4 - a b c^4 + 3 b^2 c^4 + c^6)) : :

X(29211) lies on these lines: {30, 511}

X(29211) = isogonal conjugate of X(29210)


X(29212) =  POINT POLARIS(1,-1,0,2)

Barycentrics    (b - c) (-a^3 + b^3 - 2 a b c + 2 b^2 c + 2 b c^2 + c^3) : :

X(29212) lies on these lines: {1, 25259}, {10, 4025}, {30, 511}, {1125, 3239}, {3634, 7658}, {3960, 4522}, {4122, 4378}, {4147, 21192}, {4391, 20517}, {4449, 7265}, {4458, 4791}, {4474, 4707}

X(29212) = isogonal conjugate of X(29213)


X(29213) =  ISOGONAL CONJUGATE OF X(29212)

Barycentrics    a^2/((b - c) (-a^3 + b^3 - 2 a b c + 2 b^2 c + 2 b c^2 + c^3)) : :

X(29213) lies on the circumcircle and these lines:

X(29213) = isogonal conjugate of X(29212)


X(29214) =  CIRCUMCIRCLE-ANTIPODE OF X(29113)

Barycentrics    a^2 (a^5 - a^3 b^2 - a^2 b^3 + b^5 - a^4 c - a^3 b c + 4 a^2 b^2 c - a b^3 c - b^4 c + 2 a^3 c^2 - 2 a^2 b c^2 - 2 a b^2 c^2 + 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) (a^5 - a^4 b + 2 a^3 b^2 - a^2 b^3 + a b^4 - 2 b^5 - a^3 b c - 2 a^2 b^2 c + 2 a b^3 c + b^4 c - a^3 c^2 + 4 a^2 b c^2 - 2 a b^2 c^2 - b^3 c^2 - a^2 c^3 - a b c^3 + 2 b^2 c^3 - b c^4 + c^5) : :

X(29214) lies on the circumcircle and these lines:

X(29214) = isogonal conjugate of X(29215)
X(29214) = circumcircle-antipode of X(29213)


X(29215) =  ISOGONAL CONJUGATE OF X(29214)

Barycentrics    1/((a^5 - a^3 b^2 - a^2 b^3 + b^5 - a^4 c - a^3 b c + 4 a^2 b^2 c - a b^3 c - b^4 c + 2 a^3 c^2 - 2 a^2 b c^2 - 2 a b^2 c^2 + 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) (a^5 - a^4 b + 2 a^3 b^2 - a^2 b^3 + a b^4 - 2 b^5 - a^3 b c - 2 a^2 b^2 c + 2 a b^3 c + b^4 c - a^3 c^2 + 4 a^2 b c^2 - 2 a b^2 c^2 - b^3 c^2 - a^2 c^3 - a b c^3 + 2 b^2 c^3 - b c^4 + c^5)) : :

X(29215) lies on these lines: {30, 511}, {1699, 29821}

X(29215) = isogonal conjugate of X(29214)


X(29216) =  POINT POLARIS(1,-1,2,0)

Barycentrics    (b - c) (-a^3 - 2 a^2 b + b^3 - 2 a^2 c + 2 b^2 c + 2 b c^2 + c^3) : :

X(29216) lies on these lines: {30, 511}, {649, 7265}, {3835, 21192}, {4010, 20517}, {4063, 25259}, {4122, 4834}, {4467, 14349}

X(29216) = isogonal conjugate of X(29217)


X(29217) =  ISOGONAL CONJUGATE OF X(29216)

Barycentrics    a^2/((b - c) (-a^3 - 2 a^2 b + b^3 - 2 a^2 c + 2 b^2 c + 2 b c^2 + c^3)) : :

X(29217) lies on the circumcircle and these lines:

X(29217) = isogonal conjugate of X(29216)


X(29218) =  CIRCUMCIRCLE-ANTIPODE OF X(29113)

Barycentrics    a^2 (a^6 + a^5 b - a^4 b^2 - 2 a^3 b^3 - a^2 b^4 + a b^5 + b^6 - a^4 c^2 - a^3 b 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 + b^3 c^3 + 2 a^2 c^4 + 2 b^2 c^4 - a c^5 - b c^5 - 2 c^6) (a^6 - a^4 b^2 + a^3 b^3 + 2 a^2 b^4 - a b^5 - 2 b^6 + a^5 c - a^3 b^2 c + a^2 b^3 c - b^5 c - a^4 c^2 + a b^3 c^2 + 2 b^4 c^2 - 2 a^3 c^3 - a b^2 c^3 + b^3 c^3 - a^2 c^4 - b^2 c^4 + a c^5 + c^6) : :

X(29218) lies on the circumcircle and these lines:

X(29218) = isogonal conjugate of X(29219)
X(29218) = circumcircle-antipode of X(29217)


X(29219) =  ISOGONAL CONJUGATE OF X(29218)

Barycentrics    1/((a^6 + a^5 b - a^4 b^2 - 2 a^3 b^3 - a^2 b^4 + a b^5 + b^6 - a^4 c^2 - a^3 b 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 + b^3 c^3 + 2 a^2 c^4 + 2 b^2 c^4 - a c^5 - b c^5 - 2 c^6) (a^6 - a^4 b^2 + a^3 b^3 + 2 a^2 b^4 - a b^5 - 2 b^6 + a^5 c - a^3 b^2 c + a^2 b^3 c - b^5 c - a^4 c^2 + a b^3 c^2 + 2 b^4 c^2 - 2 a^3 c^3 - a b^2 c^3 + b^3 c^3 - a^2 c^4 - b^2 c^4 + a c^5 + c^6)) : :

X(29219) lies on these lines: {1, 21270}, {3, 4445}, {8, 20074}, {10, 48}, {30, 511}, {184, 21072}, {1125, 20305}, {5721, 19925}, {21028, 23201}

X(29219) = isogonal conjugate of X(29218)


X(29220) =  POINT POLARIS(1,2,-1,0)

Barycentrics    (b - c) (a^3 - a^2 b + 2 b^3 - a^2 c + b^2 c + b c^2 + 2 c^3) : :

X(29220) lies on these lines: {30, 511}

X(29220) = isogonal conjugate of X(29221)


X(29221) =  ISOGONAL CONJUGATE OF X(29220)

Barycentrics    a^2/((b - c) (a^3 - a^2 b + 2 b^3 - a^2 c + b^2 c + b c^2 + 2 c^3)) : :

X(29221) lies on the circumcircle and these lines:

X(29221) = isogonal conjugate of X(29220)


X(29222) =  CIRCUMCIRCLE-ANTIPODE OF X(29113)

Barycentrics    a^2 (2 a^6 - a^5 b - 2 a^4 b^2 + 2 a^3 b^3 - 2 a^2 b^4 - a b^5 + 2 b^6 + a^4 c^2 + a^3 b 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 - b^3 c^3 + a^2 c^4 + b^2 c^4 + a c^5 + b c^5 - 4 c^6) (2 a^6 + a^4 b^2 - a^3 b^3 + a^2 b^4 + a b^5 - 4 b^6 - a^5 c + a^3 b^2 c - a^2 b^3 c + b^5 c - 2 a^4 c^2 - a b^3 c^2 + b^4 c^2 + 2 a^3 c^3 + a b^2 c^3 - b^3 c^3 - 2 a^2 c^4 + b^2 c^4 - a c^5 + 2 c^6) : :

X(29222) lies on the circumcircle and these lines:

X(29222) = isogonal conjugate of X(29223)
X(29222) = circumcircle-antipode of X(29221)


X(29223) =  ISOGONAL CONJUGATE OF X(29222)

Barycentrics    1/((2 a^6 - a^5 b - 2 a^4 b^2 + 2 a^3 b^3 - 2 a^2 b^4 - a b^5 + 2 b^6 + a^4 c^2 + a^3 b 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 - b^3 c^3 + a^2 c^4 + b^2 c^4 + a c^5 + b c^5 - 4 c^6) (2 a^6 + a^4 b^2 - a^3 b^3 + a^2 b^4 + a b^5 - 4 b^6 - a^5 c + a^3 b^2 c - a^2 b^3 c + b^5 c - 2 a^4 c^2 - a b^3 c^2 + b^4 c^2 + 2 a^3 c^3 + a b^2 c^3 - b^3 c^3 - 2 a^2 c^4 + b^2 c^4 - a c^5 + 2 c^6)) : :

X(29223) lies on these lines: {30, 511}

X(29223) = isogonal conjugate of X(29222)


X(29224) =  POINT POLARIS(1,2,0,-1)

Barycentrics    (b - c) (a^3 + 2 b^3 - a b c + b^2 c + b c^2 + 2 c^3) : :

X(29224) lies on these lines: {30, 511}, {3776, 19947}

X(29224) = isogonal conjugate of X(29225)


X(29225) =  ISOGONAL CONJUGATE OF X(29224)

Barycentrics    a^2/((b - c) (a^3 + 2 b^3 - a b c + b^2 c + b c^2 + 2 c^3)) : :

X(29225) lies on the circumcircle and these lines:

X(29225) = isogonal conjugate of X(29224)


X(29226) =  POINT POLARIS(0,0,-1,3)

Barycentrics    a (b - c) (-a b - a c + 3 b c) : :

X(29226) lies on these lines: {10, 23815}, {30, 511}, {659, 4449}, {661, 14470}, {663, 21343}, {665, 2516}, {667, 21385}, {693, 14404}, {764, 1734}, {905, 24174}, {1015, 24196}, {2254, 23765}, {2533, 4801}, {3669, 9508}, {3766, 23813}, {3777, 4041}, {3837, 4147}, {4010, 4462}, {4063, 4378}, {4367, 4498}, {4391, 19582}, {4394, 21832}, {4504, 4830}, {4724, 4879}, {4729, 23738}, {4730, 4905}, {4885, 25127}, {14407, 21348}, {17414, 28374}, {20983, 26824}, {21120, 23770}, {25636, 26854}

X(29226) = isogonal conjugate of X(29227)


X(29227) =  ISOGONAL CONJUGATE OF X(29226)

Barycentrics    a/((b - c) (a b + a c - 3 b c)) : :

X(29227) lies on the circumcircle and these lines:

X(29227) = isogonal conjugate of X(29226)


X(29228) =  CIRCUMCIRCLE-ANTIPODE OF X(29227)

Barycentrics    a (3 a^4 b - 3 a^3 b^2 - 3 a^2 b^3 + 3 a b^4 - a^4 c - b^4 c + 3 a^2 b c^2 + 3 a b^2 c^2 + a^2 c^3 - 6 a b c^3 + b^2 c^3) (a^4 b - a^2 b^3 - 3 a^4 c - 3 a^2 b^2 c + 6 a b^3 c + 3 a^3 c^2 - 3 a b^2 c^2 - b^3 c^2 + 3 a^2 c^3 - 3 a c^4 + b c^4) : :

X(29228) lies on the circumcircle and these lines:

X(29228) = isogonal conjugate of X(29229)


X(29229) =  ISOGONAL CONJUGATE OF X(29228)

Barycentrics    a/((3 a^4 b - 3 a^3 b^2 - 3 a^2 b^3 + 3 a b^4 - a^4 c - b^4 c + 3 a^2 b c^2 + 3 a b^2 c^2 + a^2 c^3 - 6 a b c^3 + b^2 c^3) (a^4 b - a^2 b^3 - 3 a^4 c - 3 a^2 b^2 c + 6 a b^3 c + 3 a^3 c^2 - 3 a b^2 c^2 - b^3 c^2 + 3 a^2 c^3 - 3 a c^4 + b c^4)) : :

X(29229) lies on these lines: {3, 15485}, {30, 511}, {573, 15492}, {991, 15178}, {1385, 1742}, {3579, 6210}, {14131, 24220}, {17508, 24309}, {18046, 22793}, {18480, 29705}

X(29229) = isogonal conjugate of X(29228)


X(29230) =  POINT POLARIS(2,-1,0,1)

Barycentrics    (b - c) (-2 a^3 + b^3 - a b c + 3 b^2 c + 3 b c^2 + c^3) : :

X(29230) lies on these lines: {30, 511}

X(29230) = isogonal conjugate of X(29231)


X(29231) =  ISOGONAL CONJUGATE OF X(29228)

Barycentrics    a^2/((b - c) (-2 a^3 + b^3 - a b c + 3 b^2 c + 3 b c^2 + c^3)) : :

X(29231) lies on the circumcircle and these lines:

X(29231) = isogonal conjugate of X(29230)


X(29232) =  POINT POLARIS(2,-1,1,0)

Barycentrics    (b - c) (-2 a^3 - a^2 b + b^3 - a^2 c + 3 b^2 c + 3 b c^2 + c^3) : :

X(29232) lies on these lines: {30, 511}, {667, 3700}, {1960, 4990}, {3239, 6050}, {4467, 21301}, {4705, 4976}, {17069, 21260}

X(29232) = isogonal conjugate of X(29233)


X(29233) =  ISOGONAL CONJUGATE OF X(29232)

Barycentrics    a^2/((b - c) (-2 a^3 - a^2 b + b^3 - a^2 c + 3 b^2 c + 3 b c^2 + c^3)) : :

X(29233) lies on the circumcircle and these lines:

X(29233) = isogonal conjugate of X(29232)


X(29234) =  CIRCUMCIRCLE-ANTIPODE OF X(29233)

Barycentrics    a^2 (a^6 + 2 a^5 b - a^4 b^2 - 4 a^3 b^3 - a^2 b^4 + 2 a b^5 + b^6 - 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 + 2 b^3 c^3 + a^2 c^4 + b^2 c^4 - 2 a c^5 - 2 b c^5 - 2 c^6) (a^6 + 2 a^3 b^3 + a^2 b^4 - 2 a b^5 - 2 b^6 + 2 a^5 c - 2 a^3 b^2 c + 2 a^2 b^3 c - 2 b^5 c - a^4 c^2 + 2 a b^3 c^2 + b^4 c^2 - 4 a^3 c^3 - 2 a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 + 2 a c^5 + c^6) : :

X(29234) lies on the circumcircle and these lines:

X(29234) = isogonal conjugate of X(29235)
X(29234) = circumcircle-antipode of X(29233)


X(29235) =  ISOGONAL CONJUGATE OF X(29234)

Barycentrics    1/((a^6 + 2 a^5 b - a^4 b^2 - 4 a^3 b^3 - a^2 b^4 + 2 a b^5 + b^6 - 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 + 2 b^3 c^3 + a^2 c^4 + b^2 c^4 - 2 a c^5 - 2 b c^5 - 2 c^6) (a^6 + 2 a^3 b^3 + a^2 b^4 - 2 a b^5 - 2 b^6 + 2 a^5 c - 2 a^3 b^2 c + 2 a^2 b^3 c - 2 b^5 c - a^4 c^2 + 2 a b^3 c^2 + b^4 c^2 - 4 a^3 c^3 - 2 a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 + 2 a c^5 + c^6)) : :

X(29235) lies on these lines: {3, 594}, {4, 4360}, {5, 17045}, {30, 511}, {355, 4026}, {944, 5263}, {4366, 24828}, {6653, 24813}, {6676, 21072}, {11238, 17726}

X(29235) = isogonal conjugate of X(29234)


X(29236) =  POINT POLARIS(2,0,-1,1)

Barycentrics    (b - c) (-2 a^3 + a^2 b + a^2 c - a b c + 2 b^2 c + 2 b c^2) : :

X(29236) lies on these lines: {30, 511}, {649, 4774}, {659, 4474}, {663, 4800}, {693, 4922}, {1960, 4791}, {4367, 4379}, {4382, 21343}, {4391, 4448}

X(29236) = isogonal conjugate of X(29237)


X(29237) =  ISOGONAL CONJUGATE OF X(29236)

Barycentrics    a^2/((b - c) (-2 a^3 + a^2 b + a^2 c - a b c + 2 b^2 c + 2 b c^2)) : :

X(29237) lies on the circumcircle and these lines:

X(29237) = isogonal conjugate of X(29236)


X(29238) =  POINT POLARIS(2,0,1,-1)

Barycentrics    (b - c) (-2 a^3 - a^2 b - a^2 c + a b c + 2 b^2 c + 2 b c^2) : :

X(29238) lies on these lines: {30, 511}, {663, 4810}, {1577, 4782}, {2533, 4380}, {4367, 4382}, {4560, 24719}

X(29238) = isogonal conjugate of X(29239)


X(29239) =  ISOGONAL CONJUGATE OF X(29238)

Barycentrics    a^2/((b - c) (-2 a^3 - a^2 b - a^2 c + a b c + 2 b^2 c + 2 b c^2)) : :

X(29239) lies on the circumcircle and these lines:

X(29239) = isogonal conjugate of X(29238)


X(29240) =  POINT POLARIS(2,1,-1,0)

Barycentrics    (b - c) (2 a^3 - a^2 b + b^3 - a^2 c - b^2 c - b c^2 + c^3) : :

X(29240) lies on these lines: {1, 23770}, {10, 2977}, {30, 511}, {659, 10015}, {667, 7178}, {676, 1960}, {1027, 1411}, {1638, 14419}, {1639, 14431}, {2170, 2969}, {3716, 5592}, {4378, 21104}, {4427, 21272}, {4728, 14432}, {4809, 21145}, {6050, 14837}, {6545, 14413}, {6546, 14430}, {8638, 20839}, {8651, 23723}, {9131, 24809}, {9979, 24810}

X(29240) = isogonal conjugate of X(29241)


X(29241) =  ISOGONAL CONJUGATE OF X(29240)

Barycentrics    a^2/((b - c) (2 a^3 - a^2 b + b^3 - a^2 c - b^2 c - b c^2 + c^3)) : :

X(29241) lies on the circumcircle and these lines:

X(29241) = isogonal conjugate of X(29240)


X(29242) =  CIRCUMCIRCLE-ANTIPODE OF X(29141)

Barycentrics    a^2 (a^6 - 2 a^5 b - a^4 b^2 + 4 a^3 b^3 - a^2 b^4 - 2 a b^5 + b^6 + 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 - 2 b^3 c^3 + a^2 c^4 + b^2 c^4 + 2 a c^5 + 2 b c^5 - 2 c^6) (a^6 - 2 a^3 b^3 + a^2 b^4 + 2 a b^5 - 2 b^6 - 2 a^5 c + 2 a^3 b^2 c - 2 a^2 b^3 c + 2 b^5 c - a^4 c^2 - 2 a b^3 c^2 + b^4 c^2 + 4 a^3 c^3 + 2 a b^2 c^3 - 2 b^3 c^3 - a^2 c^4 - 2 a c^5 + c^6) : :

X(29242) lies on the circumcircle and these lines:

X(29242) = isogonal conjugate of X(29243)
X(29242) = circumcircle-antipode of X(29241)


X(29243) =  ISOGONAL CONJUGATE OF X(29242)

Barycentrics    1/((a^6 - 2 a^5 b - a^4 b^2 + 4 a^3 b^3 - a^2 b^4 - 2 a b^5 + b^6 + 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 - 2 b^3 c^3 + a^2 c^4 + b^2 c^4 + 2 a c^5 + 2 b c^5 - 2 c^6) (a^6 - 2 a^3 b^3 + a^2 b^4 + 2 a b^5 - 2 b^6 - 2 a^5 c + 2 a^3 b^2 c - 2 a^2 b^3 c + 2 b^5 c - a^4 c^2 - 2 a b^3 c^2 + b^4 c^2 + 4 a^3 c^3 + 2 a b^2 c^3 - 2 b^3 c^3 - a^2 c^4 - 2 a c^5 + c^6)) : :

X(29243) lies on these lines: {3, 1086}, {4, 190}, {5, 4422}, {20, 4440}, {30, 511}, {40, 24715}, {55, 3782}, {56, 24837}, {376, 903}, {381, 4370}, {382, 24844}, {485, 24842}, {486, 24843}, {631, 27191}, {673, 5759}, {944, 24841}, {946, 4432}, {1331, 2969}, {1352, 4437}, {1478, 24845}, {1479, 24846}, {1587, 24819}, {1588, 24818}, {1657, 4409}, {1766, 2161}, {2886, 3923}, {3091, 4473}, {3419, 3729}, {3428, 20992}, {3434, 24280}, {3543, 17487}, {3575, 24814}, {3663, 24929}, {3821, 6690}, {5119, 24222}, {5173, 12722}, {5691, 24821}, {5709, 16560}, {5805, 16593}, {5870, 24832}, {5871, 24831}, {6284, 24840}, {6684, 25351}, {7354, 24816}, {7384, 27949}, {9834, 24823}, {9835, 24824}, {9839, 24839}, {9873, 24825}, {11500, 24820}, {12110, 24815}, {12113, 24830}, {12114, 24826}, {12115, 24847}, {12116, 24848}, {12717, 21375}, {17044, 24279}, {17738, 19542}

X(29243) = isogonal conjugate of X(29242)


X(29244) =  POINT POLARIS(2,1,-1,0)

Barycentrics    (b - c) (2 a^3 + b^3 - a b c - b^2 c - b c^2 + c^3) : :

X(29244) lies on these lines: {30, 511}, {4782, 7178}

X(29244) = isogonal conjugate of X(29245)


X(29245) =  ISOGONAL CONJUGATE OF X(29244)

Barycentrics    a^2/((b - c) (2 a^3 + b^3 - a b c - b^2 c - b c^2 + c^3)) : :

X(29245) lies on the circumcircle and these lines:

X(29245) = isogonal conjugate of X(29244)


X(29246) =  POINT POLARIS(-1,0,2,2)

Barycentrics    (b - c) (-a^3 + 2 a^2 b + 2 a^2 c + 2 a b c + b^2 c + b c^2) : :

X(29246) lies on these lines: {30, 511}, {663, 21146}, {2533, 4724}, {4040, 4874}, {4462, 4774}, {4490, 21302}, {4801, 4879}, {4822, 24719}

X(29246) = isogonal conjugate of X(29247)


X(29247) =  ISOGONAL CONJUGATE OF X(29246)

Barycentrics    a^2/((b - c) (-a^3 + 2 a^2 b + 2 a^2 c + 2 a b c + b^2 c + b c^2)) : :

X(29247) lies on the circumcircle and these lines:

X(29247) = isogonal conjugate of X(29246)


X(29248) =  POINT POLARIS(-1,2,0,2)

Barycentrics    (b - c) (-a^3 + 2 b^3 + 2 a b c + 3 b^2 c + 3 b c^2 + 2 c^3) : :

X(29248) lies on these lines: {30, 511}

X(29248) = isogonal conjugate of X(29249)


X(29249) =  ISOGONAL CONJUGATE OF X(29248)

Barycentrics    a^2/((b - c) (-a^3 + 2 b^3 + 2 a b c + 3 b^2 c + 3 b c^2 + 2 c^3)) : :

X(29249) lies on the circumcircle and these lines:

X(29249) = isogonal conjugate of X(29248)


X(29250) =  POINT POLARIS(-1,2,2,0)

Barycentrics    (b - c) (-a^3 + 2 a^2 b + 2 b^3 + 2 a^2 c + 3 b^2 c + 3 b c^2 + 2 c^3) : :

X(29250) lies on these lines: {30, 511}

X(29250) = isogonal conjugate of X(29251)


X(29251) =  ISOGONAL CONJUGATE OF X(29250)

Barycentrics    a^2/((b - c) (-a^3 + 2 a^2 b + 2 b^3 + 2 a^2 c + 3 b^2 c + 3 b c^2 + 2 c^3)) : :

X(29251) lies on the circumcircle and these lines:

X(29251) = isogonal conjugate of X(29250)


X(29252) =  POINT POLARIS(0,1,2,2)

Barycentrics    (b - c) (-2 a^2 b + b^3 - 2 a^2 c - 2 a b c + b^2 c + b c^2 + c^3) : :

X(29252) lies on these lines: {30, 511}, {4983, 16892}, {7265, 21146}, {22037, 23789}

X(29252) = isogonal conjugate of X(29253)


X(29253) =  ISOGONAL CONJUGATE OF X(29252)

Barycentrics    a^2/((b - c) (-2 a^2 b + b^3 - 2 a^2 c - 2 a b c + b^2 c + b c^2 + c^3)) : :

X(29253) lies on the circumcircle and these lines:

X(29253) = isogonal conjugate of X(29252)


X(29254) =  CIRCUMCIRCLE-ANTIPODE OF X(29253)

Barycentrics    a^2 (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 - b^4 c^2 - 2 a^2 b c^3 - 2 a b^2 c^3 + 2 a^2 c^4 + 4 a b c^4 + 2 b^2 c^4 - 2 c^6) (a^6 - a^4 b^2 + 2 a^2 b^4 - 2 b^6 - 2 a^4 b c - 2 a^2 b^3 c + 4 a b^4 c - a^4 c^2 + 2 a^3 b c^2 - 2 a b^3 c^2 + 2 b^4 c^2 + 2 a^2 b c^3 - a^2 c^4 - 2 a b c^4 - b^2 c^4 + c^6) : :

X(29254) lies on the circumcircle and these lines:

X(29254) = isogonal conjugate of X(29255)
X(29254) = circumcircle-antipode of X(29253)


X(29255) =  ISOGONAL CONJUGATE OF X(29254)

Barycentrics    1/((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 - b^4 c^2 - 2 a^2 b c^3 - 2 a b^2 c^3 + 2 a^2 c^4 + 4 a b c^4 + 2 b^2 c^4 - 2 c^6) (a^6 - a^4 b^2 + 2 a^2 b^4 - 2 b^6 - 2 a^4 b c - 2 a^2 b^3 c + 4 a b^4 c - a^4 c^2 + 2 a^3 b c^2 - 2 a b^3 c^2 + 2 b^4 c^2 + 2 a^2 b c^3 - a^2 c^4 - 2 a b c^4 - b^2 c^4 + c^6)) : :

X(29255) lies on these lines: {30, 511}

X(29255) = isogonal conjugate of X(29254)


X(29256) =  POINT POLARIS(0,2,-1,2)

Barycentrics    (b - c) (-a^2 b + 2 b^3 - a^2 c + 2 a b c + 2 b^2 c + 2 b c^2 + 2 c^3) : :

X(29256) lies on these lines: {30, 511}

X(29256) = isogonal conjugate of X(29257)


X(29257) =  ISOGONAL CONJUGATE OF X(29256)

Barycentrics    a^2/((b - c) (-a^2 b + 2 b^3 - a^2 c + 2 a b c + 2 b^2 c + 2 b c^2 + 2 c^3)) : :

X(29257) lies on the circumcircle and these lines:

X(29257) = isogonal conjugate of X(29256)


X(29258) =  CIRCUMCIRCLE-ANTIPODE OF X(29257)

Barycentrics    a^2 (2 a^6 - 2 a^4 b^2 - 2 a^2 b^4 + 2 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 + b^4 c^2 + 2 a^2 b c^3 + 2 a b^2 c^3 + a^2 c^4 - 4 a b c^4 + b^2 c^4 - 4 c^6) (2 a^6 + a^4 b^2 + a^2 b^4 - 4 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^3 b c^2 + 2 a b^3 c^2 + b^4 c^2 - 2 a^2 b c^3 - 2 a^2 c^4 + 2 a b c^4 + b^2 c^4 + 2 c^6) : :

X(29258) lies on the circumcircle and these lines:

X(29258) = isogonal conjugate of X(29259)
X(29258) = circumcircle-antipode of X(29257)


X(29259) =  ISOGONAL CONJUGATE OF X(29258)

Barycentrics    1/((2 a^6 - 2 a^4 b^2 - 2 a^2 b^4 + 2 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 + b^4 c^2 + 2 a^2 b c^3 + 2 a b^2 c^3 + a^2 c^4 - 4 a b c^4 + b^2 c^4 - 4 c^6) (2 a^6 + a^4 b^2 + a^2 b^4 - 4 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^3 b c^2 + 2 a b^3 c^2 + b^4 c^2 - 2 a^2 b c^3 - 2 a^2 c^4 + 2 a b c^4 + b^2 c^4 + 2 c^6)) : :

X(29259) lies on these lines: {30, 511}

X(29259) = isogonal conjugate of X(29258)


X(29260) =  POINT POLARIS(0,2,2,-1)

Barycentrics    (b - c) (2 a^2 b + 2 b^3 + 2 a^2 c - a b c + 2 b^2 c + 2 b c^2 + 2 c^3) : :

X(29260) lies on these lines: {30, 511}

X(29260) = isogonal conjugate of X(29261)


X(29261) =  ISOGONAL CONJUGATE OF X(29260)

Barycentrics    a^2/((b - c) (2 a^2 b + 2 b^3 + 2 a^2 c - a b c + 2 b^2 c + 2 b c^2 + 2 c^3)) : :

X(29261) lies on the circumcircle and these lines:

X(29261) = isogonal conjugate of X(29260)


X(29262) =  CIRCUMCIRCLE-ANTIPODE OF X(29261)

Barycentrics    a^2 (2 a^6 - 2 a^4 b^2 - 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 + 4 a^4 c^2 + 4 b^4 c^2 - a^2 b c^3 - a b^2 c^3 - 2 a^2 c^4 + 2 a b c^4 - 2 b^2 c^4 - 4 c^6) (2 a^6 + 4 a^4 b^2 - 2 a^2 b^4 - 4 b^6 - a^4 b c - a^2 b^3 c + 2 a b^4 c - 2 a^4 c^2 + a^3 b c^2 - a b^3 c^2 - 2 b^4 c^2 + a^2 b c^3 - 2 a^2 c^4 - a b c^4 + 4 b^2 c^4 + 2 c^6) : :

X(29262) lies on the circumcircle and these lines:

X(29262) = isogonal conjugate of X(29263)
X(29262) = circumcircle-antipode of X(29261)


X(29263) =  ISOGONAL CONJUGATE OF X(29262)

Barycentrics    1/((2 a^6 - 2 a^4 b^2 - 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 + 4 a^4 c^2 + 4 b^4 c^2 - a^2 b c^3 - a b^2 c^3 - 2 a^2 c^4 + 2 a b c^4 - 2 b^2 c^4 - 4 c^6) (2 a^6 + 4 a^4 b^2 - 2 a^2 b^4 - 4 b^6 - a^4 b c - a^2 b^3 c + 2 a b^4 c - 2 a^4 c^2 + a^3 b c^2 - a b^3 c^2 - 2 b^4 c^2 + a^2 b c^3 - 2 a^2 c^4 - a b c^4 + 4 b^2 c^4 + 2 c^6)) : :

X(29263) lies on these lines: {20, 19836}, {30, 511}, {382, 24309}, {4353, 6284}

X(29263) = isogonal conjugate of X(29262)


X(29264) =  POINT POLARIS(2,-1,0,2)

Barycentrics    (b - c) (-2 a^3 + b^3 - 2 a b c + 3 b^2 c + 3 b c^2 + c^3) : :

X(29264) lies on these lines: {30, 511}, {4922, 7265}

X(29264) = isogonal conjugate of X(29265)


X(29265) =  ISOGONAL CONJUGATE OF X(29264)

Barycentrics    a^2/((b - c) (-2 a^3 + b^3 - 2 a b c + 3 b^2 c + 3 b c^2 + c^3)) : :

X(29265) lies on the circumcircle and these lines:

X(29265) = isogonal conjugate of X(29264)


X(29266) =  POINT POLARIS(2,-1,2,0)

Barycentrics    (b - c) (-2 a^3 - 2 a^2 b + b^3 - 2 a^2 c + 3 b^2 c + 3 b c^2 + c^3) : :

X(29266) lies on these lines: {30, 511}

X(29266) = isogonal conjugate of X(29267)


X(29267) =  ISOGONAL CONJUGATE OF X(29266)

Barycentrics    a^2/((b - c) (-2 a^3 - 2 a^2 b + b^3 - 2 a^2 c + 3 b^2 c + 3 b c^2 + c^3)) : :

X(29267) lies on the circumcircle and these lines:

X(29267) = isogonal conjugate of X(29266)


X(29268) =  POINT POLARIS(2,0,-1,2)

Barycentrics    (b - c) (-2 a^3 + a^2 b + a^2 c - 2 a b c + 2 b^2 c + 2 b c^2) : :

X(29268) lies on these lines: {30, 511}, {667, 4474}, {1019, 4774}, {1577, 4922}, {1960, 4391}, {4560, 4770}, {14419, 21052}

X(29268) = isogonal conjugate of X(29269)


X(29269) =  ISOGONAL CONJUGATE OF X(29268)

Barycentrics    a^2/((b - c) (-2 a^3 + a^2 b + a^2 c - 2 a b c + 2 b^2 c + 2 b c^2)) : :

X(29269) lies on the circumcircle and these lines:

X(29269) = isogonal conjugate of X(29268)


X(29270) =  POINT POLARIS(2,0,2,-1)

Barycentrics    (b - c) (-2 a^3 - 2 a^2 b - 2 a^2 c + a b c + 2 b^2 c + 2 b c^2) : :

X(29270) lies on these lines: {30, 511}, {649, 4823}, {667, 4810}, {1019, 4382}, {1577, 4380}, {3835, 24948}, {4010, 4401}, {4063, 4791}, {4106, 14838}, {4170, 4794}

X(29270) = isogonal conjugate of X(29271)


X(29271) =  ISOGONAL CONJUGATE OF X(29270)

Barycentrics    a^2/((b - c) (-2 a^3 - 2 a^2 b - 2 a^2 c + a b c + 2 b^2 c + 2 b c^2)) : :

X(29271) lies on the circumcircle and these lines:

X(29271) = isogonal conjugate of X(29270)


X(29272) =  POINT POLARIS(2,2,-1,0)

Barycentrics    (b - c) (2 a^3 - a^2 b + 2 b^3 - a^2 c + 2 c^3) : :

X(29272) lies on these lines: {30, 511}, {1960, 3801}, {4378, 8636}

X(29272) = isogonal conjugate of X(29273)


X(29273) =  ISOGONAL CONJUGATE OF X(29272)

Barycentrics    a^2/((b - c) (2 a^3 - a^2 b + 2 b^3 - a^2 c + 2 c^3)) : :

X(29273) lies on the circumcircle and these lines:

X(29273) = isogonal conjugate of X(29272)


X(29274) =  POINT POLARIS(2,0,-1,-1)

Barycentrics    (b - c) (-2 a^3 + a^2 b + a^2 c + a b c + 2 b^2 c + 2 b c^2) : :

X(29274) lies on these lines: {30, 511}, {650, 21053}, {659, 28373}, {1960, 4823}, {2533, 4782}, {4382, 4879}, {4498, 4774}, {4801, 4922}

X(29274) = isogonal conjugate of X(29275)


X(29275) =  ISOGONAL CONJUGATE OF X(29274)

Barycentrics    a^2/((b - c) (-2 a^3 + a^2 b + a^2 c + a b c + 2 b^2 c + 2 b c^2)) : :

X(29275) lies on the circumcircle and these lines:

X(29275) = isogonal conjugate of X(29272)


X(29276) =  POINT POLARIS(2,-1,0,-1)

Barycentrics    (b - c) (-2 a^3 + b^3 + a b c + 3 b^2 c + 3 b c^2 + c^3) : :

X(29276) lies on these lines: {30, 511}

X(29276) = isogonal conjugate of X(29277)


X(29277) =  ISOGONAL CONJUGATE OF X(29276)

Barycentrics    a^2/((b - c) (-2 a^3 + b^3 + a b c + 3 b^2 c + 3 b c^2 + c^3)) : :

X(29277) lies on the circumcircle and these lines:

X(29277) = isogonal conjugate of X(29276)


X(29278) =  POINT POLARIS(2,-1,-1,0)

Barycentrics    (b - c) (-a + b + c) (2 a^2 + a b + b^2 + a c + 2 b c + c^2) : :

X(29278) lies on these lines: {30, 511}, {663, 3700}, {676, 1577}, {2490, 6050}, {3004, 21301}, {4041, 4528}, {4162, 4820}, {4163, 4765}, {4467, 21302}, {4474, 21120}, {4477, 21005}, {17069, 17072}

X(29278) = isogonal conjugate of X(29279)


X(29279) =  ISOGONAL CONJUGATE OF X(29278)

Barycentrics    a^2/((b - c) (-a + b + c) (2 a^2 + a b + b^2 + a c + 2 b c + c^2)) : :

X(29279) lies on the circumcircle and these lines:

X(29279) = isogonal conjugate of X(29278)


X(29280) =  POINT POLARIS(0,2,2,-1)

Barycentrics    (b - c) (-a^2 b + 2 b^3 - a^2 c - a b c + 2 b^2 c + 2 b c^2 + 2 c^3) : :

X(29280) lies on these lines: {30, 511}, {3777, 21350}, {3801, 25259}

X(29280) = isogonal conjugate of X(29281)


X(29281) =  ISOGONAL CONJUGATE OF X(29280)

Barycentrics    a^2/((b - c) (-a^2 b + 2 b^3 - a^2 c - a b c + 2 b^2 c + 2 b c^2 + 2 c^3)) : :

X(29281) lies on the circumcircle and these lines:

X(29281) = isogonal conjugate of X(29280)


X(29282) =  CIRCUMCIRCLE-ANTIPODE OF X(29281)

Barycentrics    a^2 (2 a^6 - 2 a^4 b^2 - 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 + a^4 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 - 4 c^6) (2 a^6 + a^4 b^2 + a^2 b^4 - 4 b^6 - a^4 b c - a^2 b^3 c + 2 a b^4 c - 2 a^4 c^2 + a^3 b c^2 - a b^3 c^2 + b^4 c^2 + a^2 b c^3 - 2 a^2 c^4 - a b c^4 + b^2 c^4 + 2 c^6) : :

X(29282) lies on the circumcircle and these lines:

X(29282) = isogonal conjugate of X(29283)
X(29282) = circumcircle-antipode of X(29281)


X(29283) =  ISOGONAL CONJUGATE OF X(29282)

Barycentrics    1/((2 a^6 - 2 a^4 b^2 - 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 + a^4 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 - 4 c^6) (2 a^6 + a^4 b^2 + a^2 b^4 - 4 b^6 - a^4 b c - a^2 b^3 c + 2 a b^4 c - 2 a^4 c^2 + a^3 b c^2 - a b^3 c^2 + b^4 c^2 + a^2 b c^3 - 2 a^2 c^4 - a b c^4 + b^2 c^4 + 2 c^6)) : :

X(29283) lies on these lines: {30, 511}

X(29283) = isogonal conjugate of X(29282)


X(29284) =  POINT POLARIS(0,-1,2,-1)

Barycentrics    (b - c) (-2 a^2 b + b^3 - 2 a^2 c + a b c + b^2 c + b c^2 + c^3) : :

X(29284) lies on these lines: {30, 511}, {4147, 18004}, {6332, 9508}

X(29284) = isogonal conjugate of X(29285)


X(29285) =  ISOGONAL CONJUGATE OF X(29284)

Barycentrics    a^2/((b - c) (-2 a^2 b + b^3 - 2 a^2 c + a b c + b^2 c + b c^2 + c^3)) : :

X(29285) lies on the circumcircle and these lines:

X(29285) = isogonal conjugate of X(29284)


X(29286) =  CIRCUMCIRCLE-ANTIPODE OF X(29285)

Barycentrics    a^2 (a^6 - a^4 b^2 - a^2 b^4 + b^6 + a^4 b c - a^3 b^2 c - a^2 b^3 c + a b^4 c - a^4 c^2 - b^4 c^2 + a^2 b c^3 + a b^2 c^3 + 2 a^2 c^4 - 2 a b c^4 + 2 b^2 c^4 - 2 c^6) (a^6 - a^4 b^2 + 2 a^2 b^4 - 2 b^6 + a^4 b c + a^2 b^3 c - 2 a b^4 c - a^4 c^2 - a^3 b c^2 + a b^3 c^2 + 2 b^4 c^2 - a^2 b c^3 - a^2 c^4 + a b c^4 - b^2 c^4 + c^6) : :

X(29286) lies on the circumcircle and these lines:

X(29286) = isogonal conjugate of X(29287)
X(29286) = circumcircle-antipode of X(29285)


X(29287) =  ISOGONAL CONJUGATE OF X(29286)

Barycentrics    1/((a^6 - a^4 b^2 - a^2 b^4 + b^6 + a^4 b c - a^3 b^2 c - a^2 b^3 c + a b^4 c - a^4 c^2 - b^4 c^2 + a^2 b c^3 + a b^2 c^3 + 2 a^2 c^4 - 2 a b c^4 + 2 b^2 c^4 - 2 c^6) (a^6 - a^4 b^2 + 2 a^2 b^4 - 2 b^6 + a^4 b c + a^2 b^3 c - 2 a b^4 c - a^4 c^2 - a^3 b c^2 + a b^3 c^2 + 2 b^4 c^2 - a^2 b c^3 - a^2 c^4 + a b c^4 - b^2 c^4 + c^6)) : :

