## This is PART 2: Centers X(1001) - X(3000)

 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(1001) = MIDPOINT OF X(1) AND X(9)

Trilinears    a2 - a(b + c) - 2bc : :
X(1001) = X(1) + 3X(2) + (r/R)X(3)

Let OA be the circle tangent of side BC at its midpoint and to the circumcircle on the side of BC opposite A. Define OB and OC cyclically. X(1001) is the radical center of the circles OA, OB, OC. (Randy Hutson, 9/23/2011)

Let LA and MA be the external tangents to circles OB and OC, with LA being the farther from OA. Define LB, LC, MB, and MC cyclically. Let A' = LB∩LC and A'' = MB∩MC, and define B', B'', C', and C'' cyclically. The lines A'A'', B'B'', C'C'' concur in X(1001).

Let A* be the tangency point of OA and the circumcircle, and define B* and C* cyclically. Let A** be the radical trace of OB and OC. The lines A*A**, B*B**, C*C** concur in X(1001).

For details and relationships among X(1001), X(1), X(145), X(3361), X(3616), X(3913), and X(4719), see

Luis González, "On a Triad of Circles Tangent to the Circumcircle and the Sides at Their Midpoints," Forum Geometricorum 11 (2011) 145-154.

Let A' be the line through X(1) parallel to line BC. Let AB = A'∩AB and AC = A'∩AC. Define BC and CA cyclically, and define BA and CB cyclically. The six points AB, BC, CA, AB, BC, CA lie on a conic whose center is X(1001). (Angel Montesdeoca, April 27, 2016)

X(1001) lies on these lines:
1,6   2,11   3,142   7,21   8,344   31,940   35,474   42,748   63,354   182,692   388,452   527,551   529,1056   614,968   750,902   846,982   943,1058

X(1001) is the {X(1),X(238)}-harmonic conjugate of X(6). For a list of other harmonic conjugates of X(1001), click Tables at the top of this page.

X(1001) = midpoint of X(1) and X(9)
X(1001) = reflection of X(142) in X(1125)
X(1001) = isogonal conjugate of X(1002)
X(1001) = complement of X(2550)
X(1001) = crosssum of X(i) and X(j) for these (i,j): (116,824), (788,1015)
X(1001) = crossdifference of every pair of points on line X(513)X(665)
X(1001) = anticomplement of X(3826)
X(1001) = X(6)-of-2nd-circumperp-triangle
X(1001) = X(141)-of-hexyl-triangle
X(1001) = X(5480)-of-excentral-triangle
X(1001) = inverse-in-Feuerbach-hyperbola of X(55)

### X(1002) = ISOGONAL CONJUGATE OF X(1001)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = 1/[a2 - a(b + c) - 2bc]

X(1002) lies on these lines:
1,672   2,210   6,105   8,274   28,607   42,57   55,81   65,279   145,330   277,942

X(1002) = isogonal conjugate of X(1001)

### X(1003) = INTERCEPT OF EULER LINE AND LINE X(6)X(99)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = bc(3a4 - a2b2 - a2c2 + 2b2c2)

As a point on the Euler line, X(1003) has Shinagawa coefficients ((E + F)2 - 2S2, 3S2).

X(1003) lies on these lines: 2,3   6,99   32,538   183,187

X(1003) = crossdifference of every pair of points on line X(647)X(888)

### X(1004) = INTERCEPT OF EULER LINE AND LINE X(7)X(100)

Trilinears    a5 - 2a4(b + c) + 2a2(b3 + c3) - a(b2 + c2)2 + 2bc(b - c)(b2 - c2) : :

As a point on the Euler line, X(1004) has Shinagawa coefficients ((E+F)S2-\$abSASB\$, -(E+F)S2 +\$ab\$S2).

X(1004) lies on these lines: 2,3   7,100   46,200   63,210   65,224

### X(1005) = INTERCEPT OF EULER LINE AND LINE X(9)X(100)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = a5 - 2a4(b + c) - a3bc + a2(2b3
+ 2c3 + b2c + bc2) - a(b4 + c4 - b3c - bc3 - 4b2c2)
+ bc(b - c)(b2 - c2)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c) As a point on the Euler line, X(1005) has Shinagawa coefficients ((E-2F)S2+\$abSASB\$+\$ab\$S2, 2(E+F)S2-2\$ab\$S2).

X(1005) lies on these lines: 2,3   9,100   55,329   108,342

X(1005) = isogonal conjugate of X(1242)

### X(1006) = INTERCEPT OF EULER LINE AND LINEX(9)X(48)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = a6 - a5(b + c) - a4(2b2 + bc + 2c2)
+ 2a3(b3 + c3) + a2[b4 + c4 + 2bc(b2 + c2) + 2b2c2]
- a[(b5 + c5) - bc(b3 + c3)] - bc(b2 - c2)2
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As a point on the Euler line, X(1006) has Shinagawa coefficients (\$aSA\$ + abc, - \$aSA\$).

X(1006) lies on these lines:
1,201   2,3   9,48   35,950   36,226   54,72   238,1064   944,958   954,999

X(1006) = isogonal conjugate of X(1243)

### X(1007) = INTERCEPT OF LINES X(2)X(6) AND X(4)X(99)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = bc[a4 - 4a2(b2 + c2) + 3b4 - 2b2c2 + 3c4]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1007) lies on these lines: 2,6   4,99   305,311   315,631   316,376   317,6353

### X(1008) = INTERCEPT OF EULER LINE AND LINE X(1)X(76)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = bc[a5(b + c) + a4(b + c)2 + a3(b + c)(b2 + bc + c2)
+ a2(b2 + c2 + bc)2 + abc(b + c)3 + b2c2(b + c)2]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As a point on the Euler line, X(1008) has Shinagawa coefficients ((E + F)3 + 2\$bc\$(E + F)2 - \$abSC\$(E + F) + 2(E + F)S2, \$bc\$S2).

X(1008) lies on these lines: 1,76   2,3

### X(1009) = INTERCEPT OF EULER LINE AND LINE X(1)X(39)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = a4(b + c) + 2a3bc - a2(b3 + c3) + bc(b + c)(b2 + c2)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As a point on the Euler line, X(1009) has Shinagawa coefficients (2\$bc\$(E + F)2 + \$bcSBSC\$ - 2\$bc(SA)2\$ - 4\$bc\$S2, \$bc\$S2).

X(1009) lies on these lines: 1,39   2,3   72,672   283,1065   518,583

X(1009) = isogonal conjugate of X(1244)
X(1009) = crossdifference of every pair of points on line X(647)X(659)

### X(1010) = INTERCEPT OF EULER LINE AND LINE X(1)X(75)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = bc[a2 + (b + c)2]/(b + c)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As a point on the Euler line, X(1010) has Shinagawa coefficients ((E + F)2 + \$bc\$(E + F) + abc\$a\$, S2).

X(1010) lies on these lines:
1,75   2,3   8,81   10,58   72,894   283,1065   312,975   759,833

X(1010) = isogonal conjugate of X(1245)
X(1010) = crossdifference of every pair of points on line X(647)X(798)

### X(1011) = INTERCEPT OF EULER LINE AND LINE X(6)X(31)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = a[a3(b + c) + a2bc - a(b + c)(b2 + c2) - bc(b + c)2]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As a point on the Euler line, X(1011) has Shinagawa coefficients (E + \$bc\$, -\$bc\$).

X(1011) lies on these lines: 2,3   6,31   9,228   35,43   51,573   184,572

X(1011) = isogonal conjugate of X(1246)
X(1011) = inverse-in-orthocentroidal circle of X(3136)
X(1011) = crosssum of X(834) and X(1086)
X(1011) = crossdifference of every pair of points on line X(514)X(647)

### X(1012) = INTERCEPT OF EULER LINE AND LINE X(1)X(84)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = a7 - a5u + 2a4bc(b + c) + a3v - a(b + c)2(b2 - c2)2 - 2bc(b + c)(b2 - c2)2,
where u = 3b2 - 2bc + 3c2 and v = 3b4 + 2b2c2 + 3c4
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

As a point on the Euler line, X(1012) has Shinagawa coefficients (S2, abc\$a\$ - S2).

X(1012) lies on these lines:
1,84   2,3   7,104   40,958   55,515   56,946   63,517   268,281   516,993   954,971

X(1012) = center of circle {X(1),X(1709),PU(4)}

### X(1013) = INTERCEPT OF EULER LINE AND LINE X(7)X(108)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where
f(A,B,C) = (cos A + cos B + cos C) cos A + (sec A + sec B + sec C) cos B cos C
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(1013) has Shinagawa coefficients (\$aSA\$F, \$aSBSC\$).

X(1013) lies on these lines: 2,3   6,162   7,108   33,63   55,92   100,281

### X(1014) = ISOGONAL CONJUGATE OF X(210)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = 1/[(b + c)(b + c - a)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1014) lies on these lines: 7,21   28,279   57,77   58,269   60,757   69,404   261,552   272,1088   274,961   332,1037   759,934

X(1014) = isogonal conjugate of X(210)
X(1014) = cevapoint of X(56) and X(57)
X(1014) = X(58)-cross conjugate of X(81)
X(1014) = isotomic conjugate of X(3701)
X(1014) = trilinear pole of line X(1019)X(1429)
X(1014) = crossdifference of every pair of points on line X(3709)X(4041)

### X(1015) = EXSIMILICENTER OF MOSES CIRCLE AND INCIRCLE

Trilinears    a(b - c)2 : b(c - a)2 : c(a - b)2
Barycentrics    a2(b - c)2 : b2(c - a)2 : c2(a - b)2
X(1015) = (tan ω sin 2ω)R*X(1) - r*X(39)

The circle having center X(39) and radius 2R sin2ω, where R denotes the circumradius of triangle ABC, is here introduced as the Moses circle. It is tangent to the nine-point circle at X(115), and its internal and external centers of similitude with the incircle are X(1500) and X(1015), respectively. (based on notes from Peter J. C. Moses, 5/29/03)

X(1015) is the center of the hyperbola that passes through the points A, B, C, X(1), X(2), and X(28); also, X(1015) lies on the ellipse described at X(1125). (Randy Hutson, Hyacinthos #20179, 8/13/2011)

X(1015) lies on the Steiner inellipse, the Brocard inellipse, and these lines:
1,39   2,668   6,101   11,115   32,56   36,187   37,537   55,574   76,330   214,1100   216,1060   244,665   350,538   812,1086   1960,3122

X(1015) = midpoint of X(i) and X(j) for these (i,j): (1,291), (2,3227)
X(1015) = isogonal conjugate of X(1016)
X(1015) = complement of X(668)
X(1015) = crosspoint of X(2) and X(513)
X(1015) = crosssum of X(i) and X(j) for these (i,j): (1,1018), (2,190), (6,100), (8,644), (101,595), (345,1332)
X(1015) = crossdifference of every pair of points on line X(100)X(190)
X(1015) = bicentric difference of PU(i) for these i: 25, 27
X(1015) = PU(25)-harmonic conjugate of X(1960)
X(1015) = midpoint of PU(27)
X(1015) = center of circumconic that is locus of trilinear poles of lines through X(513)
X(1015) = perspector of circumparabola centered at X(513)
X(1015) = X(2)-Ceva conjugate of X(513)
X(1015) = projection from Steiner circumellipse to Steiner inellipse of X(3227)
X(1015) = barycentric square of X(513)
X(1015) = center of {ABC, Gemini 7}-circumconic

### X(1016) = X(1)-CROSS CONJUGATE OF X(99)

Trilinears    1/[sin A sin2(A/2) [1 - cos(B - C)]] : :
Barycentrics    (b - c)- 2 : :

X(1016) lies on these lines:
8,1083   99,813   100,667   190,514   238,519   512,660   644,666

X(1016) is the trilinear pole of line X(100)X(190), which is the tangent to the circumcircle at X(100) and to the Steiner circumellipse at X(190). This line is also the locus of trilinear poles of tangents at P to hyperbola {{A,B,C,X(2),P}}, as P moves on the Nagel line, and the locus of trilinear poles of tangents at P to hyperbola {{A,B,C,X(6),P}}, as P moves on line X(1)X(6). (Randy Hutson, October 15, 2018)

X(1016) is the Brianchon point (perspector) of the inellipse that is the barycentric square of the Nagel line. (Randy Hutson, October 15, 2018)

Let A5B5C5 and A6B6C6 be the Gemini triangles 5 and 6. Let A' be the barycentric product A5*A6 and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1016). (Randy Hutson, November 30, 2018)

X(1016) = isogonal conjugate of X(1015)
X(1016) = isotomic conjugate of X(1086)
X(1016) = cevapoint of X(i) and X(j) for these (i,j): (2,190), (6,100)
X(1016) = X(i)-cross conjugate of X(j) for these (i,j): (1,99), (2,190), (6,100)
X(1016) = anticomplement of X(6547)
X(1016) = polar conjugate of X(2969)

### X(1017) = POINT ALFIRK

Trilinears    a(b + c - 2a)2 : :

X(1017) lies on the Brocard inellipse and these lines: 6,101   44,214

X(1017) = crosssum of X(2) and X(903)
X(1017) = crossdifference of every pair of points on line X(900)X(903)
X(1017) = bicentric sum of PU(99)
X(1017) = PU(99)-harmonic conjugate of X(1960)
X(1017) = barycentric square of X(44)

### X(1018) = X(512)-CROSS CONJUGATE OF X(1)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = (b + c)/(b - c)

X(1018) is the intersection, other than the vertices of the Gemini 18 triangle, of the {ABC, Gemini 18}-circumconic and {Gemini 17, Gemini 18}-circumconic. (Randy Hutson, November 30, 2018)

X(1018) lies on these lines:
1,39   9,80   40,728   63,544   99,813   100,101   163,643   190,646   346,573   519,672   664,1025

X(1018) = isogonal conjugate of X(1019)
X(1018) = X(512)-cross conjugate of X(1)
X(1018) = crosspoint of X(100) and X(190)
X(1018) = crosssum of X(513) and X(649)
X(1018) = crossdifference of every pair of points on line X(244)X(659)
X(1018) = trilinear pole of line X(37)X(42)

### X(1019) = X(99)-CEVA CONJUGATE OF X(1)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = (b - c)/(b + c)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

Evans conjectured that X(1), X(484), X(1276), X(1277) are concyclic, and reported that Paul Yiu confirmed this conjecture and noted that the center of this circle is X(1019). (Lawrence Evans, 2/24/2003)

X(1019) lies on these lines:
1,512   36,238   58,1027   81,1022   99,813   239,514   759,840

X(1019) = isogonal conjugate of X(1018)
X(1019) = isotomic conjugate of X(4033)
X(1019) = anticomplement of X(4129)
X(1019) = X(99)-Ceva conjugate of X(1)
X(1019) = crosssum of X(i) and X(j) for these (i,j): (37,661), (42,798), (649,1100), (822,836)
X(1019) = crossdifference of every pair of points on line X(37)X(42)
X(1019) = trilinear pole of line X(244)X(659)
X(1019) = bicentric difference of PU(160)

### X(1020) = POINT ALGEDI

Trilinears    (cos B + cos C)/(cos B - cos C) : :

Let X be a point on the Euler line. Let P and U be the 1st and 2nd bicentrics of X. As X varies, the bicentric sum of P and U trace the line X(42)X(65), of which X(1020) is the trilinear pole. (Randy Hutson, March 25, 2016)

X(1020) lies on these lines:
1,185   57,1086   101,651   108,109   190,658   269,292   347,573   648,1021

X(1020) = isogonal conjugate of X(1021)
X(1020) = crosssum of X(650) and X(652)
X(1020) = trilinear pole of line X(42)X(65)
X(1020) = crosspoint of X(651) and X(653)

### X(1021) = ISOGONAL CONJUGATE OF X(1020)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = (cos B - cos C)/(cos B + cos C)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1021) lies on these lines:
1,647   239,514   243,522   333,1024   521,650   648,1020

X(1021) = isogonal conjugate of X(1020)
X(1021) = crosspoint of X(81) and X(162)
X(1021) = crosssum of X(i) and X(j) for these (i,j): (37,656), (65,661), (73,822), (647,1425), (649,1104)
X(1021) = crossdifference of every pair of points on line X(42)X(65)

### X(1022) = INTERSECTION OF LINES X(1)X(513) AND X(2)X(514)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = (b - c)/(2a - b - c)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1022) lies on these lines:
1,513   2,514   81,1019   89,649   105,106   291,876   812,903

X(1022) = isogonal conjugate of X(1023)
X(1022) = crossdifference of every pair of points on line X(44)X(678)

### X(1023) = INTERSECTION OF LINES X(1)X(6) AND X(100)X(101)

Trilinears    (2a - b - c)/(b - c) : :

X(1023) lies on these lines: 1,6   100,101   813,898

X(1023) = isogonal conjugate of X(1022)
X(1023) = crossdifference of every pair of points on line X(244)X(513)
X(1023) = bicentric difference of PU(28)
X(1023) = PU(28)-harmonic conjugate of X(1)
X(1023) = trilinear pole of line X(44)X(678)

### X(1024) = INTERSECTION OF LINES X(6)X(513) ANDX(9)X(522)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = (b - c)(b + c - a)/[b2 + c2 - a(b + c)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1024) lies on these lines:
6,513   9,522   55,650   57,649   333,1021   673,812

X(1024) = isogonal conjugate of X(1025)
X(1024) = crossdifference of every pair of points on line X(241)X(518)

### X(1025) = INTERSECTION OF LINES X(2)X(7) AND X(100)X(109)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = [b2 + c2 - a(b + c)]/[(b - c)(b + c - a)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1025) lies on these lines:
2,7   56,1083   100,109   190,658   644,934   664,1018   813,927

X(1025) = isogonal conjugate of X(1024)

### X(1026) = INTERSECTION OF LINES X(1)X(2) AND X(100)X(101)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = [b2 + c2 - a(b + c)]/(b - c)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1026) lies on these lines: 1,2   55,1083   100,101   664,668   666,1027

X(1026) = isogonal conjugate of X(1027)
X(1026) = crosssum of X(513) and X(659)
X(1026) = crossdifference of every pair of points on line X(244)X(649)

### X(1027) = INTERSECTION OF LINES X(1)X(514) AND X(6)X(513)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = (b - c)/[b2 + c2 - a(b + c)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1027) lies on these lines: 1,514   6,513   56,667   58,1019   105,106   292,659   666,1026

X(1027) = isogonal conjugate of X(1026)
X(1027) = crosssum of X(659) and X(1279)
X(1027) = crossdifference of every pair of points on line X(518)X(672)

### X(1028) = POINT ALGENIB

Trilinears        1/A2 : 1/B2 : 1/C2
Barycentrics  a/A2 : b/B2 : c/C2

X(1028) = isogonal conjugate of X(1085)

### X(1029) = CYCLOCEVIAN CONJUGATE OF X(1)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = 1/[a + 2(a + b + c) cos A]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1029) lies on these lines: 10,191   115,593   319,321

X(1029) = isogonal conjugate of X(1030)
X(1029) = isotomic conjugate of X(2895)
X(1029) = X(i)-cross conjugate of X(j) for these (i,j): (79,7), (81,2)
X(1029) = cyclocevian conjugate of X(1)