X(29287) lies on these lines: {30, 511}, {355, 894}, {944, 6646}, {1385, 4357}, {3655, 17254}, {5711, 8757}, {5750, 9956}, {5810, 26487}

X(29287) = isogonal conjugate of X(29286)


X(29288) =  POINT POLARIS(0,-1,-1,2)

Barycentrics    (b - c) (a^2 b + b^3 + a^2 c - 2 a b c + b^2 c + b c^2 + c^3) : :

X(29288) lies on these lines: {30, 511}, {1577, 23770}, {2530, 4808}, {2533, 13259}, {2977, 14838}, {3004, 4705}, {3776, 17072}, {3801, 10015}, {4041, 16892}, {4834, 4897}, {4992, 18004}, {6050, 11068}, {20504, 21124}, {21108, 21119}

X(29288) = isogonal conjugate of X(29289)


X(29289) =  ISOGONAL CONJUGATE OF X(29288)

Barycentrics    a^2/((b - c) (a^2 b + b^3 + a^2 c - 2 a b c + b^2 c + b c^2 + c^3)) : :

X(29289) lies on the circumcircle and these lines:

X(29289) = isogonal conjugate of X(29288)


X(29290) =  CIRCUMCIRCLE-ANTIPODE OF X(29289)

Barycentrics    a^2 (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 + 2 a^4 c^2 + 2 b^4 c^2 - 2 a^2 b c^3 - 2 a b^2 c^3 - a^2 c^4 + 4 a b c^4 - b^2 c^4 - 2 c^6) (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 - a^4 c^2 + 2 a^3 b c^2 - 2 a b^3 c^2 - b^4 c^2 + 2 a^2 b c^3 - a^2 c^4 - 2 a b c^4 + 2 b^2 c^4 + c^6) : :

X(29290) lies on the circumcircle and these lines:

X(29290) = isogonal conjugate of X(29291)
X(29290) = circumcircle-antipode of X(29289)


X(29291) =  ISOGONAL CONJUGATE OF X(29290)

Barycentrics    1/((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 + 2 a^4 c^2 + 2 b^4 c^2 - 2 a^2 b c^3 - 2 a b^2 c^3 - a^2 c^4 + 4 a b c^4 - b^2 c^4 - 2 c^6) (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 - a^4 c^2 + 2 a^3 b c^2 - 2 a b^3 c^2 - b^4 c^2 + 2 a^2 b c^3 - a^2 c^4 - 2 a b c^4 + 2 b^2 c^4 + c^6)) : :

X(29291) lies on these lines: {3, 25914}, {4, 4429}, {5, 24309}, {30, 511}, {990, 12699}, {1486, 25365}, {1766, 17340}, {1770, 12723}, {3579, 12618}, {3663, 15171}, {4292, 12722}, {4319, 24701}, {4353, 15172}, {6211, 24828}, {6284, 24248}, {11677, 24320}, {12610, 22793}

X(29291) = isogonal conjugate of X(29290)


X(29292) =  POINT POLARIS(-1,2,0,-1)

Barycentrics    (b - c) (-a^3 + 2 b^3 - a b c + 3 b^2 c + 3 b c^2 + 2 c^3) : :

X(29292) lies on these lines: {30, 511}, {4467, 4808}

X(29292) = isogonal conjugate of X(29293)


X(29293) =  ISOGONAL CONJUGATE OF X(29292)

Barycentrics    a^2/((b - c) (-a^3 + 2 b^3 - a b c + 3 b^2 c + 3 b c^2 + 2 c^3)) : :

X(29293) lies on the circumcircle and these lines:

X(29293) = isogonal conjugate of X(29292)


X(29294) =  POINT POLARIS(-1,2,-1,0)

Barycentrics    (b - c) (-a^3 - a^2 b + 2 b^3 - a^2 c + 3 b^2 c + 3 b c^2 + 2 c^3) : :

X(29294) lies on these lines: {30, 511}, {3700, 20517}, {4522, 21192}

X(29294) = isogonal conjugate of X(29295)


X(29295) =  ISOGONAL CONJUGATE OF X(29296)

Barycentrics    a^2/((b - c) (-a^3 - a^2 b + 2 b^3 - a^2 c + 3 b^2 c + 3 b c^2 + 2 c^3)) : :

X(29295) lies on the circumcircle and these lines:

X(29295) = isogonal conjugate of X(29294)


X(29296) =  CIRCUMCIRCLE-ANTIPODE OF X(29295)

Barycentrics    a^2 (2 a^6 + a^5 b - 2 a^4 b^2 - 2 a^3 b^3 - 2 a^2 b^4 + a b^5 + 2 b^6 + a^4 c^2 - a^3 b 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 + b^3 c^3 + a^2 c^4 + b^2 c^4 - a c^5 - b c^5 - 4 c^6) (2 a^6 + a^4 b^2 + a^3 b^3 + a^2 b^4 - a b^5 - 4 b^6 + a^5 c - a^3 b^2 c + a^2 b^3 c - b^5 c - 2 a^4 c^2 + a b^3 c^2 + b^4 c^2 - 2 a^3 c^3 - a b^2 c^3 + b^3 c^3 - 2 a^2 c^4 + b^2 c^4 + a c^5 + 2 c^6) : :

X(29296) lies on the circumcircle and these lines:

X(29296) = isogonal conjugate of X(29297)
X(29296) = circumcircle-antipode of X(29295)


X(29297) =  ISOGONAL CONJUGATE OF X(29296)

Barycentrics    1/((2 a^6 + a^5 b - 2 a^4 b^2 - 2 a^3 b^3 - 2 a^2 b^4 + a b^5 + 2 b^6 + a^4 c^2 - a^3 b 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 + b^3 c^3 + a^2 c^4 + b^2 c^4 - a c^5 - b c^5 - 4 c^6) (2 a^6 + a^4 b^2 + a^3 b^3 + a^2 b^4 - a b^5 - 4 b^6 + a^5 c - a^3 b^2 c + a^2 b^3 c - b^5 c - 2 a^4 c^2 + a b^3 c^2 + b^4 c^2 - 2 a^3 c^3 - a b^2 c^3 + b^3 c^3 - 2 a^2 c^4 + b^2 c^4 + a c^5 + 2 c^6)) : :

X(29297) lies on these lines: {30, 511}

X(29297) = isogonal conjugate of X(29296)


X(29298) =  POINT POLARIS(-1,0,2,-1)

Barycentrics    (b - c) (-a^3 + 2 a^2 b + 2 a^2 c - a b c + b^2 c + b c^2) : :

X(29298) lies on these lines: {1, 2533}, {8, 4705}, {30, 511}, {145, 17166}, {663, 10459}, {667, 16158}, {1019, 4922}, {1577, 4774}, {2530, 21302}, {3632, 4824}, {4367, 4761}, {4560, 4730}, {4807, 9508}, {4978, 21343}, {16737, 25303}

X(29298) = isogonal conjugate of X(29299)


X(29299) =  ISOGONAL CONJUGATE OF X(29300)

Barycentrics    a^2/((b - c) (-a^3 + 2 a^2 b + 2 a^2 c - a b c + b^2 c + b c^2)) : :

X(29299) lies on the circumcircle and these lines:

X(29299) = isogonal conjugate of X(29298)


X(29300) =  CIRCUMCIRCLE-ANTIPODE OF X(29299)

Barycentrics    a^2 (2 a^4 b^2 + a^3 b^3 - 2 a^2 b^4 - a b^5 + a^5 c - a^4 b c - a^3 b^2 c + 2 a b^4 c - b^5 c + a^3 b c^2 - 2 b^4 c^2 - 2 a^3 c^3 + a^2 b c^3 - a b^2 c^3 + b^3 c^3 - a b c^4 + 2 b^2 c^4 + a c^5) (a^5 b - 2 a^3 b^3 + a b^5 - 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 - a b^3 c^2 + 2 b^4 c^2 + a^3 c^3 + b^3 c^3 - 2 a^2 c^4 + 2 a b c^4 - 2 b^2 c^4 - a c^5 - b c^5) : :

X(29300) lies on the circumcircle and these lines:

X(29300) = isogonal conjugate of X(29301)
X(29300) = circumcircle-antipode of X(29299)


X(29301) =  ISOGONAL CONJUGATE OF X(29300)

Barycentrics    1/((2 a^4 b^2 + a^3 b^3 - 2 a^2 b^4 - a b^5 + a^5 c - a^4 b c - a^3 b^2 c + 2 a b^4 c - b^5 c + a^3 b c^2 - 2 b^4 c^2 - 2 a^3 c^3 + a^2 b c^3 - a b^2 c^3 + b^3 c^3 - a b c^4 + 2 b^2 c^4 + a c^5) (a^5 b - 2 a^3 b^3 + a b^5 - 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 - a b^3 c^2 + 2 b^4 c^2 + a^3 c^3 + b^3 c^3 - 2 a^2 c^4 + 2 a b c^4 - 2 b^2 c^4 - a c^5 - b c^5)) : :

X(29301) lies on these lines: {30, 511}, {79, 256}, {182, 24728}, {191, 6211}, {314, 10308}, {576, 24257}, {1045, 1048}, {1756, 4459}, {3098, 3923}, {3579, 17351}, {3649, 24231}, {3650, 4480}, {3818, 4655}, {3821, 19130}, {4672, 5092}, {4696, 11684}, {4887, 11544}, {9955, 17235}, {12699, 17276}, {17182, 17628}

X(29301) = isogonal conjugate of X(29300)


X(29302) =  POINT POLARIS(-1,0,-1,2)

Barycentrics    (b - c) (-a^3 - a^2 b - a^2 c + 2 a b c + b^2 c + b c^2) : :

X(29302) lies on these lines: {30, 511}, {649, 4978}, {693, 4063}, {1019, 4380}, {1125, 6050}, {1577, 4382}, {3776, 21192}, {4106, 4129}, {4170, 4724}, {4391, 21385}, {4401, 4830}, {4705, 24719}, {4834, 21146}, {9508, 23815}, {14349, 17494}, {14838, 28374}, {20517, 23770}

X(29302) = isogonal conjugate of X(29303)


X(29303) =  ISOGONAL CONJUGATE OF X(29302)

Barycentrics    a^2/((b - c) (-a^3 - a^2 b - a^2 c + 2 a b c + b^2 c + b c^2)) : :

X(29303) lies on the circumcircle and these lines:

X(29303) = isogonal conjugate of X(29302)


X(29304) =  POINT POLARIS(-1,1,2,0)

Barycentrics    (b - c) (a^3 - 2 a^2 b + b^3 - 2 a^2 c + c^3) : :

X(29304) lies on these lines: {30, 511}, {663, 4707}, {1125, 21188}, {3801, 4775}, {3904, 4905}, {4142, 4794}, {4401, 5592}

X(29304) = isogonal conjugate of X(29305)


X(29305) =  ISOGONAL CONJUGATE OF X(29306)

Barycentrics    a^2/((b - c) (a^3 - 2 a^2 b + b^3 - 2 a^2 c + c^3)) : :

X(29305) lies on the circumcircle and these lines:

X(29305) = isogonal conjugate of X(29304)


X(29306) =  CIRCUMCIRCLE-ANTIPODE OF X(29305)

Barycentrics    a^2 (a^6 - a^5 b - a^4 b^2 + 2 a^3 b^3 - a^2 b^4 - a b^5 + b^6 - a^4 c^2 + a^3 b 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 - b^3 c^3 + 2 a^2 c^4 + 2 b^2 c^4 + a c^5 + b c^5 - 2 c^6) (a^6 - a^4 b^2 - a^3 b^3 + 2 a^2 b^4 + a b^5 - 2 b^6 - a^5 c + a^3 b^2 c - a^2 b^3 c + b^5 c - a^4 c^2 - a b^3 c^2 + 2 b^4 c^2 + 2 a^3 c^3 + a b^2 c^3 - b^3 c^3 - a^2 c^4 - b^2 c^4 - a c^5 + c^6) : :

X(29306) lies on the circumcircle and these lines:

X(29306) = isogonal conjugate of X(29307)
X(29306) = circumcircle-antipode of X(29305)


X(29307) =  ISOGONAL CONJUGATE OF X(29306)

Barycentrics    1/((a^6 - a^5 b - a^4 b^2 + 2 a^3 b^3 - a^2 b^4 - a b^5 + b^6 - a^4 c^2 + a^3 b 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 - b^3 c^3 + 2 a^2 c^4 + 2 b^2 c^4 + a c^5 + b c^5 - 2 c^6) (a^6 - a^4 b^2 - a^3 b^3 + 2 a^2 b^4 + a b^5 - 2 b^6 - a^5 c + a^3 b^2 c - a^2 b^3 c + b^5 c - a^4 c^2 - a b^3 c^2 + 2 b^4 c^2 + 2 a^3 c^3 + a b^2 c^3 - b^3 c^3 - a^2 c^4 - b^2 c^4 - a c^5 + c^6)) : :

X(29307) lies on these lines: {3, 7232}, {10, 24332}, {30, 511}, {573, 1759}, {946, 5398}, {3664, 3665}, {4416, 5016}, {4655, 24309}, {5264, 13407}, {5757, 24220}, {11442, 21072}

X(29307) = isogonal conjugate of X(29306)


X(29308) =  CIRCUMCIRCLE-ANTIPODE OF X(8708)

Barycentrics    a (2 a^4 b - 2 a^3 b^2 - 2 a^2 b^3 + 2 a b^4 + a^4 c + b^4 c + 2 a^2 b c^2 + 2 a b^2 c^2 - a^2 c^3 - 4 a b c^3 - b^2 c^3) (a^4 b - a^2 b^3 + 2 a^4 c + 2 a^2 b^2 c - 4 a b^3 c - 2 a^3 c^2 + 2 a b^2 c^2 - b^3 c^2 - 2 a^2 c^3 + 2 a c^4 + b c^4) : :

X(29308) lies on the circumcircle and these lines:

X(29308) = isogonal conjugate of X(29309)
X(29308) = circumcircle-antipode of X(8708)


X(29309) =  ISOGONAL CONJUGATE OF X(29308)

Barycentrics    a/((2 a^4 b - 2 a^3 b^2 - 2 a^2 b^3 + 2 a b^4 + a^4 c + b^4 c + 2 a^2 b c^2 + 2 a b^2 c^2 - a^2 c^3 - 4 a b c^3 - b^2 c^3) (a^4 b - a^2 b^3 + 2 a^4 c + 2 a^2 b^2 c - 4 a b^3 c - 2 a^3 c^2 + 2 a b^2 c^2 - b^3 c^2 - 2 a^2 c^3 + 2 a c^4 + b c^4)) : :

X(29309) lies on these lines: {3, 23374}, {30, 511}, {40, 13731}, {45, 573}, {946, 15489}, {970, 12699}, {991, 1482}, {1742, 7982}, {3664, 9957}, {3885, 17364}, {4416, 10914}, {5903, 21746}, {6210, 7991}, {6361, 10441}, {12109, 15171}, {14810, 24309}, {22791, 24220}

X(29309) = isogonal conjugate of X(29308)


X(29310) =  CIRCUMCIRCLE-ANTIPODE OF X(6013)

Barycentrics    a (a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + 2 a^4 c + 2 b^4 c + a^2 b c^2 + a b^2 c^2 - 2 a^2 c^3 - 2 a b c^3 - 2 b^2 c^3) (2 a^4 b - 2 a^2 b^3 + a^4 c + a^2 b^2 c - 2 a b^3 c - a^3 c^2 + a b^2 c^2 - 2 b^3 c^2 - a^2 c^3 + a c^4 + 2 b c^4) : :

X(29310) lies on the circumcircle and these lines:

X(29310) = isogonal conjugate of X(29311)
X(29310) = circumcircle-antipode of X(6013)


X(29311) =  ISOGONAL CONJUGATE OF X(29310)

Barycentrics    a/((a^4 b - a^3 b^2 - a^2 b^3 + a b^4 + 2 a^4 c + 2 b^4 c + a^2 b c^2 + a b^2 c^2 - 2 a^2 c^3 - 2 a b c^3 - 2 b^2 c^3) (2 a^4 b - 2 a^2 b^3 + a^4 c + a^2 b^2 c - 2 a b^3 c - a^3 c^2 + a b^2 c^2 - 2 b^3 c^2 - a^2 c^3 + a c^4 + 2 b c^4)) : :

X(29311) lies on these lines: {1, 573}, {2, 10439}, {3, 4497}, {8, 10435}, {10, 3781}, {30, 511}, {40, 991}, {42, 1764}, {65, 3664}, {165, 1002}, {181, 21334}, {200, 12555}, {386, 10476}, {572, 4649}, {946, 5752}, {970, 1125}, {995, 9549}, {1350, 24309}, {1458, 20367}, {1463, 4887}, {1469, 3663}, {1730, 25941}, {1738, 3792}, {1742, 7991}, {1766, 3751}, {2051, 3741}, {3057, 21746}, {3720, 21363}, {3755, 4259}, {3789, 3817}, {3869, 4416}, {4061, 17617}, {4847, 26893}, {5267, 24253}, {5745, 22276}, {5795, 22299}, {6210, 7174}, {6738, 29957}, {6744, 12109}, {7235, 24209}, {9535, 10453}, {10434, 17018}, {10445, 10477}, {10459, 11521}, {10465, 20018}, {10882, 19767}, {12527, 16980}, {15488, 19925}, {17182, 24996}, {18250, 23841}

X(29311) = isogonal conjugate of X(29310)


X(29312) =  POINT POLARIS(0,1,0,2)

Barycentrics    (b - c) (b^3 + 2 a b c + b^2 c + b c^2 + c^3) : :

X(29312) lies on these lines: {30, 511}, {693, 18015}, {764, 16892}, {2530, 21124}, {3762, 4122}, {3801, 4978}, {3954, 4024}, {4391, 18003}, {4707, 21146}, {17141, 17161}, {19947, 21212}, {20963, 22383}

X(29312) = isogonal conjugate of X(29313)


X(29313) =  ISOGONAL CONJUGATE OF X(29312)

Barycentrics    a^2/((b - c) (b^3 + 2 a b c + b^2 c + b c^2 + c^3)) : :

X(29313) lies on the circumcircle and these lines:

X(29313) = isogonal conjugate of X(29312)


X(29314) =  CIRCUMCIRCLE-ANTIPODE OF X(29313)

Barycentrics    a^2 (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 + b^4 c^2 + 2 a^2 b c^3 + 2 a b^2 c^3 - 4 a b c^4 - 2 c^6) (a^6 + a^4 b^2 - 2 b^6 + 2 a^4 b c + 2 a^2 b^3 c - 4 a b^4 c - a^4 c^2 - 2 a^3 b c^2 + 2 a b^3 c^2 - 2 a^2 b c^3 - a^2 c^4 + 2 a b c^4 + b^2 c^4 + c^6) : :

X(29314) lies on the circumcircle and these lines:

X(29314) = isogonal conjugate of X(29315)
X(29314) = circumcircle-antipode of X(29313)


X(29315) =  ISOGONAL CONJUGATE OF X(29314)

Barycentrics    1/((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 + b^4 c^2 + 2 a^2 b c^3 + 2 a b^2 c^3 - 4 a b c^4 - 2 c^6) (a^6 + a^4 b^2 - 2 b^6 + 2 a^4 b c + 2 a^2 b^3 c - 4 a b^4 c - a^4 c^2 - 2 a^3 b c^2 + 2 a b^3 c^2 - 2 a^2 b c^3 - a^2 c^4 + 2 a b c^4 + b^2 c^4 + c^6)) : :

X(29315) lies on these lines: {30, 511}, {3745, 24210}

X(29315) = isogonal conjugate of X(29314)


X(29316) =  CIRCUMCIRCLE-ANTIPODE OF X(7953)

Barycentrics    a^2 (a^6 - a^4 b^2 - a^2 b^4 + b^6 + 3 a^4 c^2 + 3 b^4 c^2 - 2 a^2 c^4 - 2 b^2 c^4 - 2 c^6) (a^6 + 3 a^4 b^2 - 2 a^2 b^4 - 2 b^6 - a^4 c^2 - 2 b^4 c^2 - a^2 c^4 + 3 b^2 c^4 + c^6) : :

X(29316) lies on the circumcircle and these lines:

X(29316) = isogonal conjugate of X(29315)
X(29316) = circumcircle-antipode of X(29313)


X(29317) =  ISOGONAL CONJUGATE OF X(29316)

Barycentrics    1/((a^6 - a^4 b^2 - a^2 b^4 + b^6 + 3 a^4 c^2 + 3 b^4 c^2 - 2 a^2 c^4 - 2 b^2 c^4 - 2 c^6) (a^6 + 3 a^4 b^2 - 2 a^2 b^4 - 2 b^6 - a^4 c^2 - 2 b^4 c^2 - a^2 c^4 + 3 b^2 c^4 + c^6)) : :

X(29317) lies on these lines: {3, 7889}, {4, 3096}, {5, 14810}, {6, 1657}, {20, 182}, {23, 5972}, {26, 10182}, {30, 511}, {110, 20063}, {115, 2076}, {125, 5189}, {141, 3627}, {143, 17712}, {159, 22802}, {184, 20062}, {376, 10168}, {382, 1350}, {428, 3819}, {548, 3589}, {549, 25565}, {550, 5092}, {575, 15704}, {576, 3529}, {597, 15686}, {599, 15684}, {620, 5103}, {858, 6723}, {1351, 17800}, {1352, 3146}, {1428, 4316}, {1513, 6721}, {1691, 6781}, {1843, 18560}, {2330, 4324}, {2549, 5039}, {3056, 10483}, {3094, 7747}, {3534, 5085}, {3543, 10519}, {3581, 20417}, {3618, 17538}, {3763, 3843}, {3830, 10516}, {5017, 7748}, {5050, 15681}, {5059, 6776}, {5093, 15685}, {5259, 9840}, {5477, 15514}, {5562, 16658}, {5651, 7519}, {5899, 14643}, {5907, 16654}, {5943, 7667}, {5999, 6036}, {6240, 12294}, {6288, 15321}, {6688, 10691}, {6756, 13348}, {6793, 10313}, {7391, 21243}, {7500, 9306}, {7553, 15644}, {7574, 7687}, {7690, 21736}, {7728, 12584}, {7734, 10219}, {7753, 22728}, {7765, 12212}, {7802, 18906}, {7830, 24256}, {8597, 19662}, {9301, 10991}, {9655, 10387}, {9751, 14492}, {9971, 18564}, {10295, 15473}, {10625, 13419}, {11001, 20423}, {11179, 15520}, {11225, 21969}, {11592, 13163}, {11745, 17704}, {12024, 13142}, {12041, 20301}, {12083, 18388}, {12085, 23049}, {12103, 18583}, {12121, 19140}, {12900, 25338}, {13403, 19161}, {13598, 16657}, {13619, 19128}, {14449, 18128}, {14677, 25328}, {14790, 23325}, {14893, 20582}, {15140, 25556}, {15687, 25561}, {18382, 20299}, {18438, 18565}, {20300, 25563}, {22051, 23060}, {23061, 24981}

X(29317) = isogonal conjugate of X(29316)


X(29318) =  POINT POLARIS(0,2,0,1)

Barycentrics    (b - c) (2 b^3 + a b c + 2 b^2 c + 2 b c^2 + 2 c^3) : :

X(29318) lies on these lines: {30, 511}, {3801, 4823}, {4122, 4791}, {4951, 14431}, {8045, 20517}, {14430, 21130}

X(29318) = isogonal conjugate of X(29319)


X(29319) =  ISOGONAL CONJUGATE OF X(29318)

Barycentrics    a^2/((b - c) (2 b^3 + a b c + 2 b^2 c + 2 b c^2 + 2 c^3)) : :

X(29319) lies on the circumcircle and these lines:

X(29319) = isogonal conjugate of X(29318)


X(29320) =  CIRCUMCIRCLE-ANTIPODE OF X(29319)

Barycentrics    a^2 (2 a^6 - 2 a^4 b^2 - 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 + 2 b^4 c^2 + a^2 b c^3 + a b^2 c^3 - 2 a b c^4 - 4 c^6) (2 a^6 + 2 a^4 b^2 - 4 b^6 + a^4 b c + a^2 b^3 c - 2 a b^4 c - 2 a^4 c^2 - a^3 b c^2 + a b^3 c^2 - a^2 b c^3 - 2 a^2 c^4 + a b c^4 + 2 b^2 c^4 + 2 c^6) : :

X(29320) lies on the circumcircle and these lines:

X(29320) = isogonal conjugate of X(29321)
X(29320) = circumcircle-antipode of X(29319)


X(29321) =  ISOGONAL CONJUGATE OF X(29320)

Barycentrics    1/((2 a^6 - 2 a^4 b^2 - 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 + 2 b^4 c^2 + a^2 b c^3 + a b^2 c^3 - 2 a b c^4 - 4 c^6) (2 a^6 + 2 a^4 b^2 - 4 b^6 + a^4 b c + a^2 b^3 c - 2 a b^4 c - 2 a^4 c^2 - a^3 b c^2 + a b^3 c^2 - a^2 b c^3 - 2 a^2 c^4 + a b c^4 + 2 b^2 c^4 + 2 c^6)) : :

X(29321) lies on these lines: {30, 511}

X(29321) = isogonal conjugate of X(29320)


X(29322) =  CIRCUMCIRCLE-ANTIPODE OF X(7954)

Barycentrics    a^2 (2 a^6 - 2 a^4 b^2 - 2 a^2 b^4 + 2 b^6 + 3 a^4 c^2 + 3 b^4 c^2 - a^2 c^4 - b^2 c^4 - 4 c^6) (2 a^6 + 3 a^4 b^2 - a^2 b^4 - 4 b^6 - 2 a^4 c^2 - b^4 c^2 - 2 a^2 c^4 + 3 b^2 c^4 + 2 c^6) : :

X(29322) lies on the circumcircle and these lines:

X(29322) = isogonal conjugate of X(29323)
X(29322) = circumcircle-antipode of X(7954)


X(29323) =  ISOGONAL CONJUGATE OF X(29322)

Barycentrics    1/((2 a^6 - 2 a^4 b^2 - 2 a^2 b^4 + 2 b^6 + 3 a^4 c^2 + 3 b^4 c^2 - a^2 c^4 - b^2 c^4 - 4 c^6) (2 a^6 + 3 a^4 b^2 - a^2 b^4 - 4 b^6 - 2 a^4 c^2 - b^4 c^2 - 2 a^2 c^4 + 3 b^2 c^4 + 2 c^6)) : :

X(29323) lies on these lines: {4, 5092}, {6, 5073}, {20, 3818}, {23, 15059}, {30, 511}, {141, 15704}, {182, 382}, {376, 25561}, {381, 17508}, {428, 6688}, {550, 21167}, {575, 3146}, {1350, 17800}, {1352, 3529}, {1495, 5189}, {1657, 3098}, {2916, 14130}, {3534, 10516}, {3543, 14561}, {3589, 3853}, {3627, 19130}, {3763, 15696}, {3830, 5085}, {4048, 7842}, {5050, 15684}, {5097, 14912}, {5476, 15682}, {5907, 16658}, {6194, 14458}, {6756, 17704}, {7387, 23325}, {7540, 16836}, {7553, 9729}, {9967, 18562}, {9969, 14641}, {10168, 15687}, {10182, 23335}, {10263, 11232}, {10519, 15683}, {10721, 19140}, {11178, 15681}, {11550, 20062}, {11572, 12087}, {11574, 18563}, {11695, 17712}, {12022, 13598}, {12295, 20301}, {13331, 14537}, {13348, 13419}, {14893, 25565}, {15578, 17714}, {15640, 20423}, {15691, 20582}, {21850, 22330}

X(29323) = isogonal conjugate of X(29322)


X(29324) =  POINT POLARIS(1,0,0,2)

Barycentrics    (b - c) (-a^3 - 2 a b c + b^2 c + b c^2) : :

X(29324) lies on these lines: {30, 511}, {659, 4462}, {663, 4922}, {667, 3762}, {905, 21051}, {1491, 17496}, {1577, 4378}, {2533, 4474}, {3669, 3837}, {3716, 4504}, {3733, 6133}, {3777, 21222}, {3960, 21260}, {4010, 4449}, {4147, 9508}, {4148, 4394}, {4367, 4391}, {4382, 23780}, {4448, 8643}, {4490, 4560}, {6332, 18004}, {21117, 23781}, {25127, 27854}

X(29324) = isogonal conjugate of X(29325)


X(29325) =  ISOGONAL CONJUGATE OF X(29324)

Barycentrics    a^2/((b - c) (-a^3 - 2 a b c + b^2 c + b c^2)) : :

X(29325) lies on the circumcircle and these lines:

X(29325) = isogonal conjugate of X(29324)


X(29326) =  CIRCUMCIRCLE-ANTIPODE OF X(29325)

Barycentrics    a^2 (a^3 b^3 - a b^5 + a^5 c - 2 a^4 b c - a^3 b^2 c - a^2 b^3 c + 4 a b^4 c - b^5 c + 2 a^3 b c^2 - a b^3 c^2 - 2 a^3 c^3 + 2 a^2 b c^3 - a b^2 c^3 + b^3 c^3 - 2 a b c^4 + a c^5) (a^5 b - 2 a^3 b^3 + a b^5 - 2 a^4 b c + 2 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 + a^3 c^3 - a^2 b c^3 - a b^2 c^3 + b^3 c^3 + 4 a b c^4 - a c^5 - b c^5) : :

X(29326) lies on the circumcircle and these lines:

X(29326) = isogonal conjugate of X(29327)
X(29326) = circumcircle-antipode of X(29325)


X(29327) =  ISOGONAL CONJUGATE OF X(29326)

Barycentrics    1/((a^3 b^3 - a b^5 + a^5 c - 2 a^4 b c - a^3 b^2 c - a^2 b^3 c + 4 a b^4 c - b^5 c + 2 a^3 b c^2 - a b^3 c^2 - 2 a^3 c^3 + 2 a^2 b c^3 - a b^2 c^3 + b^3 c^3 - 2 a b c^4 + a c^5) (a^5 b - 2 a^3 b^3 + a b^5 - 2 a^4 b c + 2 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 + a^3 c^3 - a^2 b c^3 - a b^2 c^3 + b^3 c^3 + 4 a b c^4 - a c^5 - b c^5)) : :

X(29327) lies on these lines: {3, 3685}, {5, 1738}, {30, 511}, {986, 5722}, {1266, 22791}, {3717, 5690}, {7171, 10476}, {7193, 24410}, {12589, 24248}, {17628, 24996}

X(29327) = isogonal conjugate of X(29326)


X(29328) =  POINT POLARIS(1,0,2,0)

Barycentrics    (b - c) (-a^3 - 2 a^2 b - 2 a^2 c + b^2 c + b c^2) : :

X(29328) lies on these lines: {30, 511}, {71, 20979}, {649, 4010}, {650, 4806}, {659, 4380}, {667, 4170}, {693, 4784}, {798, 6133}, {905, 4992}, {1491, 20295}, {1577, 4834}, {1839, 7649}, {2254, 24719}, {2517, 23794}, {2977, 14321}, {3572, 17458}, {3716, 4782}, {3835, 9508}, {3837, 4106}, {4057, 8053}, {4382, 21146}, {4790, 7662}, {4804, 4979}, {4813, 4824}, {4840, 7199}, {4897, 23770}, {15419, 17217}, {20291, 20294}

X(29328) = isogonal conjugate of X(29329)


X(29329) =  ISOGONAL CONJUGATE OF X(29328)

Barycentrics    a^2/((b - c) (-a^3 - 2 a^2 b - 2 a^2 c + b^2 c + b c^2)) : :

X(29329) lies on the circumcircle and these lines:

X(29329) = isogonal conjugate of X(29328)


X(29330) =  CIRCUMCIRCLE-ANTIPODE OF X(29329)

Barycentrics    a^2 (-2 a^4 b^2 + a^3 b^3 + 2 a^2 b^4 - a b^5 + a^5 c - a^3 b^2 c + a^2 b^3 c - b^5 c + a b^3 c^2 + 2 b^4 c^2 - 2 a^3 c^3 - a b^2 c^3 + b^3 c^3 - 2 b^2 c^4 + a c^5) (a^5 b - 2 a^3 b^3 + a b^5 - 2 a^4 c^2 - a^3 b c^2 - a b^3 c^2 - 2 b^4 c^2 + a^3 c^3 + a^2 b c^3 + a b^2 c^3 + b^3 c^3 + 2 a^2 c^4 + 2 b^2 c^4 - a c^5 - b c^5) : :

X(29330) lies on the circumcircle and these lines:

X(29330) = isogonal conjugate of X(29331)
X(29330) = circumcircle-antipode of X(29329)


X(29331) =  ISOGONAL CONJUGATE OF X(29330)

Barycentrics    1/((-2 a^4 b^2 + a^3 b^3 + 2 a^2 b^4 - a b^5 + a^5 c - a^3 b^2 c + a^2 b^3 c - b^5 c + a b^3 c^2 + 2 b^4 c^2 - 2 a^3 c^3 - a b^2 c^3 + b^3 c^3 - 2 b^2 c^4 + a c^5) (a^5 b - 2 a^3 b^3 + a b^5 - 2 a^4 c^2 - a^3 b c^2 - a b^3 c^2 - 2 b^4 c^2 + a^3 c^3 + a^2 b c^3 + a b^2 c^3 + b^3 c^3 + 2 a^2 c^4 + 2 b^2 c^4 - a c^5 - b c^5)) : :

X(29331) lies on these lines: {3, 239}, {4, 6542}, {5, 3912}, {20, 20016}, {30, 511}, {140, 3008}, {242, 17976}, {320, 24833}, {381, 17310}, {631, 29590}, {1656, 17266}, {3526, 29607}, {3943, 24828}, {15973, 30059}, {16086, 30225}, {16377, 26639}, {17160, 24813}, {17374, 24827}, {20072, 24817}, {24281, 30117}

X(29331) = isogonal conjugate of X(29330)


X(29332) =  POINT POLARIS(1,2,0,0)

Barycentrics    (b - c) (a^3 + 2 b^3 + b^2 c + b c^2 + 2 c^3) : :

X(29332) lies on these lines: {30, 511}, {3801, 4874}, {6133, 21121}

X(29332) = isogonal conjugate of X(29333)


X(29333) =  ISOGONAL CONJUGATE OF X(29332)

Barycentrics    a^2/((b - c) (a^3 + 2 b^3 + b^2 c + b c^2 + 2 c^3)) : :

X(29333) lies on the circumcircle and these lines:

X(29333) = isogonal conjugate of X(29332)


X(29334) =  CIRCUMCIRCLE-ANTIPODE OF X(29333)

Barycentrics    a^2 (2 a^6 - a^5 b - 2 a^4 b^2 + 2 a^3 b^3 - 2 a^2 b^4 - a b^5 + 2 b^6 + 2 a^4 c^2 + a^3 b c^2 + a b^3 c^2 + 2 b^4 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 - 4 c^6) (2 a^6 + 2 a^4 b^2 - a^3 b^3 + a b^5 - 4 b^6 - a^5 c + a^3 b^2 c - a^2 b^3 c + b^5 c - 2 a^4 c^2 - a b^3 c^2 + 2 a^3 c^3 + a b^2 c^3 - b^3 c^3 - 2 a^2 c^4 + 2 b^2 c^4 - a c^5 + 2 c^6) : :

X(29334) lies on the circumcircle and these lines:

X(29334) = isogonal conjugate of X(29335)
X(29334) = circumcircle-antipode of X(29333)


X(29335) =  ISOGONAL CONJUGATE OF X(29334)

Barycentrics    1/((2 a^6 - a^5 b - 2 a^4 b^2 + 2 a^3 b^3 - 2 a^2 b^4 - a b^5 + 2 b^6 + 2 a^4 c^2 + a^3 b c^2 + a b^3 c^2 + 2 b^4 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 - 4 c^6) (2 a^6 + 2 a^4 b^2 - a^3 b^3 + a b^5 - 4 b^6 - a^5 c + a^3 b^2 c - a^2 b^3 c + b^5 c - 2 a^4 c^2 - a b^3 c^2 + 2 a^3 c^3 + a b^2 c^3 - b^3 c^3 - 2 a^2 c^4 + 2 b^2 c^4 - a c^5 + 2 c^6)) : :

X(29335) lies on these lines: {3, 17291}, {30, 511}

X(29335) = isogonal conjugate of X(29334)


X(29336) =  POINT POLARIS(2,1,0,0)

Barycentrics    (b - c) (2 a^3 + b^3 - b^2 c - b c^2 + c^3) : :

X(29336) lies on these lines: {30, 511}

X(29336) = isogonal conjugate of X(29337)


X(29337) =  ISOGONAL CONJUGATE OF X(29336)

Barycentrics    a^2/((b - c) (2 a^3 + b^3 - b^2 c - b c^2 + c^3)) : :

X(29337) lies on the circumcircle and these lines:

X(29337) = isogonal conjugate of X(29336)


X(29338) =  CIRCUMCIRCLE-ANTIPODE OF X(29337)

Barycentrics    a^2 (a^6 - 2 a^5 b - a^4 b^2 + 4 a^3 b^3 - a^2 b^4 - 2 a b^5 + b^6 + a^4 c^2 + 2 a^3 b c^2 + 2 a b^3 c^2 + b^4 c^2 - 2 a^3 c^3 - 2 a^2 b c^3 - 2 a b^2 c^3 - 2 b^3 c^3 + 2 a c^5 + 2 b c^5 - 2 c^6) (a^6 + a^4 b^2 - 2 a^3 b^3 + 2 a b^5 - 2 b^6 - 2 a^5 c + 2 a^3 b^2 c - 2 a^2 b^3 c + 2 b^5 c - a^4 c^2 - 2 a b^3 c^2 + 4 a^3 c^3 + 2 a b^2 c^3 - 2 b^3 c^3 - a^2 c^4 + b^2 c^4 - 2 a c^5 + c^6) : :

X(29338) lies on the circumcircle and these lines:

X(29338) = isogonal conjugate of X(29339)
X(29338) = circumcircle-antipode of X(29337)


X(29339) =  ISOGONAL CONJUGATE OF X(29338)

Barycentrics    1/((a^6 - 2 a^5 b - a^4 b^2 + 4 a^3 b^3 - a^2 b^4 - 2 a b^5 + b^6 + a^4 c^2 + 2 a^3 b c^2 + 2 a b^3 c^2 + b^4 c^2 - 2 a^3 c^3 - 2 a^2 b c^3 - 2 a b^2 c^3 - 2 b^3 c^3 + 2 a c^5 + 2 b c^5 - 2 c^6) (a^6 + a^4 b^2 - 2 a^3 b^3 + 2 a b^5 - 2 b^6 - 2 a^5 c + 2 a^3 b^2 c - 2 a^2 b^3 c + 2 b^5 c - a^4 c^2 - 2 a b^3 c^2 + 4 a^3 c^3 + 2 a b^2 c^3 - 2 b^3 c^3 - a^2 c^4 + b^2 c^4 - 2 a c^5 + c^6)) : :

X(29339) lies on these lines: {30, 511}

X(29339) = isogonal conjugate of X(29338)


X(29340) =  POINT POLARIS(2,0,1,0)

Barycentrics    (b - c) (-2 a^3 - a^2 b - a^2 c + 2 b^2 c + 2 b c^2) : :

X(29340) lies on these lines: {1, 4810}, {30, 511}, {668, 4427}, {1015, 3120}, {1635, 14431}, {1960, 4010}, {4378, 4382}, {4728, 14419}, {4782, 4791}, {14422, 21297}

X(29340) = isogonal conjugate of X(29341)


X(29341) =  ISOGONAL CONJUGATE OF X(29340)

Barycentrics    a^2/((b - c) (-2 a^3 - a^2 b - a^2 c + 2 b^2 c + 2 b c^2)) : :

X(29341) lies on the circumcircle and these lines:

X(29341) = isogonal conjugate of X(29340)


X(29342) =  CIRCUMCIRCLE-ANTIPODE OF X(29341)