### X(1030) = ISOGONAL CONJUGATE OF X(1029)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = a + 2(a + b + c) cos A
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1030) lies on these lines:
3,6   35,37   36,1100   45,198   55,199   100,594

X(1030) = isogonal conjugate of X(1029)
X(1030) = X(i)-Ceva conjugate of X(j) for these (i,j): (35,55), (37,6)
X(1030) = crosspoint of X(100) and X(249)
X(1030) = crosssum of X(115) and X(513)

### X(1031) = CYCLOCEVIAN CONJUGATE OF X(6)

Trilinears    bc/(b4 + c4 - a4 + b2c2 + c2a2 + a2b2) : :     (M. Iliev, 5/13/07)

X(1031) lies on this line: 141,384

X(1031) = isotomic conjugate of X(2896)
X(1031) = X(83)-cross conjugate of X(2)
X(1031) = cyclocevian conjugate of X(6)
X(1031) = barycentric product of PU(137)

### X(1032) = CYCLOCEVIAN CONJUGATE OF X(20)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = 1/[a(w2u2 + u2v2 - v2w2) + 2uvw(au + bv + cw) cos A],
where u : v : w = X(20)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1032) = perpsector of triangle ABC and the pedal triangle of X(3346)

X(1032) lies on this line: 20,394

X(1032) = isogonal conjugate of X(1033)
X(1032) = X(4)-cross conjugate of X(69)
X(1032) = cyclocevian conjugate of X(20)
X(1032) = anticomplement of X(3343)

### X(1033) = X(3)-CEVA CONJUGATE OF X(25)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = a(w2u2 + u2v2 - v2w2) + 2uvw(au + bv + cw) cos A,
where u : v : w = X(20)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1033) lies on these lines: 6,64   19,56   25,393   55,204

X(1033) = isogonal conjugate of X(1032)
X(1033) = X(3)-Ceva conjugate of X(25)

### X(1034) = X(4)-CROSS CONJUGATE OF X(8)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = 1/[a(w2u2 + u2v2 - v2w2) + 2uvw(au + bv + cw) cos A],
where u : v : w = X(329)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1034) lies on these lines: 2,271   20,78

X(1034) = perpsector of triangle ABC and the pedal triangle of X(3345)

X(1034) = isogonal conjugate of X(1035)
X(1034) = X(i)-cross conjugate of X(j) for these (i,j): (4,8), (282,2)
X(1034) = cyclocevian conjugate of X(329)
X(1034) = anticomplement of X(3342)

### X(1035) = ISOGONAL CONJUGATE OF X(1034)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = a(w2u2 + u2v2 - v2w2) + 2uvw(au + bv + cw) cos A,
where u : v : w = X(329)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1035) lies on these lines: 3,223   6,603   25,34   55,64   222,581

X(1035) = isogonal conjugate of X(1034)
X(1035) = X(i)-Ceva conjugate of X(j) for these (i,j): (3,56), (223,6)

### X(1036) = ISOGONAL CONJUGATE OF X(388)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = 1/(1 + cos B cos C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1036) lies on these lines:
1,25   3,31   4,1065   21,332   29,497   41,219   55,78   56,77   73,1037   282,380   581,947   1058,1067   1059,1066

X(1036) = isogonal conjugate of X(388)

### X(1037) = CEVAPOINT OF X(55) AND X(56)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = 1/(1 - cos B cos C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1037) lies on these lines:
1,1041   3,1066   4,1067   29,388   48,949   55,77   56,78   73,1036   219,604   332,1014   916,1069   1056,1065   1057,1064

X(1037) = isogonal conjugate of X(497)
X(1037) = cevapoint of X(55) and X(56)

### X(1038) = INTERSECTION OF LINES X(1)X(3) AND X(2)X(34)

Trilinears    cos A + cos A cos B cos C : :
Trilinears    1 + sec B sec C : :

X(1038) lies on these lines:
1,3   2,34   4,1076   9,478   20,33   21,1041   38,1106   63,201   69,73   72,222   172,577   221,960   223,936   225,377   226,975   278,443   388,612   1068,1074

X(1038) is the {X(1),X(3)}-harmonic conjugate of X(1040). For a list of other harmonic conjugates of X(1038), click Tables at the top of this page.

X(1038) = isogonal conjugate of X(1039)

### X(1039) = ISOGONAL CONJUGATE OF X(1038)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = 1/(cos A + cos A cos B cos C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1039) lies on these lines:
1,25   4,1096   7,34   8,33   9,607   21,1040   29,314   65,1041   943,968

X(1039) = isogonal conjugate of X(1038)

### X(1040) = INTERSECTION OF LINES X(1)X(3) AND X(2)X(33)

Trilinears    cos A - cos A cos B cos C : :
Trilinears    1 - sec B sec C : :

X(1040) lies on these lines:
1,3   2,33   4,1074   20,34   21,1039   63,212   78,345   226,990   243,1096   497,614   1068,1076

X(1040) = isogonal conjugate of X(1041)
X(1040) = crosspoint of X(i) and X(j) for these (i,j): (21,332), (77,78)
X(1040) = crosssum of X(33) and X(34)
X(1040) = homothetic center of intangents triangle and mid-triangle of orthic and dual of orthic triangles

### X(1041) = ISOGONAL CONJUGATE OF X(1040)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = 1/(cos A - cos A cos B cos C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1041) lies on these lines:
1,1037   7,33   8,34   9,608   19,294   21,1038   65,1039

X(1041) = isogonal conjugate of X(1040)
X(1041) = cevapoint of X(33) and X(34)

### X(1042) = CROSSPOINT OF X(1) AND X(64)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = (1 - cos A)(cos B + cos C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1042) lies on these lines:
1,7   31,56   34,207   42,65   57,959   241,960   517,1066   604,608   741,934   942,1064

X(1042) = isogonal conjugate of X(1043)
X(1042) = crosspoint of X(i) and X(j) for these (i,j): (1,64), (34,56)
X(1042) = crosssum of X(i) and X(j) for these (i,j): (1,20), (8,78), (200,346)
X(1042) = crossdifference of every pair of points on line X(657)X(1021)

### X(1043) = CEVAPOINT OF X(1) AND X(20)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = 1/[(1 - cos A)(cos B + cos C)]
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1043) lies on these lines:
1,75   8,21   20,64   27,306   29,33   58,519   72,190   81,145   99,103   200,341   220,346   239,1104   280,285   283,643   286,322

X(1043) = isogonal conjugate of X(1042)
X(1043) = anticomplement of X(1834)
X(1043) = X(314)-Ceva conjugate of X(333)
X(1043) = cevapoint of X(i) and X(j) for these (i,j): (1,20), (8,78), (200,346)

### X(1044) = X(64)-CEVA CONJUGATE OF X(1)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = cos B + cos C - cos A + cos B cos C - cos A cos B - cos A cos C
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1044) lies on these lines: 1,7   43,46

X(1044) = X(64)-Ceva conjugate of X(1)

### X(1045) = X(42)-CEVA CONJUGATE OF X(1)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = -u2 + v2 + w2 + vw + wu + uv, u : v : w = X(2)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1045) lies on these lines: 1,75   6,2665   9,43   40,511   42,894   190,872   192,869

X(1045) = X(42)-Ceva conjugate of X(1)
X(1045) = excentral-isogonal conjugate of X(1764)
X(1045) = excentral-isotomic conjugate of X(20)
X(1045) = perspector of trilinear obverse triangle of X(2) and unary cofactor triangle of trilinear N-obverse triangle of X(2)

### X(1046) = X(65)-CEVA CONJUGATE OF X(1)

Trilinears    -u2 + v2 + w2 + vw + wu + uv, where u : v : w = X(3)

Let La be the line parallel to the Brocard axis of BCI and passing through the A-excenter. Define Lb, Lc cyclically. The lines La, Lb, Lc concur in X(1046). (Randy Hutson, January 29, 2018)

X(1046) lies on these lines:
1,21   4,2648   6,986   10,894   40,511   43,46   57,978   72,171   238,942   484,1048

X(1046) = reflection of X(i) in X(j) for these (i,j): (1,58), (1330,10)
X(1046) = isogonal conjugate of X(1247)
X(1046) = X(65)-Ceva conjugate of X(1)
X(1046) = excentral-isogonal conjugate of X(20)

### X(1047) = X(73)-CEVA CONJUGATE OF X(1)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = -u2 + v2 + w2 + vw + wu + uv, u : v : w = X(4)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1047) lies on these lines: 1,29   43,46

X(1047) = isogonal conjugate of X(1248)
X(1047) = X(73)-Ceva conjugate of X(1)

### X(1048) = POINT ALGIEBA

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = -u2 + v2 + w2 + vw + wu + uv, u : v : w = X(5)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1048) lies on these lines: 1,564   484,1046

### X(1049) = TRILINEAR PURE ANGLES CENTER

Trilinears        A : B : C
Barycentrics  aA : bB : cC

X(1049) = isogonal conjugate of X(1077)

### X(1050) = POINT ALGOL

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = -u2 + v2 + w2 + vw + wu + uv, u : v : w = X(8)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1050) lies on these lines: 1,341   40,978

### X(1051) = POINT ALGORAB

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = 3a2 + b2 + c2 + bc + 5ab + 5ac      (M. Iliev, 5/13/07)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1051) lies on these lines: 1,748   6,846   81,1054   165,572

### X(1052) = X(244)-CEVA CONJUGATE OF X(1)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = -u2 + v2 + w2 + vw + wu + uv, u : v : w = X(100)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1052) lies on these lines: 1,765   238,517   513,1054

X(1052) = X(244)-Ceva conjugate of X(1)

### X(1053) = POINT ALHENA

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = -u2 + v2 + w2 + vw + wu + uv, u : v : w = X(101)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1053) lies on these lines: 1,1110   238,517   905,1054

### X(1054) = 6th SHARYGIN POINT

Trilinears    b2 + c2 - a2 - 3bc + ab + ac : :    (M. Iliev, 5/13/07)

See the description at X(1281).

X(1054) lies on the Bevan circle and these lines:
1,88   2,846   43,57   46,978   81,1051   105,165   474,986   513,1052   905,1053

X(1054) = reflection of X(1) in X(106)
X(1054) = inverse-in-circumcircle of X(1283)
X(1054) = X(1)-Hirst inverse of X(244)
X(1054) = crosssum of PU(33)
X(1054) = intersection of tangents at PU(34) to conic {{A,B,C,PU(34)}}
X(1054) = crosspoint of PU(34)
X(1054) = trilinear pole wrt excentral triangle of line X(1)X(6)
X(1054) = excentral isogonal conjugate of X(3667)
X(1054) = X(107) of excentral triangle

### X(1055) = POINT ALIOTH

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = a(b2 + c2 - 2a2 + ab + ac - 2bc)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1055) lies on these lines: 6,41   36,101   106,919   187,237   609,995

X(1055) = isogonal conjugate of X(1121)
X(1055) = crosssum of X(2) and X(527)
X(1055) = crossdifference of every pair of points on line X(2)X(522)
X(1055) = inverse-in-Parry-isodynamic-circle of X(5075); see X(2)

### X(1056) = POINT ALCYONE

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = 2 + cos B cos C
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1056) = 4(R/r)*X(1) + 3X(2) - 2X(3)

X(1056) lies on these lines:
1,4   2,495   7,517   8,443   29,1059   30,390   55,376   56,631   145,377   329,392   355,938   529,1001   1037,1065

X(1056) = isogonal conjugate of X(1057)

X(1056) is the {X(1),X(4)}-harmonic conjugate of X(1058). For a list of other harmonic conjugates of X(1056), click Tables at the top of this page.

### X(1057) = ISOGONAL CONJUGATE OF X(1056)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = 1/(2 + cos B cos C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1057) lies on these lines:
29,1058   73,1059   77,999   78,392   497,1065   1037,1064

X(1057) = isogonal conjugate of X(1056)

### X(1058) = POINT ALKALUROPS

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = 2 - cos B cos C
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)
X(1058) = 4(R/r)*X(1) - 3X(2) + 2X(3)

X(1058) lies on these lines:
1,4   2,496   3,390   8,392   20,999   29,1057   55,631   56,376   149,377   452,956   517,938   942,962   943,1001   1036,1067

X(1058) = isogonal conjugate of X(1059)

### X(1059) = ISOGONAL CONJUGATE OF X(1058)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = 1/(2 - cos B cos C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1059) lies on these lines: 29,1056   73,1057   78,999   388,1067   1036,1066

X(1059) = isogonal conjugate of X(1058)

### X(1060) = INTERSECTION OF LINES X(1)X(3) AND X(5)X(34)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = 2 + sec B sec C
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1060) lies on these lines:
1,3   5,34   21,1063   30,33   68,73   72,394   141,997   201,255   216,1015   222,912   377,1068   495,612   601,774   976,1066

X(1060) = isogonal conjugate of X(1061)
X(1060) = homothetic center of 2nd Euler triangle and anti-tangential midarc triangle

### X(1061) = ISOGONAL CONJUGATE OF X(1060)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = 1/(2 + sec B sec C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1061) lies on these lines:
1,24   8,406   21,1062   33,80   34,79   65,1063

X(1061) = isogonal conjugate of X(1060)

### X(1062) = INTERSECTION OF LINES X(1)X(3) AND X(5)X(33)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = 2 - sec B sec C
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1062) lies on these lines:
1,3   5,33   21,1061   30,34   394,1069   496,614   602,774

X(1062) = isogonal conjugate of X(1063)
X(1062) = homothetic center of intangents triangle and 2nd Euler triangle

### X(1063) = ISOGONAL CONJUGATE OF X(1062)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = 1/(2 - sec B sec C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1063) lies on these lines:
1,378   8,475   21,1060   33,79   34,80   65,1061

X(1063) = isogonal conjugate of X(1062)

### X(1064) = INTERSECTION OF LINES X(1)X(4) AND X(3)X(31)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = 1 + cos A (cos B + cos C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1064) lies on these lines:
1,4   3,31   38,912   40,386   42,517   102,112   104,256   238,1006   631,978   942,1042   991,995   1037,1057

X(1064) = isogonal conjugate of X(1065)
X(1064) = crosssum of X(1) and X(1478)

### X(1065) = ISOGONAL CONJUGATE OF X(1064)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = 1/[1 + cos A (cos B + cos C)]
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1065) lies on these lines:
3,388   4,1036   102,226   283,1010   284,515   497,1057   1037,1056

X(1065) = isogonal conjugate of X(1064)
X(1065) = polar conjugate of X(30687)

### X(1066) = HASTINGS POINT

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = 1 - cos A (cos B + cos C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1066) lies on these lines: 1,4   3,1037   42,942   222,601   517,1042   774,912   947,951   976,1060   1036,1059

X(1066) = isogonal conjugate of X(1067)
X(1066) = crosssum of X(1) and X(1479)

### X(1067) = ISOGONAL CONJUGATE OF X(1066)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = 1/[1 - cos A (cos B + cos C)]
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1067) lies on these lines:
3,496   4,1037   388,1059   946,951   947,950   1036,1058

X(1067) = isogonal conjugate of X(1066)

### X(1068) = X(46)-CROSS CONJUGATE OF X(4)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = 1 - sec A (cos B + cos C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1068) lies on these lines:
1,4   8,860   24,108   92,406   155,651   281,451   318,475   377,1060   429,495   1038,1074   1040,1076

X(1068) = isogonal conjugate of X(1069)
X(1068) = X(158)-Ceva conjugate of X(4)
X(1068) = X(46)-cross conjugate of X(4)

### X(1069) = X(255)-CROSS CONJUGATE OF X(3)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = 1/[1 - sec A (cos B + cos C)]
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1069) lies on these lines: 1,90   11,68   394,1062   496,613   916,1037

X(1069) = isogonal conjugate of X(1068)
X(1069) = X(90)-Ceva conjugate of X(3)
X(1069) = X(255)-cross conjugate of X(3)

### X(1070) = POINT ALKES

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = 1 + cos B cos C (cos B + cos C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1070) lies on these lines: 1,4   55,1076   56,1074

### X(1071) = INTERSECTION OF LINES X(1)X(84) AND X(4)X(7)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = (b2 + c2 - a2)[a3(b + c) - (b - c)2(a2 + a(b + c) - (b + c)2)][a3 - (b + c)(a2 + a(b + c) - (b - c)2)]
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1071) appears in Hyacinthos message #3849, Paul Yiu, Sept. 19, 2001.

If you have The Geometer's Sketchpad, you can view X(1071).

X(1071) lies on these lines: 1,84   4,7   6,63   10,2801   20,145   21,104   27,1871   198,1741   227,1735   355,377   412,1872   496,1519   774,1458   910,1729   1210,1532   1317,1364

X(1071) = reflection of X(i) in X(j) for these (i,j): (4,942)   (72,3)
X(1071) = crosspoint of X(7) and X(63)
X(1071) = crosssum of X(i) and X(j) for these (i,j): (1,1777)   (19,55)   (25,2331)

### X(1072) = POINT ALNILAM

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = 1 - cos2B cos C - cos B cos2C
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1072) lies on these lines: 1,4   55,1074   56,1076

### X(1073) = X(6)-CROSS CONJUGATE OF X(3)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = (cot A)/(cos A - cos B cos C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1073) lies on the Thomson cubic and these lines:
1,3341   2,253   3,64   4,3350   6,3343   9,223   57,3351   222,268

X(1073) = isogonal conjugate of X(1249)
X(1073) = X(253)-Ceva conjugate of X(64)
X(1073) = cevapoint of X(6) and X(64)
X(1073) = X(i)-cross conjugate of X(j) for these (i,j): (6,3), (185,69)
X(1073) = crosssum of X(6) and X(1033)

### X(1074) = POINT ALNITAK

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = cos A + cos2B cos C + cos B cos2C
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1074) lies on these lines: 1,224   3,225   4,1040   55,1072   56,1070   1038,1068

### X(1075) = X(3)-CEVA CONJUGATE OF X(4)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = cos B cos C (cos2C cos2A + cos2A cos2B - cos2B cos2C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1075) lies on the McCay cubic and these lines: 4,51   155,450   216,631   243,920   648,1092

X(1075) = eigencenter of cevian triangle of X(3)
X(1075) = eigencenter of anticevian triangle of X(4)
X(1075) = X(3)-Ceva conjugate of X(4)
X(1075) = X(155)-Hirst inverse of X(450)

### X(1076) = POINT ALPHARD

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = cos A - cos2B cos C - cos B cos2C
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1076) lies on these lines: 3,225   4,1038   55,1070   56,1072   1040,1068

### X(1077) = ISOGONAL CONJUGATE OF X(1049)

Trilinears        1/A : 1/B : 1/C
Barycentrics   a/A : b/B : c/C

X(1077) = isogonal conjugate of X(1049).