Barycentrics    a^2 (-a^4 b^2 + 2 a^3 b^3 + a^2 b^4 - 2 a b^5 + 2 a^5 c - 2 a^3 b^2 c + 2 a^2 b^3 c - 2 b^5 c + 2 a b^3 c^2 + b^4 c^2 - 4 a^3 c^3 - 2 a b^2 c^3 + 2 b^3 c^3 - b^2 c^4 + 2 a c^5) (2 a^5 b - 4 a^3 b^3 + 2 a b^5 - a^4 c^2 - 2 a^3 b c^2 - 2 a b^3 c^2 - b^4 c^2 + 2 a^3 c^3 + 2 a^2 b c^3 + 2 a b^2 c^3 + 2 b^3 c^3 + a^2 c^4 + b^2 c^4 - 2 a c^5 - 2 b c^5) : :

X(29342) lies on the circumcircle and these lines:

X(29342) = isogonal conjugate of X(29343)
X(29342) = circumcircle-antipode of X(29341)


X(29343) =  ISOGONAL CONJUGATE OF X(29342)

Barycentrics    1/((-a^4 b^2 + 2 a^3 b^3 + a^2 b^4 - 2 a b^5 + 2 a^5 c - 2 a^3 b^2 c + 2 a^2 b^3 c - 2 b^5 c + 2 a b^3 c^2 + b^4 c^2 - 4 a^3 c^3 - 2 a b^2 c^3 + 2 b^3 c^3 - b^2 c^4 + 2 a c^5) (2 a^5 b - 4 a^3 b^3 + 2 a b^5 - a^4 c^2 - 2 a^3 b c^2 - 2 a b^3 c^2 - b^4 c^2 + 2 a^3 c^3 + 2 a^2 b c^3 + 2 a b^2 c^3 + 2 b^3 c^3 + a^2 c^4 + b^2 c^4 - 2 a c^5 - 2 b c^5)) : :

X(29343) lies on these lines: {3, 17119}, {30, 511}

X(29343) = isogonal conjugate of X(29342)


X(29344) =  POINT POLARIS(2,0,0,1)

Barycentrics    (b - c) (-2 a^3 - a b c + 2 b^2 c + 2 b c^2) : :

X(29344) lies on these lines: {30, 511}, {667, 4791}, {4063, 4474}, {4367, 4823}, {4391, 4401}, {4774, 4834}

X(29344) = isogonal conjugate of X(29345)


X(29345) =  ISOGONAL CONJUGATE OF X(29344)

Barycentrics    a^2/((b - c) (-2 a^3 - a b c + 2 b^2 c + 2 b c^2)) : :

X(29345) lies on the circumcircle and these lines:

X(29345) = isogonal conjugate of X(29344)


X(29346) =  CIRCUMCIRCLE-ANTIPODE OF X(29345)

Barycentrics    a^2 (2 a^3 b^3 - 2 a b^5 + 2 a^5 c - a^4 b c - 2 a^3 b^2 c + a^2 b^3 c + 2 a b^4 c - 2 b^5 c + a^3 b c^2 + a b^3 c^2 - 4 a^3 c^3 + a^2 b c^3 - 2 a b^2 c^3 + 2 b^3 c^3 - a b c^4 + 2 a c^5) (2 a^5 b - 4 a^3 b^3 + 2 a b^5 - a^4 b c + a^3 b^2 c + a^2 b^3 c - a b^4 c - 2 a^3 b c^2 - 2 a b^3 c^2 + 2 a^3 c^3 + a^2 b c^3 + a b^2 c^3 + 2 b^3 c^3 + 2 a b c^4 - 2 a c^5 - 2 b c^5) : :

X(29346) lies on the circumcircle and these lines:

X(29346) = isogonal conjugate of X(29347)
X(29346) = circumcircle-antipode of X(29345)


X(29347) =  ISOGONAL CONJUGATE OF X(29346)

Barycentrics    1/((2 a^3 b^3 - 2 a b^5 + 2 a^5 c - a^4 b c - 2 a^3 b^2 c + a^2 b^3 c + 2 a b^4 c - 2 b^5 c + a^3 b c^2 + a b^3 c^2 - 4 a^3 c^3 + a^2 b c^3 - 2 a b^2 c^3 + 2 b^3 c^3 - a b c^4 + 2 a c^5) (2 a^5 b - 4 a^3 b^3 + 2 a b^5 - a^4 b c + a^3 b^2 c + a^2 b^3 c - a b^4 c - 2 a^3 b c^2 - 2 a b^3 c^2 + 2 a^3 c^3 + a^2 b c^3 + a b^2 c^3 + 2 b^3 c^3 + 2 a b c^4 - 2 a c^5 - 2 b c^5)) : :

X(29347) lies on these lines: {30, 511}, {1284, 24209}, {1764, 4365}, {2051, 4970}, {3218, 13244}, {3993, 24220}, {4021, 15950}, {5530, 7951}, {10434, 28605}, {10447, 10882}, {17355, 24269}

X(29347) = isogonal conjugate of X(29346)


X(29348) =  CIRCUMCIRCLE-ANTIPODE OF X(898)

Barycentrics    a (2 a^4 b - 2 a^3 b^2 - 2 a^2 b^3 + 2 a b^4 - a^4 c - b^4 c + 2 a^2 b c^2 + 2 a b^2 c^2 + a^2 c^3 - 4 a b c^3 + b^2 c^3) (a^4 b - a^2 b^3 - 2 a^4 c - 2 a^2 b^2 c + 4 a b^3 c + 2 a^3 c^2 - 2 a b^2 c^2 - b^3 c^2 + 2 a^2 c^3 - 2 a c^4 + b c^4) : :

X(29348) lies on the circumcircle and these lines:

X(29348) = isogonal conjugate of X(29349)
X(29348) = circumcircle-antipode of X(898)


X(29349) =  ISOGONAL CONJUGATE OF X(29348)

Barycentrics    a/((2 a^4 b - 2 a^3 b^2 - 2 a^2 b^3 + 2 a b^4 - a^4 c - b^4 c + 2 a^2 b c^2 + 2 a b^2 c^2 + a^2 c^3 - 4 a b c^3 + b^2 c^3) (a^4 b - a^2 b^3 - 2 a^4 c - 2 a^2 b^2 c + 4 a b^3 c + 2 a^3 c^2 - 2 a b^2 c^2 - b^3 c^2 + 2 a^2 c^3 - 2 a c^4 + b c^4)) : :

X(29349) lies on these lines: {1, 4014}, {3, 16686}, {4, 6335}, {8, 4499}, {30, 511}, {40, 9355}, {149, 3937}, {165, 2108}, {573, 12034}, {991, 10246}, {1015, 24289}, {1742, 3576}, {3271, 24715}, {3664, 5049}, {3681, 4450}, {3784, 9580}, {5092, 24309}, {5902, 21746}, {8757, 12912}

X(29349) = isogonal conjugate of X(29348)


X(29350) =  POINT POLARIS(0,0,2,-1)

Barycentrics    a (b - c) (-2 a b - 2 a c + b c) : :

X(29350) lies on these lines: {1, 649}, {8, 20295}, {10, 3835}, {30, 511}, {40, 15599}, {65, 3676}, {145, 26853}, {386, 29487}, {551, 14474}, {659, 4775}, {660, 6633}, {663, 4063}, {665, 30234}, {667, 4879}, {693, 4761}, {876, 14421}, {960, 4521}, {984, 14437}, {1002, 1022}, {1019, 4449}, {1125, 4507}, {1491, 4730}, {1635, 3250}, {1734, 4729}, {1960, 4782}, {2533, 4823}, {3295, 23865}, {3616, 27013}, {3617, 26798}, {3679, 14433}, {3783, 4893}, {3803, 4162}, {3869, 4468}, {3878, 11068}, {3899, 6546}, {3919, 21204}, {4010, 4791}, {4040, 4498}, {4041, 14349}, {4129, 4147}, {4170, 4391}, {4367, 4834}, {4378, 4784}, {4490, 4983}, {4560, 5216}, {4647, 20909}, {4658, 18200}, {4724, 21385}, {4807, 17072}, {4832, 21348}, {4992, 21260}, {7287, 21272}, {8583, 25955}, {9780, 27138}, {16830, 26277}, {17217, 20907}, {19853, 27293}, {21056, 24087}, {21197, 24176}, {23655, 29807}

X(29350) = isogonal conjugate of X(29351)
X(29350) = crossdifference of every pair of points on line X(6)X(750)


X(29351) =  ISOGONAL CONJUGATE OF X(29350)

Barycentrics    a/((b - c) (2 a b + 2 a c - b c)) : :

X(29351) lies on the circumcircle and these lines:

X(29351) = isogonal conjugate of X(29350)
X(29351) = trilinear pole of line X(6)X(750)
X(29351) = Ψ(X(6), X(750))


X(29352) =  CIRCUMCIRCLE-ANTIPODE OF X(29351)

Barycentrics    a (a^3 b - 2 a^2 b^2 + a b^3 - 2 a^3 c + a^2 b c + 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^3 b - 2 a^2 b^2 - a^3 c - a^2 b c + 2 a b^2 c + 2 a^2 c^2 - a b c^2 - 2 b^2 c^2 - a c^3 + 2 b c^3) : :

X(29352) lies on the circumcircle and these lines:

X(29352) = isogonal conjugate of X(29353)
X(29352) = circumcircle-antipode of X(29351)


X(29353) =  ISOGONAL CONJUGATE OF X(29352)

Barycentrics    a/((a^3 b - 2 a^2 b^2 + a b^3 - 2 a^3 c + a^2 b c + 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^3 b - 2 a^2 b^2 - a^3 c - a^2 b c + 2 a b^2 c + 2 a^2 c^2 - a b c^2 - 2 b^2 c^2 - a c^3 + 2 b c^3)) : :

X(29353) lies on these lines: {3, 4471}, {6, 24309}, {30, 511}, {43, 165}, {354, 1122}, {970, 12512}, {991, 995}, {1266, 25048}, {1633, 2323}, {2262, 17668}, {2325, 4553}, {2340, 21362}, {3000, 20367}, {3008, 3271}, {3056, 3663}, {3146, 12435}, {3681, 4416}, {3729, 25304}, {3730, 24708}, {3784, 11019}, {3817, 3840}, {3888, 3912}, {4014, 4887}, {4430, 17364}, {4459, 24209}, {4480, 4499}, {4847, 26892}, {4890, 4909}, {5091, 8540}, {5360, 22003}, {6510, 11712}, {6743, 29958}, {7186, 24210}, {9355, 18788}, {9812, 10439}, {10164, 24494}, {10247, 24405}, {17355, 17792}, {25101, 25279}

X(29353) = isogonal conjugate of X(29352)


X(29354) =  POINT POLARIS(0,-1,0,2)

Barycentrics    (b - c) (b^3 - 2 a b c + b^2 c + b c^2 + c^3) : :

X(29354) lies on these lines: {30, 511}, {2254, 4808}, {2530, 4088}, {3762, 3801}, {3776, 21260}, {4122, 4978}, {4522, 23815}, {4705, 16892}, {17990, 23785}

X(29354) = isogonal conjugate of X(29355)


X(29355) =  ISOGONAL CONJUGATE OF X(29354)

Barycentrics    a^2/((b - c) (-b^3 + 2 a b c - b^2 c - b c^2 - c^3)) : :

X(29355) lies on the circumcircle and these lines:

X(29355) = isogonal conjugate of X(29354)


X(29356) =  CIRCUMCIRCLE-ANTIPODE OF X(29355)

Barycentrics    a^2 (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 + b^4 c^2 - 2 a^2 b c^3 - 2 a b^2 c^3 + 4 a b c^4 - 2 c^6) (a^6 + a^4 b^2 - 2 b^6 - 2 a^4 b c - 2 a^2 b^3 c + 4 a b^4 c - a^4 c^2 + 2 a^3 b c^2 - 2 a b^3 c^2 + 2 a^2 b c^3 - a^2 c^4 - 2 a b c^4 + b^2 c^4 + c^6) : :

X(29356) lies on the circumcircle and these lines:

X(29356) = isogonal conjugate of X(29357)
X(29356) = circumcircle-antipode of X(29355)


X(29357) =  ISOGONAL CONJUGATE OF X(29356)

Barycentrics    a/((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 + b^4 c^2 - 2 a^2 b c^3 - 2 a b^2 c^3 + 4 a b c^4 - 2 c^6) (a^6 + a^4 b^2 - 2 b^6 - 2 a^4 b c - 2 a^2 b^3 c + 4 a b^4 c - a^4 c^2 + 2 a^3 b c^2 - 2 a b^3 c^2 + 2 a^2 b c^3 - a^2 c^4 - 2 a b c^4 + b^2 c^4 + c^6)) : :

X(29357) lies on these lines:

X(29357) = isogonal conjugate of X(29356)


X(29358) =  POINT POLARIS(0,2,0,-1)

Barycentrics    (b - c) (2 b^3 - a b c + 2 b^2 c + 2 b c^2 + 2 c^3) : :

X(29358) lies on these lines: {30, 511}, {3801, 4791}, {4122, 4823}

X(29358) = isogonal conjugate of X(29359)


X(29359) =  ISOGONAL CONJUGATE OF X(29358)

Barycentrics    a^2/((b - c) (2 b^3 - a b c + 2 b^2 c + 2 b c^2 + 2 c^3)) : :

X(29359) lies on the circumcircle and these lines:

X(29359) = isogonal conjugate of X(29358)


X(29360) =  CIRCUMCIRCLE-ANTIPODE OF X(29359)

Barycentrics    a^2 (2 a^6 - 2 a^4 b^2 - 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 + 2 b^4 c^2 - a^2 b c^3 - a b^2 c^3 + 2 a b c^4 - 4 c^6) (2 a^6 + 2 a^4 b^2 - 4 b^6 - a^4 b c - a^2 b^3 c + 2 a b^4 c - 2 a^4 c^2 + a^3 b c^2 - a b^3 c^2 + a^2 b c^3 - 2 a^2 c^4 - a b c^4 + 2 b^2 c^4 + 2 c^6) : :

X(29360) lies on the circumcircle and these lines:

X(29360) = isogonal conjugate of X(29361)
X(29360) = circumcircle-antipode of X(29359)


X(29361) =  ISOGONAL CONJUGATE OF X(29360)

Barycentrics    a/((2 a^6 - 2 a^4 b^2 - 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 + 2 b^4 c^2 - a^2 b c^3 - a b^2 c^3 + 2 a b c^4 - 4 c^6) (2 a^6 + 2 a^4 b^2 - 4 b^6 - a^4 b c - a^2 b^3 c + 2 a b^4 c - 2 a^4 c^2 + a^3 b c^2 - a b^3 c^2 + a^2 b c^3 - 2 a^2 c^4 - a b c^4 + 2 b^2 c^4 + 2 c^6)) : :

X(29361) lies on these lines:

X(29361) = isogonal conjugate of X(29360)


X(29362) =  POINT POLARIS(-1,0,0,2)

Barycentrics    (b - c) (-a^3 + 2 a b c + b^2 c + b c^2) : :

X(29362) lies on these lines: {30, 511}, {649, 21146}, {650, 3837}, {659, 693}, {661, 24719}, {667, 4978}, {1491, 17494}, {2533, 4498}, {3261, 27855}, {3733, 7199}, {3777, 4560}, {4010, 4382}, {4106, 4806}, {4367, 4801}, {4369, 4782}, {4380, 4784}, {4468, 18004}, {4490, 21301}, {9508, 24720}, {14838, 23815}, {17496, 23765}, {18072, 18133}, {20293, 25292}, {20316, 25121}, {25009, 25926}, {28399, 30061}

X(29362) = isogonal conjugate of X(29363)


X(29363) =  ISOGONAL CONJUGATE OF X(29362)

Barycentrics    a^2/((b - c) (-a^3 + 2 a b c + b^2 c + b c^2)) : :

X(29363) lies on the circumcircle and these lines:

X(29363) = isogonal conjugate of X(29362)


X(29364) =  CIRCUMCIRCLE-ANTIPODE OF X(29363)

Barycentrics    a^2 (a^3 b^3 - a b^5 + a^5 c + 2 a^4 b c - a^3 b^2 c + 3 a^2 b^3 c - 4 a b^4 c - b^5 c - 2 a^3 b c^2 + 3 a b^3 c^2 - 2 a^3 c^3 - 2 a^2 b c^3 - a b^2 c^3 + b^3 c^3 + 2 a b c^4 + a c^5) (a^5 b - 2 a^3 b^3 + a b^5 + 2 a^4 b c - 2 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 + a^3 c^3 + 3 a^2 b c^3 + 3 a b^2 c^3 + b^3 c^3 - 4 a b c^4 - a c^5 - b c^5) : :

X(29364) lies on the circumcircle and these lines:

X(29364) = isogonal conjugate of X(29365)
X(29364) = circumcircle-antipode of X(29363)


X(29365) =  ISOGONAL CONJUGATE OF X(29364)

Barycentrics    1/((a^3 b^3 - a b^5 + a^5 c + 2 a^4 b c - a^3 b^2 c + 3 a^2 b^3 c - 4 a b^4 c - b^5 c - 2 a^3 b c^2 + 3 a b^3 c^2 - 2 a^3 c^3 - 2 a^2 b c^3 - a b^2 c^3 + b^3 c^3 + 2 a b c^4 + a c^5) (a^5 b - 2 a^3 b^3 + a b^5 + 2 a^4 b c - 2 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 + a^3 c^3 + 3 a^2 b c^3 + 3 a b^2 c^3 + b^3 c^3 - 4 a b c^4 - a c^5 - b c^5)) : :

X(29365) lies on these lines: {3, 16823}, {30, 511}, {4881, 16383}

X(29365) = isogonal conjugate of X(29364)


X(29366) =  POINT POLARIS(-1,0,2,0)

Barycentrics    (b - c) (-a^3 + 2 a^2 b + 2 a^2 c + b^2 c + b c^2) : :

X(29366) lies on these lines: {8, 4490}, {30, 511}, {663, 2533}, {667, 4761}, {693, 4879}, {1491, 21302}, {1577, 4775}, {4162, 7662}, {4449, 21146}, {4801, 21343}, {8630, 23865}

X(29366) = isogonal conjugate of X(29367)


X(29367) =  ISOGONAL CONJUGATE OF X(29366)

Barycentrics    a^2/((b - c) (-a^3 + 2 a^2 b + 2 a^2 c + b^2 c + b c^2)) : :

X(29367) lies on the circumcircle and these lines:

X(29367) = isogonal conjugate of X(29366)


X(29368) =  CIRCUMCIRCLE-ANTIPODE OF X(29367)

Barycentrics    a^2 (2 a^4 b^2 + a^3 b^3 - 2 a^2 b^4 - a b^5 + a^5 c - a^3 b^2 c + a^2 b^3 c - b^5 c + a b^3 c^2 - 2 b^4 c^2 - 2 a^3 c^3 - a b^2 c^3 + b^3 c^3 + 2 b^2 c^4 + a c^5) (a^5 b - 2 a^3 b^3 + a b^5 + 2 a^4 c^2 - a^3 b c^2 - a b^3 c^2 + 2 b^4 c^2 + a^3 c^3 + a^2 b c^3 + a b^2 c^3 + b^3 c^3 - 2 a^2 c^4 - 2 b^2 c^4 - a c^5 - b c^5) : :

X(29368) lies on the circumcircle and these lines:

X(29368) = isogonal conjugate of X(29369)
X(29368) = circumcircle-antipode of X(29367)


X(29369) =  ISOGONAL CONJUGATE OF X(29368)

Barycentrics    1/((2 a^4 b^2 + a^3 b^3 - 2 a^2 b^4 - a b^5 + a^5 c - a^3 b^2 c + a^2 b^3 c - b^5 c + a b^3 c^2 - 2 b^4 c^2 - 2 a^3 c^3 - a b^2 c^3 + b^3 c^3 + 2 b^2 c^4 + a c^5) (a^5 b - 2 a^3 b^3 + a b^5 + 2 a^4 c^2 - a^3 b c^2 - a b^3 c^2 + 2 b^4 c^2 + a^3 c^3 + a^2 b c^3 + a b^2 c^3 + b^3 c^3 - 2 a^2 c^4 - 2 b^2 c^4 - a c^5 - b c^5)) : :

X(29369) lies on these lines: {3, 894}, {4, 6646}, {5, 4357}, {30, 511}, {140, 5750}, {381, 17254}, {986, 5725}, {1656, 17326}, {3927, 4385}, {4654, 17591}, {6147, 24231}, {6211, 26921}, {7009, 22161}, {7330, 10476}, {10446, 20430}

X(29369) = isogonal conjugate of X(29368)


X(29370) =  POINT POLARIS(-1,2,0,0)

Barycentrics    (b - c) (-a^3 + 2 b^3 + 3 b^2 c + 3 b c^2 + 2 c^3) : :

X(29370) lies on these lines: {2, 4951}, {30, 511}, {4122, 4809}

X(29370) = isogonal conjugate of X(29371)


X(29371) =  ISOGONAL CONJUGATE OF X(29370)

Barycentrics    a^2/((b - c) (-a^3 + 2 b^3 + 3 b^2 c + 3 b c^2 + 2 c^3)) : :

X(29371) lies on the circumcircle and these lines:

X(29371) = isogonal conjugate of X(29370)


X(29372) =  CIRCUMCIRCLE-ANTIPODE OF X(29371)

Barycentrics    a^2 (2 a^6 + a^5 b - 2 a^4 b^2 - 2 a^3 b^3 - 2 a^2 b^4 + a b^5 + 2 b^6 + 2 a^4 c^2 - a^3 b c^2 - a b^3 c^2 + 2 b^4 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 - 4 c^6) (2 a^6 + 2 a^4 b^2 + a^3 b^3 - a b^5 - 4 b^6 + a^5 c - a^3 b^2 c + a^2 b^3 c - b^5 c - 2 a^4 c^2 + a b^3 c^2 - 2 a^3 c^3 - a b^2 c^3 + b^3 c^3 - 2 a^2 c^4 + 2 b^2 c^4 + a c^5 + 2 c^6) : :

X(29372) lies on the circumcircle and these lines:

X(29372) = isogonal conjugate of X(29373)
X(29372) = circumcircle-antipode of X(29371)


X(29373) =  ISOGONAL CONJUGATE OF X(29372)

Barycentrics    1/((2 a^6 + a^5 b - 2 a^4 b^2 - 2 a^3 b^3 - 2 a^2 b^4 + a b^5 + 2 b^6 + 2 a^4 c^2 - a^3 b c^2 - a b^3 c^2 + 2 b^4 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 - 4 c^6) (2 a^6 + 2 a^4 b^2 + a^3 b^3 - a b^5 - 4 b^6 + a^5 c - a^3 b^2 c + a^2 b^3 c - b^5 c - 2 a^4 c^2 + a b^3 c^2 - 2 a^3 c^3 - a b^2 c^3 + b^3 c^3 - 2 a^2 c^4 + 2 b^2 c^4 + a c^5 + 2 c^6)) : :

X(29373) lies on these lines: {3, 17292}, {30, 511}

X(29373) = isogonal conjugate of X(29372)


X(29374) =  ISOGONAL CONJUGATE OF X(1768)

Barycentrics    b c /(a^5 - a^4 (b + c) - a^3 (2 b^2 - 5 b c + 2 c^2) + 2 a^2 (b - c)^2 (b + c) + a (b - c)^2 (b^2 - b c + c^2) - b^5 + b^4 c + b c^4 - c^5) : :

See Randy Hutson, Hyacinthos 28698.

X(29374) lies on these lines: {484,1785}, {516,5080}, {517,1456}, {522,1768}, {910,5537}, {1325,5538}, {5536,22464} et al

X(29374) = isogonal conjugate of X(1768)
X(29374) = [X(4)-Ceva conjugate of X(110)]-of-excentral triangle






leftri  Collineation mappings involving Gemini triangle 89: X(29375) - X(29433)  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 89, as in centers X(28375)-X(29433). Then

m(X) = (b + c) (a^2 + b c) x + b(c - a) y - c (b - a) z : :

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, December 6, 2018)

underbar




X(29375) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^3 b - a^2 b^2 + a^3 c + 3 a b^2 c - a^2 c^2 + 3 a b c^2 - 2 b^2 c^2 : :

X(29375) lies on these lines:


X(29376) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^6 b - a^2 b^5 + a^6 c - 2 a^3 b^3 c - a^2 b^4 c + 2 a b^5 c + a^2 b^3 c^2 - b^5 c^2 - 2 a^3 b c^3 + a^2 b^2 c^3 - 4 a b^3 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 + 2 a b c^5 - b^2 c^5 : :

X(29376) lies on these lines:


X(29377) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^6 b - a^2 b^5 + a^6 c - 2 a^5 b c + 2 a^3 b^3 c - a^2 b^4 c + a^2 b^3 c^2 - b^5 c^2 + 2 a^3 b c^3 + a^2 b^2 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 - b^2 c^5 : :

X(29377) lies on these lines:


X(29378) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^6 b - a^2 b^5 + a^6 c - 4 a^5 b c + 6 a^3 b^3 c - a^2 b^4 c - 2 a b^5 c + a^2 b^3 c^2 - b^5 c^2 + 6 a^3 b c^3 + a^2 b^2 c^3 + 4 a b^3 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 - 2 a b c^5 - b^2 c^5 : :

X(29378) lies on these lines:


X(29379) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^4 b - a^2 b^3 + a^4 c + a^2 b^2 c + 2 a b^3 c + a^2 b c^2 - b^3 c^2 - a^2 c^3 + 2 a b c^3 - b^2 c^3 : :

X(29379) lies on these lines:


X(29380) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a (a^3 b - a b^3 + a^3 c - 4 a^2 b c + 4 a b^2 c - b^3 c + 4 a b c^2 - a c^3 - b c^3) : :

X(29380) lies on these lines:


X(29381) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^3 b - a^2 b^2 + a^3 c - 2 a^2 b c + a b^2 c - a^2 c^2 + a b c^2 - 2 b^2 c^2 : :

X(29381) lies on these lines:


X(29382) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a (a^3 b - a b^3 + a^3 c - 2 a^2 b c + b^3 c - 4 b^2 c^2 - a c^3 + b c^3) : :

X(29382) lies on these lines:


X(29383) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^3 b - a^2 b^2 + a^3 c - 4 a^2 b c - a b^2 c - a^2 c^2 - a b c^2 - 2 b^2 c^2 : :

X(29383) lies on these lines:


X(29384) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^5 b - a^4 b^2 + a^3 b^3 - a^2 b^4 + a^5 c - 6 a^4 b c + 5 a^3 b^2 c + 2 a^2 b^3 c - 2 a b^4 c - a^4 c^2 + 5 a^3 b c^2 - 10 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 + 2 b^3 c^3 - a^2 c^4 - 2 a b c^4 - b^2 c^4 : :

X(29384) lies on these lines:


X(29385) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^8 b + a^6 b^3 - a^4 b^5 - a^2 b^7 + a^8 c + a^6 b^2 c - 2 a^5 b^3 c - a^4 b^4 c - a^2 b^6 c + 2 a b^7 c + a^6 b c^2 + a^4 b^3 c^2 - a^2 b^5 c^2 - b^7 c^2 + a^6 c^3 - 2 a^5 b c^3 + a^4 b^2 c^3 + a^2 b^4 c^3 - 2 a b^5 c^3 + b^6 c^3 - a^4 b c^4 + a^2 b^3 c^4 - a^4 c^5 - a^2 b^2 c^5 - 2 a b^3 c^5 - a^2 b c^6 + b^3 c^6 - a^2 c^7 + 2 a b c^7 - b^2 c^7 : :

X(29385) lies on these lines:


X(29386) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^8 b + a^6 b^3 - a^4 b^5 - a^2 b^7 + a^8 c + a^6 b^2 c - 2 a^5 b^3 c - a^4 b^4 c - a^2 b^6 c + 2 a b^7 c + a^6 b c^2 + a^4 b^3 c^2 - a^2 b^5 c^2 - b^7 c^2 + a^6 c^3 - 2 a^5 b c^3 + a^4 b^2 c^3 + 4 a^3 b^3 c^3 + a^2 b^4 c^3 - 2 a b^5 c^3 + b^6 c^3 - a^4 b c^4 + a^2 b^3 c^4 - a^4 c^5 - a^2 b^2 c^5 - 2 a b^3 c^5 - a^2 b c^6 + b^3 c^6 - a^2 c^7 + 2 a b c^7 - b^2 c^7 : :

X(29386) lies on these lines:


X(29387) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^6 b - a^2 b^5 + a^6 c + a^4 b^2 c + 2 a b^5 c + a^4 b c^2 - b^5 c^2 - a^2 c^5 + 2 a b c^5 - b^2 c^5 : :

X(29387) lies on these lines:


X(29388) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    b c (-2 a^3 - a^2 b - a b^2 - a^2 c - 4 a b c + b^2 c - a c^2 + b c^2) : :

X(29388) lies on these lines:


X(29389) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^6 b - a^5 b^2 + a^3 b^4 - a^2 b^5 + a^6 c - 2 a^5 b c + a^4 b^2 c - a^3 b^3 c - a^2 b^4 c + 2 a b^5 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^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^2 c^5 + 2 a b c^5 - b^2 c^5 : :

X(29389) lies on these lines:


X(29390) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^4 b^2 - a^3 b^3 + 2 a^4 b c + 2 a^2 b^3 c + a^4 c^2 + 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 : :

X(29390) lies on these lines:


X(29391) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^4 b^2 - a^3 b^3 + 2 a^4 b c + 2 a^2 b^3 c + a^4 c^2 - 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 : :

X(29391) lies on these lines:


X(29392) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    2 a^4 b - 2 a^2 b^3 + 2 a^4 c - 2 a^3 b c + a^2 b^2 c + 3 a b^3 c + a^2 b c^2 - 4 a b^2 c^2 - b^3 c^2 - 2 a^2 c^3 + 3 a b c^3 - b^2 c^3 : :

X(29392) lies on these lines:


X(29393) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^4 b - a^2 b^3 + a^4 c - 4 a^3 b c - a^2 b^2 c - a^2 b c^2 - 8 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3 : :

X(29393) lies on these lines:


X(29394) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^5 b - a^4 b^2 + a^3 b^3 - a^2 b^4 + a^5 c - 2 a^4 b c + a^3 b^2 c - 2 a^2 b^3 c + 2 a b^4 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^3 c^3 - 2 a^2 b c^3 - a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 + 2 a b c^4 - b^2 c^4 : :

X(29394) lies on these lines:


X(29395) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    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 - b^3 c^2 - a^2 c^3 - b^2 c^3 : :

X(29395) lies on these lines:


X(29396) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    b c (-2 a^3 + 3 a^2 b - a b^2 + 3 a^2 c + b^2 c - a c^2 + b c^2) : :

X(29396) lies on these lines:


X(29397) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    b c (-a^4 b + 2 a^3 b^2 - a^2 b^3 - a^4 c + a^2 b^2 c + 2 a^3 c^2 + a^2 b c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3) : :

X(29397) lies on these lines:


X(29398) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    (b + c) (-a^2 - a b - a c + b c) (a^3 - a b^2 + 3 a b c + b^2 c - a c^2 + b c^2) : :

X(29398) lies on these lines:


X(29399) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    (b + c) (-a^4 + a^2 b^2 - 6 a^2 b c - 2 a b^2 c + a^2 c^2 - 2 a b c^2 + b^2 c^2) : :

X(29399) lies on these lines:


X(29400) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^3 b - a^2 b^2 + a^3 c - a^2 b c + 2 a b^2 c - a^2 c^2 + 2 a b c^2 - 2 b^2 c^2 : :

X(28400) lies on these lines:


X(29401) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^4 b - a^2 b^3 + a^4 c + 2 a^3 b c - 2 a^2 b^2 c + 3 a b^3 c - 2 a^2 b c^2 - 2 b^3 c^2 - a^2 c^3 + 3 a b c^3 - 2 b^2 c^3 : :

X(28401) lies on these lines:


X(29402) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    (a + b) (b - c) (a + c) (a^2 b + a^2 c - 2 a b c + b^2 c + b c^2) : :

X(28402) lies on these lines:


X(29403) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    (b - c) (-a^5 b - a^3 b^3 - a^5 c - 2 a^3 b^2 c + 3 a^2 b^3 c - 2 a^3 b c^2 + 3 a^2 b^2 c^2 - 3 a b^3 c^2 - a^3 c^3 + 3 a^2 b c^3 - 3 a b^2 c^3 + b^3 c^3) : :

X(28403) lies on these lines:


X(29404) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    b c (b - c) (3 a^3 - 2 a^2 b + a b^2 - 2 a^2 c + a b c + b^2 c + a c^2 + b c^2) : :

X(28404) lies on these lines:


X(29405) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^5 b - a^2 b^4 + a^5 c + a^3 b^2 c + 2 a b^4 c + a^3 b c^2 - 4 a^2 b^2 c^2 - b^4 c^2 - a^2 c^4 + 2 a b c^4 - b^2 c^4 : :

X(28405) lies on these lines:


X(29406) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^3 b - a^2 b^2 + a^3 c + 4 a^2 b c + 7 a b^2 c - a^2 c^2 + 7 a b c^2 - 2 b^2 c^2 : :

X(28406) lies on these lines:


X(29407) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^6 b - a^2 b^5 + a^6 c - a^5 b c - a^2 b^4 c + a b^5 c + a^2 b^3 c^2 - b^5 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^2 c^5 + a b c^5 - b^2 c^5 : :

X(28407) lies on these lines:


X(29408) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    (b + c) (-a^6 + a^2 b^4 + 2 a^3 b^2 c - 2 a b^4 c + 2 a^3 b c^2 + 3 a^2 b^2 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) : :

X(28408) lies on these lines:


X(29409) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^8 b + a^6 b^3 - a^4 b^5 - a^2 b^7 + a^8 c + a^6 b^2 c - 2 a^5 b^3 c - a^4 b^4 c - a^2 b^6 c + 2 a b^7 c + a^6 b c^2 + a^4 b^3 c^2 - a^2 b^5 c^2 - b^7 c^2 + a^6 c^3 - 2 a^5 b c^3 + a^4 b^2 c^3 + 8 a^3 b^3 c^3 + a^2 b^4 c^3 - 2 a b^5 c^3 + b^6 c^3 - a^4 b c^4 + a^2 b^3 c^4 - a^4 c^5 - a^2 b^2 c^5 - 2 a b^3 c^5 - a^2 b c^6 + b^3 c^6 - a^2 c^7 + 2 a b c^7 - b^2 c^7 : :

X(28409) lies on these lines:


X(29410) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    (b + c) (-a^8 - 2 a^7 b - a^6 b^2 + a^4 b^4 + 2 a^3 b^5 + a^2 b^6 - 2 a^7 c + 2 a^6 b c + 2 a^5 b^2 c - 4 a^4 b^3 c + 2 a^3 b^4 c + 2 a^2 b^5 c - 2 a b^6 c - a^6 c^2 + 2 a^5 b c^2 - a^4 b^2 c^2 - 6 a^3 b^3 c^2 + a^2 b^4 c^2 + 4 a b^5 c^2 + b^6 c^2 - 4 a^4 b c^3 - 6 a^3 b^2 c^3 - 4 a^2 b^3 c^3 - 2 a b^4 c^3 + a^4 c^4 + 2 a^3 b c^4 + a^2 b^2 c^4 - 2 a b^3 c^4 - 2 b^4 c^4 + 2 a^3 c^5 + 2 a^2 b c^5 + 4 a b^2 c^5 + a^2 c^6 - 2 a b c^6 + b^2 c^6) : :

X(28410) lies on these lines:


X(29411) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    (b + c) (-a^9 - a^8 b - 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^8 c - 2 a^7 b c + a^6 b^2 c + 2 a^5 b^3 c + a^4 b^4 c + 2 a^3 b^5 c - a^2 b^6 c - 2 a b^7 c - a^7 c^2 + a^6 b c^2 - a^5 b^2 c^2 - 5 a^4 b^3 c^2 + a^3 b^4 c^2 + 3 a^2 b^5 c^2 + a b^6 c^2 + b^7 c^2 - a^6 c^3 + 2 a^5 b c^3 - 5 a^4 b^2 c^3 - 12 a^3 b^3 c^3 - a^2 b^4 c^3 + 2 a b^5 c^3 - b^6 c^3 + a^5 c^4 + a^4 b c^4 + a^3 b^2 c^4 - a^2 b^3 c^4 - 2 a b^4 c^4 + a^4 c^5 + 2 a^3 b c^5 + 3 a^2 b^2 c^5 + 2 a b^3 c^5 + a^3 c^6 - a^2 b c^6 + a b^2 c^6 - b^3 c^6 + a^2 c^7 - 2 a b c^7 + b^2 c^7) : :

X(28411) lies on these lines:


X(29412) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    (b + c) (-a^9 + a^8 b + 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^8 c - 2 a^7 b c - a^6 b^2 c + 6 a^5 b^3 c + 3 a^4 b^4 c - 2 a^3 b^5 c - 3 a^2 b^6 c - 2 a b^7 c + a^7 c^2 - a^6 b c^2 - 5 a^5 b^2 c^2 + a^4 b^3 c^2 + 5 a^3 b^4 c^2 - a^2 b^5 c^2 - a b^6 c^2 + b^7 c^2 - a^6 c^3 + 6 a^5 b c^3 + a^4 b^2 c^3 - 8 a^3 b^3 c^3 + 3 a^2 b^4 c^3 + 2 a b^5 c^3 - 3 b^6 c^3 + a^5 c^4 + 3 a^4 b c^4 + 5 a^3 b^2 c^4 + 3 a^2 b^3 c^4 + 2 a b^4 c^4 + 2 b^5 c^4 - a^4 c^5 - 2 a^3 b c^5 - a^2 b^2 c^5 + 2 a b^3 c^5 + 2 b^4 c^5 - a^3 c^6 - 3 a^2 b c^6 - a b^2 c^6 - 3 b^3 c^6 + a^2 c^7 - 2 a b c^7 + b^2 c^7) : :

X(28412) lies on these lines:


X(29413) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    3 a^6 b - 3 a^2 b^5 + 3 a^6 c - 2 a^5 b c - 2 a^3 b^3 c - 3 a^2 b^4 c + 4 a b^5 c + 3 a^2 b^3 c^2 - 3 b^5 c^2 - 2 a^3 b c^3 + 3 a^2 b^2 c^3 - 8 a b^3 c^3 + 3 b^4 c^3 - 3 a^2 b c^4 + 3 b^3 c^4 - 3 a^2 c^5 + 4 a b c^5 - 3 b^2 c^5 : :

X(28413) lies on these lines:


X(29414) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^6 b - a^2 b^5 + a^6 c - 2 a^5 b c + 2 a^3 b^3 c - a^2 b^4 c + 4 a^3 b^2 c^2 + 5 a^2 b^3 c^2 - b^5 c^2 + 2 a^3 b c^3 + 5 a^2 b^2 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 - b^2 c^5 : :

X(28414) lies on these lines:


X(29415) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^6 b - a^2 b^5 + a^6 c + 2 a^3 b^3 c - a^2 b^4 c + 2 a b^5 c + a^2 b^3 c^2 - b^5 c^2 + 2 a^3 b c^3 + a^2 b^2 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 + 2 a b c^5 - b^2 c^5 : :

X(28415) lies on these lines:


X(29416) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^4 b^2 - a^3 b^3 - 3 a^3 b^2 c - a b^4 c + a^4 c^2 - 3 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 - a b c^4 + b^2 c^4 : :

X(28416) lies on these lines:


X(29417) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a (a^4 b + a^3 b^2 - a^2 b^3 - a b^4 + a^4 c - 2 a^2 b^2 c + b^4 c + a^3 c^2 - 2 a^2 b c^2 + 12 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(28417) lies on these lines:


X(29418) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a (a^4 b + a^3 b^2 - a^2 b^3 - a b^4 + a^4 c - 2 a^2 b^2 c + b^4 c + a^3 c^2 - 2 a^2 b c^2 + 4 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(28418) lies on these lines:


X(29419) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^6 b - a^2 b^5 + a^6 c + a^4 b^2 c + 4 a^3 b^3 c + 2 a b^5 c + a^4 b c^2 - b^5 c^2 + 4 a^3 b c^3 + 4 a b^3 c^3 - a^2 c^5 + 2 a b c^5 - b^2 c^5 : :

X(28419) lies on these lines:


X(29420) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    (a - b) (a - c) (b + c) (a^4 + a^3 b + a^2 b^2 + a b^3 + a^3 c - b^3 c + a^2 c^2 + 3 b^2 c^2 + a c^3 - b c^3) : :

X(28420) lies on these lines:


X(29421) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    (a - b) (a - c) (a^3 b + a b^3 + a^3 c - b^3 c + 2 b^2 c^2 + a c^3 - b c^3) : :

X(28421) lies on these lines:


X(29422) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    (a - b) (a - c) (a^4 b + a b^4 + a^4 c + a^2 b^2 c - a b^3 c - b^4 c + a^2 b c^2 + b^3 c^2 - a b c^3 + b^2 c^3 + a c^4 - b c^4) : :

X(28422) lies on these lines:


X(29423) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    b c (-2 a^3 + a^2 b - a b^2 + a^2 c - 2 a b c + b^2 c - a c^2 + b c^2) : :

X(28423) lies on these lines:


X(29424) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    (b - c) (a^4 - 4 a^3 b + a^2 b^2 - 4 a^3 c + 8 a^2 b c - 4 a b^2 c + 2 b^3 c + a^2 c^2 - 4 a b c^2 + b^2 c^2 + 2 b c^3) : :

X(28424) lies on these lines:


X(29425) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    b c (-a^4 b - a^2 b^3 - a^4 c + a^2 b^2 c + a^2 b c^2 - 2 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3) : :

X(28425) lies on these lines:


X(29426) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a (b - c) (a^3 b + a^2 b^2 + a^3 c + a^2 b c - 2 a b^2 c + a^2 c^2 - 2 a b c^2 + 3 b^2 c^2) : :

X(28426) lies on these lines:


X(29427) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    b c (b - c) (-a^3 + 2 a^2 b + a b^2 + 2 a^2 c - 3 a b c + b^2 c + a c^2 + b c^2) : :

X(28427) lies on these lines:


X(29428) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    (b - c) (-a^5 b - a^3 b^3 - a^5 c + a^4 b c - a^3 b^2 c + 3 a^2 b^3 c - a^3 b c^2 - a b^3 c^2 - a^3 c^3 + 3 a^2 b c^3 - a b^2 c^3 + b^3 c^3) : :

X(28428) lies on these lines:


X(29429) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a (a^3 b - a b^3 + a^3 c - 2 a^2 b c + 4 a b^2 c + b^3 c + 4 a b c^2 - a c^3 + b c^3) : :

X(28429) lies on these lines:


X(29430) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    2 a^5 b + a^4 b^2 - a^3 b^3 - 2 a^2 b^4 + 2 a^5 c - a^3 b^2 c + 3 a b^4 c + a^4 c^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 - 2 a^2 c^4 + 3 a b c^4 - b^2 c^4 : :

X(28430) lies on these lines:


X(29431) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^4 b^2 - a^3 b^3 - 3 a^3 b^2 c - a b^4 c + a^4 c^2 - 3 a^3 b c^2 - 3 a b^3 c^2 + b^4 c^2 - a^3 c^3 - 3 a b^2 c^3 - a b c^4 + b^2 c^4 : :

X(28431) lies on these lines:


X(29432) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 89

Barycentrics    a^5 b - a^2 b^4 + a^5 c + a^3 b^2 c + 2 a b^4 c + a^3 b c^2 - b^4 c^2 - a^2 c^4 + 2 a b c^4 - b^2 c^4 : :

X(28432) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 90: X(29433) - X(29494)  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 90, as in centers X(28433)-X(29494). Then

m(X) = (b + c) (a^2 + b c) x - b (a + c)^2 y - c (a + b)^2 z : :

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, December 7, 2018)

underbar




X(29433) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^3 b-a^2 b^2+a^3 c-a b^2 c-a^2 c^2-a b c^2-2 b^2 c^2 : :

X(29433) lies on these lines:


X(29434) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^6 b - a^2 b^5 + a^6 c + 2 a^3 b^3 c - a^2 b^4 c - 2 a b^5 c + a^2 b^3 c^2 - b^5 c^2 + 2 a^3 b c^3 + a^2 b^2 c^3 + 4 a b^3 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 - 2 a b c^5 - b^2 c^5 : :

X(29434) lies on these lines:


X(29435) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^6 b - a^2 b^5 + a^6 c + 2 a^5 b c - 2 a^3 b^3 c - a^2 b^4 c + a^2 b^3 c^2 - b^5 c^2 - 2 a^3 b c^3 + a^2 b^2 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 - b^2 c^5 : :

X(29435) lies on these lines:


X(29436) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^5 b - 3 a^4 b^2 + 3 a^3 b^3 - a^2 b^4 + a^5 c - 4 a^4 b c + 4 a^2 b^3 c - a b^4 c - 3 a^4 c^2 - 16 a^2 b^2 c^2 + 5 a b^3 c^2 - 2 b^4 c^2 + 3 a^3 c^3 + 4 a^2 b c^3 + 5 a b^2 c^3 + 4 b^3 c^3 - a^2 c^4 - a b c^4 - 2 b^2 c^4 : :

X(29436) lies on these lines:


X(29437) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^4 b - a^2 b^3 + a^4 c + a^2 b^2 c - 2 a b^3 c + a^2 b c^2 - b^3 c^2 - a^2 c^3 - 2 a b c^3 - b^2 c^3 : :

X(29437) lies on these lines:


X(29438) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^3 b - a^2 b^2 + a^3 c + 2 a^2 b c + a b^2 c - a^2 c^2 + a b c^2 - 2 b^2 c^2 : :

X(29438) lies on these lines:


X(29439) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a (a^3 b - a b^3 + a^3 c - 2 a^2 b c + 4 a b^2 c - 3 b^3 c + 4 a b c^2 + 4 b^2 c^2 - a c^3 - 3 b c^3) : :

X(29439) lies on these lines:


X(29440) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^3 b - a^2 b^2 + a^3 c + 4 a^2 b c + 3 a b^2 c - a^2 c^2 + 3 a b c^2 - 2 b^2 c^2 : :

X(29440) lies on these lines:


X(29441) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    (a - b) (a - c) (a^3 b + a b^3 + a^3 c + 4 a^2 b c - b^3 c + 2 b^2 c^2 + a c^3 - b c^3) : :

X(29441) lies on these lines:


X(29442) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^8 b + a^6 b^3 - a^4 b^5 - a^2 b^7 + a^8 c + a^6 b^2 c + 2 a^5 b^3 c - a^4 b^4 c - a^2 b^6 c - 2 a b^7 c + a^6 b c^2 + a^4 b^3 c^2 - a^2 b^5 c^2 - b^7 c^2 + a^6 c^3 + 2 a^5 b c^3 + a^4 b^2 c^3 + a^2 b^4 c^3 + 2 a b^5 c^3 + b^6 c^3 - a^4 b c^4 + a^2 b^3 c^4 - a^4 c^5 - a^2 b^2 c^5 + 2 a b^3 c^5 - a^2 b c^6 + b^3 c^6 - a^2 c^7 - 2 a b c^7 - b^2 c^7 : :

X(29442) lies on these lines:


X(29443) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^8 b + a^6 b^3 - a^4 b^5 - a^2 b^7 + a^8 c + a^6 b^2 c + 2 a^5 b^3 c - a^4 b^4 c - a^2 b^6 c - 2 a b^7 c + a^6 b c^2 + a^4 b^3 c^2 - a^2 b^5 c^2 - b^7 c^2 + a^6 c^3 + 2 a^5 b c^3 + a^4 b^2 c^3 - 4 a^3 b^3 c^3 + a^2 b^4 c^3 + 2 a b^5 c^3 + b^6 c^3 - a^4 b c^4 + a^2 b^3 c^4 - a^4 c^5 - a^2 b^2 c^5 + 2 a b^3 c^5 - a^2 b c^6 + b^3 c^6 - a^2 c^7 - 2 a b c^7 - b^2 c^7 : :

X(29443) lies on these lines:


X(29444) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^5 b-a^2 b^4+a^5 c+a^3 b^2 c-2 a b^4 c+a^3 b c^2-b^4 c^2-a^2 c^4-2 a b c^4-b^2 c^4 : :

X(29444) lies on these lines:


X(29445) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^6 b - a^2 b^5 + a^6 c + a^4 b^2 c - 2 a b^5 c + a^4 b c^2 - b^5 c^2 - a^2 c^5 - 2 a b c^5 - b^2 c^5 : :

X(29445) lies on these lines:


X(29446) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    b c (-2 a^3 + 3 a^2 b - a b^2 + 3 a^2 c + 4 a b c + b^2 c - a c^2 + b c^2) : :

X(29446) lies on these lines:


X(29447) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    b c (-a^2 + a b + a c + b c) (a^2 b - a b^2 + a^2 c + 2 a b c + b^2 c - a c^2 + b c^2) : :

X(29447) lies on these lines:


X(29448) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^4 b^2 - a^3 b^3 + 2 a^4 b c - 2 a^2 b^3 c + a^4 c^2 + 2 a^2 b^2 c^2 - 3 a b^3 c^2 - a^3 c^3 - 2 a^2 b c^3 - 3 a b^2 c^3 - 2 b^3 c^3 : :

X(29448) lies on these lines:


X(29449) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^4 b^2 - a^3 b^3 + 2 a^4 b c - 2 a^2 b^3 c + a^4 c^2 + 6 a^2 b^2 c^2 - 3 a b^3 c^2 - a^3 c^3 - 2 a^2 b c^3 - 3 a b^2 c^3 - 2 b^3 c^3 : :

X(29449) lies on these lines:


X(29450) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    2 a^4 b - 2 a^2 b^3 + 2 a^4 c - 2 a^3 b c + 5 a^2 b^2 c - 5 a b^3 c + 5 a^2 b c^2 + 4 a b^2 c^2 - b^3 c^2 - 2 a^2 c^3 - 5 a b c^3 - b^2 c^3 : :

X(29450) lies on these lines:


X(29451) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^4 b - a^2 b^3 + a^4 c - 4 a^3 b c + 7 a^2 b^2 c - 4 a b^3 c + 7 a^2 b c^2 + 8 a b^2 c^2 + b^3 c^2 - a^2 c^3 - 4 a b c^3 + b^2 c^3 : :

X(29451) lies on these lines:


X(29452) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^5 b - a^4 b^2 + a^3 b^3 - a^2 b^4 + a^5 c - 2 a^4 b c + a^3 b^2 c + 2 a^2 b^3 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 + a^3 c^3 + 2 a^2 b c^3 + 3 a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 - 2 a b c^4 - b^2 c^4 : :

X(29452) lies on these lines:


X(29453) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    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 - b^3 c^2 - a^2 c^3 - b^2 c^3 : :

X(29453) lies on these lines:


X(29454) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    b c (-a^4 b - 2 a^3 b^2 - a^2 b^3 - a^4 c + a^2 b^2 c - 2 a^3 c^2 + a^2 b c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3) : :

X(29454) lies on these lines:


X(29455) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^3 b - a^2 b^2 + a^3 c + a^2 b c - a^2 c^2 - 2 b^2 c^2 : :

X(29455) lies on these lines:


X(29456) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^4 b - a^2 b^3 + a^4 c + 2 a^3 b c + 2 a^2 b^2 c - a b^3 c + 2 a^2 b c^2 - 2 b^3 c^2 - a^2 c^3 - a b c^3 - 2 b^2 c^3 : :

X(29456) lies on these lines:


X(29457) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    (b - c) (a^4 b + a^3 b^2 + a^4 c + 4 a^3 b c + a b^3 c + a^3 c^2 + 4 a b^2 c^2 + b^3 c^2 + a b c^3 + b^2 c^3) : :

X(29457) lies on these lines:


X(29458) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    (b - c) (-a^5 b - a^3 b^3 - a^5 c - 2 a^3 b^2 c - a^2 b^3 c - 2 a^3 b c^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 + b^3 c^3) : :

X(29458) lies on these lines:


X(29459) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^5 b - a^2 b^4 + a^5 c + a^3 b^2 c - 2 a b^4 c + a^3 b c^2 + 4 a^2 b^2 c^2 - b^4 c^2 - a^2 c^4 - 2 a b c^4 - b^2 c^4 : :

X(29459) lies on these lines:


X(29460) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^3 b - a^2 b^2 + a^3 c - 4 a^2 b c - 5 a b^2 c - a^2 c^2 - 5 a b c^2 - 2 b^2 c^2 : :

X(29460) lies on these lines:


X(29461) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^6 b - a^2 b^5 + a^6 c + a^5 b c - a^2 b^4 c - a b^5 c + a^2 b^3 c^2 - b^5 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^2 c^5 - a b c^5 - b^2 c^5 : :

X(29461) lies on these lines:


X(29462) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^8 b + a^6 b^3 - a^4 b^5 - a^2 b^7 + a^8 c + a^6 b^2 c + 2 a^5 b^3 c - a^4 b^4 c - a^2 b^6 c - 2 a b^7 c + a^6 b c^2 + a^4 b^3 c^2 - a^2 b^5 c^2 - b^7 c^2 + a^6 c^3 + 2 a^5 b c^3 + a^4 b^2 c^3 - 8 a^3 b^3 c^3 + a^2 b^4 c^3 + 2 a b^5 c^3 + b^6 c^3 - a^4 b c^4 + a^2 b^3 c^4 - a^4 c^5 - a^2 b^2 c^5 + 2 a b^3 c^5 - a^2 b c^6 + b^3 c^6 - a^2 c^7 - 2 a b c^7 - b^2 c^7 : :

X(29462) lies on these lines:


X(29463) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^7 + a^5 b^2 - a^3 b^4 - a b^6 + a^5 b c + a^4 b^2 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 - 2 a^2 b^2 c^3 + 2 a b^3 c^3 - a^3 c^4 + a b^2 c^4 - a b c^5 + b^2 c^5 - a c^6 - b c^6 : :

X(29463) lies on these lines:


X(29464) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^7 - 2 a^6 b + a^5 b^2 - a^3 b^4 + 2 a^2 b^5 - a b^6 - 2 a^6 c - 3 a^5 b c + a^4 b^2 c + 4 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 - 4 a^2 b^3 c^2 + a b^4 c^2 + 3 b^5 c^2 + 4 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 + 2 a^2 b c^4 + a b^2 c^4 - 2 b^3 c^4 + 2 a^2 c^5 - a b c^5 + 3 b^2 c^5 - a c^6 - b c^6 : :

X(29464) lies on these lines:


X(29465) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    3 a^6 b - 3 a^2 b^5 + 3 a^6 c + 2 a^5 b c + 2 a^3 b^3 c - 3 a^2 b^4 c - 4 a b^5 c + 3 a^2 b^3 c^2 - 3 b^5 c^2 + 2 a^3 b c^3 + 3 a^2 b^2 c^3 + 8 a b^3 c^3 + 3 b^4 c^3 - 3 a^2 b c^4 + 3 b^3 c^4 - 3 a^2 c^5 - 4 a b c^5 - 3 b^2 c^5 : :

X(29465) lies on these lines:


X(29466) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^6 b - a^2 b^5 + a^6 c + 2 a^5 b c - 2 a^3 b^3 c - a^2 b^4 c - 4 a^3 b^2 c^2 - 3 a^2 b^3 c^2 - b^5 c^2 - 2 a^3 b c^3 - 3 a^2 b^2 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 - b^2 c^5 : :

X(29466) lies on these lines:


X(29467) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^6 b - a^2 b^5 + a^6 c - 2 a^3 b^3 c - a^2 b^4 c - 2 a b^5 c + a^2 b^3 c^2 - b^5 c^2 - 2 a^3 b c^3 + a^2 b^2 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 - 2 a b c^5 - b^2 c^5 : :

X(29467) lies on these lines:


X(29468) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^6 b - a^2 b^5 + a^6 c + 4 a^5 b c + 2 a^4 b^2 c - 8 a^3 b^3 c - a^2 b^4 c + 2 a b^5 c + 2 a^4 b c^2 - a^2 b^3 c^2 - b^5 c^2 - 8 a^3 b c^3 - a^2 b^2 c^3 - 8 a b^3 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 + 2 a b c^5 - b^2 c^5 : :

X(29468) lies on these lines:


X(29469) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^6 b - a^2 b^5 + a^6 c + a^4 b^2 c + a^3 b^3 c - a^2 b^4 c - 2 a b^5 c + a^4 b c^2 - b^5 c^2 + a^3 b c^3 + 2 a b^3 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 - 2 a b c^5 - b^2 c^5 : :

X(29469) lies on these lines:


X(29470) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^4 b^2 - a^3 b^3 + a^3 b^2 c - a b^4 c + a^4 c^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 + a b^2 c^3 - a b c^4 + b^2 c^4 : :

X(29470) lies on these lines:


X(29471) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^6 b - a^2 b^5 + a^6 c + 2 a^4 b^2 c - a^2 b^4 c - 2 a b^5 c + 2 a^4 b c^2 - a^2 b^3 c^2 - b^5 c^2 - a^2 b^2 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 - 2 a b c^5 - b^2 c^5 : :

X(29471) lies on these lines:


X(29472) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a (a^4 b + a^3 b^2 - a^2 b^3 - a b^4 + a^4 c + 2 a^2 b^2 c - 3 b^4 c + a^3 c^2 + 2 a^2 b c^2 - 4 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3 - a c^4 - 3 b c^4) : :

X(29472) lies on these lines:


X(29473) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^4 - a b^3 + a^2 b c - b^3 c - a c^3 - b c^3 : :

X(29473) lies on these lines:


X(29474) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a (a^4 b + a^3 b^2 - a^2 b^3 - a b^4 + a^4 c + 2 a^2 b^2 c - 3 b^4 c + a^3 c^2 + 2 a^2 b c^2 + 4 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3 - a c^4 - 3 b c^4) : :

X(29474) lies on these lines:


X(29475) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^10 b - a^2 b^9 + a^10 c + 2 a^9 b c + a^8 b^2 c - a^6 b^4 c - 2 a^5 b^5 c - a^4 b^6 c + a^8 b c^2 - b^9 c^2 - a^6 b c^4 + a^2 b^5 c^4 - 2 a^5 b c^5 + a^2 b^4 c^5 + b^6 c^5 - a^4 b c^6 + b^5 c^6 - a^2 c^9 - b^2 c^9 : :

X(29475) lies on these lines:


X(29476) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^6 b - a^2 b^5 + a^6 c + a^4 b^2 c - 4 a^3 b^3 c - 2 a b^5 c + a^4 b c^2 - b^5 c^2 - 4 a^3 b c^3 - 4 a b^3 c^3 - a^2 c^5 - 2 a b c^5 - b^2 c^5 : :

X(29476) lies on these lines:


X(29477) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    b c (-2 a^5 - a^4 b + 3 a^3 b^2 + a^2 b^3 - a b^4 - a^4 c + b^4 c + 3 a^3 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) : :

X(29477) lies on these lines:


X(29478) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^5 b^3 - a^4 b^4 - a^5 b^2 c + a^3 b^4 c - a^5 b c^2 + 2 a^4 b^2 c^2 + 3 a^2 b^4 c^2 + a^5 c^3 - a b^4 c^3 - a^4 c^4 + a^3 b c^4 + 3 a^2 b^2 c^4 - a b^3 c^4 - 2 b^4 c^4 : :

X(29478) lies on these lines:


X(29479) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^4 - a^3 b + a^2 b^2 - a b^3 - a^3 c - b^3 c + a^2 c^2 + 3 b^2 c^2 - a c^3 - b c^3 : :

X(29479) lies on these lines:


X(29480) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^5 b - a^4 b^2 + a^3 b^3 - a^2 b^4 + a^5 c - 2 a^4 b c + a^3 b^2 c + 2 a^2 b^3 c - 2 a b^4 c - a^4 c^2 + a^3 b c^2 - 6 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 + 2 b^3 c^3 - a^2 c^4 - 2 a b c^4 - b^2 c^4 : :

X(29480) lies on these lines:


X(29481) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^6 - a^5 b + a^2 b^4 - a b^5 - 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^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(29481) lies on these lines:


X(29482) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    (a - b) (a - c) (a^4 b + a^3 b^2 + a^2 b^3 + a b^4 + a^4 c + 6 a^3 b c + 5 a^2 b^2 c + a b^3 c - b^4 c + a^3 c^2 + 5 a^2 b c^2 + 4 a b^2 c^2 + 2 b^3 c^2 + a^2 c^3 + a b c^3 + 2 b^2 c^3 + a c^4 - b c^4) : :

X(29482) lies on these lines:


X(29483) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^4 b - a^2 b^3 + a^4 c - 4 a^3 b c + 3 a^2 b^2 c - 4 a b^3 c + 3 a^2 b c^2 + 4 a b^2 c^2 + b^3 c^2 - a^2 c^3 - 4 a b c^3 + b^2 c^3 : :

X(29483) lies on these lines:


X(29484) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    b c (-2 a^3 + a^2 b - a b^2 + a^2 c + 2 a b c + b^2 c - a c^2 + b c^2) : :

X(29484) lies on these lines:


X(29485) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    (b - c) (a^4 - 4 a^3 b + a^2 b^2 - 4 a^3 c + 2 b^3 c + a^2 c^2 + b^2 c^2 + 2 b c^3) : :

X(29485) lies on these lines:


X(29486) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    b c (-a^4 b - a^2 b^3 - a^4 c + a^2 b^2 c + a^2 b c^2 + 2 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3) : :

X(29486) lies on these lines:


X(29487) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a (b - c) (a^3 b + a^2 b^2 + a^3 c + a^2 b c + 2 a b^2 c + a^2 c^2 + 2 a b c^2 - b^2 c^2) : :

X(29487) lies on these lines:


X(29488) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    b c (b - c) (3 a^3 - 2 a^2 b + a b^2 - 2 a^2 c + 5 a b c + b^2 c + a c^2 + b c^2) : :

X(29488) lies on these lines:


X(29489) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    (b - c) (-a^5 b - a^3 b^3 - a^5 c + a^4 b c - a^3 b^2 c - a^2 b^3 c - a^3 b c^2 + 3 a b^3 c^2 - a^3 c^3 - a^2 b c^3 + 3 a b^2 c^3 + b^3 c^3) : :

X(29489) lies on these lines:


X(29490) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^5 - a^4 b + 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 - 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(29490) lies on these lines:


X(29491) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    b c (a^4 b - 4 a^3 b^2 - a^2 b^3 + a^4 c + 4 a^3 b c + 3 a^2 b^2 c - 4 a^3 c^2 + 3 a^2 b c^2 - 4 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3) : :

X(29491) lies on these lines:


X(29492) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a (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(29492) lies on these lines:


X(29493) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    2 a^5 b + a^4 b^2 - a^3 b^3 - 2 a^2 b^4 + 2 a^5 c + 3 a^3 b^2 c - 5 a b^4 c + a^4 c^2 + 3 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 + a b^2 c^3 - 2 a^2 c^4 - 5 a b c^4 - b^2 c^4 : :

X(29493) lies on these lines:


X(29494) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 90

Barycentrics    a^4 b^2 - a^3 b^3 + a^3 b^2 c - a b^4 c + a^4 c^2 + a^3 b c^2 + 8 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 b c^4 + b^2 c^4 : :

X(29494) lies on these lines:


X(29495) =  (name pending)

Barycentrics    a^14 b^2-5 a^12 b^4+11 a^10 b^6-15 a^8 b^8+15 a^6 b^10-11 a^4 b^12+5 a^2 b^14-b^16+a^14 c^2-10 a^12 b^2 c^2+25 a^10 b^4 c^2-26 a^8 b^6 c^2+7 a^6 b^8 c^2+14 a^4 b^10 c^2-17 a^2 b^12 c^2+6 b^14 c^2-5 a^12 c^4+25 a^10 b^2 c^4-26 a^8 b^4 c^4+5 a^6 b^6 c^4-4 a^4 b^8 c^4+21 a^2 b^10 c^4-16 b^12 c^4+11 a^10 c^6-26 a^8 b^2 c^6+5 a^6 b^4 c^6+2 a^4 b^6 c^6-9 a^2 b^8 c^6+26 b^10 c^6-15 a^8 c^8+7 a^6 b^2 c^8-4 a^4 b^4 c^8-9 a^2 b^6 c^8-30 b^8 c^8+15 a^6 c^10+14 a^4 b^2 c^10+21 a^2 b^4 c^10+26 b^6 c^10-11 a^4 c^12-17 a^2 b^2 c^12-16 b^4 c^12+5 a^2 c^14+6 b^2 c^14-c^16 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28708.

X(29495) lies on this line: {5,195}


X(29496) =  35TH HATZIPOLAKIS-MOSES-EULER POINT

Barycentrics    a^14 b^2-5 a^12 b^4+11 a^10 b^6-15 a^8 b^8+15 a^6 b^10-11 a^4 b^12+5 a^2 b^14-b^16+a^14 c^2-10 a^12 b^2 c^2+17 a^10 b^4 c^2-2 a^8 b^6 c^2-17 a^6 b^8 c^2+22 a^4 b^10 c^2-17 a^2 b^12 c^2+6 b^14 c^2-5 a^12 c^4+17 a^10 b^2 c^4-10 a^8 b^4 c^4+5 a^6 b^6 c^4-12 a^4 b^8 c^4+21 a^2 b^10 c^4-16 b^12 c^4+11 a^10 c^6-2 a^8 b^2 c^6+5 a^6 b^4 c^6+2 a^4 b^6 c^6-9 a^2 b^8 c^6+26 b^10 c^6-15 a^8 c^8-17 a^6 b^2 c^8-12 a^4 b^4 c^8-9 a^2 b^6 c^8-30 b^8 c^8+15 a^6 c^10+22 a^4 b^2 c^10+21 a^2 b^4 c^10+26 b^6 c^10-11 a^4 c^12-17 a^2 b^2 c^12-16 b^4 c^12+5 a^2 c^14+6 b^2 c^14-c^16 : :

See Antreas Hatzipolakis and Peter Moses, Hyacinthos 28708.

X(X29496) lies on these lines: {2,3}, {511,14140}, {5663,16337}, {13391,16336}






leftri  Collineation mappings involving Gemini triangle 91: X(29497) - X(29551)  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 91, as in centers X(28497)-X(29551). Then

m(X) = (a^2 b + a^2 c - 2 a b c + b^2 c + b c^2) x -b (a^2 + c^2) y - c (a^2 + b^2) z : :

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, December 8, 2018)

underbar




X(29497) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a (a^3 b - a b^3 + a^3 c - 3 a^2 b c + 2 a b^2 c + 2 a b c^2 - 2 b^2 c^2 - a c^3) : :

X(29497) lies on these lines:


X(29498) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^8 b + a^6 b^3 - a^4 b^5 - a^2 b^7 + a^8 c - 2 a^7 b c + a^6 b^2 c - a^4 b^4 c + 2 a^3 b^5 c - a^2 b^6 c + a^6 b c^2 + a^4 b^3 c^2 - a^2 b^5 c^2 - b^7 c^2 + a^6 c^3 + a^4 b^2 c^3 + a^2 b^4 c^3 + b^6 c^3 - a^4 b c^4 + a^2 b^3 c^4 - a^4 c^5 + 2 a^3 b c^5 - a^2 b^2 c^5 - a^2 b c^6 + b^3 c^6 - a^2 c^7 - b^2 c^7 : :

X(29498) lies on these lines:


X(29499) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^8 b + a^6 b^3 - a^4 b^5 - a^2 b^7 + a^8 c - 2 a^7 b c + a^6 b^2 c - a^4 b^4 c + 2 a^3 b^5 c - a^2 b^6 c + a^6 b c^2 + a^4 b^3 c^2 - a^2 b^5 c^2 - b^7 c^2 + a^6 c^3 + a^4 b^2 c^3 - 2 a^3 b^3 c^3 + a^2 b^4 c^3 + b^6 c^3 - a^4 b c^4 + a^2 b^3 c^4 - a^4 c^5 + 2 a^3 b c^5 - a^2 b^2 c^5 - a^2 b c^6 + b^3 c^6 - a^2 c^7 - b^2 c^7 : :

X(29499) lies on these lines:


X(29500) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^5 b - a^2 b^4 + a^5 c - 2 a^4 b c + a^3 b^2 c + a^3 b c^2 - b^4 c^2 - a^2 c^4 - b^2 c^4 : :

X(29500) lies on these lines:


X(29501) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^6 b - a^2 b^5 + a^6 c - 2 a^5 b c + a^4 b^2 c + a^4 b c^2 - b^5 c^2 - a^2 c^5 - b^2 c^5 : :

X(29501) lies on these lines:


X(29502) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^4 b^2 - a^3 b^3 + 2 a^4 b c - 2 a^3 b^2 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 b^2 c^3 - 2 b^3 c^3 : :

X(29502) lies on these lines:


X(29503) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^4 b^2 - a^3 b^3 + 2 a^4 b c - 2 a^3 b^2 c + a^4 c^2 - 2 a^3 b c^2 + 4 a^2 b^2 c^2 - a b^3 c^2 - a^3 c^3 - a b^2 c^3 - 2 b^3 c^3 : :

X(29503) lies on these lines:


X(29504) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    2 a^4 b - 2 a^2 b^3 + 2 a^4 c - 6 a^3 b c + 5 a^2 b^2 c - a b^3 c + 5 a^2 b c^2 - b^3 c^2 - 2 a^2 c^3 - a b c^3 - b^2 c^3 : :

X(29504) lies on these lines:


X(29505) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^4 b - a^2 b^3 + a^4 c - 6 a^3 b c + 7 a^2 b^2 c - 2 a b^3 c + 7 a^2 b c^2 + b^3 c^2 - a^2 c^3 - 2 a b c^3 + b^2 c^3 : :

X(29505) lies on these lines:


X(29506) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^5 b - a^4 b^2 + a^3 b^3 - a^2 b^4 + a^5 c - 4 a^4 b c + 3 a^3 b^2 c - a^4 c^2 + 3 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 + a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 - b^2 c^4 : :

X(29506) lies on these lines:


X(29507) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^4 b - a^2 b^3 + a^4 c - a^3 b c + a^2 b^2 c + a b^3 c + a^2 b c^2 - b^3 c^2 - a^2 c^3 + a b c^3 - b^2 c^3 : :

X(29507) lies on these lines:


X(29508) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^5 b + a^4 b^2 - a^3 b^3 - a^2 b^4 + 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 - a^2 c^4 - b^2 c^4 : :

X(29508) lies on these lines:


X(29509) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    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 - b^3 c^2 - a^2 c^3 - b^2 c^3 : :

X(29509) lies on these lines:


X(29510) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    2 a^3 b - 2 a^2 b^2 + 2 a^3 c - 3 a^2 b c + 3 a b^2 c - 2 a^2 c^2 + 3 a b c^2 - 4 b^2 c^2 : :

X(29510) lies on these lines:


X(29511) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    (b + c) (-a^4 + a^2 b^2 - a^2 b c - a b^2 c + a^2 c^2 - a b c^2 + 2 b^2 c^2) : :

X(29511) lies on these lines:


X(29512) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    (b - c) (b + c) (a^4 + a^3 b + a^3 c - 2 a^2 b c + a b^2 c + a b c^2 + b^2 c^2) : :

X(29512) lies on these lines:


X(29513) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    (b - c) (-a^5 b - a^3 b^3 - a^5 c + 2 a^4 b c - 2 a^3 b^2 c + a^2 b^3 c - 2 a^3 b c^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 + b^3 c^3) : :

X(29513) lies on these lines:


X(29514) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^5 b - a^2 b^4 + a^5 c - 2 a^4 b c + a^3 b^2 c + a^3 b c^2 + 2 a^2 b^2 c^2 - b^4 c^2 - a^2 c^4 - b^2 c^4 : :

X(29514) lies on these lines:


X(29515) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    2 a^6 b - 2 a^2 b^5 + 2 a^6 c - 3 a^5 b c + 2 a^3 b^3 c - 2 a^2 b^4 c + a b^5 c + 2 a^2 b^3 c^2 - 2 b^5 c^2 + 2 a^3 b c^3 + 2 a^2 b^2 c^3 - 2 a b^3 c^3 + 2 b^4 c^3 - 2 a^2 b c^4 + 2 b^3 c^4 - 2 a^2 c^5 + a b c^5 - 2 b^2 c^5 : :

X(29515) lies on these lines:


X(29516) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^6 b - a^2 b^5 + a^6 c - 2 a^5 b c + 2 a^3 b^3 c - a^2 b^4 c + 2 a^3 b^2 c^2 + 3 a^2 b^3 c^2 - b^5 c^2 + 2 a^3 b c^3 + 3 a^2 b^2 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 - b^2 c^5 : :

X(29516) lies on these lines:


X(29517) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^8 b + a^6 b^3 - a^4 b^5 - a^2 b^7 + a^8 c - 2 a^7 b c + a^6 b^2 c - a^4 b^4 c + 2 a^3 b^5 c - a^2 b^6 c + a^6 b c^2 + a^4 b^3 c^2 - a^2 b^5 c^2 - b^7 c^2 + a^6 c^3 + a^4 b^2 c^3 - 4 a^3 b^3 c^3 + a^2 b^4 c^3 + b^6 c^3 - a^4 b c^4 + a^2 b^3 c^4 - a^4 c^5 + 2 a^3 b c^5 - a^2 b^2 c^5 - a^2 b c^6 + b^3 c^6 - a^2 c^7 - b^2 c^7 : :

X(29517) lies on these lines:


X(29518) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^8 b + 2 a^7 b^2 + a^6 b^3 - a^4 b^5 - 2 a^3 b^6 - a^2 b^7 + a^8 c + 2 a^7 b c + a^6 b^2 c - a^4 b^4 c - 2 a^3 b^5 c - a^2 b^6 c + 2 a^7 c^2 + a^6 b c^2 - 2 a^5 b^2 c^2 + a^4 b^3 c^2 - a^2 b^5 c^2 - b^7 c^2 + a^6 c^3 + a^4 b^2 c^3 - a^2 b^4 c^3 - b^6 c^3 - a^4 b c^4 - a^2 b^3 c^4 + 2 b^5 c^4 - a^4 c^5 - 2 a^3 b c^5 - a^2 b^2 c^5 + 2 b^4 c^5 - 2 a^3 c^6 - a^2 b c^6 - b^3 c^6 - a^2 c^7 - b^2 c^7 : :

X(29518) lies on these lines:


X(29519) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^9 b + 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^9 c + a^7 b^2 c + 2 a^6 b^3 c - a^5 b^4 c - a^3 b^6 c - 2 a^2 b^7 c + a^8 c^2 + a^7 b c^2 + a^5 b^3 c^2 - a^3 b^5 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 - 2 a^4 b^3 c^3 - a^3 b^4 c^3 - a b^6 c^3 + a^6 c^4 - a^5 b c^4 - a^3 b^3 c^4 - 2 a^2 b^4 c^4 + 2 a b^5 c^4 + b^6 c^4 - a^5 c^5 - a^3 b^2 c^5 + 2 a b^4 c^5 - a^4 c^6 - a^3 b 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 - b^2 c^8 : :

X(29519) lies on these lines:


X(29520) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^9 b - 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^9 c - 4 a^8 b c - 3 a^7 b^2 c + 2 a^6 b^3 c - a^5 b^4 c + 4 a^4 b^5 c + 3 a^3 b^6 c - 2 a^2 b^7 c - a^8 c^2 - 3 a^7 b c^2 + 3 a^5 b^3 c^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 + 3 a^5 b^2 c^3 - 6 a^4 b^3 c^3 - a^3 b^4 c^3 + 2 a^2 b^5 c^3 - a b^6 c^3 + 2 b^7 c^3 + a^6 c^4 - a^5 b c^4 - a^3 b^3 c^4 - 2 a^2 b^4 c^4 + 2 a b^5 c^4 + b^6 c^4 - a^5 c^5 + 4 a^4 b c^5 + a^3 b^2 c^5 + 2 a^2 b^3 c^5 + 2 a b^4 c^5 - 4 b^5 c^5 + a^4 c^6 + 3 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 + 2 b^3 c^7 - a^2 c^8 - b^2 c^8 : :

X(29520) lies on these lines:


X(29521) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    3 a^6 b - 3 a^2 b^5 + 3 a^6 c - 5 a^5 b c + 4 a^3 b^3 c - 3 a^2 b^4 c + a b^5 c + 3 a^2 b^3 c^2 - 3 b^5 c^2 + 4 a^3 b c^3 + 3 a^2 b^2 c^3 - 2 a b^3 c^3 + 3 b^4 c^3 - 3 a^2 b c^4 + 3 b^3 c^4 - 3 a^2 c^5 + a b c^5 - 3 b^2 c^5 : :

X(295212) lies on these lines:


X(29522) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^6 b - a^2 b^5 + a^6 c - a^5 b c - a^2 b^4 c + a b^5 c - 2 a^3 b^2 c^2 - a^2 b^3 c^2 - b^5 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^2 c^5 + a b c^5 - b^2 c^5 : :

X(29522) lies on these lines:


X(29523) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^6 b - a^2 b^5 + a^6 c - 2 a^5 b c - a^2 b^4 c + a^2 b^3 c^2 - b^5 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^2 c^5 - b^2 c^5 : :

X(29523) lies on these lines:


X(29524) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^6 b - a^2 b^5 + a^6 c + 2 a^4 b^2 c - 4 a^3 b^3 c - a^2 b^4 c + 2 a b^5 c + 2 a^4 b c^2 - 4 a^3 b^2 c^2 + 3 a^2 b^3 c^2 - b^5 c^2 - 4 a^3 b c^3 + 3 a^2 b^2 c^3 - 8 a b^3 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 + 2 a b c^5 - b^2 c^5 : :

X(29524) lies on these lines:


X(29525) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^6 b - a^2 b^5 + a^6 c - 2 a^5 b c - a^4 b^2 c + 3 a^3 b^3 c - a^2 b^4 c - a^4 b c^2 + 2 a^3 b^2 c^2 - b^5 c^2 + 3 a^3 b c^3 + 2 a b^3 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 - b^2 c^5 : :

X(29525) lies on these lines:


X(29526) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^6 b - a^2 b^5 + a^6 c - 2 a^5 b c + a^4 b^2 c + a^3 b^3 c - a^2 b^4 c + a^4 b c^2 - 2 a^3 b^2 c^2 + 2 a^2 b^3 c^2 - b^5 c^2 + a^3 b c^3 + 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^2 c^5 - b^2 c^5 : :

X(29526) lies on these lines:


X(29527) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^4 b^2 - a^3 b^3 - a^3 b^2 c + 2 a^2 b^3 c - a b^4 c + a^4 c^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 b c^4 + b^2 c^4 : :

X(29527) lies on these lines:


X(29528) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^6 b - a^2 b^5 + a^6 c - 2 a^5 b c + 2 a^4 b^2 c - a^2 b^4 c + 2 a^4 b c^2 - 4 a^3 b^2 c^2 + 3 a^2 b^3 c^2 - b^5 c^2 + 3 a^2 b^2 c^3 - 4 a b^3 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 - b^2 c^5 : :

X(29528) lies on these lines:


X(29529) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a (a^4 b + a^3 b^2 - a^2 b^3 - a b^4 + 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) : :

X(29529) lies on these lines:


X(29530) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^6 b + a^5 b^2 - a^3 b^4 - a^2 b^5 + a^6 c - a^2 b^4 c + a^5 c^2 + a^2 b^3 c^2 - a b^4 c^2 - b^5 c^2 + a^2 b^2 c^3 - 2 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 - a^2 c^5 - b^2 c^5 : :

X(29530) lies on these lines:


X(29531) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a (a^4 b + a^3 b^2 - a^2 b^3 - a b^4 + a^4 c - 2 a^3 b c + 2 a b^3 c - b^4 c + a^3 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 - a c^4 - b c^4) : :

X(29531) lies on these lines:


X(29532) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(66), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^10 b - a^2 b^9 + a^10 c - a^9 b c + a^8 b^2 c - a^6 b^4 c - a^4 b^6 c + a b^9 c + a^8 b c^2 - b^9 c^2 - a^6 b c^4 + a^2 b^5 c^4 + a^2 b^4 c^5 - 2 a b^5 c^5 + b^6 c^5 - a^4 b c^6 + b^5 c^6 - a^2 c^9 + a b c^9 - b^2 c^9 : :

X(29532) lies on these lines:


X(29533) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(82), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    (a^2 + b^2) (a^2 + c^2) (a^3 b - b^4 + a^3 c - 2 a^2 b c + a b^2 c + a b c^2 - 2 b^2 c^2 - c^4) : :

X(29533) lies on these lines:


X(29534) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    (a^2 + b^2) (a^2 + c^2) (a^2 b - b^3 + a^2 c - 2 a b c - c^3) : :

X(29534) lies on these lines:


X(29535) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    b c (-2 a^5 + a^4 b - a^3 b^2 + 3 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 - a^3 c^2 - 2 a^2 b c^2 + 6 a b^2 c^2 - b^3 c^2 + 3 a^2 c^3 - 2 a b c^3 - b^2 c^3 - a c^4 + b c^4) : :

X(29535) lies on these lines:


X(29536) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^5 b^3 - a^4 b^4 - a^5 b^2 c - 2 a^4 b^3 c + 3 a^3 b^4 c - a^5 b c^2 + 6 a^4 b^2 c^2 - 2 a^3 b^3 c^2 - a^2 b^4 c^2 + a^5 c^3 - 2 a^4 b c^3 - 2 a^3 b^2 c^3 + 2 a^2 b^3 c^3 + a b^4 c^3 - a^4 c^4 + 3 a^3 b c^4 - a^2 b^2 c^4 + a b^3 c^4 - 2 b^4 c^4 : :

X(29536) lies on these lines:


X(29537) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^5 b + a^4 b^2 - a^3 b^3 - a^2 b^4 + a^5 c - 4 a^4 b c + a^3 b^2 c + 4 a^2 b^3 c - 2 a b^4 c + a^4 c^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 + 4 a^2 b c^3 - a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 - 2 a b c^4 + b^2 c^4 : :

X(29537) lies on these lines:


X(29538) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^6 b - a^2 b^5 + 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 - a^4 b c^2 + 2 a^2 b^3 c^2 - b^5 c^2 + 2 a^3 b c^3 + 2 a^2 b^2 c^3 - 2 a b^3 c^3 + 2 b^4 c^3 - 2 a^2 b c^4 + 2 b^3 c^4 - a^2 c^5 - b^2 c^5 : :

X(29538) lies on these lines:


X(29539) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^5 b - a^4 b^2 + a^3 b^3 - a^2 b^4 + a^5 c - 4 a^4 b c + 3 a^3 b^2 c - a^4 c^2 + 3 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 + a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 - b^2 c^4 : :

X(29539) lies on these lines:


X(29540) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^8 b - a^6 b^3 + a^4 b^5 - a^2 b^7 + a^8 c - 2 a^7 b c + 2 a^5 b^3 c - a^4 b^4 c - a^4 b^3 c^2 + 2 a^2 b^5 c^2 - b^7 c^2 - a^6 c^3 + 2 a^5 b c^3 - a^4 b^2 c^3 - 2 a^3 b^3 c^3 - a^4 b c^4 + b^5 c^4 + a^4 c^5 + 2 a^2 b^2 c^5 + b^4 c^5 - a^2 c^7 - b^2 c^7 : :

X(29540) lies on these lines:


X(29541) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^4 b - a^2 b^3 + a^4 c - 6 a^3 b c + 5 a^2 b^2 c - 2 a b^3 c + 5 a^2 b 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 : :

X(29541) lies on these lines:


X(29542) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    b c (-2 a^3 + 2 a^2 b - a b^2 + 2 a^2 c - a b c + b^2 c - a c^2 + b c^2) : :

X(29542) lies on these lines:


X(29543) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    2 a^6 b - 2 a^2 b^5 + 2 a^6 c - 4 a^5 b c + a^4 b^2 c + 2 a^3 b^3 c - a^2 b^4 c + a^4 b c^2 + a^2 b^3 c^2 - 2 b^5 c^2 + 2 a^3 b c^3 + a^2 b^2 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - 2 a^2 c^5 - 2 b^2 c^5 : :

X(29543) lies on these lines:


X(29544) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    b c (-a^4 b + a^3 b^2 - a^2 b^3 - 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) : :

X(29544) lies on these lines:


X(29545) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a (b - c) (a^3 b + a^2 b^2 + a^3 c - a^2 b c + a^2 c^2 + b^2 c^2) : :

X(29545) lies on these lines:


X(29546) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    (b - c) (-a^5 b - a^3 b^3 - a^5 c + 3 a^4 b c - a^3 b^2 c + a^2 b^3 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 + b^3 c^3) : :

X(29546) lies on these lines:


X(29547) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    b c (a^4 b - 2 a^3 b^2 - a^2 b^3 + a^4 c - 2 a^3 b c + 5 a^2 b^2 c - 2 a^3 c^2 + 5 a^2 b c^2 - 6 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3) : :

X(29547) lies on these lines:


X(29548) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(669), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    (b - c) (-a^6 b^2 - a^4 b^4 - 2 a^6 b c + 2 a^5 b^2 c - 2 a^4 b^3 c - a^6 c^2 + 2 a^5 b c^2 - 3 a^4 b^2 c^2 - 2 a^4 b c^3 - a^4 c^4 + b^4 c^4) : :

X(29548) lies on these lines:


X(29549) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(896), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    2 a^5 b + a^4 b^2 - a^3 b^3 - 2 a^2 b^4 + 2 a^5 c - 4 a^4 b c + a^3 b^2 c + 2 a^2 b^3 c - a b^4 c + a^4 c^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 - 2 a^2 c^4 - a b c^4 - b^2 c^4 : :

X(29549) lies on these lines:


X(29550) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^4 b^2 - a^3 b^3 - a^3 b^2 c + 2 a^2 b^3 c - a b^4 c + a^4 c^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 + 2 a^2 b c^3 - a b^2 c^3 - a b c^4 + b^2 c^4 : :

X(29550) lies on these lines:


X(29551) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1150), WHERE A'B'C' = GEMINI TRIANGLE 91

Barycentrics    a^5 b + a^4 b^2 - a^3 b^3 - a^2 b^4 + a^5 c + a^3 b^2 c + 2 a^2 b^3 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^3 c^3 + 2 a^2 b c^3 + a b^2 c^3 - 2 b^3 c^3 - a^2 c^4 - b^2 c^4 : :

X(29551) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 92: X(29552) - X(29568)  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 92, as in centers X(28552)-X(29568). Then

m(X) = (a^2 b + a^2 c + 2 a b c + b^2 c + b c^2) x - b (a^2 + c^2) y - c (a^2 + b^2) z : :

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, December 8, 2018)

underbar




X(29552) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 92

Barycentrics    a (a^3 b - a b^3 + a^3 c - a^2 b c + 2 a b^2 c - 2 b^3 c + 2 a b c^2 + 2 b^2 c^2 - a c^3 - 2 b c^3) : :

X(29552) lies on these lines:


X(29553) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 92

Barycentrics    a^8 b + a^6 b^3 - a^4 b^5 - a^2 b^7 + a^8 c + 2 a^7 b c + a^6 b^2 c - a^4 b^4 c - 2 a^3 b^5 c - a^2 b^6 c + a^6 b c^2 + a^4 b^3 c^2 - a^2 b^5 c^2 - b^7 c^2 + a^6 c^3 + a^4 b^2 c^3 + a^2 b^4 c^3 + b^6 c^3 - a^4 b c^4 + a^2 b^3 c^4 - a^4 c^5 - 2 a^3 b c^5 - a^2 b^2 c^5 - a^2 b c^6 + b^3 c^6 - a^2 c^7 - b^2 c^7 : :

X(29553) lies on these lines:


X(29554) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 92

Barycentrics    a^8 b + a^6 b^3 - a^4 b^5 - a^2 b^7 + a^8 c + 2 a^7 b c + a^6 b^2 c - a^4 b^4 c - 2 a^3 b^5 c - a^2 b^6 c + a^6 b c^2 + a^4 b^3 c^2 - a^2 b^5 c^2 - b^7 c^2 + a^6 c^3 + a^4 b^2 c^3 + 2 a^3 b^3 c^3 + a^2 b^4 c^3 + b^6 c^3 - a^4 b c^4 + a^2 b^3 c^4 - a^4 c^5 - 2 a^3 b c^5 - a^2 b^2 c^5 - a^2 b c^6 + b^3 c^6 - a^2 c^7 - b^2 c^7 : :

X(29554) lies on these lines:


X(29555) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 92

Barycentrics    a^5 b - a^2 b^4 + a^5 c + 2 a^4 b c + a^3 b^2 c + a^3 b c^2 - b^4 c^2 - a^2 c^4 - b^2 c^4 : :

X(29555) lies on these lines:


X(29556) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 92

Barycentrics    a^6 b - a^2 b^5 + a^6 c + 2 a^5 b c + a^4 b^2 c + a^4 b c^2 - b^5 c^2 - a^2 c^5 - b^2 c^5 : :

X(29556) lies on these lines:


X(29557) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 92

Barycentrics    a^4 b^2 - a^3 b^3 + 2 a^4 b c + 2 a^3 b^2 c + a^4 c^2 + 2 a^3 b c^2 - a b^3 c^2 - a^3 c^3 - a b^2 c^3 - 2 b^3 c^3 : :

X(29557) lies on these lines:


X(29558) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 92

Barycentrics    a^4 b - a^2 b^3 + a^4 c + a^3 b c + a^2 b^2 c - a b^3 c + a^2 b c^2 - b^3 c^2 - a^2 c^3 - a b c^3 - b^2 c^3 : :

X(29558) lies on these lines:


X(29559) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 92

Barycentrics    a^4 b - a^2 b^3 + a^4 c + 2 a^3 b c + 3 a^2 b^2 c + 3 a^2 b c^2 + 2 a b^2 c^2 - b^3 c^2 - a^2 c^3 - b^2 c^3 : :

X(29559) lies on these lines:


X(29560) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 92

Barycentrics    2 a^3 b - 2 a^2 b^2 + 2 a^3 c + 3 a^2 b c + a b^2 c - 2 a^2 c^2 + a b c^2 - 4 b^2 c^2 : :

X(29560) lies on these lines:


X(29561) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 92

Barycentrics    a^4 b - a^2 b^3 + a^4 c + 4 a^3 b c + a b^3 c - 2 a b^2 c^2 - 2 b^3 c^2 - a^2 c^3 + a b c^3 - 2 b^2 c^3 : :

X(29561) lies on these lines:


X(29562) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 92

Barycentrics    a^5 b - a^2 b^4 + a^5 c + 2 a^4 b c + a^3 b^2 c + a^3 b c^2 - 2 a^2 b^2 c^2 - b^4 c^2 - a^2 c^4 - b^2 c^4 : :

X(29562) lies on these lines:


X(29563) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 92

Barycentrics    2 a^6 b - 2 a^2 b^5 + 2 a^6 c + 3 a^5 b c - 2 a^3 b^3 c - 2 a^2 b^4 c - a b^5 c + 2 a^2 b^3 c^2 - 2 b^5 c^2 - 2 a^3 b c^3 + 2 a^2 b^2 c^3 + 2 a b^3 c^3 + 2 b^4 c^3 - 2 a^2 b c^4 + 2 b^3 c^4 - 2 a^2 c^5 - a b c^5 - 2 b^2 c^5 : :

X(29563) lies on these lines:


X(29564) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 92

Barycentrics    a^6 b - a^2 b^5 + a^6 c + 2 a^5 b c - 2 a^3 b^3 c - a^2 b^4 c - 2 a^3 b^2 c^2 - a^2 b^3 c^2 - b^5 c^2 - 2 a^3 b c^3 - a^2 b^2 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 - b^2 c^5 : :

X(29564) lies on these lines:


X(29565) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 92

Barycentrics    3 a^6 b - 3 a^2 b^5 + 3 a^6 c + 5 a^5 b c - 4 a^3 b^3 c - 3 a^2 b^4 c - a b^5 c + 3 a^2 b^3 c^2 - 3 b^5 c^2 - 4 a^3 b c^3 + 3 a^2 b^2 c^3 + 2 a b^3 c^3 + 3 b^4 c^3 - 3 a^2 b c^4 + 3 b^3 c^4 - 3 a^2 c^5 - a b c^5 - 3 b^2 c^5 : :

X(29565) lies on these lines:


X(29566) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 92

Barycentrics    a^6 b - a^2 b^5 + a^6 c + a^5 b c - a^2 b^4 c - a b^5 c + 2 a^3 b^2 c^2 + 3 a^2 b^3 c^2 - b^5 c^2 + 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^2 c^5 - a b c^5 - b^2 c^5 : :

X(29566) lies on these lines:


X(29567) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 92

Barycentrics    a^6 b - a^2 b^5 + a^6 c + 2 a^5 b c - a^2 b^4 c + a^2 b^3 c^2 - b^5 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^2 c^5 - b^2 c^5 : :

X(29567) lies on these lines:


X(29568) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 92

Barycentrics    (a^2 + b^2) (a^2 + c^2) (a^2 b - b^3 + a^2 c + 2 a b c - c^3) : :

X(29568) lies on these lines:






leftri  Points Capella(h,j,k): X(29569) - X(29630)  rightri

Definition: Point Capella(h,j,k) = f(h,j,k,a,b,c) : f(h,j,k,b,c,a) : f(h,j,k,c,a,b) (barycentrics), where

f(h,j,k,a,b,c) = h a^2 + j (b^2 + c^2) + k b c + (h - j + k)(a b + a c),

where h, j, k are real numbers, not all zero. These points lie on the line X(1)X(2). (Clark Kimberling, December 8, 2018)

underbar




X(29569) =  POINT CAPELLA(1,-1,1)

Barycentrics    a^2 + 3 a b - b^2 + 3 a c + b c - c^2 : :

X(29569) lies on these lines:


X(29570) =  POINT CAPELLA(2,0,1)

Barycentrics    2 a^2 + 3 a b + 3 a c + b c : :

X(29570) lies on these lines:


X(29571) =  POINT CAPELLA(0,1,-2)

Barycentrics    -3 a b + b^2 - 3 a c - 2 b c + c^2 : :

X(29571) lies on these lines:


X(29572) =  POINT CAPELLA(0,2,-1)

Barycentrics    -3 a b + 2 b^2 - 3 a c - b c + 2 c^2 : :

X(29572) lies on these lines:


X(29573) =  POINT CAPELLA(1,-2,0)

Barycentrics    a^2 + 3 a b - 2 b^2 + 3 a c - 2 c^2 : :

X(29573) lies on these lines:


X(29574) =  POINT CAPELLA(2,-1,0)

Barycentrics    2 a^2 + 3 a b - b^2 + 3 a c - c^2 : :

X(29574) lies on these lines:


X(29575) =  POINT CAPELLA(1,-2,1)

Barycentrics    a^2 + 4 a b - 2 b^2 + 4 a c + b c - 2 c^2 : :

X(29575) lies on these lines:


X(29576) =  POINT CAPELLA(0,1,3)

Barycentrics    2 a b + b^2 + 2 a c + 3 b c + c^2 : :

X(29576) lies on these lines:


X(29577) =  POINT CAPELLA(0,3,1)

Barycentrics    -2 a b + 3 b^2 - 2 a c + b c + 3 c^2 : :

X(29577) lies on these lines:


X(29578) =  POINT CAPELLA(1,0,3)

Barycentrics    a^2 + 4 a b + 4 a c + 3 b c : :

X(29578) lies on these lines:


X(29579) =  POINT CAPELLA(1,3,0)

Barycentrics    a^2 - 2 a b + 3 b^2 - 2 a c + 3 c^2 : :

X(29579) lies on these lines:


X(29580) =  POINT CAPELLA(3,0,1)

Barycentrics    3 a^2 + 4 a b + 4 a c + b c : :

X(29580) lies on these lines:


X(29581) =  POINT CAPELLA(0,1,-3)

Barycentrics    -4 a b + b^2 - 4 a c - 3 b c + c^2 : :

X(29581) lies on these lines:


X(29582) =  POINT CAPELLA(0,3,-1)

Barycentrics    -4 a b + 3 b^2 - 4 a c - b c + 3 c^2 : :

X(29582) lies on these lines:


X(29583) =  POINT CAPELLA(1,-3,0)

Barycentrics    a^2 + 4 a b - 3 b^2 + 4 a c - 3 c^2 : :

X(29583) lies on these lines:


X(29584) =  POINT CAPELLA(3,0,-1)

Barycentrics    3 a^2 + 2 a b + 2 a c - b c : :

X(29584) lies on these lines:


X(29585) =  POINT CAPELLA(3,-1,0)

Barycentrics    3 a^2 + 4 a b - b^2 + 4 a c - c^2 : :

X(29585) lies on these lines:


X(29586) =  POINT CAPELLA(3,1,1)

Barycentrics    3 a^2 + 3 a b + b^2 + 3 a c + b c + c^2 : :

X(29586) lies on these lines:


X(29587) =  POINT CAPELLA(1,3,1)

Barycentrics    a^2 - a b + 3 b^2 - a c + b c + 3 c^2 : :

X(29587) lies on these lines:


X(29588) =  POINT CAPELLA(-3,1,1)

Barycentrics    -3 a^2 - 3 a b + b^2 - 3 a c + b c + c^2 : :

X(29588) lies on these lines:


X(29589) =  POINT CAPELLA(1,-3,1)

Barycentrics    a^2 + 5 a b - 3 b^2 + 5 a c + b c - 3 c^2 : :

X(29589) lies on these lines:


X(29590) =  POINT CAPELLA(3,1,-3)

Barycentrics    3 a^2 - a b + b^2 - a c - 3 b c + c^2 : :

X(29590) lies on these lines:


X(29591) =  POINT CAPELLA(1,3,3)

Barycentrics    a^2 + a b + 3 b^2 + a c + 3 b c + 3 c^2 : :

X(29591) lies on these lines:


X(29592) =  POINT CAPELLA(3,1,3)

Barycentrics    3 a^2 + 5 a b + b^2 + 5 a c + 3 b c + c^2 : :

X(29592) lies on these lines:


X(29593) =  POINT CAPELLA(0,2,3)

Barycentrics    a b + 2 b^2 + a c + 3 b c + 2 c^2 : :

X(29593) lies on these lines:


X(29594) =  POINT CAPELLA(0,3,2)

Barycentrics    -a b + 3 b^2 - a c + 2 b c + 3 c^2 : :

X(29594) lies on these lines:


X(29595) =  POINT CAPELLA(2,0,3)

Barycentrics    2 a^2 + 5 a b + 5 a c + 3 b c : :

X(29595) lies on these lines:


X(29596) =  POINT CAPELLA(2,3,0)

Barycentrics    2 a^2 - a b + 3 b^2 - a c + 3 c^2 : :

X(29596) lies on these lines:


X(29597) =  POINT CAPELLA(3,0,2)

Barycentrics    3 a^2 + 5 a b + 5 a c + 2 b c : :

X(29597) lies on these lines:


X(29598) =  POINT CAPELLA(3,2,0)

Barycentrics    3 a^2 + a b + 2 b^2 + a c + 2 c^2 : :

X(29598) lies on these lines:


X(29599) =  POINT CAPELLA(0,2,-3)

Barycentrics    -5 a b + 2 b^2 - 5 a c - 3 b c + 2 c^2 : :

X(29599) lies on these lines:


X(29600) =  POINT CAPELLA(0,3,-2)

Barycentrics    -5 a b + 3 b^2 - 5 a c - 2 b c + 3 c^2 : :

X(29600) lies on these lines:


X(29601) =  POINT CAPELLA(2,-3,0)

Barycentrics    2 a^2 + 5 a b - 3 b^2 + 5 a c - 3 c^2 : :

X(29601) lies on these lines:


X(29602) =  POINT CAPELLA(3,-2,0)

Barycentrics    3 a^2 + 5 a b - 2 b^2 + 5 a c - 2 c^2 : :

X(29602) lies on these lines:


X(29603) =  POINT CAPELLA(3,2,2)

Barycentrics    3 a^2 + 3 a b + 2 b^2 + 3 a c + 2 b c + 2 c^2 : :

X(29603) lies on these lines:


X(29604) =  POINT CAPELLA(2,3,2)

Barycentrics    2 a^2 + a b + 3 b^2 + a c + 2 b c + 3 c^2 : :

X(29604) lies on these lines:


X(29605) =  POINT CAPELLA(-3,2,2)

Barycentrics    -3 a^2 - 3 a b + 2 b^2 - 3 a c + 2 b c + 2 c^2 : :

X(29605) lies on these lines:


X(29606) =  POINT CAPELLA(2,-3,2)

Barycentrics    2 a^2 + 7 a b - 3 b^2 + 7 a c + 2 b c - 3 c^2 : :

X(29606) lies on these lines:


X(29607) =  POINT CAPELLA(3,2,-3)

Barycentrics    3 a^2 - 2 a b + 2 b^2 - 2 a c - 3 b c + 2 c^2 : :

X(29607) lies on these lines:


X(29608) =  POINT CAPELLA(2,3,3)

Barycentrics    2 a^2 + 2 a b + 3 b^2 + 2 a c + 3 b c + 3 c^2 : :

X(29608) lies on these lines:


X(29609) =  POINT CAPELLA(3,2,3)

Barycentrics    3 a^2 + 4 a b + 2 b^2 + 4 a c + 3 b c + 2 c^2 : :

X(29609) lies on these lines:


X(29610) =  POINT CAPELLA(1,2,3)

Barycentrics    a^2 + 2 a b + 2 b^2 + 2 a c + 3 b c + 2 c^2 : :

X(29610) lies on these lines:


X(29611) =  POINT CAPELLA(1,3,2)

Barycentrics    a^2 + 3 b^2 + 2 b c + 3 c^2 : :

X(29611) lies on these lines:


X(29612) =  POINT CAPELLA(2,1,3)

Barycentrics    2 a^2 + 4 a b + b^2 + 4 a c + 3 b c + c^2 : :

X(29612) lies on these lines:


X(29613) =  POINT CAPELLA(2,3,1)

Barycentrics    2 a^2 + 3 b^2 + b c + 3 c^2 : :

X(29613) lies on these lines:


X(29614) =  POINT CAPELLA(3,2,1)

Barycentrics    3 a^2 + 2 a b + 2 b^2 + 2 a c + b c + 2 c^2 : :

X(29614) lies on these lines:


X(29615) =  POINT CAPELLA(-1,2,3)

Barycentrics    -a^2 + 2 b^2 + 3 b c + 2 c^2 : :

X(29615) lies on these lines:


X(29616) =  POINT CAPELLA(-1,3,2)

Barycentrics    -a^2 - 2 a b + 3 b^2 - 2 a c + 2 b c + 3 c^2 : :

X(29616) lies on these lines:


X(29617) =  POINT CAPELLA(-2,1,3)

Barycentrics    -2 a^2 + b^2 + 3 b c + c^2 : :

X(29617) lies on these lines:


X(29618) =  POINT CAPELLA(-2,3,1)

Barycentrics    -2 a^2 - 4 a b + 3 b^2 - 4 a c + b c + 3 c^2 : :

X(29618) lies on these lines:


X(29619) =  POINT CAPELLA(-3,2,1)

Barycentrics    -3 a^2 - 4 a b + 2 b^2 - 4 a c + b c + 2 c^2 : :

X(29619) lies on these lines:


X(29620) =  POINT CAPELLA(1,-2,3)

Barycentrics    a^2 + 6 a b - 2 b^2 + 6 a c + 3 b c - 2 c^2 : :

X(29620) lies on these lines:


X(29621) =  POINT CAPELLA(1,-3,2)

Barycentrics    a^2 + 6 a b - 3 b^2 + 6 a c + 2 b c - 3 c^2 : :

X(29621) lies on these lines:


X(29622) =  POINT CAPELLA(2,-1,3)

Barycentrics    2 a^2 + 6 a b - b^2 + 6 a c + 3 b c - c^2 : :

X(29622) lies on these lines:


X(29623) =  POINT CAPELLA(2,-3,1)

Barycentrics    2 a^2 + 6 a b - 3 b^2 + 6 a c + b c - 3 c^2 : :

X(29623) lies on these lines:


X(29624) =  POINT CAPELLA(3,-1,2)

Barycentrics    3 a^2 + 6 a b - b^2 + 6 a c + 2 b c - c^2 : :

X(29624) lies on these lines:


X(29625) =  POINT CAPELLA(3,-2,1)

Barycentrics    3 a^2 + 6 a b - 2 b^2 + 6 a c + b c - 2 c^2 : :

X(29625) lies on these lines:


X(29626) =  POINT CAPELLA(1,2,-3)

Barycentrics    a^2 - 4 a b + 2 b^2 - 4 a c - 3 b c + 2 c^2 : :

X(29626) lies on these lines:


X(29627) =  POINT CAPELLA(1,3,-2)

Barycentrics    a^2 - 4 a b + 3 b^2 - 4 a c - 2 b c + 3 c^2 : :

X(29627) lies on these lines:


X(29628) =  POINT CAPELLA(2,1,-3)

Barycentrics    2 a^2 - 2 a b + b^2 - 2 a c - 3 b c + c^2 : :

X(29628) lies on these lines:


X(29629) =  POINT CAPELLA(2,3,-1)

Barycentrics    2 a^2 - 2 a b + 3 b^2 - 2 a c - b c + 3 c^2 : :

X(29629) lies on these lines:


X(29630) =  POINT CAPELLA(3,2,-1)

Barycentrics    3 a^2 + 2 b^2 - b c + 2 c^2 : :

X(29630) lies on these lines:






leftri  Points Castor(h,j,k,p,q): X(29631) - X(29690)  rightri

Definition: Point Castor(h,j,k,p,q,a,b,c) = f(h,j,k,p,q,a,b,c) : f(h,j,k,p,q,b,c,a) : f(h,j,k,p,q,c,a,b) (barycentrics), where

f(h,j,k,p,q,a,b,c) = h (a^3 + b^3 + c^3) + j (a^2 b + b^2 c + c^2 a + a^2 c + b^2 a + c^2 b) + k (a b c) + a (p (a^2 + b^2 + c^2) + q (b c + c a + a b)),

where h, j, k, p, q are real numbers, not all zero. These points lie on the line X(1)X(2). (Clark Kimberling, December 9, 2018)

underbar




X(29631) =  POINT CASTOR(1,0,0,0,1)

Barycentrics    a^3 + a^2 b + b^3 + a^2 c + a b c + c^3 : :

X(29631) lies on these lines:


X(29632) =  POINT CASTOR(1,0,0,0,-1)

Barycentrics    a^3 - a^2 b + b^3 - a^2 c - a b c + c^3 : :

X(29632) lies on these lines:


X(29633) =  POINT CASTOR(1,1,0,0,1)

Barycentrics    a^3 + 2 a^2 b + a b^2 + b^3 + 2 a^2 c + a b c + b^2 c + a c^2 + b c^2 + c^3 : :

X(29633) lies on these lines:


X(29634) =  POINT CASTOR(1,0,1,1,0)

Barycentrics    2 a^3 + a b^2 + b^3 + a b c + a c^2 + c^3 : :

X(29634) lies on these lines:


X(29635) =  POINT CASTOR(1,0,1,0,1)

Barycentrics    a^3 + a^2 b + b^3 + a^2 c + 2 a b c + c^3 : :

X(29635) lies on these lines:


X(29636) =  POINT CASTOR(1,0,0,1,1)

Barycentrics    2 a^3 + a^2 b + a b^2 + b^3 + a^2 c + a b c + a c^2 + c^3 : :

X(29636) lies on these lines:


X(29637) =  POINT CASTOR(1,1,0,0,-1)

Barycentrics    a^3 + a b^2 + b^3 - a b c + b^2 c + a c^2 + b c^2 + c^3 : :

X(29637) lies on these lines:


X(29638) =  POINT CASTOR(1,0,0,1,-1)

Barycentrics    2 a^3 - a^2 b + a b^2 + b^3 - a^2 c - a b c + a c^2 + c^3 : :

X(29638) lies on these lines:


X(29639) =  POINT CASTOR(1,-1,0,-1,0)

Barycentrics    -a^2 b - 2 a b^2 + b^3 - a^2 c - b^2 c - 2 a c^2 - b c^2 + c^3 : :

X(29639) lies on these lines:


X(29640) =  POINT CASTOR(1,-1,0,0,-1)

Barycentrics    a^3 - 2 a^2 b - a b^2 + b^3 - 2 a^2 c - a b c - b^2 c - a c^2 - b c^2 + c^3 : :

X(29640) lies on these lines:


X(29641) =  POINT CASTOR(1,0,-1,-1,0)

Barycentrics    -a b^2 + b^3 - a b c - a c^2 + c^3 : :

X(29641) lies on these lines:


X(29642) =  POINT CASTOR(1,0,-1,0,-1)

Barycentrics    a^3 - a^2 b + b^3 - a^2 c - 2 a b c + c^3 : :

X(29642) lies on these lines:


X(29643) =  POINT CASTOR(1,0,0,-1,-1)

Barycentrics    -a^2 b - a b^2 + b^3 - a^2 c - a b c - a c^2 + c^3 : :

X(29643) lies on these lines:


X(29644) =  POINT CASTOR(0,1,1,1,1)

Barycentrics    a^3 + 2 a^2 b + 2 a b^2 + 2 a^2 c + 2 a b c + b^2 c + 2 a c^2 + b c^2 : :

X(29644) lies on these lines:


X(29645) =  POINT CASTOR(1,0,1,1,1)

Barycentrics    2 a^3 + a^2 b + a b^2 + b^3 + a^2 c + 2 a b c + a c^2 + c^3 : :

X(29645) lies on these lines:


X(29646) =  POINT CASTOR(1,1,0,1,1)

Barycentrics    2 a^3 + 2 a^2 b + 2 a b^2 + b^3 + 2 a^2 c + a b c + b^2 c + 2 a c^2 + b c^2 + c^3 : :

X(29646) lies on these lines:


X(29647) =  POINT CASTOR(1,1,1,0,1)

Barycentrics    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(29647) lies on these lines:


X(29648) =  POINT CASTOR(1,1,1,1,0)

Barycentrics    2 a^3 + a^2 b + 2 a b^2 + b^3 + a^2 c + a b c + b^2 c + 2 a c^2 + b c^2 + c^3 : :

X(29648) lies on these lines:


X(29649) =  POINT CASTOR(0,-1,1,1,1)

Barycentrics    a^3 + 2 a b c - b^2 c - b c^2 : :

X(29649) lies on these lines:


X(29650) =  POINT CASTOR(0,1,-1,1,1)

Barycentrics    a^3 + 2 a^2 b + 2 a b^2 + 2 a^2 c + b^2 c + 2 a c^2 + b c^2 : :

X(29650) lies on these lines:


X(29651) =  POINT CASTOR(0,1,1,-1,1)

Barycentrics    -a^3 + 2 a^2 b + 2 a^2 c + 2 a b c + b^2 c + b c^2 : :

X(29651) lies on these lines:


X(29652) =  POINT CASTOR(0,1,1,1,-1)

Barycentrics    a^3 + 2 a b^2 + b^2 c + 2 a c^2 + b c^2 : :

X(29652) lies on these lines:


X(29653) =  POINT CASTOR(-1,0,1,1,1)

Barycentrics    (b + c) (a^2 + a b - b^2 + a c + b c - c^2) : :

X(29653) lies on these lines:


X(29654) =  POINT CASTOR(1,0,-1,1,1)

Barycentrics    2 a^3 + a^2 b + a b^2 + b^3 + a^2 c + a c^2 + c^3 : :

X(29654) lies on these lines:


X(29655) =  POINT CASTOR(1,0,1,-1,1)

Barycentrics    a^2 b - a b^2 + b^3 + a^2 c + 2 a b c - a c^2 + c^3 : :

X(29655) lies on these lines:


X(29656) =  POINT CASTOR(1,0,1,1,-1)

Barycentrics    2 a^3 - a^2 b + a b^2 + b^3 - a^2 c + a c^2 + c^3 : :

X(29656) lies on these lines:


X(29657) =  POINT CASTOR(-1,1,0,1,1)

Barycentrics    2 a^2 b + 2 a b^2 - b^3 + 2 a^2 c + a b c + b^2 c + 2 a c^2 + b c^2 - c^3 : :

X(29657) lies on these lines:


X(29658) =  POINT CASTOR(1,-1,0,1,1)

Barycentrics    2 a^3 + b^3 + a b c - b^2 c - b c^2 + c^3 : :

X(29658) lies on these lines:


X(29659) =  POINT CASTOR(1,1,0,-1,1)

Barycentrics    2 a^2 b + b^3 + 2 a^2 c + a b c + b^2 c + b c^2 + c^3 : :

X(29659) lies on these lines:


X(29660) =  POINT CASTOR(1,1,0,1,-1)

Barycentrics    2 a^3 + 2 a b^2 + b^3 - a b c + b^2 c + 2 a c^2 + b c^2 + c^3 : :

X(29660) lies on these lines:


X(29661) =  POINT CASTOR(-1,1,1,0,1)

Barycentrics    -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(29661) lies on these lines:


X(29662) =  POINT CASTOR(1,-1,1,0,1)

Barycentrics    a^3 - a b^2 + b^3 + 2 a b c - b^2 c - a c^2 - b c^2 + c^3 : :

X(29662) lies on these lines:


X(29663) =  POINT CASTOR(1,1,-1,0,1)

Barycentrics    a^3 + 2 a^2 b + a b^2 + b^3 + 2 a^2 c + b^2 c + a c^2 + b c^2 + c^3 : :

X(29663) lies on these lines:


X(29664) =  POINT CASTOR(-1,1,1,1,0)

Barycentrics    a^2 b + 2 a b^2 - b^3 + a^2 c + a b c + b^2 c + 2 a c^2 + b c^2 - c^3 : :

X(29664) lies on these lines:


X(29665) =  POINT CASTOR(1,-1,1,1,0)

Barycentrics    2 a^3 - a^2 b + b^3 - a^2 c + a b c - b^2 c - b c^2 + c^3 : :

X(29665) lies on these lines:


X(29666) =  POINT CASTOR(1,1,-1,1,0)

Barycentrics    2 a^3 + a^2 b + 2 a b^2 + b^3 + a^2 c - a b c + b^2 c + 2 a c^2 + b c^2 + c^3 : :

X(29666) lies on these lines:


X(29667) =  POINT CASTOR(1,1,1,-1,0)

Barycentrics    a^2 b + b^3 + a^2 c + a b c + b^2 c + b c^2 + c^3 : :

X(29667) lies on these lines:


X(29668) =  POINT CASTOR(0,-1,1,-1,1)

Barycentrics    -a^3 - 2 a b^2 + 2 a b c - b^2 c - 2 a c^2 - b c^2 : :

X(29668) lies on these lines:


X(29669) =  POINT CASTOR(1,2,1,-2,1)

Barycentrics    -a^3 + 3 a^2 b + b^3 + 3 a^2 c + 2 a b c + 2 b^2 c + 2 b c^2 + c^3 : :

X(29669) lies on these lines:


X(29670) =  POINT CASTOR(0,1,-1,-1,1)

Barycentrics    -a^3 + 2 a^2 b + 2 a^2 c + b^2 c + b c^2 : :

X(29670) lies on these lines:


X(29671) =  POINT CASTOR(-1,0,-1,1,1)

Barycentrics    a^2 b + a b^2 - b^3 + a^2 c + a c^2 - c^3 : :

X(29671) lies on these lines:


X(29672) =  POINT CASTOR(-1,0,1,-1,1)

Barycentrics    -2 a^3 + a^2 b - a b^2 - b^3 + a^2 c + 2 a b c - a c^2 - c^3 : :

X(29672) lies on these lines:


X(29673) =  POINT CASTOR(-1,0,1,1,-1)

Barycentrics    -a^2 b + a b^2 - b^3 - a^2 c + a c^2 - c^3 : :

X(29673) lies on these lines:


X(29674) =  POINT CASTOR(-1,-1,0,1,1)

Barycentrics    -b^3 + a b c - b^2 c - b c^2 - c^3 : :

X(29674) lies on these lines:


X(29675) =  POINT CASTOR(-1,1,0,-1,1)

Barycentrics    -2 a^3 + 2 a^2 b - b^3 + 2 a^2 c + a b c + b^2 c + b c^2 - c^3 : :

X(29675) lies on these lines:


X(29676) =  POINT CASTOR(-1,1,0,1,-1)

Barycentrics    2 a b^2 - b^3 - a b c + b^2 c + 2 a c^2 + b c^2 - c^3 : :

X(29676) lies on these lines:


X(29677) =  POINT CASTOR(-1,-1,1,0,1)

Barycentrics    -a^3 - a b^2 - b^3 + 2 a b c - b^2 c - a c^2 - b c^2 - c^3 : :

X(29677) lies on these lines:


X(29678) =  POINT CASTOR(-1,1,-1,0,1)

Barycentrics    -a^3 + 2 a^2 b + a b^2 - b^3 + 2 a^2 c + b^2 c + a c^2 + b c^2 - c^3 : :

X(29678) lies on these lines:


X(29679) =  POINT CASTOR(-1,-1,1,1,0)

Barycentrics    -a^2 b - b^3 - a^2 c + a b c - b^2 c - b c^2 - c^3 : :

X(29679) lies on these lines:


X(29680) =  POINT CASTOR(-1,1,-1,1,0)

Barycentrics    a^2 b + 2 a b^2 - b^3 + a^2 c - a b c + b^2 c + 2 a c^2 + b c^2 - c^3 : :

X(29680) lies on these lines:


X(29681) =  POINT CASTOR(-1,1,1,-1,0)

Barycentrics    -2 a^3 + a^2 b - b^3 + a^2 c + a b c + b^2 c + b c^2 - c^3 : :

X(29681) lies on these lines:


X(29682) =  POINT CASTOR(-1,1,1,1,1)

Barycentrics    2 a^2 b + 2 a b^2 - b^3 + 2 a^2 c + 2 a b c + b^2 c + 2 a c^2 + b c^2 - c^3 : :

X(29682) lies on these lines:


X(29683) =  POINT CASTOR(1,-1,1,1,1)

Barycentrics    2 a^3 + b^3 + 2 a b c - b^2 c - b c^2 + c^3 : :

X(29683) lies on these lines:


X(29684) =  POINT CASTOR(1,1,-1,1,1)

Barycentrics    2 a^3 + 2 a^2 b + 2 a b^2 + b^3 + 2 a^2 c + b^2 c + 2 a c^2 + b c^2 + c^3 : :

X(29684) lies on these lines:


X(29685) =  POINT CASTOR(1,1,1,-1,1)

Barycentrics    2 a^2 b + b^3 + 2 a^2 c + 2 a b c + b^2 c + b c^2 + c^3 : :

X(29685) lies on these lines:


X(29686) =  POINT CASTOR(1,1,1,1,-1)

Barycentrics    2 a^3 + 2 a b^2 + b^3 + b^2 c + 2 a c^2 + b c^2 + c^3 : :

X(29686) lies on these lines:


X(29687) =  POINT CASTOR(-1,-1,1,1,1)

Barycentrics    -b^3 + 2 a b c - b^2 c - b c^2 - c^3 : :

X(29687) lies on these lines:


X(29688) =  POINT CASTOR(-1,1,-1,1,1)

Barycentrics    2 a^2 b + 2 a b^2 - b^3 + 2 a^2 c + b^2 c + 2 a c^2 + b c^2 - c^3 : :

X(29688) lies on these lines:


X(29689) =  POINT CASTOR(-1,1,1,-1,1)

Barycentrics    -2 a^3 + 2 a^2 b - b^3 + 2 a^2 c + 2 a b c + b^2 c + b c^2 - c^3 : :

X(29689) lies on these lines:


X(29690) =  POINT CASTOR(-1,1,1,1,-1)

Barycentrics    2 a b^2 - b^3 + b^2 c + 2 a c^2 + b c^2 - c^3 : :

X(29690) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 93: X(29691) - X(29741)  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 93, as in centers X(28691)-X(29741). Then

m(X) = (a^2 b + a^2 c - 2 a b c + b^2 c + b c^2) x - b (a - c)^2) y - c (a - b)^2 z : :

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, December 9, 2018)

underbar




X(29691) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^3 b - a^2 b^2 + a^3 c - 2 a^2 b c + 3 a b^2 c - a^2 c^2 + 3 a b c^2 - 2 b^2 c^2 : :

X(29691) lies on these lines:


X(29692) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^6 b - a^2 b^5 + a^6 c - 2 a^5 b c - a^2 b^4 c + 2 a b^5 c + a^2 b^3 c^2 - b^5 c^2 + a^2 b^2 c^3 - 4 a b^3 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 + 2 a b c^5 - b^2 c^5 : :

X(29692) lies on these lines:


X(29693) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^6 b - a^2 b^5 + a^6 c - 3 a^5 b c + 2 a^3 b^3 c - a^2 b^4 c + a b^5 c + a^2 b^3 c^2 - b^5 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^2 c^5 + a b c^5 - b^2 c^5 : :

X(29693) lies on these lines:


X(29694) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^6 b - a^2 b^5 + a^6 c - 4 a^5 b c + 4 a^3 b^3 c - a^2 b^4 c + a^2 b^3 c^2 - b^5 c^2 + 4 a^3 b c^3 + a^2 b^2 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 - b^2 c^5 : :

X(29694) lies on these lines:


X(29695) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^4 b - a^2 b^3 + a^4 c - 2 a^3 b c + a^2 b^2 c + 2 a b^3 c + a^2 b c^2 - b^3 c^2 - a^2 c^3 + 2 a b c^3 - b^2 c^3 : :

X(29695) lies on these lines:


X(29696) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a (a^3 b - a b^3 + a^3 c - 5 a^2 b c + 4 a b^2 c + 4 a b c^2 - 2 b^2 c^2 - a c^3) : :

X(29696) lies on these lines:


X(29697) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^3 b - a^2 b^2 + a^3 c - 3 a^2 b c + 2 a b^2 c - a^2 c^2 + 2 a b c^2 - 2 b^2 c^2 : :

X(29697) lies on these lines:


X(29698) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a (a^3 b - a b^3 + a^3 c - 4 a^2 b c + 2 a b^2 c + b^3 c + 2 a b c^2 - 4 b^2 c^2 - a c^3 + b c^3) : :

X(29698) lies on these lines:


X(29699) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^3 b - a^2 b^2 + a^3 c - 4 a^2 b c + a b^2 c - a^2 c^2 + a b c^2 - 2 b^2 c^2 : :

X(29699) lies on these lines:


X(29700) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^5 b - a^4 b^2 + a^3 b^3 - a^2 b^4 + a^5 c - 6 a^4 b c + 5 a^3 b^2 c - a^4 c^2 + 5 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 + 2 b^3 c^3 - a^2 c^4 - b^2 c^4 : :

X(29700) lies on these lines:


X(29701) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^8 b + a^6 b^3 - a^4 b^5 - a^2 b^7 + a^8 c - 2 a^7 b c + a^6 b^2 c - 2 a^5 b^3 c - a^4 b^4 c + 2 a^3 b^5 c - a^2 b^6 c + 2 a b^7 c + a^6 b c^2 + a^4 b^3 c^2 - a^2 b^5 c^2 - b^7 c^2 + a^6 c^3 - 2 a^5 b c^3 + a^4 b^2 c^3 + a^2 b^4 c^3 - 2 a b^5 c^3 + b^6 c^3 - a^4 b c^4 + a^2 b^3 c^4 - a^4 c^5 + 2 a^3 b c^5 - a^2 b^2 c^5 - 2 a b^3 c^5 - a^2 b c^6 + b^3 c^6 - a^2 c^7 + 2 a b c^7 - b^2 c^7 : :

X(29701) lies on these lines:


X(29702) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^8 b + a^6 b^3 - a^4 b^5 - a^2 b^7 + a^8 c - 2 a^7 b c + a^6 b^2 c - 2 a^5 b^3 c - a^4 b^4 c + 2 a^3 b^5 c - a^2 b^6 c + 2 a b^7 c + a^6 b c^2 + a^4 b^3 c^2 - a^2 b^5 c^2 - b^7 c^2 + a^6 c^3 - 2 a^5 b c^3 + a^4 b^2 c^3 + 2 a^3 b^3 c^3 + a^2 b^4 c^3 - 2 a b^5 c^3 + b^6 c^3 - a^4 b c^4 + a^2 b^3 c^4 - a^4 c^5 + 2 a^3 b c^5 - a^2 b^2 c^5 - 2 a b^3 c^5 - a^2 b c^6 + b^3 c^6 - a^2 c^7 + 2 a b c^7 - b^2 c^7 : :

X(29702) lies on these lines:


X(29703) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^5 b - a^2 b^4 + a^5 c - 2 a^4 b c + a^3 b^2 c + 2 a b^4 c + a^3 b c^2 - b^4 c^2 - a^2 c^4 + 2 a b c^4 - b^2 c^4 : :

X(29703) lies on these lines:


X(29704) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^6 b - a^2 b^5 + a^6 c - 2 a^5 b c + a^4 b^2 c + 2 a b^5 c + a^4 b c^2 - b^5 c^2 - a^2 c^5 + 2 a b c^5 - b^2 c^5 : :

X(29704) lies on these lines:


X(29705) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    b c (-2 a^3 + a^2 b - a b^2 + a^2 c - 4 a b c + b^2 c - a c^2 + b c^2) : :

X(29705) lies on these lines:


X(29706) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    b c (-a^4 b - a^2 b^3 - a^4 c + a^2 b^2 c + a^2 b c^2 - 4 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3) : :

X(29706) lies on these lines:


X(29707) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    (a b + a c - b c) (a^3 b - a^2 b^2 + a^3 c + a b^2 c - a^2 c^2 + a b c^2 + 2 b^2 c^2) : :

X(29707) lies on these lines:


X(29708) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^4 b^2 - a^3 b^3 + 2 a^4 b c - 2 a^3 b^2 c + 2 a^2 b^3 c + a^4 c^2 - 2 a^3 b 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 : :

X(29708) lies on these lines:


X(29709) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    2 a^4 b - 2 a^2 b^3 + 2 a^4 c - 6 a^3 b c + 3 a^2 b^2 c + 3 a b^3 c + 3 a^2 b c^2 - 4 a b^2 c^2 - b^3 c^2 - 2 a^2 c^3 + 3 a b c^3 - b^2 c^3 : :

X(29709) lies on these lines:


X(29710) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^4 b - a^2 b^3 + a^4 c - 6 a^3 b c + 3 a^2 b^2 c + 3 a^2 b c^2 - 8 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3 : :

X(29710) lies on these lines:


X(29711) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^4 b - a^2 b^3 + a^4 c - 3 a^3 b c + a^2 b^2 c + a b^3 c + a^2 b c^2 - b^3 c^2 - a^2 c^3 + a b c^3 - b^2 c^3 : :

X(29711) lies on these lines:


X(29712) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    b c (-2 a^3 + 3 a^2 b - a b^2 + 3 a^2 c - 2 a b c + b^2 c - a c^2 + b c^2) : :

X(29712) lies on these lines:


X(29713) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    b c (-a^4 b + 2 a^3 b^2 - a^2 b^3 - a^4 c + a^2 b^2 c + 2 a^3 c^2 + a^2 b c^2 - 2 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3) : :

X(29713) lies on these lines:


X(29714) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^4 b - a^2 b^3 + a^4 c - 2 a^3 b c + 3 a^2 b^2 c + 2 a b^3 c + 3 a^2 b 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 : :

X(29714) lies on these lines:


X(29715) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    2 a^3 b - 2 a^2 b^2 + 2 a^3 c - 5 a^2 b c + 5 a b^2 c - 2 a^2 c^2 + 5 a b c^2 - 4 b^2 c^2 : :

X(29715) lies on these lines:


X(29716) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^4 b - a^2 b^3 + a^4 c - 2 a^2 b^2 c + 3 a b^3 c - 2 a^2 b c^2 + 2 a b^2 c^2 - 2 b^3 c^2 - a^2 c^3 + 3 a b c^3 - 2 b^2 c^3 : :

X(29716) lies on these lines:


X(29717) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(661), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    (b - c) (a^4 b + a^3 b^2 + a^4 c - 2 a^2 b^2 c + a b^3 c + a^3 c^2 - 2 a^2 b c^2 + b^3 c^2 + a b c^3 + b^2 c^3) : :

X(29717) lies on these lines:


X(29718) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    (b - c) (-a^5 b - a^3 b^3 - a^5 c + 2 a^4 b c - 2 a^3 b^2 c + 3 a^2 b^3 c - 2 a^3 b c^2 + 3 a^2 b^2 c^2 - 3 a b^3 c^2 - a^3 c^3 + 3 a^2 b c^3 - 3 a b^2 c^3 + b^3 c^3) : :

X(29718) lies on these lines:


X(29719) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^5 b - a^2 b^4 + a^5 c - 2 a^4 b c + a^3 b^2 c + 2 a b^4 c + a^3 b c^2 - 2 a^2 b^2 c^2 - b^4 c^2 - a^2 c^4 + 2 a b c^4 - b^2 c^4 : :

X(29719) lies on these lines:


X(29720) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^3 b - a^2 b^2 + a^3 c + 5 a b^2 c - a^2 c^2 + 5 a b c^2 - 2 b^2 c^2 : :

X(29720) lies on these lines:


X(29721) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    2 a^6 b - 2 a^2 b^5 + 2 a^6 c - 5 a^5 b c + 2 a^3 b^3 c - 2 a^2 b^4 c + 3 a b^5 c + 2 a^2 b^3 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 - 2 a^2 b c^4 + 2 b^3 c^4 - 2 a^2 c^5 + 3 a b c^5 - 2 b^2 c^5 : :

X(29721) lies on these lines:


X(29722) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^6 b - a^2 b^5 + a^6 c - 2 a^5 b c - a^2 b^4 c + 2 a b^5 c - 2 a^3 b^2 c^2 - a^2 b^3 c^2 - b^5 c^2 - a^2 b^2 c^3 - 4 a b^3 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 + 2 a b c^5 - b^2 c^5 : :

X(29722) lies on these lines:


X(29723) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^8 b + a^6 b^3 - a^4 b^5 - a^2 b^7 + a^8 c - 2 a^7 b c + a^6 b^2 c - 2 a^5 b^3 c - a^4 b^4 c + 2 a^3 b^5 c - a^2 b^6 c + 2 a b^7 c + a^6 b c^2 + a^4 b^3 c^2 - a^2 b^5 c^2 - b^7 c^2 + a^6 c^3 - 2 a^5 b c^3 + a^4 b^2 c^3 + 4 a^3 b^3 c^3 + a^2 b^4 c^3 - 2 a b^5 c^3 + b^6 c^3 - a^4 b c^4 + a^2 b^3 c^4 - a^4 c^5 + 2 a^3 b c^5 - a^2 b^2 c^5 - 2 a b^3 c^5 - a^2 b c^6 + b^3 c^6 - a^2 c^7 + 2 a b c^7 - b^2 c^7 : :

X(29723) lies on these lines:


X(29724) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^8 b + 2 a^7 b^2 + a^6 b^3 - a^4 b^5 - 2 a^3 b^6 - a^2 b^7 + a^8 c + 2 a^7 b c - 3 a^6 b^2 c - 2 a^5 b^3 c + 3 a^4 b^4 c - 2 a^3 b^5 c - a^2 b^6 c + 2 a b^7 c + 2 a^7 c^2 - 3 a^6 b c^2 - 6 a^5 b^2 c^2 + 5 a^4 b^3 c^2 + 4 a^3 b^4 c^2 - a^2 b^5 c^2 - b^7 c^2 + a^6 c^3 - 2 a^5 b c^3 + 5 a^4 b^2 c^3 + 8 a^3 b^3 c^3 - a^2 b^4 c^3 - 2 a b^5 c^3 - b^6 c^3 + 3 a^4 b c^4 + 4 a^3 b^2 c^4 - a^2 b^3 c^4 + 2 b^5 c^4 - a^4 c^5 - 2 a^3 b c^5 - a^2 b^2 c^5 - 2 a b^3 c^5 + 2 b^4 c^5 - 2 a^3 c^6 - a^2 b c^6 - b^3 c^6 - a^2 c^7 + 2 a b c^7 - b^2 c^7 : :

X(29724) lies on these lines:


X(29725) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^9 b + 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^9 c + a^7 b^2 c - 3 a^5 b^4 c - a^3 b^6 c + 2 a b^8 c + a^8 c^2 + a^7 b c^2 - 4 a^6 b^2 c^2 - a^5 b^3 c^2 + 4 a^4 b^4 c^2 - a^3 b^5 c^2 + a b^7 c^2 - b^8 c^2 + a^7 c^3 - a^5 b^2 c^3 + 6 a^4 b^3 c^3 + 7 a^3 b^4 c^3 - 2 a^2 b^5 c^3 - 3 a b^6 c^3 + a^6 c^4 - 3 a^5 b c^4 + 4 a^4 b^2 c^4 + 7 a^3 b^3 c^4 - 2 a^2 b^4 c^4 + b^6 c^4 - a^5 c^5 - a^3 b^2 c^5 - 2 a^2 b^3 c^5 - a^4 c^6 - a^3 b c^6 - 3 a b^3 c^6 + b^4 c^6 - a^3 c^7 + a b^2 c^7 - a^2 c^8 + 2 a b c^8 - b^2 c^8 : :

X(29725) lies on these lines:


X(29726) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^9 b - 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^9 c - 4 a^8 b c + a^7 b^2 c + 4 a^6 b^3 c - 7 a^5 b^4 c + 3 a^3 b^6 c + 2 a b^8 c - a^8 c^2 + a^7 b c^2 + 4 a^6 b^2 c^2 + a^5 b^3 c^2 - 4 a^4 b^4 c^2 - 3 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 + 4 a^6 b c^3 + a^5 b^2 c^3 - 6 a^4 b^3 c^3 + 3 a^3 b^4 c^3 - 3 a b^6 c^3 + 2 b^7 c^3 + a^6 c^4 - 7 a^5 b c^4 - 4 a^4 b^2 c^4 + 3 a^3 b^3 c^4 - 2 a^2 b^4 c^4 + b^6 c^4 - a^5 c^5 - 3 a^3 b^2 c^5 - 4 b^5 c^5 + a^4 c^6 + 3 a^3 b c^6 + 2 a^2 b^2 c^6 - 3 a b^3 c^6 + b^4 c^6 + a^3 c^7 + a b^2 c^7 + 2 b^3 c^7 - a^2 c^8 + 2 a b c^8 - b^2 c^8 : :

X(29726) lies on these lines:


X(29727) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    3 a^6 b - 3 a^2 b^5 + 3 a^6 c - 7 a^5 b c + 2 a^3 b^3 c - 3 a^2 b^4 c + 5 a b^5 c + 3 a^2 b^3 c^2 - 3 b^5 c^2 + 2 a^3 b c^3 + 3 a^2 b^2 c^3 - 10 a b^3 c^3 + 3 b^4 c^3 - 3 a^2 b c^4 + 3 b^3 c^4 - 3 a^2 c^5 + 5 a b c^5 - 3 b^2 c^5 : :

X(29727) lies on these lines:


X(29728) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^6 b - a^2 b^5 + a^6 c - 3 a^5 b c + 2 a^3 b^3 c - a^2 b^4 c + a b^5 c + 2 a^3 b^2 c^2 + 3 a^2 b^3 c^2 - b^5 c^2 + 2 a^3 b c^3 + 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^2 c^5 + a b c^5 - b^2 c^5 : :

X(29728) lies on these lines:


X(29729) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^6 b - a^2 b^5 + a^6 c - 2 a^5 b c + 2 a^3 b^3 c - a^2 b^4 c + 2 a b^5 c + a^2 b^3 c^2 - b^5 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^2 c^5 + 2 a b c^5 - b^2 c^5 : :

X(29729) lies on these lines:


X(29730) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^6 b - a^2 b^5 + a^6 c - 4 a^5 b c + 2 a^4 b^2 c + 2 a^3 b^3 c - a^2 b^4 c + 2 a^4 b c^2 - 4 a^3 b^2 c^2 + 7 a^2 b^3 c^2 - b^5 c^2 + 2 a^3 b c^3 + 7 a^2 b^2 c^3 - 4 a b^3 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 - b^2 c^5 : :

X(29730) lies on these lines:


X(29731) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    (a - b) (a - c) (a^4 b + a^3 b^2 + a^2 b^3 + a b^4 + a^4 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 + 2 b^3 c^2 + a^2 c^3 + a b c^3 + 2 b^2 c^3 + a c^4 - b c^4) : :

X(29731) lies on these lines:


X(29732) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    (a - b) (a - c) (a^3 b + a b^3 + a^3 c - 2 a^2 b c - b^3 c + 2 b^2 c^2 + a c^3 - b c^3) : :

X(29732) lies on these lines:


X(29733) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    (a - b) (a - c) (a^4 b + a b^4 + a^4 c - 2 a^3 b c + a^2 b^2 c - a b^3 c - b^4 c + a^2 b c^2 + b^3 c^2 - a b c^3 + b^2 c^3 + a c^4 - b c^4) : :

X(29733) lies on these lines:


X(29734) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    (a - b) (a - c) (a^6 b + a^5 b^2 + a^2 b^5 + a b^6 + a^6 c + a^3 b^3 c - a b^5 c - b^6 c + a^5 c^2 - a b^4 c^2 + a^3 b c^3 + 2 a b^3 c^3 + b^4 c^3 - a b^2 c^4 + b^3 c^4 + a^2 c^5 - a b c^5 + a c^6 - b c^6) : :

X(29734) lies on these lines:


X(29735) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^6 b - a^2 b^5 + a^6 c - 4 a^5 b c - a^4 b^2 c + 4 a^3 b^3 c - 2 a^2 b^4 c - a^4 b c^2 + 2 a^2 b^3 c^2 - b^5 c^2 + 4 a^3 b c^3 + 2 a^2 b^2 c^3 - 4 a b^3 c^3 + 2 b^4 c^3 - 2 a^2 b c^4 + 2 b^3 c^4 - a^2 c^5 - b^2 c^5 : :

X(29735) lies on these lines:


X(29736) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    b c (-a^4 b + a^3 b^2 - a^2 b^3 - a^4 c + a^2 b^2 c + 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) : :

X(29736) lies on these lines:


X(29737) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    b c (b - c) (-a^6 + a^2 b^4 + 2 a^2 b^3 c + 3 a^2 b^2 c^2 - 4 a b^3 c^2 + b^4 c^2 + 2 a^2 b c^3 - 4 a b^2 c^3 + 2 b^3 c^3 + a^2 c^4 + b^2 c^4) : :

X(29737) lies on these lines:


X(29738) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a (b - c) (a^3 b + a^2 b^2 + a^3 c - a^2 b c - 2 a b^2 c + a^2 c^2 - 2 a b c^2 + 3 b^2 c^2) : :

X(29738) lies on these lines:


X(29739) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    b c (b - c) (a^3 + a b^2 - 3 a b c + b^2 c + a c^2 + b c^2) : :

X(29739) lies on these lines:


X(29740) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a (a^3 b - a b^3 + a^3 c - 4 a^2 b c + 4 a b^2 c + b^3 c + 4 a b c^2 - 2 b^2 c^2 - a c^3 + b c^3) : :

X(29740) lies on these lines:


X(29741) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 93

Barycentrics    a^4 b^2 - a^3 b^3 - 3 a^3 b^2 c + 2 a^2 b^3 c - a b^4 c + a^4 c^2 - 3 a^3 b c^2 + 2 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 - a b c^4 + b^2 c^4 : :

X(29741) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 94: X(29742) - X(29813)  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 94, as in centers X(29742)-X(29813). Then

m(X) = (a^2 b + a^2 c + 2 a b c + b^2 c + b c^2) x - b (a + c)^2 y - c (a + b^2) z : :

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, December 10, 2018)

underbar




X(29742) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^3 b - a^2 b^2 + a^3 c + 2 a^2 b c - a b^2 c - a^2 c^2 - a b c^2 - 2 b^2 c^2 : :

X(29742) lies on these lines:


X(29743) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^6 b - a^2 b^5 + a^6 c + 2 a^5 b c - a^2 b^4 c - 2 a b^5 c + a^2 b^3 c^2 - b^5 c^2 + a^2 b^2 c^3 + 4 a b^3 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 - 2 a b c^5 - b^2 c^5 : :

X(29743) lies on these lines:


X(29744) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^6 b - a^2 b^5 + a^6 c + 3 a^5 b c - 2 a^3 b^3 c - a^2 b^4 c - a b^5 c + a^2 b^3 c^2 - b^5 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^2 c^5 - a b c^5 - b^2 c^5 : :

X(29744) lies on these lines:


X(29745) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^6 b - a^2 b^5 + a^6 c + 4 a^5 b c - 4 a^3 b^3 c - a^2 b^4 c + a^2 b^3 c^2 - b^5 c^2 - 4 a^3 b c^3 + a^2 b^2 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 - b^2 c^5 : :

X(29745) lies on these lines:


X(29746) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^4 b - a^2 b^3 + a^4 c + 2 a^3 b c + a^2 b^2 c - 2 a b^3 c + a^2 b c^2 - b^3 c^2 - a^2 c^3 - 2 a b c^3 - b^2 c^3 : :

X(29746) lies on these lines:


X(29747) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a (a^3 b - a b^3 + a^3 c + a^2 b c - 2 b^3 c + 2 b^2 c^2 - a c^3 - 2 b c^3) : :

X(29747) lies on these lines:


X(29748) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^3 b - a^2 b^2 + a^3 c + 3 a^2 b c - a^2 c^2 - 2 b^2 c^2 : :

X(29748) lies on these lines:


X(29749) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a (a^3 b - a b^3 + a^3 c + 2 a b^2 c - 3 b^3 c + 2 a b c^2 + 4 b^2 c^2 - a c^3 - 3 b c^3) : :

X(29749) lies on these lines:


X(29750) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^3 b - a^2 b^2 + a^3 c + 4 a^2 b c + a b^2 c - a^2 c^2 + a b c^2 - 2 b^2 c^2 : :

X(29750) lies on these lines:


X(29751) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^5 b - a^4 b^2 + a^3 b^3 - a^2 b^4 + a^5 c + 2 a^4 b c - 3 a^3 b^2 c - a^4 c^2 - 3 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 + a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 - b^2 c^4 : :

X(29751) lies on these lines:


X(29752) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^8 b + a^6 b^3 - a^4 b^5 - a^2 b^7 + a^8 c + 2 a^7 b c + a^6 b^2 c + 2 a^5 b^3 c - a^4 b^4 c - 2 a^3 b^5 c - a^2 b^6 c - 2 a b^7 c + a^6 b c^2 + a^4 b^3 c^2 - a^2 b^5 c^2 - b^7 c^2 + a^6 c^3 + 2 a^5 b c^3 + a^4 b^2 c^3 + a^2 b^4 c^3 + 2 a b^5 c^3 + b^6 c^3 - a^4 b c^4 + a^2 b^3 c^4 - a^4 c^5 - 2 a^3 b c^5 - a^2 b^2 c^5 + 2 a b^3 c^5 - a^2 b c^6 + b^3 c^6 - a^2 c^7 - 2 a b c^7 - b^2 c^7 : :

X(29752) lies on these lines:


X(29753) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^8 b + a^6 b^3 - a^4 b^5 - a^2 b^7 + a^8 c + 2 a^7 b c + a^6 b^2 c + 2 a^5 b^3 c - a^4 b^4 c - 2 a^3 b^5 c - a^2 b^6 c - 2 a b^7 c + a^6 b c^2 + a^4 b^3 c^2 - a^2 b^5 c^2 - b^7 c^2 + a^6 c^3 + 2 a^5 b c^3 + a^4 b^2 c^3 - 2 a^3 b^3 c^3 + a^2 b^4 c^3 + 2 a b^5 c^3 + b^6 c^3 - a^4 b c^4 + a^2 b^3 c^4 - a^4 c^5 - 2 a^3 b c^5 - a^2 b^2 c^5 + 2 a b^3 c^5 - a^2 b c^6 + b^3 c^6 - a^2 c^7 - 2 a b c^7 - b^2 c^7 : :

X(29753) lies on these lines:


X(29754) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^5 b - a^2 b^4 + a^5 c + 2 a^4 b c + a^3 b^2 c - 2 a b^4 c + a^3 b c^2 - b^4 c^2 - a^2 c^4 - 2 a b c^4 - b^2 c^4 : :

X(29754) lies on these lines:


X(29755) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^6 b - a^2 b^5 + a^6 c + 2 a^5 b c + a^4 b^2 c - 2 a b^5 c + a^4 b c^2 - b^5 c^2 - a^2 c^5 - 2 a b c^5 - b^2 c^5 : :

X(29755) lies on these lines:


X(29756) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    b c (-2 a^3 + a^2 b - a b^2 + a^2 c + 4 a b c + b^2 c - a c^2 + b c^2) : :

X(29756) lies on these lines:


X(29757) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    b c (-a^4 b - a^2 b^3 - a^4 c + a^2 b^2 c + a^2 b c^2 + 4 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3) : :

X(29757) lies on these lines:


X(29758) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^4 b^2 - a^3 b^3 + 2 a^4 b c + 2 a^3 b^2 c - 2 a^2 b^3 c + a^4 c^2 + 2 a^3 b c^2 + 2 a^2 b^2 c^2 - 3 a b^3 c^2 - a^3 c^3 - 2 a^2 b c^3 - 3 a b^2 c^3 - 2 b^3 c^3 : :

X(29758) lies on these lines:


X(29759) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^4 b^2 - a^3 b^3 + 2 a^4 b c + 2 a^3 b^2 c - 2 a^2 b^3 c + a^4 c^2 + 2 a^3 b c^2 + 4 a^2 b^2 c^2 - 3 a b^3 c^2 - a^3 c^3 - 2 a^2 b c^3 - 3 a b^2 c^3 - 2 b^3 c^3 : :

X(29759) lies on these lines:


X(29760) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    2 a^4 b - 2 a^2 b^3 + 2 a^4 c + 2 a^3 b c + 3 a^2 b^2 c - 5 a b^3 c + 3 a^2 b c^2 + 4 a b^2 c^2 - b^3 c^2 - 2 a^2 c^3 - 5 a b c^3 - b^2 c^3 : :

X(29760) lies on these lines:


X(29761) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^4 b - a^2 b^3 + a^4 c - 2 a^3 b c + 3 a^2 b^2 c - 4 a b^3 c + 3 a^2 b c^2 + 8 a b^2 c^2 + b^3 c^2 - a^2 c^3 - 4 a b c^3 + b^2 c^3 : :

X(29761) lies on these lines:


X(29762) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^5 b - a^4 b^2 + a^3 b^3 - a^2 b^4 + a^5 c - a^3 b^2 c + 2 a^2 b^3 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 + a^3 c^3 + 2 a^2 b c^3 + 3 a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 - 2 a b c^4 - b^2 c^4 : :

X(29762) lies on these lines:


X(29763) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^4 b - a^2 b^3 + a^4 c + 3 a^3 b c + a^2 b^2 c - a b^3 c + a^2 b c^2 - b^3 c^2 - a^2 c^3 - a b c^3 - b^2 c^3 : :

X(29763) lies on these lines:


X(29764) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    b c (-2 a^3 - a^2 b - a b^2 - a^2 c + 2 a b c + b^2 c - a c^2 + b c^2) : :

X(29764) lies on these lines:


X(29765) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    b c (-a^4 b - 2 a^3 b^2 - a^2 b^3 - a^4 c + a^2 b^2 c - 2 a^3 c^2 + a^2 b c^2 + 2 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3) : :

X(29765) lies on these lines:


X(29766) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    (a + b) (a + c) (a^3 b - a b^3 + a^3 c + 2 a^2 b c - b^3 c - 2 b^2 c^2 - a c^3 - b c^3) : :

X(29766) lies on these lines:


X(29767) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    (a + b) (a + c) (a^2 b - a b^2 + a^2 c - b^2 c - a c^2 - b c^2) : :

X(29767) lies on these lines:


X(29768) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    2 a^3 b - 2 a^2 b^2 + 2 a^3 c + 5 a^2 b c - a b^2 c - 2 a^2 c^2 - a b c^2 - 4 b^2 c^2 : :

X(29768) lies on these lines:


X(29769) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^4 b - a^2 b^3 + a^4 c + 4 a^3 b c + 2 a^2 b^2 c - a b^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 : :

X(29769) lies on these lines:


X(29770) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    (b - c) (-a^5 b - a^3 b^3 - a^5 c - 2 a^4 b c - 2 a^3 b^2 c - a^2 b^3 c - 2 a^3 b c^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 + b^3 c^3) : :

X(29770) lies on these lines:


X(29771) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    b c (b - c) (-a^3 + 2 a^2 b + a b^2 + 2 a^2 c + 3 a b c + b^2 c + a c^2 + b c^2) : :

X(29771) lies on these lines:


X(29772) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^5 b - a^2 b^4 + a^5 c + 2 a^4 b c + a^3 b^2 c - 2 a b^4 c + a^3 b c^2 + 2 a^2 b^2 c^2 - b^4 c^2 - a^2 c^4 - 2 a b c^4 - b^2 c^4 : :

X(29772) lies on these lines:


X(29773) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    (a^2 - a b - a c - b c) (a b + a c + 2 b c) : :

X(29773) lies on these lines:


X(29774) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    2 a^6 b - 2 a^2 b^5 + 2 a^6 c + 5 a^5 b c - 2 a^3 b^3 c - 2 a^2 b^4 c - 3 a b^5 c + 2 a^2 b^3 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 - 2 a^2 b c^4 + 2 b^3 c^4 - 2 a^2 c^5 - 3 a b c^5 - 2 b^2 c^5 : :

X(29774) lies on these lines:


X(29775) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    (a + b) (a + c) (a^4 b - a^3 b^2 + a^2 b^3 - a b^4 + a^4 c - 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 - a c^4 - b c^4) : :

X(29775) lies on these lines:


X(29776) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^8 b + a^6 b^3 - a^4 b^5 - a^2 b^7 + a^8 c + 2 a^7 b c + a^6 b^2 c + 2 a^5 b^3 c - a^4 b^4 c - 2 a^3 b^5 c - a^2 b^6 c - 2 a b^7 c + a^6 b c^2 + a^4 b^3 c^2 - a^2 b^5 c^2 - b^7 c^2 + a^6 c^3 + 2 a^5 b c^3 + a^4 b^2 c^3 - 4 a^3 b^3 c^3 + a^2 b^4 c^3 + 2 a b^5 c^3 + b^6 c^3 - a^4 b c^4 + a^2 b^3 c^4 - a^4 c^5 - 2 a^3 b c^5 - a^2 b^2 c^5 + 2 a b^3 c^5 - a^2 b c^6 + b^3 c^6 - a^2 c^7 - 2 a b c^7 - b^2 c^7 : :

X(29776) lies on these lines:


X(29777) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    (a + b) (a + c) (a^6 b + a^5 b^2 - a^2 b^5 - a b^6 + a^6 c + 4 a^5 b c + 3 a^4 b^2 c - 2 a^3 b^3 c - 3 a^2 b^4 c - 2 a b^5 c - b^6 c + a^5 c^2 + 3 a^4 b c^2 - 4 a^3 b^2 c^2 + a b^4 c^2 - b^5 c^2 - 2 a^3 b c^3 + 4 a b^3 c^3 + 2 b^4 c^3 - 3 a^2 b c^4 + a b^2 c^4 + 2 b^3 c^4 - a^2 c^5 - 2 a b c^5 - b^2 c^5 - a c^6 - b c^6) : :

X(29777) lies on these lines:


X(29778) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    (a + b) (a + c) (a^7 b + a^5 b^3 - a^3 b^5 - a b^7 + a^7 c + 2 a^6 b c + 2 a^5 b^2 c + a^4 b^3 c - a^3 b^4 c - 2 a^2 b^5 c - 2 a b^6 c - b^7 c + 2 a^5 b c^2 + 2 a^4 b^2 c^2 - 2 a^3 b^3 c^2 - 2 a^2 b^4 c^2 + a^5 c^3 + a^4 b c^3 - 2 a^3 b^2 c^3 + 3 a b^4 c^3 + b^5 c^3 - a^3 b c^4 - 2 a^2 b^2 c^4 + 3 a b^3 c^4 - a^3 c^5 - 2 a^2 b c^5 + b^3 c^5 - 2 a b c^6 - a c^7 - b c^7) : :

X(29778) lies on these lines:


X(29779) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    (a + b) (a + c) (a^7 b - 2 a^6 b^2 + a^5 b^3 - a^3 b^5 + 2 a^2 b^6 - a b^7 + a^7 c - 2 a^6 b c - 4 a^5 b^2 c + a^4 b^3 c + 3 a^3 b^4 c + 2 a^2 b^5 c - b^7 c - 2 a^6 c^2 - 4 a^5 b c^2 + 2 a^4 b^2 c^2 + 2 a^3 b^3 c^2 - 2 a^2 b^4 c^2 + 2 a b^5 c^2 + 2 b^6 c^2 + a^5 c^3 + a^4 b c^3 + 2 a^3 b^2 c^3 - 4 a^2 b^3 c^3 - a b^4 c^3 + b^5 c^3 + 3 a^3 b c^4 - 2 a^2 b^2 c^4 - a b^3 c^4 - 4 b^4 c^4 - a^3 c^5 + 2 a^2 b c^5 + 2 a b^2 c^5 + b^3 c^5 + 2 a^2 c^6 + 2 b^2 c^6 - a c^7 - b c^7) : :

X(29779) lies on these lines:


X(29780) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    3 a^6 b - 3 a^2 b^5 + 3 a^6 c + 7 a^5 b c - 2 a^3 b^3 c - 3 a^2 b^4 c - 5 a b^5 c + 3 a^2 b^3 c^2 - 3 b^5 c^2 - 2 a^3 b c^3 + 3 a^2 b^2 c^3 + 10 a b^3 c^3 + 3 b^4 c^3 - 3 a^2 b c^4 + 3 b^3 c^4 - 3 a^2 c^5 - 5 a b c^5 - 3 b^2 c^5 : :

X(29780) lies on these lines:


X(29781) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^6 b - a^2 b^5 + a^6 c + 3 a^5 b c - 2 a^3 b^3 c - a^2 b^4 c - a b^5 c - 2 a^3 b^2 c^2 - a^2 b^3 c^2 - b^5 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^2 c^5 - a b c^5 - b^2 c^5 : :

X(29781) lies on these lines:


X(29782) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^6 b - a^2 b^5 + a^6 c + 2 a^5 b c - 2 a^3 b^3 c - a^2 b^4 c - 2 a b^5 c + a^2 b^3 c^2 - b^5 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^2 c^5 - 2 a b c^5 - b^2 c^5 : :

X(29782) lies on these lines:


X(29783) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(12), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^6 b - a^2 b^5 + a^6 c + 4 a^5 b c + 2 a^4 b^2 c - 6 a^3 b^3 c - a^2 b^4 c + 2 a^4 b c^2 + 4 a^3 b^2 c^2 - a^2 b^3 c^2 - b^5 c^2 - 6 a^3 b c^3 - a^2 b^2 c^3 - 4 a b^3 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 - b^2 c^5 : :

X(29783) lies on these lines:


X(29784) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(35), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^6 b - a^2 b^5 + a^6 c + 2 a^5 b c - a^4 b^2 c + a^3 b^3 c - a^2 b^4 c - 2 a b^5 c - a^4 b c^2 - 2 a^3 b^2 c^2 + 2 a^2 b^3 c^2 - b^5 c^2 + a^3 b c^3 + 2 a^2 b^2 c^3 + 6 a b^3 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 - 2 a b c^5 - b^2 c^5 : :

X(29784) lies on these lines:


X(29785) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(36), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^6 b - a^2 b^5 + a^6 c + 2 a^5 b c + a^4 b^2 c - a^3 b^3 c - a^2 b^4 c - 2 a b^5 c + a^4 b c^2 + 2 a^3 b^2 c^2 - b^5 c^2 - a^3 b c^3 + 2 a b^3 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 - 2 a b c^5 - b^2 c^5 : :

X(29785) lies on these lines:


X(29786) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(38), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^4 b^2 - a^3 b^3 + a^3 b^2 c - 2 a^2 b^3 c - a b^4 c + a^4 c^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 b c^4 + b^2 c^4 : :

X(29786) lies on these lines:


X(29787) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(56), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^6 b - a^2 b^5 + a^6 c + 2 a^5 b c + 2 a^4 b^2 c - 2 a^3 b^3 c - a^2 b^4 c - 2 a b^5 c + 2 a^4 b c^2 + 4 a^3 b^2 c^2 - a^2 b^3 c^2 - b^5 c^2 - 2 a^3 b c^3 - a^2 b^2 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - a^2 c^5 - 2 a b c^5 - b^2 c^5 : :

X(29787) lies on these lines:


X(29788) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(57), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a (a^4 b + a^3 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 - 3 b^4 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 - a c^4 - 3 b c^4) : :

X(29788) lies on these lines:


X(29789) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(58), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    (a + b) (a + c) (a^4 b - a b^4 + a^4 c + 2 a^3 b c + a^2 b^2 c - a b^3 c - b^4 c + a^2 b c^2 - b^3 c^2 - a b c^3 - b^2 c^3 - a c^4 - b c^4) : :

X(29789) lies on these lines:


X(29790) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(63), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a (a^4 b + a^3 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 - 3 b^4 c + a^3 c^2 + 2 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 - a c^4 - 3 b c^4) : :

X(29790) lies on these lines:


X(29791) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(65), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    2 a^5 b^2 - 2 a^3 b^4 + 2 a^5 b c + 3 a^4 b^2 c + a^3 b^3 c - 5 a^2 b^4 c - a b^5 c + 2 a^5 c^2 + 3 a^4 b c^2 - 2 a b^4 c^2 + b^5 c^2 + a^3 b c^3 + 2 a b^3 c^3 - b^4 c^3 - 2 a^3 c^4 - 5 a^2 b c^4 - 2 a b^2 c^4 - b^3 c^4 - a b c^5 + b^2 c^5 : :

X(29791) lies on these lines:


X(29792) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(83), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^6 b - a^2 b^5 + a^6 c + 2 a^5 b c + a^4 b^2 c - 2 a^3 b^3 c - 2 a b^5 c + a^4 b c^2 - b^5 c^2 - 2 a^3 b c^3 - 2 a b^3 c^3 - a^2 c^5 - 2 a b c^5 - b^2 c^5 : :

X(29792) lies on these lines:


X(29793) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(85), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    b c (-2 a^5 - a^4 b + 3 a^3 b^2 + a^2 b^3 - a b^4 - a^4 c - 2 a^3 b c + 2 a b^3 c + b^4 c + 3 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 - a c^4 + b c^4) : :

X(29793) lies on these lines:


X(29794) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(87), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^5 b^3 - a^4 b^4 - a^5 b^2 c + 2 a^4 b^3 c + a^3 b^4 c - a^5 b c^2 - 2 a^4 b^2 c^2 + 3 a^2 b^4 c^2 + a^5 c^3 + 2 a^4 b c^3 - 2 a^2 b^3 c^3 - a b^4 c^3 - a^4 c^4 + a^3 b c^4 + 3 a^2 b^2 c^4 - a b^3 c^4 - 2 b^4 c^4 : :

X(29794) lies on these lines:


X(29795) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(88), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^5 b + a^4 b^2 - a^3 b^3 - a^2 b^4 + a^5 c + a^3 b^2 c - 2 a^2 b^3 c - 4 a b^4 c + a^4 c^2 + a^3 b c^2 - 4 a^2 b^2 c^2 + 5 a b^3 c^2 + b^4 c^2 - a^3 c^3 - 2 a^2 b c^3 + 5 a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 - 4 a b c^4 + b^2 c^4 : :

X(29795) lies on these lines:


X(29796) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(99), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    (a + b) (a + c) (a^4 b - a^3 b^2 + a^2 b^3 - a b^4 + a^4 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 + 2 b^3 c^2 + a^2 c^3 - a b c^3 + 2 b^2 c^3 - a c^4 - b c^4) : :

X(29796) lies on these lines:


X(29797) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(100), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^5 b - a^4 b^2 + a^3 b^3 - a^2 b^4 + a^5 c - a^3 b^2 c + 2 a^2 b^3 c - 2 a b^4 c - a^4 c^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 + 2 a^2 b c^3 + 3 a b^2 c^3 + 2 b^3 c^3 - a^2 c^4 - 2 a b c^4 - b^2 c^4 : :

X(29797) lies on these lines:


X(29798) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(101), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^6 b - a^5 b^2 + a^3 b^4 - a^2 b^5 + a^6 c - 2 a^3 b^3 c + 3 a^2 b^4 c - 2 a b^5 c - a^5 c^2 - a^2 b^3 c^2 + 3 a b^4 c^2 - b^5 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^3 c^4 + 3 a^2 b c^4 + 3 a b^2 c^4 + b^3 c^4 - a^2 c^5 - 2 a b c^5 - b^2 c^5 : :

X(29798) lies on these lines:


X(29799) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(110), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    (a + b) (a + c) (a^6 b - a^5 b^2 + a^2 b^5 - a b^6 + a^6 c - a^3 b^3 c + a b^5 c - b^6 c - a^5 c^2 + a b^4 c^2 - a^3 b c^3 - 2 a b^3 c^3 + b^4 c^3 + a b^2 c^4 + b^3 c^4 + a^2 c^5 + a b c^5 - a c^6 - b c^6) : :

X(29799) lies on these lines:


X(29800) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(115), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^6 b - a^2 b^5 + a^6 c + 4 a^5 b c - a^4 b^2 c - 4 a^3 b^3 c - 2 a^2 b^4 c - a^4 b c^2 + 2 a^2 b^3 c^2 - b^5 c^2 - 4 a^3 b c^3 + 2 a^2 b^2 c^3 + 4 a b^3 c^3 + 2 b^4 c^3 - 2 a^2 b c^4 + 2 b^3 c^4 - a^2 c^5 - b^2 c^5 : :

X(29800) lies on these lines:


X(29801) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(190), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^4 b - a^2 b^3 + a^4 c - 2 a^3 b c + a^2 b^2 c - 4 a b^3 c + a^2 b c^2 + 6 a b^2 c^2 + b^3 c^2 - a^2 c^3 - 4 a b c^3 + b^2 c^3 : :

X(29801) lies on these lines:


X(29802) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(192), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    b c (-2 a^3 - a b^2 + 3 a b c + b^2 c - a c^2 + b c^2) : :

X(29802) lies on these lines:


X(29803) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(900), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    (b - c) (a^4 - 4 a^3 b + a^2 b^2 - 4 a^3 c - 4 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 b c^3) : :

X(29803) lies on these lines:


X(29804) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(187), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    2 a^6 b - 2 a^2 b^5 + 2 a^6 c + 4 a^5 b c + a^4 b^2 c - a^2 b^4 c - 4 a b^5 c + a^4 b c^2 + a^2 b^3 c^2 - 2 b^5 c^2 + a^2 b^2 c^3 + 4 a b^3 c^3 + b^4 c^3 - a^2 b c^4 + b^3 c^4 - 2 a^2 c^5 - 4 a b c^5 - 2 b^2 c^5 : :

X(29804) lies on these lines:


X(29805) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(194), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    b c (-a^4 b - a^3 b^2 - a^2 b^3 - a^4 c + a^2 b^2 c - 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) : :

X(29805) lies on these lines:


X(29806) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(647), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    b c (b - c) (-a^6 + a^2 b^4 + 2 a^2 b^3 c + 3 a^2 b^2 c^2 + 4 a b^3 c^2 + b^4 c^2 + 2 a^2 b c^3 + 4 a b^2 c^3 + 2 b^3 c^3 + a^2 c^4 + b^2 c^4) : :

X(29806) lies on these lines:


X(29807) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a (b - c) (a^3 b + a^2 b^2 + a^3 c + 3 a^2 b c + 2 a b^2 c + a^2 c^2 + 2 a b c^2 - b^2 c^2) : :

X(29807) lies on these lines:


X(29808) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    b c (b - c) (a^3 + a b^2 + 5 a b c + b^2 c + a c^2 + b c^2) : :

X(29808) lies on these lines:


X(29809) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(659), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    (b - c) (-a^5 b - a^3 b^3 - a^5 c - a^4 b c - a^3 b^2 c - a^2 b^3 c - a^3 b c^2 + 2 a^2 b^2 c^2 + 3 a b^3 c^2 - a^3 c^3 - a^2 b c^3 + 3 a b^2 c^3 + b^3 c^3) : :

X(29809) lies on these lines:


X(29810) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(662), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    (a + b) (a + c) (a^5 b - a^4 b^2 + a^2 b^4 - a b^5 + a^5 c - a^2 b^3 c + a b^4 c - b^5 c - a^4 c^2 - 2 a^2 b^2 c^2 + a b^3 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 - a c^5 - b c^5) : :

X(29810) lies on these lines:


X(29811) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(668), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    b c (a^4 b - 4 a^3 b^2 - a^2 b^3 + a^4 c + 6 a^3 b c + a^2 b^2 c - 4 a^3 c^2 + a^2 b c^2 - 2 a b^2 c^2 + b^3 c^2 - a^2 c^3 + b^2 c^3) : :

X(29811) lies on these lines:


X(29812) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(894), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a (a^3 b - a b^3 + a^3 c - 3 b^3 c + 2 b^2 c^2 - a c^3 - 3 b c^3) : :

X(29812) lies on these lines:


X(29813) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(984), WHERE A'B'C' = GEMINI TRIANGLE 94

Barycentrics    a^4 b^2 - a^3 b^3 + a^3 b^2 c - 2 a^2 b^3 c - a b^4 c + a^4 c^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 - 2 a^2 b c^3 + a b^2 c^3 - a b c^4 + b^2 c^4 : :

X(29813) lies on these lines:


X(29814) =  POINT CASTOR(0,0,1,0,2)

Barycentrics    a (2 a b + 2 a c + 3 b c) : :

Points Castor are defined in the preamble just before X(29631).