### X(1078) = INTERSECTION OF LINES X(2)X(32) AND X(3)X(76)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = bc(b2c2 + c2a2 + a2b2 - a4)
Barycentrics  b2c2 + c2a2 + a2b2 - a4 : c2a2 + a2b2 + b2c2 - b4 : a2b2 + b2c2 + c2a2 - c4

X(1078) lies on these lines:
2,32   3,76   5,316   24,264   35,350   39,385   54,69   140,325   186,1235   187,384   194,574   274,404   298,619   279,618   302,635   303,636

X(1078) = isotomic conjugate of X(3613)
X(1078) = anticomplement of X(1506)
X(1078) = X(249)-Ceva conjugate of X(99)

### X(1079) = TRILINEAR SQUARE OF X(46)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos B + cos C - cos A)2
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1079) lies on these lines: 1,4   46,1406   77,498   484,1103   651,920

### X(1080) = INTERCEPT OF EULER LINE AND LINE (13)X(98)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = csc(B - C) [sin 2B cos(C - ω) sin(C - π/3) - sin 2C cos(B - ω) sin(B - π/3)]
Trilinears        g(a,b,c) : g(b,c,a) : g(c,a,b),
where g(a,b,c) = bc[3(a2 + b2 + c2)(a2 - b2 + c2)(a2 + b2 - c2) + 64(31/23]    (M. Iliev, 4/12/07)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(1080) has Shinagawa coefficients (31/2S, 3E + 3F).

Coordinates for X(1080) are obtained from those of X(383) by changing π/3 to - π/3; contributed by Edward Brisse.

X(1080) lies on these lines: 2,3   13,98   14,262   183,622   298,511   325,621

X(1080) = inverse-in-orthocentroidal-circle of X(383)

### X(1081) = POINT ALPHECCA

Trilinears       sec(A/2) csc(A/2 - π/3) : sec(B/2) csc(B/2 - π/3) : sec(C/2) csc(C/2 - π/3)
Barycentrics  sin A sec(A/2) csc(A/2 - π/3) : sin B sec(B/2) csc(B/2 - π/3) : sin C sec(C/2) csc(C/2 - π/3)

Coordinates for X(1081) are obtained from those of X(554) by changing π/3 to - π/3; contributed by Edward Brisse.

X(1081) lies on these lines: 1,30   7,559   13,226   75,298

X(1081) = isogonal conjugate of X(1250)

### X(1082) = POINT ALPHERATZ

Trilinears       (sec A/2) sin(A/2 - π/3) : (sec B/2) sin(B/2 - π/3) : (sec C/2) sin(C/2 - π/3)
Barycentrics  (sin A/2) sin(A/2 - π/3) : (sin B/2) sin(B/2 - π/3) : (sin C/2) sin(C/2 - π/3)

Coordinates for X(1082) are obtained from those of X(559) by changing π/3 to - π/3; contributed by Edward Brisse.

X(1082) lies on these lines: 1,3   7,554   13,226   298,319

X(1082) = isogonal conjugate of X(1251)

### X(1083) = MIDPOINT OF X(105) AND X(644)

Trilinears    a4 - a3(b + c) - a2bc + 2abc(b + c) - bc(b2 + c2): :L

X(1083) lies on a circle related to the 1st and 2nd Brocard points; Hyacinthos message #4053, Paul Yiu, Oct. 4, 2001. X(1083) lies on the Brocard circle.

X(1083) lies on the Brocard circle, the circle O(1,3), and these lines:
1,6   3,667   8,1016   55,1026   56,1025   105,644   840,898

X(1083) = midpoint of X(105) and X(644)
X(1083) = inverse-in-circumcircle of X(667)
X(1083) = X(6)-Hirst inverse of X(518)
X(1083) = X(105)-of-1st-Brocard triangle
X(1083) = X(105)-of-X(1)-Brocard triangle
X(1083) = X(112)-of-1st-Montesdeoca-bisector-triangle
X(1083) = X(112)-of-2nd-Montesdeoca-bisector-triangle
X(1083) = similicenter of 1st and 2nd Montesdeoca bisector triangles

### X(1084) = CROSSPOINT OF X(2) AND X(512)

Trilinears        a3(b2 - c2)2 : b3(c2 - a2)2 : c3(a2 - b2)2
Barycentrics  a4(b2 - c2)2 : b4(c2 - a2)2 : c4(a2 - b2)2

Let f(a,b,c) = a3(b2 - c2)2. Then the line
f(a,b,c)x + f(b,c,a)y + f(c,a,b)z = 0 is tangent to the circumcircle at X(99).

Randy Hutson observed (9/23/2011) that the centers of homothety of the Lucas(L:W) homothetic triangles and triangle ABC form a circumconic which passes through the points X(493), X(494), X(588), and X(589). Indeed the perspectors are given by barycentric coordinates

a2/[a2 + (L/W)S] : b2/[b2 + (L/W)S] : c2/[c2 + (L/W)S],

and the conic is the isogonal conjugate of the line X(2)X(6). Thus, X(2), X(6), and dozens of other named points lie on the conic; click Tables at the top of ETC, select CENTRAL LINES, and scroll to #15.

X(1084) is the center of the hyperbola H = {{A,B,C,X(2),X(6)}}, which is tangent to Brocard axis at X(6) and to line X(2)X(39) at X(2). Also, H is the locus of the trilinear pole of a line parallel to Lemoine axis (i.e. lines that pass through X(512)), and H is the isotomic conjugate of line the X(2)X(39). (Randy Hutson, July 20, 2016)

X(1084) lies on the Steiner inellipse and these lines: 2,670   6,694   39,597   115,804   351,865

X(1084) = midpoint of X(i) and X(j) for these (i,j): (6,694),(2,3228)
X(1084) = complement of X(670)
X(1084) = crosspoint of X(2) and X(512)
X(1084) = crosssum of X(i) and X(j) for these (i,j): (6,99), (76,670), (799,873)
X(1084) = crossdifference of every pair of points on line X(99)X(670)
X(1084) = center of the circumconic {{A,BN,C,X(2),X(6)}}
X(1084) = projection from Steiner circumellipse to Steiner inellipse of X(3228)
X(1084) = perspector of circumconic centered at X(512) (parabola {{A,B,C,X(512),X(669)}})
X(1084) = intersection of trilinear polars of X(512) and X(669)
X(1084) = X(2)-Ceva conjugate of X(512)
X(1084) = perspector of ABC and the medial triangle of the cevian triangle of X(512)
X(1084) = perspector of unary cofactor triangles of 3rd, 5th and 6th Brocard triangles
X(1084) = barycentric square of X(512)

### X(1085) = ISOGONAL CONJUGATE OF X(1028)

Trilinears        A2 : B2 : C2
Barycentrics  A2 sin A : B2 sin B : C2 sin C

X(1085) = isogonal conjugate of X(1028)

### X(1086) = CENTER OF HYPERBOLA {{A,B,C,X(2),X(7)|}

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = bc(b - c)2
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b - c)2

The line f(a,b,c)x + f(b,c,a)y + f(c,a,b)z = 0 is tangent to the circumcircle at X(101). Also, X(1086) is the point of tangency of the Steiner inscribed ellipse with the line tangent to the nine-point circle and the incircle. (Paul Yiu, #4197, 11/24/01).

X(1086) = center of circumconic that is locus of trilinear poles of lines parallel to Gergonne line (i.e. lines that pass through X(514)). This conic is the isotomic conjugate of the Nagel line. (Randy Hutson, September 14, 2016)

Let A7B7C7 and A8B8C8 be the Gemini triangles 7 and 8. Let A' be the barycentric product A7*A8 and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1086). (Randy Hutson, November 30, 2018)

X(1086) lies on the Steiner inellipse and these lines:
1,528   2,45   6,7   8,599   10,537   11,244   37,142   44,527   53,273   57,1020   75,141   115,116   220,277   239,320   812,1015   918,1111

X(1086) = midpoint of X(i) and X(j) for these (i,j): (2,903), (7,673), (75,335), (239,320)
X(1086) = isogonal conjugate of X(1252)
X(1086) = isotomic conjugate of X(1016)
X(1086) = complement of X(190)
X(1086) = crosspoint of X(2) and X(514)
X(1086) = crosssum of X(i) and X(j) for these (i,j): (6,101), (9,1018), (32,692), (219,906)
X(1086) = crossdifference of every pair of points on line X(101)X(692)
X(1086) = perspector of circumconic centered at X(514)
X(1086) = X(2)-Ceva conjugate of X(514)
X(1086) = projection from Steiner circumellipse to Steiner inellipse of X(903)
X(1086) = trilinear pole of line X(764)X(1647)
X(1086) = trilinear pole wrt medial triangle of Nagel line
X(1086) = anticomplement of X(4422)
X(1086) = barycentric product X(5997)*X(5998)
X(1086) = barycentric square of X(514)

### X(1087) = TRILINEAR SQUARE OF X(5)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = cos2(B - C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1087) lies on these lines: 1,564   31,91   92,255

### X(1088) = TRILINEAR SQUARE OF X(7)

Trilinears    sec4(A/2) : :

Let A'B'C' be the cross-triangle of the inner and outer Soddy triangles. Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1088). (Randy Hutson, December 10, 2016)

X(1088) is the Brianchon point (perspector) of the inellipse that is the trilinear square of the Gergonne line. The center of this inellipse is X(11019). (Randy Hutson, October 15, 2018)

X(1088) lies on these lines:
2,85   7,354   57,658   75,3668   86,269   234,555   272,1014   305,341   675,934

X(1088) = isogonal conjugate of X(1253)
X(1088) = isotomic conjugate of X(200)
X(1088) = X(7)-cross conjugate of X(85)
X(1088) = cevapoint of X(7) and X(279)

### X(1089) = TRILINEAR SQUARE OF X(10)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = b2c2(b + c)2
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1089) lies on these lines:
1,312   8,80   10,321   76,334   190,191   200,318   244,596   345,498   594,762   740,872

X(1089) = isogonal conjugate of X(849)
X(1089) = isotomic conjugate of X(757)
X(1089) = crosspoint of X(313) and X(321)

X(1089) = trilinear product of vertices of outer Garcia triangle

### X(1090) = TRILINEAR SQUARE OF X(11)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = [1 - cos(B - C)]2
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1090) lies on these lines: 5,1091   11,523

### X(1091) = TRILINEAR SQUARE OF X(12)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = [1 + cos(B - C)]2
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1091) lies on these lines: 5,1090   12,1109

### X(1092) = TRILINEAR CUBE OF X(3)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = cos3A
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1092) lies on these lines:
2,578   3,49   4,801   20,110   24,511   54,69   68,125   140,343   156,550   450,1093   648,1075

X(1092) = reflection of X(1204) in X(3)
X(1092) = isogonal conjugate of X(1093)

### X(1093) = TRILINEAR CUBE OF X(4)

Trilinears    sec3A : :
X(1093) lies on these lines:
3,1105   4,51   5,264   24,107   155,648   158,225   393,800   403,847   436,578   450,1092

X(1093) = isogonal conjugate of X(1092)
X(1093) = isotomic conjugate of X(3964)
X(1093) = X(235)-cross conjugate of X(4)
X(1093) = polar conjugate of X(394)
X(1093) = X(978)-of-orthic-triangle if ABC is acute

### X(1094) = TRILINEAR SQUARE OF X(15)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = sin2(A + π/3)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1094) lies on these lines: 15,36   48,163

### X(1095) = TRILINEAR SQUARE OF X(16)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = sin2(A - π/3)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1095) lies on these lines: 16,36   48,163

### X(1096) = TRILINEAR SQUARE OF X(19)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = tan2A
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1096) lies on these lines:
1,29   4,1039   19,31   33,42   34,207   63,240   107,741   213,607   243,1040   278,614   281,612

X(1096) = isogonal conjugate of X(326)
X(1096) = X(158)-Ceva conjugate of X(19)
X(1096) = crosssum of X(394) and X(1259)

### X(1097) = TRILINEAR SQUARE OF X(20)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = (cos A - cos B cos C)2
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1097) lies on these lines: 1,75   31,775

### X(1098) = TRILINEAR SQUARE OF X(21)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = 1/(cos B + cos C)2
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1098) lies on these lines:
3,662   8,643   21,60   29,270   58,86   65,409   81,1104

X(1098) = isogonal conjugate of X(1254)
X(1098) = cevapoint of X(i) and X(j) for these (i,j): (1,411), (21,283)

### X(1099) = TRILINEAR SQUARE OF X(30)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = (cos A - 2 cos B cos C)2
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1099) lies on the inellipse centered at X(10) and on these lines: 1,564   75,811   162,255

### X(1100) = COMPLEMENT OF X(319)

Trilinears    2a + b + c : 2b + c + a : 2c + a + b    (M. Iliev, 5/13/2007)
Trilinears    ar + S : br + S : cr + S;   (C. Lozada, 9/07/2013)
Barycentrics   a(2a + b + c) : b(2b + c + a) : c(2c + a + b)

X(1100) is the midpoint of the bicentric pair y : z : x and z : x : y, where x : y : z = X(37)

X(1100) lies on these lines:
1,6   2,319   36,1030   48,354   65,604   71,583   81,593   86,239   214,1015   284,501   517,572   519,594   536,894   820,836

X(1100) is the {X(1),X(6)}-harmonic conjugate of X(37). For a list of other harmonic conjugates of X(1100), click Tables at the top of this page.

X(1100) = isogonal conjugate of X(1255)
X(1100) = complement of X(319)
X(1100) = crosspoint of X(i) and X(j) for these (i,j): (1,81), (2,79)
X(1100) = crosssum of X(i) and X(j) for these (i,j): (1,37), (6,35), (559,1082)
X(1100) = crossdifference of every pair of points on line X(484)X(513)
X(1100) = bicentric sum of PU(31)
X(1100) = midpoint of PU(31)
X(1100) = reflection of X(3775) in X(1125)
X(1100) = X(1)-Ceva conjugate of X(1962)
X(1100) = X(1962)-Ceva conjugate of X(1125) wrt incentral triangle
X(1100) = perspector wrt incentral triangle of bicevian conic of X(1) and X(2)

### X(1101) = TRILINEAR SQUARE OF X(110)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = csc2(B - C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1101) lies on these lines: 59,60   163,798   656,662

X(1101) = isogonal conjugate of X(1109)
X(1101) = cevapoint of X(i) and X(j) for these (i,j): (31,163), (60,110)
X(1101) = X(i)-cross conjugate of X(j) for these (i,j): (31,163), (47,162)

### X(1102) = TRILINEAR CUBE OF X(63)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = cot3A
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1102) lies on these lines: 63,304   255,326

### X(1103) = TRILINEAR SQUARE OF X(40)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = (cos B + cos C - cos A - 1)2
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1103) lies on these lines: 1,2   31,937   40,221   46,269   165,255

X(1103) = isogonal conjugate of X(1256)

### X(1104) = CROSSPOINT OF X(1) AND X(28)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = 2avw + cv2 + bw2 + u(bv + cw), u : v : w = X(72)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1104) is the midpoint of the bicentric pair y : z : x and z : x : y, where x : y : z = X(72)

X(1104) lies on these lines:
1,6   11,429   25,34   31,65   32,910   58,942   81,1098   105,961   210,976   229,593   239,1043   440,950   517,580   581,995

X(1104) = isogonal conjugate of X(1257)
X(1104) = crosspoint of X(i) and X(j) for these (i,j): (1,28), (81,269)
X(1104) = crosssum of X(i) and X(j) for these (i,j): (1,72), (37,200)

### X(1105) = CEVAPOINT OF X(3) AND X(4)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = (sec A)/(cos2B + cos2C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

Let A'B'C' be the cevian triangle of X(3). Let LA be the reflection of the line B'C' in the line BC, and define LB and LC cyclically. Let A'' = LB∩LC, and define B'' and C'' cyclically. The lines AA'', BB'', CC'' concur in X(1105). (Randy Hutson, 9/23/2011)

X(1105) lies on these lines: 3,1093   4,801   20,393   185,648   225,412   243,411   378,847

X(1105) = isogonal conjugate of X(185)
X(1105) = cevapoint of X(3) and X(4)

### X(1106) = TRILINEAR SQUARE OF X(56)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = (1 - cos A)2
= sin4A/2 : sin4B/2 : sin4C/2

Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A) f(A,B,C)

X(1106) lies on these lines:
3,1037   7,987   31,56   32,604   34,244   36,255   38,1038   57,961   58,269   77,988   279,985   388,750   601,999   651,978   727,934

X(1106) = isogonal conjugate of X(341)

### X(1107) = CROSSPOINT OF X(1) AND X(274)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2) + bc(b + c)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1107) =(tan ω sin 2ω)R*X(10) + r*X(39)
X(1107) = X(1) + X(8) + (cot ω csc 2ω)(2r/R)*X(39)

X(1107) = midpoint of the bicentric pair y : z : x and z : x : y, where x : y : z = X(213)
X(1107) = insimilicenter of the Spieker and (1/2)-Moses circle. [The (1/2)-Moses circle is described at X(1575).]

X(1107) lies on these lines: 1,6   2,330   10,39   32,993   75,194   210,869   239,257

X(1107) = isogonal conjugate of X(1258)
X(1107) = isotomic conjugate of X(1221)
X(1107) = complement of X(1909)
X(1107) = crosspoint of X(i) and X(j) for these (i,j): (1,274), (2,256), (81,87)
X(1107) = crosssum of X(i) and X(j) for these (i,j): (1,213), (6,171), (37,43)

### X(1108) = CROSSPOINT OF X(2) AND X(84)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2avw + cv2 + bw2 + u(bv + cw),
where u : v : w = X(219)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1108) is the midpoint of the bicentric pair y : z : x and z : x : y, where x : y : z = X(219)

X(1108) lies on these lines: 1,6   2,322   19,56   104,112   241,347   278,393   517,579

X(1108) = complement of X(322)
X(1108) = crosspoint of X(i) and X(j) for these (i,j): (1,278), (2,84)
X(1108) = crosssum of X(i) and X(j) for these (i,j): (1,219), (6,40)

### X(1109) = TRILINEAR SQUARE OF X(523)

Trilinears    sin2(B - C) : :

X(1109): Let A'B'C' be the Feuerbach triangle. Let La be the trilinear polar of A', and define Lb, Lc cyclically. Let A" = Lb∩Lc, B" = Lc∩La, C" = La∩Lb. The lines AA", BB", CC" concur in X(1109). (Randy Hutson, December 26, 2015)

Let A'B'C' be the Feuerbach triangle. Let Ma be the tangent to conic {{A,B,C,B',C'}} at A, and define Mb and Mc cyclically. Let A* = Mb∩Mc, B* = Mc∩Ma, C* = Ma∩Mb. The lines AA*, BB*, CC* concur in X(1109); see also X(523). (Randy Hutson, December 26, 2015)

Let A'B'C' be the Feuerbach triangle. Let Ab = BC∩C'A', and define Bc and Ca cyclically. Let Ac = BC∩A'B', and define Ba and Cb cyclically. The points Ab, Ac, Bc, Ba, Ca, Cb lie on an ellipse, denoted by E. Let A" be the intersection of the tangents to E at Ba and Ca, and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1109). (Randy Hutson, December 26, 2015)

Let A'B'C' be the Feuerbach triangle. Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1109). (Randy Hutson, December 26, 2015)

Let La be the A-extraversion of line X(2610)X(4024) (the trilinear polar of X(12)), and define Lb and Lc cyclically. Let A' = Lb ∩ Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1109). (Randy Hutson, January 29, 2018)

Let F be the Feuerbach point, X(11), and FaFbFc be the Feuerbach triangle (the extraversion triangle of X(11)). Let A' be the trilinear product F*Fa, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1109). (Randy Hutson, January 29, 2018)