X(29814) lies on these lines:


X(29815) =  POINT CASTOR(0,0,1,2,0)

Barycentrics    a (2 a^2 + 2 b^2 + b c + 2 c^2) : :

X(29815) lies on these lines:


X(29816) =  POINT CASTOR(0,0,1,2,1)

Barycentrics    a (2 a^2 + a b + 2 b^2 + a c + 2 b c + 2 c^2) : :

X(29816) lies on these lines:


X(29817) =  POINT CASTOR(0,0,1,-1,2)

Barycentrics    a (a^2 - 2 a b + b^2 - 2 a c - 3 b c + c^2) : :

X(29817) lies on these lines:


X(29818) =  POINT CASTOR(0,0,1,-2,1)

Barycentrics    a (2 a^2 - a b + 2 b^2 - a c - 2 b c + 2 c^2) : :

X(29818) lies on these lines:


X(29819) =  POINT CASTOR(0,0,1,-2,-1)

Barycentrics    a (2 a^2 + a b + 2 b^2 + a c + 2 c^2) : :

X(29819) lies on these lines:


X(29820) =  POINT CASTOR(0,0,2,-1,1)

Barycentrics    a (a^2 - a b + b^2 - a c - 3 b c + c^2) : :

X(29820) lies on these lines:


X(29821) =  POINT CASTOR(0,0,2,-1,-1)

Barycentrics    a (a^2 + a b + b^2 + a c - b c + c^2) : :

X(29821) lies on these lines:


X(29822) =  POINT CASTOR(0,1,0,0,2)

Barycentrics    (b + c) (3 a^2 + a b + a c + b c) : :

X(29822) lies on these lines:


X(29823) =  POINT CASTOR(0,1,0,2,0)

Barycentrics    2 a^3 + a^2 b + 3 a b^2 + a^2 c + b^2 c + 3 a c^2 + b c^2 : :

X(29823) lies on these lines:


X(29824) =  POINT CASTOR(0,1,0,0,-2)

Barycentrics    -a^2 b + a b^2 - a^2 c - 2 a b c + b^2 c + a c^2 + b c^2 : :

X(29824) lies on these lines:


X(29825) =  POINT CASTOR(0,2,0,0,1)

Barycentrics    3 a^2 b + 2 a b^2 + 3 a^2 c + a b c + 2 b^2 c + 2 a c^2 + 2 b c^2 : :

X(29825) lies on these lines:


X(29826) =  POINT CASTOR(0,2,0,1,0)

Barycentrics    a^3 + 2 a^2 b + 3 a b^2 + 2 a^2 c + 2 b^2 c + 3 a c^2 + 2 b c^2 : :

X(29826) lies on these lines:


X(29827) =  POINT CASTOR(0,2,0,0,-1)

Barycentrics    a^2 b + 2 a b^2 + a^2 c - a b c + 2 b^2 c + 2 a c^2 + 2 b c^2 : :

X(29827) lies on these lines:


X(29828) =  POINT CASTOR(0,2,0,-1,0)

Barycentrics    -a^3 + 2 a^2 b + a b^2 + 2 a^2 c + 2 b^2 c + a c^2 + 2 b c^2 : :

X(29828) lies on these lines:


X(29829) =  POINT CASTOR(1,0,0,0,2)

Barycentrics    a^3 + 2 a^2 b + b^3 + 2 a^2 c + 2 a b c + c^3 : :

X(29829) lies on these lines:


X(29830) =  POINT CASTOR(1,0,0,0,-2)

Barycentrics    a^3 - 2 a^2 b + b^3 - 2 a^2 c - 2 a b c + c^3 : :

X(29830) lies on these lines:


X(29831) =  POINT CASTOR(1,0,0,2,0)

Barycentrics    3 a^3 + 2 a b^2 + b^3 + 2 a c^2 + c^3 : :

X(29831) lies on these lines:


X(29832) =  POINT CASTOR(1,0,0,-2,0)

Barycentrics    -a^3 - 2 a b^2 + b^3 - 2 a c^2 + c^3 : :

X(29832) lies on these lines:


X(29833) =  POINT CASTOR(1,0,0,1,2)

Barycentrics    2 a^3 + 2 a^2 b + a b^2 + b^3 + 2 a^2 c + 2 a b c + a c^2 + c^3 : :

X(29833) lies on these lines:


X(29834) =  POINT CASTOR(1,0,0,2,1)

Barycentrics    3 a^3 + a^2 b + 2 a b^2 + b^3 + a^2 c + a b c + 2 a c^2 + c^3 : :

X(29834) lies on these lines:


X(29835) =  POINT CASTOR(1,0,0,-1,2)

Barycentrics    2 a^2 b - a b^2 + b^3 + 2 a^2 c + 2 a b c - a c^2 + c^3 : :

X(29835) lies on these lines:


X(29836) =  POINT CASTOR(1,0,0,2,-1)

Barycentrics    3 a^3 - a^2 b + 2 a b^2 + b^3 - a^2 c - a b c + 2 a c^2 + c^3 : :

X(29836) lies on these lines:


X(29837) =  POINT CASTOR(1,0,1,0,2)

Barycentrics    a^3 + 2 a^2 b + b^3 + 2 a^2 c + 3 a b c + c^3 : :

X(29837) lies on these lines:


X(29838) =  POINT CASTOR(1,0,1,2,0)

Barycentrics    3 a^3 + 2 a b^2 + b^3 + a b c + 2 a c^2 + c^3 : :

X(29838) lies on these lines:


X(29839) =  POINT CASTOR(1,0,1,0,-2)

Barycentrics    a^3 - 2 a^2 b + b^3 - 2 a^2 c - a b c + c^3 : :

X(29839) lies on these lines:


X(29840) =  POINT CASTOR(1,0,1,-2,0)

Barycentrics    -a^3 - 2 a b^2 + b^3 + a b c - 2 a c^2 + c^3 : :

X(29840) lies on these lines:


X(29841) =  POINT CASTOR(1,0,1,1,2)

Barycentrics    2 a^3 + 2 a^2 b + a b^2 + b^3 + 2 a^2 c + 3 a b c + a c^2 + c^3 : :

X(29841) lies on these lines:


X(29842) =  POINT CASTOR(1,0,1,2,1)

Barycentrics    3 a^3 + a^2 b + 2 a b^2 + b^3 + a^2 c + 2 a b c + 2 a c^2 + c^3 : :

X(29842) lies on these lines:


X(29843) =  POINT CASTOR(1,0,1,-1,2)

Barycentrics    2 a^2 b - a b^2 + b^3 + 2 a^2 c + 3 a b c - a c^2 + c^3 : :

X(29843) lies on these lines:


X(29844) =  POINT CASTOR(1,0,1,-2,1)

Barycentrics    -a^3 + a^2 b - 2 a b^2 + b^3 + a^2 c + 2 a b c - 2 a c^2 + c^3 : :

X(29844) lies on these lines:


X(29845) =  POINT CASTOR(1,0,2,0,1)

Barycentrics    a^3 + a^2 b + b^3 + a^2 c + 3 a b c + c^3 : :

X(29845) lies on these lines:


X(29846) =  POINT CASTOR(1,0,2,0,-1)

Barycentrics    a^3 - a^2 b + b^3 - a^2 c + a b c + c^3 : :

X(29846) lies on these lines:


X(29847) =  POINT CASTOR(1,0,2,1,1)

Barycentrics    2 a^3 + a^2 b + a b^2 + b^3 + a^2 c + 3 a b c + a c^2 + c^3 : :

X(29847) lies on these lines:


X(29848) =  POINT CASTOR(1,0,2,1,-1)

Barycentrics    2 a^3 - a^2 b + a b^2 + b^3 - a^2 c + a b c + a c^2 + c^3 : :

X(29848) lies on these lines:


X(29849) =  POINT CASTOR(1,0,2,-1,-1)

Barycentrics    -a^2 b - a b^2 + b^3 - a^2 c + a b c - a c^2 + c^3 : :

X(29849) lies on these lines:


X(29850) =  POINT CASTOR(1,0,-2,0,1)

Barycentrics    a^3 + a^2 b + b^3 + a^2 c - a b c + c^3 : :

X(29850) lies on these lines:


X(29851) =  POINT CASTOR(1,0,-2,0,-1)

Barycentrics    a^3 - a^2 b + b^3 - a^2 c - 3 a b c + c^3 : :

X(29851) lies on these lines:


X(29852) =  POINT CASTOR(1,0,-2,1,1)

Barycentrics    2 a^3 + a^2 b + a b^2 + b^3 + a^2 c - a b c + a c^2 + c^3 : :

X(29852) lies on these lines:


X(29853) =  POINT CASTOR(1,0,-2,1,-1)

Barycentrics    2 a^3 - a^2 b + a b^2 + b^3 - a^2 c - 3 a b c + a c^2 + c^3 : :

X(29853) lies on these lines:


X(29854) =  POINT CASTOR(1,0,-2,-1,-1)

Barycentrics    -a^2 b - a b^2 + b^3 - a^2 c - 3 a b c - a c^2 + c^3 : :

X(29854) lies on these lines:


X(29855) =  POINT CASTOR(2,0,0,1,0)

Barycentrics    3 a^3 + a b^2 + 2 b^3 + a c^2 + 2 c^3 : :

X(29855) lies on these lines:


X(29856) =  POINT CASTOR(2,0,0,0,1)

Barycentrics    2 a^3 + a^2 b + 2 b^3 + a^2 c + a b c + 2 c^3 : :

X(29856) lies on these lines:


X(29857) =  POINT CASTOR(2,0,0,-1,0)

Barycentrics    a^3 - a b^2 + 2 b^3 - a c^2 + 2 c^3 : :

X(29857) lies on these lines:


X(29858) =  POINT CASTOR(2,0,0,0,-1)

Barycentrics    2 a^3 - a^2 b + 2 b^3 - a^2 c - a b c + 2 c^3 : :

X(29858) lies on these lines:


X(29859) =  POINT CASTOR(2,0,0,1,1)

Barycentrics    3 a^3 + a^2 b + a b^2 + 2 b^3 + a^2 c + a b c + a c^2 + 2 c^3 : :

X(29859) lies on these lines:


X(29860) =  POINT CASTOR(2,0,0,1,-1)

Barycentrics    3 a^3 - a^2 b + a b^2 + 2 b^3 - a^2 c - a b c + a c^2 + 2 c^3 : :

X(29860) lies on these lines:


X(29861) =  POINT CASTOR(2,0,0,-1,1)

Barycentrics    a^3 + a^2 b - a b^2 + 2 b^3 + a^2 c + a b c - a c^2 + 2 c^3 : :

X(29861) lies on these lines:


X(29862) =  POINT CASTOR(2,0,0,-1,-1)

Barycentrics    a^3 - a^2 b - a b^2 + 2 b^3 - a^2 c - a b c - a c^2 + 2 c^3 : :

X(29862) lies on these lines:


X(29863) =  POINT CASTOR(2,0,1,0,1)

Barycentrics    2 a^3 + a^2 b + 2 b^3 + a^2 c + 2 a b c + 2 c^3 : :

X(29863) lies on these lines:


X(29864) =  POINT CASTOR(2,0,1,0,2)

Barycentrics    2 a^3 + 2 a^2 b + 2 b^3 + 2 a^2 c + 3 a b c + 2 c^3 : :

X(29864) lies on these lines:


X(29865) =  POINT CASTOR(2,0,1,0,-1)

Barycentrics    2 a^3 - a^2 b + 2 b^3 - a^2 c + 2 c^3 : :

X(29865) lies on these lines:


X(29866) =  POINT CASTOR(2,0,1,0,-2)

Barycentrics    2 a^3 - 2 a^2 b + 2 b^3 - 2 a^2 c - a b c + 2 c^3 : :

X(29866) lies on these lines:


X(29867) =  POINT CASTOR(2,0,-1,0,1)

Barycentrics    2 a^3 + a^2 b + 2 b^3 + a^2 c + 2 c^3 : :

X(29867) lies on these lines:


X(29868) =  POINT CASTOR(2,0,-1,0,2)

Barycentrics    2 a^3 + 2 a^2 b + 2 b^3 + 2 a^2 c + a b c + 2 c^3 : :

X(29868) lies on these lines:


X(29869) =  POINT CASTOR(2,0,-1,0,-1)

Barycentrics    2 a^3 - a^2 b + 2 b^3 - a^2 c - 2 a b c + 2 c^3 : :

X(29869) lies on these lines:


X(29870) =  POINT CASTOR(2,0,-1,0,-2)

Barycentrics    2 a^3 - 2 a^2 b + 2 b^3 - 2 a^2 c - 3 a b c + 2 c^3 : :

X(29870) lies on these lines:


X(29871) =  POINT CASTOR(2,0,-1,1,0)

Barycentrics    3 a^3 + a b^2 + 2 b^3 - a b c + a c^2 + 2 c^3 : :

X(29871) lies on these lines:


X(29872) =  POINT CASTOR(2,0,1,-1,0)

Barycentrics    a^3 - a b^2 + 2 b^3 + a b c - a c^2 + 2 c^3 : :

X(29872) lies on these lines:


X(29873) =  POINT CASTOR(2,0,-1,-1,0)

Barycentrics    a^3 - a b^2 + 2 b^3 - a b c - a c^2 + 2 c^3 : :

X(29873) lies on these lines:


X(29874) =  POINT CASTOR(2,0,1,1,0)

Barycentrics    3 a^3 + a b^2 + 2 b^3 + a b c + a c^2 + 2 c^3 : :

X(29874) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 95: X(29875) - X(29894)  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 95, as in centers X(29875)-X(29894). Then

m(X) = b c (a^2 - 2 b c)(4 a^2 - b c) x + 2 a c (a^2 - 2 b c)(c^2 - 2 a b) y + 2 a b (a^2 - 2 b c)(c^2 - 2 a b) z : :

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, December 11, 2018)

underbar




X(29875) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 95

Barycentrics    (a^2 - 2 b c) (-4 a^5 + 4 a b^4 + a^3 b c + 2 a^2 b^2 c - a b^3 c + 2 b^4 c + 2 a^2 b c^2 - 8 a b^2 c^2 - 2 b^3 c^2 - a b c^3 - 2 b^2 c^3 + 4 a c^4 + 2 b c^4) : :

X(29875) lies on these lines:


X(29876) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 95

Barycentrics    (a^2 - 2 b c) (-2 a^5 b^3 + 4 a^3 b^5 - 2 a b^7 + 12 a^6 b c - 8 a^4 b^3 c - 4 a^2 b^5 c - a^4 b^2 c^2 + b^6 c^2 - 2 a^5 c^3 - 8 a^4 b c^3 + 8 a^2 b^3 c^3 + 2 a b^4 c^3 + 2 a b^3 c^4 - 2 b^4 c^4 + 4 a^3 c^5 - 4 a^2 b c^5 + b^2 c^6 - 2 a c^7) : :

X(29876) lies on these lines:


X(29877) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 95

Barycentrics    (a^2 - 2 b c) (-2 a^5 b^3 + 4 a^3 b^5 - 2 a b^7 + 8 a^6 b c - 8 a^4 b^3 c + 2 a^3 b^3 c^2 - a^2 b^4 c^2 + 2 a b^5 c^2 + b^6 c^2 - 2 a^5 c^3 - 8 a^4 b c^3 + 2 a^3 b^2 c^3 - a^2 b^2 c^4 - 2 b^4 c^4 + 4 a^3 c^5 + 2 a b^2 c^5 + b^2 c^6 - 2 a c^7) : :

X(29877) lies on these lines:


X(29878) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 95

Barycentrics    (a^2 - 2 b c) (4 a^3 - 4 a b^2 - a b c + 2 b^2 c - 4 a c^2 + 2 b c^2) : :

X(29878) lies on these lines:


X(29879) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 95

Barycentrics    (a^2 - 2 b c) (-4 a^3 + 6 a b^2 + 2 b^3 - 7 a b c - 3 b^2 c + 6 a c^2 - 3 b c^2 + 2 c^3) : :

X(29879) lies on these lines:


X(29880) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 95

Barycentrics    (a^2 - 2 b c) (2 a^2 b^3 + 2 a b^4 - 8 a^3 b c - b^3 c^2 + 2 a^2 c^3 - b^2 c^3 + 2 a c^4) : :

X(29880) lies on these lines:


X(29881) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 95

Barycentrics    (a^2 - 2 b c) (-4 a^7 - 4 a^5 b^2 + 4 a^3 b^4 + 4 a b^6 + a^5 b c + 2 a^4 b^2 c - a b^5 c + 2 b^6 c - 4 a^5 c^2 + 2 a^4 b c^2 - 4 a b^4 c^2 - 2 b^5 c^2 + 4 a^3 c^4 - 4 a b^2 c^4 - a b c^5 - 2 b^2 c^5 + 4 a c^6 + 2 b c^6) : :

X(29881) lies on these lines:


X(29882) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 95

Barycentrics    (a^2 - 2 b c) (-4 a^7 - 4 a^5 b^2 + 4 a^3 b^4 + 4 a b^6 + a^5 b c + 2 a^4 b^2 c - 2 a^2 b^4 c - a b^5 c + 2 b^6 c - 4 a^5 c^2 + 2 a^4 b c^2 + 4 a^3 b^2 c^2 - 4 a b^4 c^2 - 2 b^5 c^2 + a b^3 c^3 + 4 a^3 c^4 - 2 a^2 b c^4 - 4 a b^2 c^4 - a b c^5 - 2 b^2 c^5 + 4 a c^6 + 2 b c^6) : :

X(29882) lies on these lines:


X(29883) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 95

Barycentrics    (a^2 - 2 b c) (4 a^4 - 4 a b^3 - a^2 b c + 4 b^2 c^2 - 4 a c^3) : :

X(29883) lies on these lines:


X(29884) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 95

Barycentrics    (a^2 - 2 b c) (4 a^5 - 4 a b^4 - a^3 b c + 2 b^3 c^2 + 2 b^2 c^3 - 4 a c^4) : :

X(29884) lies on these lines:


X(29885) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 95

Barycentrics    (a^2 - 2 b c) (2 a b^2 + 2 b^3 - 8 a b c - b^2 c + 2 a c^2 - b c^2 + 2 c^3) : :

X(29885) lies on these lines:


X(29886) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 95

Barycentrics    b c (a^2 - 2 b c) (2 a^2 b - a b^2 + 2 b^3 + 2 a^2 c - 8 a b c - a c^2 + 2 c^3) : :

X(29886) lies on these lines:


X(29887) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 95

Barycentrics    (a^2 - 2 b c) (2 a^2 b^4 - 4 a^3 b^2 c - 4 a^3 b c^2 + 4 a^2 b^2 c^2 - b^3 c^3 + 2 a^2 c^4) : :

X(29887) lies on these lines:


X(29888) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 95

Barycentrics    (a^2 - 2 b c) (2 a^3 b^5 - 4 a^4 b^3 c - 4 a^4 b c^3 + 4 a^2 b^3 c^3 - b^4 c^4 + 2 a^3 c^5) : :

X(29888) lies on these lines:


X(29889) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 95

Barycentrics    (a^2 - 2 b c) (2 a^5 b^3 - 4 a^3 b^5 + 2 a b^7 - 20 a^6 b c + 8 a^4 b^3 c + 12 a^2 b^5 c + 3 a^4 b^2 c^2 + 4 a^3 b^3 c^2 - 2 a^2 b^4 c^2 + 4 a b^5 c^2 - b^6 c^2 + 2 a^5 c^3 + 8 a^4 b c^3 + 4 a^3 b^2 c^3 - 24 a^2 b^3 c^3 - 6 a b^4 c^3 - 2 a^2 b^2 c^4 - 6 a b^3 c^4 + 2 b^4 c^4 - 4 a^3 c^5 + 12 a^2 b c^5 + 4 a b^2 c^5 - b^2 c^6 + 2 a c^7) : :

X(29889) lies on these lines:


X(29890) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 95

Barycentrics    (a^2 - 2 b c) (4 a^5 - 2 a^2 b^3 - 6 a b^4 + 3 a^3 b c + 2 a^2 b^2 c - a b^3 c - 2 b^4 c + 2 a^2 b c^2 + 9 a b^2 c^2 + 3 b^3 c^2 - 2 a^2 c^3 - a b c^3 + 3 b^2 c^3 - 6 a c^4 - 2 b c^4) : :

X(29890) lies on these lines:


X(29891) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 95

Barycentrics    (a^2 - 2 b c) (-4 a^7 - 4 a^5 b^2 + 4 a^3 b^4 + 4 a b^6 + a^5 b c + 2 a^4 b^2 c - 4 a^2 b^4 c - a b^5 c + 2 b^6 c - 4 a^5 c^2 + 2 a^4 b c^2 + 8 a^3 b^2 c^2 - 4 a b^4 c^2 - 2 b^5 c^2 + 2 a b^3 c^3 + 4 a^3 c^4 - 4 a^2 b c^4 - 4 a b^2 c^4 - a b c^5 - 2 b^2 c^5 + 4 a c^6 + 2 b c^6) : :

X(29891) lies on these lines:


X(29892) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 95

Barycentrics    (a^2 - 2 b c) (-2 a^5 b^3 + 4 a^3 b^5 - 2 a b^7 + 28 a^6 b c - 8 a^4 b^3 c - 20 a^2 b^5 c - 5 a^4 b^2 c^2 - 8 a^3 b^3 c^2 + 4 a^2 b^4 c^2 - 8 a b^5 c^2 + b^6 c^2 - 2 a^5 c^3 - 8 a^4 b c^3 - 8 a^3 b^2 c^3 + 40 a^2 b^3 c^3 + 10 a b^4 c^3 + 4 a^2 b^2 c^4 + 10 a b^3 c^4 - 2 b^4 c^4 + 4 a^3 c^5 - 20 a^2 b c^5 - 8 a b^2 c^5 + b^2 c^6 - 2 a c^7) : :

X(29892) lies on these lines:


X(29893) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 95

Barycentrics    (a^2 - 2 b c) (-2 a^5 b^3 + 4 a^3 b^5 - 2 a b^7 + 12 a^6 b c - 8 a^4 b^3 c + 4 a^3 b^4 c - 9 a^4 b^2 c^2 - 8 a^3 b^3 c^2 + 4 a^2 b^4 c^2 + b^6 c^2 - 2 a^5 c^3 - 8 a^4 b c^3 - 8 a^3 b^2 c^3 + 6 a^2 b^3 c^3 + 4 a^3 b c^4 + 4 a^2 b^2 c^4 - 2 b^4 c^4 + 4 a^3 c^5 + b^2 c^6 - 2 a c^7) : :

X(29893) lies on these lines:


X(29894) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 95

Barycentrics    (a^2 - 2 b c) (2 a^3 b^5 + 4 a^6 b c - 4 a^4 b^3 c - 4 a^2 b^5 c - a^4 b^2 c^2 - 4 a^4 b c^3 + 4 a^2 b^3 c^3 + 2 a b^4 c^3 + 2 a b^3 c^4 - b^4 c^4 + 2 a^3 c^5 - 4 a^2 b c^5) : :

X(29894) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 96: X(29895) - X(29917)  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 96, as in centers X(29895)-X(29917). Then

m(X) = b c (4 a^2 - b c) (a^2 + 2 b c) (a^2 + 2 b c) x -2 a c(a^2 + 2 b c) (c^2 + 2 a b) y - 2 a b (a^2 + 2 b c)( b^2 + 2 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(29895) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 96

Barycentrics    (a^2 + 2 b c) (4 a^2 - 4 a b - 2 b^2 - 4 a c - b c - 2 c^2) : :

X(29895) lies on these lines:


X(29896) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 96

Barycentrics    (a^2 + 2 b c) (4 a^5 - 4 a b^4 - a^3 b c + 2 a^2 b^2 c + a b^3 c + 2 b^4 c + 2 a^2 b c^2 + 8 a b^2 c^2 - 2 b^3 c^2 + a b c^3 - 2 b^2 c^3 - 4 a c^4 + 2 b c^4) : :

X(29896) lies on these lines:


X(29897) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 96

Barycentrics    (a^2 + 2 b c) (2 a^5 b^3 - 4 a^3 b^5 + 2 a b^7 + 12 a^6 b c - 8 a^4 b^3 c - 4 a^2 b^5 c - a^4 b^2 c^2 + b^6 c^2 + 2 a^5 c^3 - 8 a^4 b c^3 + 8 a^2 b^3 c^3 - 2 a b^4 c^3 - 2 a b^3 c^4 - 2 b^4 c^4 - 4 a^3 c^5 - 4 a^2 b c^5 + b^2 c^6 + 2 a c^7) : :

X(29897) lies on these lines:


X(29898) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 96

Barycentrics    (a^2 + 2 b c) (2 a^5 b^3 - 4 a^3 b^5 + 2 a b^7 + 8 a^6 b c - 8 a^4 b^3 c - 2 a^3 b^3 c^2 - a^2 b^4 c^2 - 2 a b^5 c^2 + b^6 c^2 + 2 a^5 c^3 - 8 a^4 b c^3 - 2 a^3 b^2 c^3 - a^2 b^2 c^4 - 2 b^4 c^4 - 4 a^3 c^5 - 2 a b^2 c^5 + b^2 c^6 + 2 a c^7) : :

X(29898) lies on these lines:


X(29899) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 96

Barycentrics    (a^2 + 2 b c) (4 a^3 - 4 a b^2 - a b c - 2 b^2 c - 4 a c^2 - 2 b c^2) : :

X(29899) lies on these lines:


X(29900) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 96

Barycentrics    (a^2 + 2 b c) (4 a^3 - 2 a b^2 + 2 b^3 + 7 a b c - b^2 c - 2 a c^2 - b c^2 + 2 c^3) : :

X(29900) lies on these lines:


X(29901) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 96

Barycentrics    (a^2 + 2 b c) (2 a^2 b^3 + 2 a b^4 + 8 a^3 b c + b^3 c^2 + 2 a^2 c^3 + b^2 c^3 + 2 a c^4) : :

X(29901) lies on these lines:


X(29902) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 96

Barycentrics    (a^2 + 2 b c) (4 a^7 + 4 a^5 b^2 - 4 a^3 b^4 - 4 a b^6 - a^5 b c + 2 a^4 b^2 c + a b^5 c + 2 b^6 c + 4 a^5 c^2 + 2 a^4 b c^2 + 4 a b^4 c^2 - 2 b^5 c^2 - 4 a^3 c^4 + 4 a b^2 c^4 + a b c^5 - 2 b^2 c^5 - 4 a c^6 + 2 b c^6) : :

X(29902) lies on these lines:


X(29903) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 96

Barycentrics    (a^2 + 2 b c) (4 a^7 + 4 a^5 b^2 - 4 a^3 b^4 - 4 a b^6 - a^5 b c + 2 a^4 b^2 c - 2 a^2 b^4 c + a b^5 c + 2 b^6 c + 4 a^5 c^2 + 2 a^4 b c^2 - 4 a^3 b^2 c^2 + 4 a b^4 c^2 - 2 b^5 c^2 - a b^3 c^3 - 4 a^3 c^4 - 2 a^2 b c^4 + 4 a b^2 c^4 + a b c^5 - 2 b^2 c^5 - 4 a c^6 + 2 b c^6) : :

X(29903) lies on these lines:


X(29904) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 96

Barycentrics    (a^2 + 2 b c) (4 a^4 - 4 a b^3 - a^2 b c - 4 b^2 c^2 - 4 a c^3) : :

X(29904) lies on these lines:


X(29905) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 96

Barycentrics    (a^2 + 2 b c) (4 a^5 - 4 a b^4 - a^3 b c - 2 b^3 c^2 - 2 b^2 c^3 - 4 a c^4) : :

X(29905) lies on these lines:


X(29906) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(37), WHERE A'B'C' = GEMINI TRIANGLE 96

Barycentrics    (a^2 + 2 b c) (2 a b^2 + 2 b^3 + 8 a b c + b^2 c + 2 a c^2 + b c^2 + 2 c^3) : :

X(29906) lies on these lines:


X(29907) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 96

Barycentrics    b c (a^2 + 2 b c) (2 a^2 b + a b^2 + 2 b^3 + 2 a^2 c + 8 a b c + a c^2 + 2 c^3) : :

X(29907) lies on these lines:


X(29908) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 96

Barycentrics    (2 a - b - c) (2 a + 2 b - c) (2 a - b + 2 c) (a^2 + 2 b c) : :

X(29908) lies on these lines:


X(29909) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 96

Barycentrics    (a^2 + 2 b c) (2 a^2 b^4 + 4 a^3 b^2 c + 4 a^3 b c^2 - 4 a^2 b^2 c^2 + b^3 c^3 + 2 a^2 c^4) : :

X(29909) lies on these lines:


X(29910) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 96

Barycentrics    (a^2 + 2 b c) (2 a^3 b^5 + 4 a^4 b^3 c + 4 a^4 b c^3 - 4 a^2 b^3 c^3 + b^4 c^4 + 2 a^3 c^5) : :

X(29910) lies on these lines:


X(29911) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 96

Barycentrics    (a^2 + 2 b c) (2 a^5 b^3 - 4 a^3 b^5 + 2 a b^7 + 20 a^6 b c - 8 a^4 b^3 c - 12 a^2 b^5 c - 3 a^4 b^2 c^2 + 4 a^3 b^3 c^2 + 2 a^2 b^4 c^2 + 4 a b^5 c^2 + b^6 c^2 + 2 a^5 c^3 - 8 a^4 b c^3 + 4 a^3 b^2 c^3 + 24 a^2 b^3 c^3 - 6 a b^4 c^3 + 2 a^2 b^2 c^4 - 6 a b^3 c^4 - 2 b^4 c^4 - 4 a^3 c^5 - 12 a^2 b c^5 + 4 a b^2 c^5 + b^2 c^6 + 2 a c^7) : :

X(29911) lies on these lines:


X(29912) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 96

Barycentrics    (a^2 + 2 b c) (4 a^5 + 2 a^2 b^3 - 2 a b^4 + 3 a^3 b c + 6 a^2 b^2 c + 3 a b^3 c + 2 b^4 c + 6 a^2 b c^2 + 9 a b^2 c^2 - b^3 c^2 + 2 a^2 c^3 + 3 a b c^3 - b^2 c^3 - 2 a c^4 + 2 b c^4) : :

X(29912) lies on these lines:


X(29913) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 96

Barycentrics    (a^2 + 2 b c) (4 a^7 + 4 a^5 b^2 - 4 a^3 b^4 - 4 a b^6 - a^5 b c + 2 a^4 b^2 c - 4 a^2 b^4 c + a b^5 c + 2 b^6 c + 4 a^5 c^2 + 2 a^4 b c^2 - 8 a^3 b^2 c^2 + 4 a b^4 c^2 - 2 b^5 c^2 - 2 a b^3 c^3 - 4 a^3 c^4 - 4 a^2 b c^4 + 4 a b^2 c^4 + a b c^5 - 2 b^2 c^5 - 4 a c^6 + 2 b c^6) : :

X(29913) lies on these lines:


X(29914) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 96

Barycentrics    (a^2 + 2 b c) (2 a^5 b^3 - 4 a^3 b^5 + 2 a b^7 + 28 a^6 b c - 8 a^4 b^3 c - 20 a^2 b^5 c - 5 a^4 b^2 c^2 + 8 a^3 b^3 c^2 + 4 a^2 b^4 c^2 + 8 a b^5 c^2 + b^6 c^2 + 2 a^5 c^3 - 8 a^4 b c^3 + 8 a^3 b^2 c^3 + 40 a^2 b^3 c^3 - 10 a b^4 c^3 + 4 a^2 b^2 c^4 - 10 a b^3 c^4 - 2 b^4 c^4 - 4 a^3 c^5 - 20 a^2 b c^5 + 8 a b^2 c^5 + b^2 c^6 + 2 a c^7) : :

X(29914) lies on these lines:


X(29915) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 96

Barycentrics    (a^2 + 2 b c) (2 a^5 b^3 - 4 a^3 b^5 + 2 a b^7 + 12 a^6 b c - 8 a^4 b^3 c - 4 a^3 b^4 c - 8 a^2 b^5 c - 9 a^4 b^2 c^2 - 8 a^3 b^3 c^2 - 4 a^2 b^4 c^2 + b^6 c^2 + 2 a^5 c^3 - 8 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 - 4 a^3 b c^4 - 4 a^2 b^2 c^4 - 4 a b^3 c^4 - 2 b^4 c^4 - 4 a^3 c^5 - 8 a^2 b c^5 + b^2 c^6 + 2 a c^7) : :

X(29915) lies on these lines:


X(29916) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 96

Barycentrics    (a^2 + 2 b c) (-2 a^3 b^5 + 4 a^6 b c - 4 a^4 b^3 c - 4 a^2 b^5 c - a^4 b^2 c^2 - 4 a^4 b c^3 + 4 a^2 b^3 c^3 - 2 a b^4 c^3 - 2 a b^3 c^4 - b^4 c^4 - 2 a^3 c^5 - 4 a^2 b c^5) : :

X(29916) lies on these lines:


X(29917) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 96

Barycentrics    (a^2 + 2 b c) (4 a^4 - 2 a b^3 + 3 a^2 b c - 3 b^2 c^2 - 2 a c^3) : :

X(29917) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 97: X(29918) - X(29935)  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 97, as in centers X(29918)-X(29935). Then

m(X) = (b + c) (b^2 + c^2 - a^2 - b c) x - b (a + b) (b + c) y - c (a + c) (b + 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(29918) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 97

Barycentrics    a (b^2 - b c + c^2) (a^4 + a^2 b^2 - 3 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) : :

X(29918) lies on these lines:


X(29919) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 97

Barycentrics    a (b^2 - b c + c^2) (a^7 - a^3 b^4 - a^5 b c + a^4 b^2 c + a b^5 c - b^6 c + a^4 b c^2 - a^3 b^2 c^2 + a^2 b^3 c^2 - a b^4 c^2 + a^2 b^2 c^3 - 2 a b^3 c^3 + b^4 c^3 - a^3 c^4 - a b^2 c^4 + b^3 c^4 + a b c^5 - b c^6) : :

X(29919) lies on these lines:


X(29920) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 97

Barycentrics    (b^2 - b c + c^2) (-a^8 - a^7 b - a^6 b^2 + 2 a^5 b^3 + a^4 b^4 - a^3 b^5 + a^2 b^6 - a^7 c + 3 a^6 b c - a^5 b^2 c - 2 a^4 b^3 c + a^3 b^4 c - a^2 b^5 c + a b^6 c - a^6 c^2 - a^5 b c^2 - 3 a^4 b^2 c^2 + 2 a^3 b^3 c^2 - a^2 b^4 c^2 - a b^5 c^2 + b^6 c^2 + 2 a^5 c^3 - 2 a^4 b c^3 + 2 a^3 b^2 c^3 + 2 a^2 b^3 c^3 + a^4 c^4 + a^3 b c^4 - a^2 b^2 c^4 - 2 b^4 c^4 - a^3 c^5 - a^2 b c^5 - a b^2 c^5 + a^2 c^6 + a b c^6 + b^2 c^6) : :

X(29920) lies on these lines:


X(29921) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 97

Barycentrics    (b^2 - b c + c^2) (-a^7 b - a^6 b^2 + 2 a^5 b^3 - a^3 b^5 + a^2 b^6 - a^7 c + 2 a^6 b c - 2 a^4 b^3 c + a^3 b^4 c - a^6 c^2 - 4 a^4 b^2 c^2 + 3 a^3 b^3 c^2 - 2 a^2 b^4 c^2 - a b^5 c^2 + b^6 c^2 + 2 a^5 c^3 - 2 a^4 b c^3 + 3 a^3 b^2 c^3 + a b^4 c^3 + 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 b^2 c^5 + a^2 c^6 + b^2 c^6) : :

X(29921) lies on these lines:


X(29922) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 97

Barycentrics    a (b^2 - b c + c^2) (a^5 + a^3 b^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 + a b^2 c^2 + a b c^3 - b c^4) : :