X(1109) lies on these lines:
1,564   11,523   12,1091   31,92   75,799   91,255

X(1109) = isogonal conjugate of X(1101)
X(1109) = crosspoint of X(12) and X(523)
X(1109) = crosssum of X(i) and X(j) for these (i,j): (31,163), (60,110)
X(1109) = trilinear product X(11)*X(12)
X(1109) = antipode of X(4736) in the inellipse centered at X(10)
X(1109) = reflection of X(4736) in X(10)
X(1109) = crossdifference of every pair of points on line X(163)X(1983)
X(1109) = bicentric difference of PU(73)
X(1109) = PU(73)-harmonic conjugate of X(2624)
X(1109) = polar conjugate of isotomic conjugate of X(20902)

### X(1110) = TRILINEAR SQUARE OF X(101)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = [a/(b - c)]2
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

X(1110) lies on these lines:
1,1053   36,59   101,663   249,849   667,692

X(1110) = isogonal conjugate of X(1111)
X(1110) = X(i)-cross conjugate of X(j) for these (i,j): (32,163), (41,101)
X(1110) = crosssum of X(11) and X(1086)

### X(1111) = TRILINEAR SQUARE OF X(514)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = [(b - c)/a]2
Trilinears    squared distance from A to line X(1)X(6) : :
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = af(a,b,c)

Let A7B7C7 and A8B8C8 be the Gemini triangles 7 and 8. Let A' be the trilinear product A7*A8 and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1111). (Randy Hutson, November 30, 2018)

Let A7B7C7 and A8B8C8 be Gemini triangles 7 and 8, resp. Let A' be the intersection of the tangent to the {ABC, Gemini 7}-circumconic at A7 and the tangent to the {ABC, Gemini 8}-circumconic at A8. Define B' and C' cyclically. The lines AA', BB', CC' concur in X(1111). (Randy Hutson, January 15, 2019)

X(1111) lies the inellipse centered at X(10) and on these lines:
1,85   7,80   75,537   76,334   269,273   348,499   918,1086

X(1111) = isogonal conjugate of X(1110)
X(1111) = isotomic conjugate of X(765)
X(1111) = crosssum of X(31) and X(692)
X(1111) = antipode of X(4712) in inellipse centered at X(10)
X(1111) = reflection of X(4712) in X(10)
X(1111) = trilinear product X(5997)*X(5998)
X(1111) = perspector of side- and vertex-triangles of Gemini triangles 7 and 8
X(1111) = trilinear product of vertices of Gemini triangle 7
X(1111) = trilinear product of vertices of Gemini triangle 8

### X(1112) = CROSSPOINT OF X(4) AND X(250)

Trilinears    a[a4(b2 + c2) - 2a2(b4 + c4) + b6 + c6 ]/(b2 + c2 - a2) : :
Trilinears    sec A (1 + cos 3A cos(B - C)) : :

If you have The Geometer's Sketchpad, you can view X(1112).

X(1112) is the center of the hyperbola that passes through the vertices of the cevian triangles of X(4) and X(648), and also through the centers X(i) for I = 4, 113, 155, 193. (Paul Yiu, Oct. 16, 2001, as contributing editor for Clark Kimberling, "Conics associated with a cevian nest," Forum Geometricorum 1 (2001) 141-150; see Example 2.)

X(1112) is X(11)-of-the-orthic-triangle if ABC is acute. (Peter Moses, July 7, 2009)

X(1112) lies on these lines:
4,94   6,1177   25,110   51,125   52,113   389,974   428,542   468,511

X(1112) = reflection of X(974) in X(389)
X(1112) = crosspoint of X(4) and X(250)
X(1112) = crosssum of X(3) and X(125)
X(1112) = inverse-in-polar-circle of X(3448)
X(1112) = inverse-in-orthosymmedial-circle of X(427)

### X(1113) = 1st EULER-LINE-CIRCUMCIRCLE INTERSECTION

Trilinears    (R - d)cos A - 2R cos B cos C : (R - d)cos B - 2R cos C cos A : (R - d)cos C - 2R cos A cos B, where R = circumradius, d = distance |OH| between X(3) and X(4). (Joe Goggins, 2002)
Trilinears    (1 - J) cos A - 2 cos B cos C : : , where J = |OH|/R = (1/abc)[S(6) - S(2,4) + 3a2b2c2]1/2, where S(6) = a6 + b6 + c6, and S(2,4) = a2b4 + a2c4 + b2c4 + b2a4 + c2a4 + c2b4 (Peter J. C. Moses, 10/2/03)
Barycentrics    2RSBSC + (|OH| - R)a2SA : : , where |OH| = distance between X(3) and X(4), and R = circumradius (Peter J. C. Moses, 3/2003; cf. X(1313), X(1314))
Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) - a^2*(-a^2 + b^2 + c^2)*(1 - J) : :
X(1113) = 3X(2) + (-3 + |OH|/R)*X(3) = X(1113) = (- 1 + |OH|/R)*X(3) + X(4)

As a point on the Euler line, X(1113) has Shinagawa coefficients (R - |OH|, -3R + |OH|).

X(1113) is a point of intersection of the Euler line and the circumcircle. The other is X(1114). Of the two, X(1113) is the one closer to X(4).

X(1113) is one of 2 points P such that P is the circumcircle-antipode of Λ (trilinear polar of P); the other is X(1114). (Randy Hutson, November 2, 2017)

If you have The Geometer's Sketchpad, you can view X(1113) and X(1114).

X(1113) lies on these lines:
{1, 2100}, {2, 3}, {6, 2104}, {11, 10781}, {54, 14374}, {74, 2575}, {98, 2593}, {100, 2580}, {101, 2576}, {108, 2586}, {109, 1822}, {110, 2574}, {111, 8106}, {112, 8105}, {165, 2101}, {187, 8426}, {511, 2105}, {517, 2103}, {691, 9173}, {759, 2589}, {1495, 13415}, {2249, 2579}, {2777, 14500}, {5840, 10782}, {5972, 14499}, {10287, 10686}

X(1113) = reflection of X(i) in X(j) for these (i,j): (4,1312), (1114,3)
X(1113) = isogonal conjugate of X(2574)
X(1113) = anticomplement of X(1313)
X(1113) = X(250)-Ceva conjugate of X(1114)
X(1113) = trilinear product X(110)*X(1823)
X(1113) = trilinear pole of line X(6)X(1345) (the major axis of the orthic inconic)
X(1113) = Ψ(X(6), X(1345))
X(1113) = pole wrt polar circle of trilinear polar of X(2592) (line X(523)X(1313))
X(1113) = polar conjugate of X(2592)
X(1113) = inverse-in-polar-circle of X(1313)
X(1113) = Thomson isogonal conjugate of X(2575)
X(1113) = Lucas isogonal conjugate of X(2575)

### X(1114) = 2nd EULER-LINE-CIRCUMCIRCLE INTERSECTION

Trilinears    (R + d)cos A - 2R cos B cos C : (R + d)cos B - 2R cos C cos A : (R + d)cos C - 2R cos A cos B, where R = circumradius, d = distance between X(3) and X(4). (Joe Goggins, 2002)
Trilinears    (1 + J) cos A - 2 cos B cos C, where J = |OH|/R; see X(1113)
Barycentrics    2RSBSC - (|OH| +R)a2SA : : , where |OH| = distance between X(3) and X(4), and R = circumradius (Peter J. C. Moses, 3/2003)
Barycentrics    (a^2 + b^2 - c^2)*(a^2 - b^2 + c^2) - a^2*(-a^2 + b^2 + c^2)*(1 + J) : :
X(1114) = 3X(2) - (3 + |OH|/R)*X(3) = - (1 + |OH|/R)*X(3) + X(4)

As a point on the Euler line, X(1114) has Shinagawa coefficients (R + |OH|, -3R - |OH|).

X(1114) is a point of intersection of the Euler line and the circumcircle. Its antipode is X(1113).

X(1114) is one of 2 points P such that P is the circumcircle-antipode of Λ (trilinear polar of P); the other is X(1113). (Randy Hutson, November 2, 2017)

X(1114) lies on these lines:
{1, 2101}, {2, 3}, {6, 2105}, {11, 10782}, {54, 14375}, {74, 2574}, {98, 2592}, {100, 2581}, {101, 2577}, {108, 2587}, {109, 1823}, {110, 2575}, {111, 8105}, {112, 8106}, {165, 2100}, {187, 8427}, {511, 2104}, {517, 2102}, {691, 9174}, {759, 2588}, {1379, 14899}, {1495, 13414}, {2249, 2578}, {2777, 14499}, {5840, 10781}, {5972, 14500}, {10288, 10687}

X(1114) = reflection of X(i) in X(j) for these (i,j): (4,1313), (1113,3)
X(1114) = isogonal conjugate of X(2575)
X(1114) = anticomplement of X(1312)
X(1114) = X(250)-Ceva conjugate of X(1113)

X(1114) = trilinear product X(110)*X(1822)

X(1114) = trilinear pole of line X(6)X(1344) (the minor axis of the orthic inconic)
X(1114) = Ψ(X(6), X(1344))
X(1114) = pole wrt polar circle of trilinear polar of X(2593) (line X(523)X(1312))
X(1114) = polar conjugate of X(2593)
X(1114) = inverse-in-polar-circle of X(1312)
X(1114) = Thomson isogonal conjugate of X(2574)
X(1114) = Lucas isogonal conjugate of X(2574)

Centers 1115 - 1150
were added to ETC on 1/10/03.