X(29922) lies on these lines:


X(29923) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 97

Barycentrics    a (b^2 - b c + c^2) (a^9 + a^7 b^2 - a^5 b^4 - a^3 b^6 - 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^3 b^4 c^2 + a^2 b^5 c^2 - a b^6 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 + b^5 c^4 + a^3 b c^5 + 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^2 c^6 + a b c^7 - b c^8) : :

X(29923) lies on these lines:


X(29924) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 97

Barycentrics    a (b^2 - b c + c^2) (a^9 + a^7 b^2 - a^5 b^4 - a^3 b^6 - 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 + 2 a^5 b^2 c^2 - a^4 b^3 c^2 + a^2 b^5 c^2 - a b^6 c^2 - a^5 b c^3 - a^4 b^2 c^3 + a^3 b^3 c^3 - a^2 b^4 c^3 - a b^5 c^3 - a^5 c^4 + a^4 b c^4 - a^2 b^3 c^4 + a b^4 c^4 + b^5 c^4 + a^3 b c^5 + 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^2 c^6 + a b c^7 - b c^8) : :

X(29924) lies on these lines:


X(29925) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 97

Barycentrics    a (b^2 - b c + c^2) (a^6 + a^4 b^2 - a^4 b c - a^2 b^3 c + a b^4 c - b^5 c + a^4 c^2 + a^2 b^2 c^2 - a^2 b c^3 + a b c^4 - b c^5) : :

X(29925) lies on these lines:


X(29926) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 97

Barycentrics    a (b^2 - b c + c^2) (a^7 + a^5 b^2 - a^5 b c - a^2 b^4 c + a b^5 c - b^6 c + a^5 c^2 + a^3 b^2 c^2 - a^2 b c^4 + a b c^5 - b c^6) : :

X(29926) lies on these lines:


X(29927) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 97

Barycentrics    (b^2 - b c + c^2) (-3 a^8 - a^7 b - a^6 b^2 + 2 a^5 b^3 + 3 a^4 b^4 - a^3 b^5 + a^2 b^6 - a^7 c + 5 a^6 b c - 3 a^5 b^2 c - 2 a^4 b^3 c + a^3 b^4 c - 3 a^2 b^5 c + 3 a b^6 c - a^6 c^2 - 3 a^5 b c^2 - a^4 b^2 c^2 + a^2 b^4 c^2 - a b^5 c^2 + b^6 c^2 + 2 a^5 c^3 - 2 a^4 b c^3 + 6 a^2 b^3 c^3 - 2 a b^4 c^3 + 3 a^4 c^4 + a^3 b c^4 + a^2 b^2 c^4 - 2 a b^3 c^4 - 2 b^4 c^4 - a^3 c^5 - 3 a^2 b c^5 - a b^2 c^5 + a^2 c^6 + 3 a b c^6 + b^2 c^6) : :

X(29927) lies on these lines:


X(29928) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 97

Barycentrics    a (b^2 - b c + c^2) (a^7 - a^3 b^4 - 2 a^5 b c + a^4 b^2 c - a^2 b^4 c + a b^5 c - b^6 c + a^4 b c^2 - a b^3 c^3 - a^3 c^4 - a^2 b c^4 + a b c^5 - b c^6) : :

X(29928) lies on these lines:


X(29929) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 97

Barycentrics    a (b^2 - b c + c^2) (a^9 + a^7 b^2 - a^5 b^4 - a^3 b^6 - 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 + 3 a^5 b^2 c^2 - 2 a^4 b^3 c^2 + a^3 b^4 c^2 + a^2 b^5 c^2 - a b^6 c^2 - a^5 b c^3 - 2 a^4 b^2 c^3 + 2 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 + a^3 b^2 c^4 - 2 a^2 b^3 c^4 + 2 a b^4 c^4 + b^5 c^4 + a^3 b c^5 + 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^2 c^6 + a b c^7 - b c^8) : :

X(29929) lies on these lines:


X(29930) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 97

Barycentrics    (b^2 - b c + c^2) (-a^10 - a^9 b - 2 a^8 b^2 - a^7 b^3 + 3 a^6 b^4 + a^5 b^5 + a^3 b^7 - a^9 c - 2 a^8 b c + a^6 b^3 c + a^2 b^7 c + a b^8 c - 2 a^8 c^2 - 2 a^6 b^2 c^2 - 4 a^5 b^3 c^2 - 2 a^3 b^5 c^2 + 2 a b^7 c^2 - a^7 c^3 + a^6 b c^3 - 4 a^5 b^2 c^3 - a^4 b^3 c^3 + 5 a^3 b^4 c^3 - a^2 b^5 c^3 + b^7 c^3 + 3 a^6 c^4 + 5 a^3 b^3 c^4 - 3 a b^5 c^4 + a^5 c^5 - 2 a^3 b^2 c^5 - a^2 b^3 c^5 - 3 a b^4 c^5 - 2 b^5 c^5 + a^3 c^7 + a^2 b c^7 + 2 a b^2 c^7 + b^3 c^7 + a b c^8) : :

X(29930) lies on these lines:


X(29931) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 97

Barycentrics    a (b^2 - b c + c^2) (a^10 + a^9 b + 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^9 c + 2 a^7 b^2 c + a^6 b^3 c - 3 a^5 b^4 c + a^4 b^5 c - a^2 b^7 c - b^9 c + a^8 c^2 + 2 a^7 b c^2 + 2 a^6 b^2 c^2 + a^5 b^3 c^2 + 2 a^4 b^4 c^2 + 2 a^3 b^5 c^2 - a b^7 c^2 - b^8 c^2 + a^7 c^3 + a^6 b c^3 + a^5 b^2 c^3 + a^4 b^3 c^3 - a^3 b^4 c^3 - a^2 b^5 c^3 - a b^6 c^3 - b^7 c^3 - a^6 c^4 - 3 a^5 b c^4 + 2 a^4 b^2 c^4 - a^3 b^3 c^4 - 4 a^2 b^4 c^4 + 2 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 - a^2 b^3 c^5 + 2 a b^4 c^5 + 4 b^5 c^5 - a^4 c^6 - a b^3 c^6 + b^4 c^6 - a^3 c^7 - a^2 b c^7 - a b^2 c^7 - b^3 c^7 - b^2 c^8 - b c^9) : :

X(29931) lies on these lines:


X(29932) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 97

Barycentrics    (b^2 - b c + c^2) (-a^11 + a^9 b^2 + a^8 b^3 - 2 a^6 b^5 + a^5 b^6 + a^4 b^7 - a^3 b^8 + 4 a^9 b c - 2 a^7 b^3 c + 2 a^6 b^4 c - 3 a^5 b^5 c - 2 a^2 b^8 c + a b^9 c + a^9 c^2 - 2 a^7 b^2 c^2 + 3 a^6 b^3 c^2 - 2 a^4 b^5 c^2 + 2 a^3 b^6 c^2 - a^2 b^7 c^2 - a b^8 c^2 + a^8 c^3 - 2 a^7 b c^3 + 3 a^6 b^2 c^3 + 7 a^5 b^3 c^3 - 2 a^4 b^4 c^3 + 3 a^2 b^6 c^3 - a b^7 c^3 - b^8 c^3 + 2 a^6 b c^4 - 2 a^4 b^3 c^4 - 2 a^3 b^4 c^4 + a b^6 c^4 - b^7 c^4 - 2 a^6 c^5 - 3 a^5 b c^5 - 2 a^4 b^2 c^5 + 2 b^6 c^5 + a^5 c^6 + 2 a^3 b^2 c^6 + 3 a^2 b^3 c^6 + a b^4 c^6 + 2 b^5 c^6 + a^4 c^7 - a^2 b^2 c^7 - a b^3 c^7 - b^4 c^7 - a^3 c^8 - 2 a^2 b c^8 - a b^2 c^8 - b^3 c^8 + a b c^9) : :

X(29932) lies on these lines:


X(29933) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 97

Barycentrics    (b^2 - b c + c^2) (-5 a^8 - a^7 b - a^6 b^2 + 2 a^5 b^3 + 5 a^4 b^4 - a^3 b^5 + a^2 b^6 - a^7 c + 7 a^6 b c - 5 a^5 b^2 c - 2 a^4 b^3 c + a^3 b^4 c - 5 a^2 b^5 c + 5 a b^6 c - a^6 c^2 - 5 a^5 b c^2 + a^4 b^2 c^2 - 2 a^3 b^3 c^2 + 3 a^2 b^4 c^2 - a b^5 c^2 + b^6 c^2 + 2 a^5 c^3 - 2 a^4 b c^3 - 2 a^3 b^2 c^3 + 10 a^2 b^3 c^3 - 4 a b^4 c^3 + 5 a^4 c^4 + a^3 b c^4 + 3 a^2 b^2 c^4 - 4 a b^3 c^4 - 2 b^4 c^4 - a^3 c^5 - 5 a^2 b c^5 - a b^2 c^5 + a^2 c^6 + 5 a b c^6 + b^2 c^6) : :

X(29933) lies on these lines:


X(29934) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 97

Barycentrics    (b^2 - b c + c^2) (-a^8 - a^7 b - a^6 b^2 + 2 a^5 b^3 + a^4 b^4 - a^3 b^5 + a^2 b^6 - a^7 c + a^6 b c - a^5 b^2 c - 2 a^4 b^3 c - a^3 b^4 c - a^2 b^5 c + a b^6 c - a^6 c^2 - a^5 b c^2 - a^4 b^2 c^2 + a^2 b^4 c^2 - a b^5 c^2 + b^6 c^2 + 2 a^5 c^3 - 2 a^4 b c^3 + 4 a^2 b^3 c^3 - 2 a b^4 c^3 + a^4 c^4 - a^3 b c^4 + a^2 b^2 c^4 - 2 a b^3 c^4 - 2 b^4 c^4 - a^3 c^5 - a^2 b c^5 - a b^2 c^5 + a^2 c^6 + a b c^6 + b^2 c^6) : :

X(29934) lies on these lines:


X(29935) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 97

Barycentrics    (b^2 - b c + c^2) (-a^8 - a^6 b^2 + a^5 b^3 + a^6 b c - a^4 b^3 c + a^3 b^4 c - a^2 b^5 c + a b^6 c - a^6 c^2 - 2 a^4 b^2 c^2 + a^3 b^3 c^2 - a^2 b^4 c^2 + a^5 c^3 - a^4 b c^3 + a^3 b^2 c^3 + a^2 b^3 c^3 + a^3 b c^4 - a^2 b^2 c^4 - b^4 c^4 - a^2 b c^5 + a b c^6) : :

X(29935) lies on these lines:






leftri  Collineation mappings involving Gemini triangle 98: X(29936) - X(29956)  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 98, as in centers X(29936)-X(29956). Then

m(X) = (b^2 + b c + c^2) (a^4 + a^2 b^2 + a^2 c^2 + b^2 c^2 - a^2 b c) x + a c (a^2 + a b + b^2) (b^2 + b c + c^2) y + a b (a^2 + a c + c^2) (b^2 + b c + c^2) 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(29936) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 98

Barycentrics    a (b^2 + b c + c^2) (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^2 c^2 + b c^3) : :

X(29936) lies on these lines:


X(29937) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 98

Barycentrics    a (b^2 + b c + c^2) (a^7 - a^3 b^4 - a^5 b c - a^4 b^2 c + a b^5 c + b^6 c - a^4 b c^2 - a^3 b^2 c^2 - a^2 b^3 c^2 - a b^4 c^2 - a^2 b^2 c^3 - 2 a b^3 c^3 - b^4 c^3 - a^3 c^4 - a b^2 c^4 - b^3 c^4 + a b c^5 + b c^6) : :

X(29937) lies on these lines:


X(29938) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 98

Barycentrics    (b^2 + b c + c^2) (-a^8 + a^7 b - a^6 b^2 - 2 a^5 b^3 + a^4 b^4 + a^3 b^5 + a^2 b^6 + a^7 c + 3 a^6 b c + a^5 b^2 c - 2 a^4 b^3 c - a^3 b^4 c - a^2 b^5 c - a b^6 c - a^6 c^2 + a^5 b c^2 - 3 a^4 b^2 c^2 - 2 a^3 b^3 c^2 - a^2 b^4 c^2 + a b^5 c^2 + b^6 c^2 - 2 a^5 c^3 - 2 a^4 b c^3 - 2 a^3 b^2 c^3 + 2 a^2 b^3 c^3 + a^4 c^4 - a^3 b c^4 - a^2 b^2 c^4 - 2 b^4 c^4 + a^3 c^5 - a^2 b c^5 + a b^2 c^5 + a^2 c^6 - a b c^6 + b^2 c^6) : :

X(29938) lies on these lines:


X(29939) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), WHERE A'B'C' = GEMINI TRIANGLE 98

Barycentrics    (b^2 + b c + c^2) (a^7 b - a^6 b^2 - 2 a^5 b^3 + a^3 b^5 + a^2 b^6 + a^7 c + 2 a^6 b c - 2 a^4 b^3 c - a^3 b^4 c - a^6 c^2 - 4 a^4 b^2 c^2 - 3 a^3 b^3 c^2 - 2 a^2 b^4 c^2 + a b^5 c^2 + b^6 c^2 - 2 a^5 c^3 - 2 a^4 b c^3 - 3 a^3 b^2 c^3 - a b^4 c^3 - 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 b^2 c^5 + a^2 c^6 + b^2 c^6) : :

X(29939) lies on these lines:


X(29940) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), WHERE A'B'C' = GEMINI TRIANGLE 98

Barycentrics    a (b^2 + b c + c^2) (a^5 + a^3 b^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 + a b^2 c^2 + a b c^3 + b c^4) : :

X(29940) lies on these lines:


X(29941) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 98

Barycentrics    a (b^2 + b c + c^2) (a^9 + a^7 b^2 - a^5 b^4 - a^3 b^6 - 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^3 b^4 c^2 - a^2 b^5 c^2 - a b^6 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 - b^5 c^4 + a^3 b c^5 - 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^2 c^6 + a b c^7 + b c^8) : :

X(29941) lies on these lines:


X(29942) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 98

Barycentrics    a (b^2 + b c + c^2) (a^9 + a^7 b^2 - a^5 b^4 - a^3 b^6 - 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 + 2 a^5 b^2 c^2 + a^4 b^3 c^2 - a^2 b^5 c^2 - a b^6 c^2 - a^5 b c^3 + a^4 b^2 c^3 + a^3 b^3 c^3 + a^2 b^4 c^3 - a b^5 c^3 - a^5 c^4 - a^4 b c^4 + a^2 b^3 c^4 + a b^4 c^4 - b^5 c^4 + a^3 b c^5 - 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^2 c^6 + a b c^7 + b c^8) : :

X(29942) lies on these lines:


X(29943) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 98

Barycentrics    a (b^2 + b c + c^2) (a^6 + a^4 b^2 - a^4 b c + a^2 b^3 c + a b^4 c + b^5 c + a^4 c^2 + a^2 b^2 c^2 + a^2 b c^3 + a b c^4 + b c^5) : :

X(29943) lies on these lines:


X(29944) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), WHERE A'B'C' = GEMINI TRIANGLE 98

Barycentrics    a (b^2 + b c + c^2) (a^7 + a^5 b^2 - a^5 b c + a^2 b^4 c + a b^5 c + b^6 c + a^5 c^2 + a^3 b^2 c^2 + a^2 b c^4 + a b c^5 + b c^6) : :

X(29944) lies on these lines:


X(29945) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 98

Barycentrics    (b^2 + b c + c^2) (a^6 - 2 a^4 b c + a b^4 c - b^3 c^3 + a b c^4) : :

X(29945) lies on these lines:


X(29946) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(693), WHERE A'B'C' = GEMINI TRIANGLE 98

Barycentrics    (b - c) (a^2 - b c) (b^2 + b c + c^2) (-a^3 b + a^2 b^2 - a^3 c - a^2 b c + a^2 c^2 + b^2 c^2) : :

X(29946) lies on these lines:


X(29947) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 98

Barycentrics    (b^2 + b c + c^2) (-3 a^8 + a^7 b - a^6 b^2 - 2 a^5 b^3 + 3 a^4 b^4 + a^3 b^5 + a^2 b^6 + a^7 c + 5 a^6 b c + 3 a^5 b^2 c - 2 a^4 b^3 c - a^3 b^4 c - 3 a^2 b^5 c - 3 a b^6 c - a^6 c^2 + 3 a^5 b c^2 - a^4 b^2 c^2 + a^2 b^4 c^2 + a b^5 c^2 + b^6 c^2 - 2 a^5 c^3 - 2 a^4 b c^3 + 6 a^2 b^3 c^3 + 2 a b^4 c^3 + 3 a^4 c^4 - a^3 b c^4 + a^2 b^2 c^4 + 2 a b^3 c^4 - 2 b^4 c^4 + a^3 c^5 - 3 a^2 b c^5 + a b^2 c^5 + a^2 c^6 - 3 a b c^6 + b^2 c^6) : :

X(29947) lies on these lines:


X(29948) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 98

Barycentrics    a (b^2 + b c + c^2) (a^7 - a^3 b^4 - 2 a^5 b c - 3 a^4 b^2 c - 2 a^3 b^3 c - a^2 b^4 c + a b^5 c + b^6 c - 3 a^4 b c^2 - 4 a^3 b^2 c^2 - 4 a^2 b^3 c^2 - 2 a b^4 c^2 - 2 a^3 b c^3 - 4 a^2 b^2 c^3 - 5 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 b c^5 + b c^6) : :

X(29948) lies on these lines:


X(29949) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 98

Barycentrics    a (b^2 + b c + c^2) (a^9 + a^7 b^2 - a^5 b^4 - a^3 b^6 - 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 + 3 a^5 b^2 c^2 + 2 a^4 b^3 c^2 + a^3 b^4 c^2 - a^2 b^5 c^2 - a b^6 c^2 - a^5 b c^3 + 2 a^4 b^2 c^3 + 2 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 + a^3 b^2 c^4 + 2 a^2 b^3 c^4 + 2 a b^4 c^4 - b^5 c^4 + a^3 b c^5 - 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^2 c^6 + a b c^7 + b c^8) : :

X(29949) lies on these lines:


X(29950) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 98

Barycentrics    (b^2 + b c + c^2) (a^10 + a^9 b + a^7 b^3 + a^6 b^4 - a^5 b^5 - 2 a^4 b^6 - a^3 b^7 + a^9 c - 2 a^8 b c - 4 a^7 b^2 c + a^6 b^3 c + 2 a^5 b^4 c + a^2 b^7 c + a b^8 c - 4 a^7 b c^2 - 2 a^6 b^2 c^2 + 6 a^5 b^3 c^2 + 6 a^4 b^4 c^2 + 2 a^3 b^5 c^2 + a^7 c^3 + a^6 b c^3 + 6 a^5 b^2 c^3 + 9 a^4 b^3 c^3 + 3 a^3 b^4 c^3 - a^2 b^5 c^3 - 2 a b^6 c^3 - b^7 c^3 + a^6 c^4 + 2 a^5 b c^4 + 6 a^4 b^2 c^4 + 3 a^3 b^3 c^4 + a b^5 c^4 - a^5 c^5 + 2 a^3 b^2 c^5 - a^2 b^3 c^5 + a b^4 c^5 + 2 b^5 c^5 - 2 a^4 c^6 - 2 a b^3 c^6 - a^3 c^7 + a^2 b c^7 - b^3 c^7 + a b c^8) : :

X(29950) lies on these lines:


X(29951) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 98

Barycentrics    a (b^2 + b c + c^2) (a^10 + a^9 b + 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^9 c - 2 a^7 b^2 c - a^6 b^3 c - a^5 b^4 c - a^4 b^5 c + a^2 b^7 c + 2 a b^8 c + b^9 c + a^8 c^2 - 2 a^7 b c^2 - 2 a^6 b^2 c^2 + 3 a^5 b^3 c^2 + 4 a^4 b^4 c^2 + 2 a^3 b^5 c^2 + a b^7 c^2 + b^8 c^2 + a^7 c^3 - a^6 b c^3 + 3 a^5 b^2 c^3 + 9 a^4 b^3 c^3 + 7 a^3 b^4 c^3 + a^2 b^5 c^3 - 3 a b^6 c^3 - b^7 c^3 - a^6 c^4 - a^5 b c^4 + 4 a^4 b^2 c^4 + 7 a^3 b^3 c^4 + 4 a^2 b^4 c^4 - b^6 c^4 - a^5 c^5 - a^4 b c^5 + 2 a^3 b^2 c^5 + a^2 b^3 c^5 - a^4 c^6 - 3 a b^3 c^6 - b^4 c^6 - a^3 c^7 + a^2 b c^7 + a b^2 c^7 - b^3 c^7 + 2 a b c^8 + b^2 c^8 + b c^9) : :

X(29951) lies on these lines:


X(29952) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 98

Barycentrics    (b^2 + b c + c^2) (-5 a^8 + a^7 b - a^6 b^2 - 2 a^5 b^3 + 5 a^4 b^4 + a^3 b^5 + a^2 b^6 + a^7 c + 7 a^6 b c + 5 a^5 b^2 c - 2 a^4 b^3 c - a^3 b^4 c - 5 a^2 b^5 c - 5 a b^6 c - a^6 c^2 + 5 a^5 b c^2 + a^4 b^2 c^2 + 2 a^3 b^3 c^2 + 3 a^2 b^4 c^2 + a b^5 c^2 + b^6 c^2 - 2 a^5 c^3 - 2 a^4 b c^3 + 2 a^3 b^2 c^3 + 10 a^2 b^3 c^3 + 4 a b^4 c^3 + 5 a^4 c^4 - a^3 b c^4 + 3 a^2 b^2 c^4 + 4 a b^3 c^4 - 2 b^4 c^4 + a^3 c^5 - 5 a^2 b c^5 + a b^2 c^5 + a^2 c^6 - 5 a b c^6 + b^2 c^6) : :

X(29952) lies on these lines:


X(29953) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 98

Barycentrics    (b^2 + b c + c^2) (-a^8 + a^7 b - a^6 b^2 - 2 a^5 b^3 + a^4 b^4 + a^3 b^5 + a^2 b^6 + a^7 c + a^6 b c - 3 a^5 b^2 c - 6 a^4 b^3 c - 3 a^3 b^4 c - a^2 b^5 c - a b^6 c - a^6 c^2 - 3 a^5 b c^2 - 9 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 + b^6 c^2 - 2 a^5 c^3 - 6 a^4 b c^3 - 8 a^3 b^2 c^3 - 4 a^2 b^3 c^3 - 2 a b^4 c^3 + a^4 c^4 - 3 a^3 b c^4 - 3 a^2 b^2 c^4 - 2 a b^3 c^4 - 2 b^4 c^4 + a^3 c^5 - a^2 b c^5 + a b^2 c^5 + a^2 c^6 - a b c^6 + b^2 c^6) : :

X(29953) lies on these lines:


X(29954) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 98

Barycentrics    (b^2 + b c + c^2) (a^8 + a^6 b^2 + a^5 b^3 - a^6 b c + a^4 b^3 c + a^3 b^4 c + a^2 b^5 c + a b^6 c + a^6 c^2 + 2 a^4 b^2 c^2 + a^3 b^3 c^2 + a^2 b^4 c^2 + a^5 c^3 + a^4 b c^3 + a^3 b^2 c^3 - a^2 b^3 c^3 + a^3 b c^4 + a^2 b^2 c^4 + b^4 c^4 + a^2 b c^5 + a b c^6) : :

X(29954) lies on these lines:


X(29955) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(649), WHERE A'B'C' = GEMINI TRIANGLE 98

Barycentrics    a (b - c) (a^2 - b c) (b^2 + b c + c^2) (a^3 + a b^2 - a b c - b^2 c + a c^2 - b c^2) : :

X(29955) lies on these lines:


X(29956) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(650), WHERE A'B'C' = GEMINI TRIANGLE 98

Barycentrics    a^2 (b - c) (a^2 + b^2 - a c - b c) (a^2 - a b - b c + c^2) (b^2 + b c + c^2) : :

X(29956) lies on these lines:


X(29957) =  X(1)X(3688)∩X(7)X(2808)

Barycentrics    a^2*((b^2+c^2)*a^4-2*(b^3+c^3)*a^3-2*b^2*c^2*a^2+2*(b^3-c^3)*(b^2-c^2)*a-(b^4+c^4+2*(b^2+c^2)*b*c)*(b-c)^2) : :

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28716.

X(29957) lies on these lines: {1, 3688}, {7, 2808}, {77, 14520}, {511, 5728}, {674, 5572}, {916, 942}, {938, 5933}, {2389, 15587}, {2810, 18412}, {3211, 9306}, {3819, 11018}, {3917, 11020}, {4253, 20793}, {6738, 29311}, {7146, 21746}, {8680, 13563}, {9440, 20683}, {13754, 15939}, {17092, 22440}


X(29958) =  COMPLEMENT OF X(23154)

Barycentrics    a^2*((b^2+c^2)*a^3+(b^2-c^2)*(b-c)*a^2-(b^4+c^4)*a-(b+c)*(b^4+c^4-2*(b^2+c^2)*b*c)): :
X(29958) = 3*X(51)-X(3868), 3*X(51)-2*X(12109), 3*X(375)-2*X(3812), 2*X(942)-3*X(5943), 3*X(3819)-4*X(5044), 3*X(3819)-2*X(11573), X(3874)-3*X(15049), 5*X(3876)-3*X(3917), 5*X(5439)-6*X(6688), 3*X(10167)-4*X(17704), 3*X(10176)-X(23156), 3*X(10202)-4*X(11695), 2*X(13369)-3*X(16836)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28716.

X(29958) lies on these lines: {1, 2810}, {2, 23154}, {51, 3868}, {63, 970}, {65, 23841}, {72, 511}, {78, 26892}, {181, 1046}, {185, 2808}, {329, 10441}, {355, 2818}, {375, 3812}, {386, 20805}, {389, 912}, {404, 3937}, {517, 12527}, {581, 20760}, {651, 1425}, {936, 3784}, {942, 5943}, {960, 8679}, {978, 1401}, {986, 23638}, {1331, 3145}, {1463, 24178}, {1757, 10822}, {1762, 7066}, {2390, 5836}, {2392, 3678}, {2842, 3754}, {3061, 23630}, {3157, 9306}, {3219, 22076}, {3732, 17499}, {3819, 5044}, {3869, 16980}, {3874, 15049}, {3876, 3917}, {3916, 15489}, {3927, 5752}, {3951, 26893}, {4339, 9309}, {4415, 18178}, {5396, 22458}, {5439, 6688}, {5462, 24475}, {5777, 5907}, {5904, 9052}, {6743, 29353}, {7078, 24320}, {7248, 11512}, {9021, 9969}, {9822, 24476}, {10110, 24474}, {10167, 17704}, {10176, 23156}, {10202, 11695}, {13369, 16836}, {13731, 21361}, {17114, 24174}, {17572, 26910}, {17768, 22300}

X(29958) = midpoint of X(i) and X(j) for these {i,j}: {185, 12528}, {3869, 16980}
X(29958) = reflection of X(i) in X(j) for these (i,j): (65, 23841), (3868, 12109), (5907, 5777), (11573, 5044), (24474, 10110), (24475, 5462), (24476, 9822)
X(29958) = complement of X(23154)
X(29958) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (51, 3868, 12109), (5044, 11573, 3819)

X(29959) =  X(49)X(575)∩X(51)X(524)

Barycentrics    a^2*(b^2+c^2)*(a^4-b^4+4*b^2*c^2-c^4) : :
X(29959) = X(6)-4*X(9822), X(6)+2*X(14913), X(69)+2*X(9969), 2*X(141)+X(1843), 4*X(141)-X(3313), 3*X(373)-2*X(597), 2*X(389)+X(15069), X(1205)-4*X(6698), 2*X(1352)+X(19161), 2*X(1843)+X(3313), X(1992)-3*X(5640), X(2979)-3*X(21356), 7*X(3090)-X(15073), 4*X(3589)-X(6467), 5*X(3618)+X(12272), 7*X(3619)-X(12220), 4*X(3628)-X(15074), 2*X(9822)+X(14913), X(9967)-4*X(24206), X(12162)-4*X(18553)

See Antreas Hatzipolakis and César Lozada, Hyacinthos 28716.

X(29959) lies on these lines: {2, 2393}, {5, 5181}, {6, 1196}, {39, 1634}, {49, 575}, {51, 524}, {67, 3521}, {69, 3060}, {110, 12039}, {141, 427}, {159, 3796}, {338, 6248}, {373, 597}, {381, 511}, {384, 1632}, {389, 15069}, {542, 9730}, {895, 16042}, {1205, 6698}, {1235, 27373}, {1350, 1597}, {1352, 7706}, {1495, 19127}, {1568, 5480}, {1992, 5640}, {1995, 8542}, {2072, 9967}, {2386, 11286}, {2781, 15030}, {2882, 8370}, {2979, 21356}, {3003, 11328}, {3090, 15073}, {3564, 5946}, {3589, 6467}, {3618, 12272}, {3619, 12220}, {3628, 15074}, {3763, 9973}, {3818, 16194}, {3819, 21358}, {5085, 11202}, {5092, 12367}, {5169, 19510}, {5421, 20794}, {5476, 14845}, {5650, 8705}, {5651, 8541}, {5890, 11180}, {5892, 11179}, {6697, 26156}, {6776, 15045}, {7669, 13335}, {8547, 22112}, {10110, 11477}, {10151, 12294}, {10602, 11284}, {10984, 15581}, {11002, 11160}, {11451, 15531}, {11645, 14855}, {11898, 13321}, {12093, 17430}, {14810, 18859}, {15060, 18358}, {15533, 21849}, {16072, 23049}, {19126, 20987}, {21513, 22143}, {21969, 22165}, {22087, 23635}

X(29959) = midpoint of X(i) and X(j) for these {i,j}: {2, 11188}, {69, 3060}, {599, 9971}, {1843, 3917}, {5890, 11180}, {5943, 14913}
X(29959) = reflection of X(i) in X(j) for these (i,j): (6, 5943), (51, 16776), (3060, 9969), (3313, 3917), (3917, 141), (5891, 11178), (5943, 9822), (11179, 5892), (15060, 18358), (16194, 3818)
X(29959) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (9306, 9813, 6), (9822, 14913, 6)






leftri  Collineation mappings involving Gemini triangle 99: X(29960) - X(30026)  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 99, as in centers X(29960)-X(30026). Then

m(X) = (a b^2 + a c^2 + b^2 c + b c^2) x - b (b c + c a + a b) y - c (b c + c a + a b) z : :

and m(X) is on the Euler line if and only if X is on the Euler line. (Clark Kimberling, December 13, 2018)

underbar




X(29960) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    -a^2 b^2 + a b^3 + b^3 c - a^2 c^2 + a c^3 + b c^3 : :

X(29960) lies on these lines:


X(29961) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(3), WHERE A'B'C' = GEMINI TRIANGLE 99

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(29961) lies on these lines:


X(29962) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(4), WHERE A'B'C' = GEMINI TRIANGLE 99

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(29962) lies on these lines:


X(29963) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(5), 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 - 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(29963) lies on these lines:


X(29964) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(6), 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 + a b c^3 + a c^4 + b c^4 : :

X(29964) lies on these lines:


X(29965) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(7), WHERE A'B'C' = GEMINI TRIANGLE 99

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(29965) lies on these lines:


X(29966) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(8), WHERE A'B'C' = GEMINI TRIANGLE 99

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(29966) lies on these lines:


X(29967) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(9), WHERE A'B'C' = GEMINI TRIANGLE 99

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 - b^3 c^2 - b^2 c^3 + a c^4 + b c^4 : :

X(29967) lies on these lines:


X(29968) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(10), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    a^2 b^2 - a b^3 + 2 a^2 b c + 2 a b^2 c - b^3 c + a^2 c^2 + 2 a b c^2 - a c^3 - b c^3 : :

X(29968) lies on these lines:


X(29969) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(11), 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^4 b c - 2 a^3 b^2 c - 2 a^2 b^3 c + 3 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 + 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(29969) lies on these lines:


X(29970) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(22), WHERE A'B'C' = GEMINI TRIANGLE 99

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(29970) lies on these lines:


X(29971) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(23), WHERE A'B'C' = GEMINI TRIANGLE 99

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(29971) lies on these lines:


X(29972) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(31), WHERE A'B'C' = GEMINI TRIANGLE 99

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(29972) lies on these lines:


X(29973) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(32), 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 + a b c^5 + a c^6 + b c^6 : :

X(29973) lies on these lines:


X(29974) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(39), WHERE A'B'C' = GEMINI TRIANGLE 99

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(29974) lies on these lines:


X(29975) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(41), WHERE A'B'C' = GEMINI TRIANGLE 99

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 - 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 - a b c^5 - b^2 c^5 + a c^6 + b c^6 : :

X(29975) lies on these lines:


X(29976) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(42), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    -a^3 b^3 + a^2 b^4 - a^3 b^2 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 + a b^2 c^3 + a^2 c^4 + 2 a b c^4 + b^2 c^4 : :

X(29976) lies on these lines:


X(29977) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(43), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    -a^3 b^3 + a^2 b^4 - a^3 b^2 c + 2 a b^4 c - 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 + a b^2 c^3 + a^2 c^4 + 2 a b c^4 + b^2 c^4 : :

X(29977) lies on these lines:


X(29978) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(44), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    -2 a^3 b^2 + 2 a b^4 - 2 a^2 b^2 c + a b^3 c + 2 b^4 c - 2 a^3 c^2 - 2 a^2 b c^2 - b^3 c^2 + a b c^3 - b^2 c^3 + 2 a c^4 + 2 b c^4 : :

X(29978) lies on these lines:


X(29979) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(45), 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 b^3 c^2 - a b c^3 - 2 b^2 c^3 + a c^4 + b c^4 : :

X(29979) lies on these lines:


X(29980) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(55), 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 + 2 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(29980) lies on these lines:


X(29981) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(69), WHERE A'B'C' = GEMINI TRIANGLE 99

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(29981) lies on these lines:


X(29982) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(75), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    b c (-2 a^2 b + a b^2 - 2 a^2 c - 2 a b c + b^2 c + a c^2 + b c^2) : :

X(29982) lies on these lines:


X(29983) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(76), WHERE A'B'C' = GEMINI TRIANGLE 99

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(29983) lies on these lines:


X(29984) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(81), 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 + 3 a b^4 c + b^5 c - a^4 c^2 - 2 a^3 b c^2 + a b^3 c^2 + b^4 c^2 - a^3 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(29984) lies on these lines:


X(29985) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(86), 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 + a b c^3 + a c^4 + b c^4 : :

X(29985) lies on these lines:


X(29986) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(145), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    -2 a^2 b^2 + 2 a b^3 - a^2 b c - a b^2 c + 2 b^3 c - 2 a^2 c^2 - a b c^2 + 2 a c^3 + 2 b c^3 : :

X(29986) lies on these lines:


X(29987) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(200), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    -a^4 b^2 + 3 a^3 b^3 - 3 a^2 b^4 + a b^5 + 2 a^3 b^2 c + a^2 b^3 c - 4 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 - 2 b^4 c^2 + 3 a^3 c^3 + a^2 b c^3 - a b^2 c^3 + 2 b^3 c^3 - 3 a^2 c^4 - 4 a b c^4 - 2 b^2 c^4 + a c^5 + b c^5 : :

X(29987) lies on these lines:


X(29988) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(239), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    -a^3 b^2 + a b^4 - 3 a^2 b^2 c + 2 a b^3 c + b^4 c - a^3 c^2 - 3 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(29988) lies on these lines:


X(29989) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(667), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    (b - c) (-a^4 b^2 - a^2 b^4 - a^3 b^2 c - 2 a^2 b^3 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 - 2 a^2 b c^3 + a b^2 c^3 + b^3 c^3 - a^2 c^4 + b^2 c^4) : :

X(29989) lies on these lines:


X(29990) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(238), WHERE A'B'C' = GEMINI TRIANGLE 99

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(29990) lies on these lines:


X(29991) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(1125), WHERE A'B'C' = GEMINI TRIANGLE 99

Barycentrics    -a^2 b^2 + a b^3 + 2 a^2 b c + 2 a b^2 c + b^3 c - a^2 c^2 + 2 a b c^2 + a c^3 + b c^3 : :

X(29991) lies on these lines:


X(29992) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(20), WHERE A'B'C' = GEMINI TRIANGLE 99

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(29992) lies on these lines:


X(29993) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(21), WHERE A'B'C' = GEMINI TRIANGLE 99

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 - 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(29993) lies on these lines:


X(29994) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(25), WHERE A'B'C' = GEMINI TRIANGLE 99

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(29994) lies on these lines:


X(29995) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(27), WHERE A'B'C' = GEMINI TRIANGLE 99

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 - 5 a^6 b^2 c - 5 a^5 b^3 c + a^4 b^4 c + 3 a^2 b^6 c + 5 a b^7 c + b^8 c - a^7 c^2 - 5 a^6 b c^2 - 6 a^5 b^2 c^2 + 4 a^4 b^3 c^2 + 5 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 - 5 a^5 b c^3 + 4 a^4 b^2 c^3 + 12 a^3 b^3 c^3 - 5 a b^5 c^3 - a^5 c^4 + a^4 b c^4 + 5 a^3 b^2 c^4 - 6 a b^4 c^4 - 3 b^5 c^4 - a^2 b^2 c^5 - 5 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 + 5 a b c^7 + 2 b^2 c^7 + a c^8 + b c^8 : :

X(29995) lies on these lines:


X(29996) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(28), WHERE A'B'C' = GEMINI TRIANGLE 99

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 - 4 a^6 b^3 c - 3 a^5 b^4 c - a^4 b^5 c + 2 a^3 b^6 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 + 2 a^3 b^5 c^2 + 2 a^2 b^6 c^2 + 3 a b^7 c^2 + b^8 c^2 - a^7 c^3 - 4 a^6 b c^3 - 3 a^5 b^2 c^3 + 8 a^4 b^3 c^3 + 7 a^3 b^4 c^3 - 2 a^2 b^5 c^3 - a b^6 c^3 - a^6 c^4 - 3 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 - 6 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 - 2 a^2 b^3 c^5 - 6 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 + 4 a^2 b c^7 + 3 a b^2 c^7 + a^2 c^8 + 3 a b c^8 + b^2 c^8 + a c^9 + b c^9 : :

X(29996) lies on these lines:


X(29997) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(29), WHERE A'B'C' = GEMINI TRIANGLE 99

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 + 4 a^6 b^3 c - a^5 b^4 c - 3 a^4 b^5 c - 2 a^2 b^7 c - a b^8 c + b^9 c - a^8 c^2 + 2 a^7 b c^2 + 8 a^6 b^2 c^2 + a^5 b^3 c^2 - 6 a^4 b^4 c^2 - 3 a b^7 c^2 - b^8 c^2 + a^7 c^3 + 4 a^6 b c^3 + a^5 b^2 c^3 - 4 a^4 b^3 c^3 - 3 a^3 b^4 c^3 + 2 a^2 b^5 c^3 + a b^6 c^3 - 2 b^7 c^3 + a^6 c^4 - a^5 b c^4 - 6 a^4 b^2 c^4 - 3 a^3 b^3 c^4 + 2 a^2 b^4 c^4 + 2 a b^5 c^4 + b^6 c^4 - a^5 c^5 - 3 a^4 b c^5 + 2 a^2 b^3 c^5 + 2 a b^4 c^5 + 2 b^5 c^5 + a^4 c^6 + a b^3 c^6 + b^4 c^6 - a^3 c^7 - 2 a^2 b c^7 - 3 a b^2 c^7 - 2 b^3 c^7 - a^2 c^8 - a b c^8 - b^2 c^8 + a c^9 + b c^9 : :

X(29997) lies on these lines:


X(29998) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(376), WHERE A'B'C' = GEMINI TRIANGLE 99

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(29998) lies on these lines:


X(29999) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(377), WHERE A'B'C' = GEMINI TRIANGLE 99

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 + 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 - b^4 c^3 - a b^2 c^4 - b^3 c^4 + a c^6 + b c^6 : :

X(29999) lies on these lines:


X(30000) =  (A,B,C,X(2); A',B',C',X(2)) COLLINEATION IMAGE OF X(384), WHERE A'B'C' = GEMINI TRIANGLE 99

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(30000) lies on these lines:



underbar

 

This is the end of PART 15: Centers X(28001) - X(30000)

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)