### X(1115) = STEINER CURVATURE CENTROID

Trilinears       (π - A)/a : (π - B)/b : (π - C)/c
Barycentrics  π - A : π - B : π - C

X(1115) is the center of mass of a point-mass system obtained by placing at vertex A a mass equal to the magnitude of the exterior angle (that's π - A) at A, and cyclically for B and C. (Peter Scales, #5528, 5/22/02)

X(1115) lies on this line: 2,360

X(1115) = complement of X(360)

### X(1116) = CENTER OF THE LESTER CIRCLE

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = bc(b2-c2)[2(a2-b2)(c2-a2) + 3R2(2a2-b2-c2) - a2(a2+b2+c2) + a4+b4+c4],
where R = (a csc A)/2 = circumradius of ABC.

Barycentrics  af(a,b,c) : bf(b,c,a): cf(c,a,b)

The Lester circle passes through the points X(3), X(5), X(13), X(14). Coordinates of the center were determined by Milorad Stevanovic (#5895, 9/20/02). The circle is described in

June Lester, "Triangles III: complex centre functions and Ceva's theorem," Aequationes Mathematicae 53 (1997) 4-35.

If you have The Geometer's Sketchpad, you can view X(1116).

X(1116) lies on these lines: 115,125   140,523

X(1116) = pole wrt orthocentroidal circle of Napoleon axis (line X(6)X(17))

### X(1117) = POINT ALRESCHA

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = n(a,b,c)/d(a,b,c),
n(a,b,c) = 5(SA)2[(SB)2 + (SC)2] - 3a2(SA)3 - 4(SB)2(SC)2 - SASBSC(2SA - SB - SC),
d(a,b,c) = 2a[4(SA)2 - b2c2][b2c2(3a2 - 8SA) + 8(SA)3],
SA = (b2 + c2 - a2)/2; SB = (c2 + a2 - b2)/2; SC = (a2 + b2 - c2)/2;
(coordinates by Edward Brisse, Peter J. C. Moses).

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

X(1117) lies on the Lester circle. See Bernard Gibert's message, Hyacinthos #5613, 5/31/02.

X(1117) on the Lester circle, the cubic K060, and these lines: {5,3470}, {30,5671}, {265,13582}, {11071,11581}

X(1117) = X(13582)-Ceva conjugate of X(11071)
X(1117) = cevapoint of X(3471) and X(5671)
X(1117) = isogonal conjugate of inverse-in-circumcircle of isogonal conjugate of X(399)
X(1117) = antigonal conjugate of X(399)
X(1117) = syngonal conjugate of X(10264)
X(1117) = barycentric product X(1272)*X(11071)
X(1117) = barycentric product X(1989)X(1272)X(13582)
X(1117) = barycentric quotient X(i)/X(j) for these {i,j}: {11071, 1138}, {11074, 3470}

### X(1118) = 1st HATZIPOLAKIS PERSPECTOR

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (1 - cos A)/cos2A
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Let A'B'C' be the intouch triangle of ABC. Let CA be the point other than C' in which the perpendicular to BC from C' meets the incircle, let BA be the point other than B' in which the perpendicular to BC from B' meets the incircle, and let A0 be the point of intersection of lines BCA and CBA. Define B0 and C0 cyclically. Then triangle A0B0C0 is perspective to ABC, and the perspector is X(1118). (Antreas Hatzipolakis, #5321, 4/30/02)

X(1118) is the trilinear product A0*B0*C0, where A0, B0, C0 are as defined above. (Randy Hutson, January 15, 2019)

X(1118) lies on the hyperbola {{A,B,C,X(4),X(19)}} these lines: 4,65   7,286   12,281   19,208   20,243   24,108   28,56   34,207   92,388

X(1118) = isogonal conjugate of X(1259)
X(1118) = isotomic conjugate of X(1264)
X(1118) = polar conjugate of X(345)

### X(1119) = 2nd HATZIPOLAKIS PERSPECTOR

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = (1 - sec A)/(1 + cos A)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Let triangle A0B0C0 be as defined for X(1118). Let A1 be the orthogonal projection of A0 onto line BC, and define B1 and C1 cyclically. Then triangle A1B1C1 is perspective to ABC, and the perspector is X(1119). (Antreas Hatzipolakis, #5321, 4/30/02)

X(1119) lies on the hyperbola {{A,B,C,X(4),X(19)}} these lines: 3,347   4,7   19,57   28,279   34,269   142,281   393,1086   579,1020   915,934

X(1119) = isogonal conjugate of X(1260)
X(1119) = isotomic conjugate of X(1265)
X(1119) = X(34)-cross conjugate of X(278)
X(1119) = polar conjugate of X(346)

### X(1120) = X(2)-BLAIKIE TRANSFORM OF X(1)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = bc/(b2 + c2 + ba + ca - 4bc)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

For the definition of Blaikie transform, see X(903).

X(1120) lies on these lines: 6,644   56,100   58,643   106,519  269,664

X(1120) = isogonal conjugate of X(1149)
X(1120) = isotomic conjugate of X(1266)
X(1120) = trilinear pole of line X(9)X(649)

### X(1121) = X(7)-BLAIKIE TRANSFORM OF X(2)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = bc/[(b + c - a)a - (a + b - c)(a + c - b)]
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

For the definition of Blaikie transform, see X(903).

X(1121) lies on the Steiner circumellipse and these lines:
2,664   8,190   29,648   99,333   312,668   519,666   903,918

X(1121) = reflection of X(i) in X(j) for these (i,j): (2,1146), (664,2)
X(1121) = isogonal conjugate of X(1055)
X(1121) = isotomic conjugate of X(527)
X(1121) = Steiner-circumellipse-antipode of X(664)
X(1121) = projection from Steiner inellipse to Steiner circumellipse of X(1146)
X(1121) = antipode of X(8) in hyperbola {{A,B,C,X(2),X(8)}}
X(1121) = trilinear pole of line X(2)X(522)

### X(1122) = 1st GRINBERG POINT

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b),    where f(a,b,c) = [a(b + c) + (b - c)2]/(b + c - a)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Let AB be the touchpoint of the A-excircle and line AB, let AC be the touchpoint of the A-excircle and line AC, and let MA be the midpoint of segment ABAC. Define MB and MC cyclically. Let A', B', C' be the touchpoints of the incircle with lines BC, CA, AB, respectively. The triangles MAMBMC and A'B'C' are perspective, and the perspector is X(1122). (Darij Grinberg, 12/28/02)

If you have The Geometer's Sketchpad, you can view X(1122).

X(1122) lies on these lines: 7,8   56,269

X(1122) = isogonal conjugate of X(1261)
X(1122) = crosspoint of X(7) and X(269)
X(1122) = crosssum of X(55) and X(200)

### X(1123) = PAASCHE POINT

Trilinears       1/(1 + sin A) : 1/(1 + sin B) : 1/(1 + sin C)
Barycentrics  (sin A)/(1 + sin A) : (sin B)/(1 + sin B) : (sin C)/(1 + sin C)

Let D and E be the congruent circles each tangent to the other and to line BC, with D also tangent to line AB and E also tangent to line CA, meeting in a point A' lying outside triangle ABC. Define B' and C' cyclically. Then A'B'C' is perspective to ABC, and the perspector is X(1123). See

Ivan Paasche, Aufgabe P 933, Praxis der Mathematik 1 (1990), page 40.

Let PA be the parabola with focus A and directrix BC, and let LA be the line of the points of intersection of PA with the segments AB and AC. Define LB and LC cyclically. Let A' = LB∩LC. The lines AA', BB', CC' concur in X(1123). (Randy Hutson, 9/23/2011)

Suppose that ABC is an acute triangle. Let oA be the circle with diameter BC. Let wA be the circle tangent to segments AB and AC and also externally tangent to oA, in point XA. Define XB and XC cyclically. The lines AXA, BXB, CXC concur in X(1123). (Tomasz Cieśla, 19 January 2013) For a related construction, see X(1336).

If you have The Geometer's Sketchpad, you can view X(1123)

X(1123) lies on these lines: 1,3069   2,586   37,158   57,482   81,1335   498,3302   499,3300   920,3068

X(1123) = isogonal conjugate of X(1124)
X(1123) = isotomic conjugate of X(1267)
X(1125) = trilinear pole of line X(4969)X(4976) (the perspectrix of ABC and Gemini triangle 12)

### X(1124) = ISOGONAL CONJUGATE OF X(1123)

Trilinears       1 + sin A : 1 + sin B : 1 + sin C
Trilinears       a(S + bc) : b(S + ca) : c(S + ab)
Barycentrics  (1 + sin A) sin A : (1 + sin B) sin B : (1 + sin C) sin C

X(1124) lies on these lines:
1,6   3,2066   11,485   12,486   35,1152   36,1151   42,494   55,372   56,371   176,651   255,605   498,615   499,590

X(1124) = isogonal conjugate of X(1123)
X(1124) = {X(1),X(6)}-harmonic conjugate of X(1335)
X(1124) = insimilicenter of incircle and 2nd Lemoine circle

### X(1125) = COMPLEMENT OF X(10)

Trilinears    (2a+b+c)/a : (a+2b+c)/b : (a+b+2c)/c
Trilinears    r + 2 R sin B sin C : :
Barycentrics  2a+b+c : a+2b+c : a+b+2c
X(1125) = X(1) + 3*X(2) = 3*X(1) + X(8) = 3*X(2) - X(10) = X(8) - 3*X(10)

The centroid of four points A,B,C,P is the complement of the complement of P with respect to triangle ABC. As an example, X(1125) is the centroid of {A,B,C,X(1)}. (Darij Grinberg, 12/28/02)

Let A' the midpoint of segment BC and let A'' be the midpoint of segment AA'. Define B'' and C'' cyclically. The triangle A''B''C'' is homothetic to ABC, and the center of homothety is X(1125).

Let I be the incenter of triangle ABC and A' the centroid of triangle BCI, and define B' and C' cyclically. The triangle A'B'C' is homothetic to ABC, and the center of homothety is X(1125).

X(1125) is the center of the ellipse which is the locus of centers of the conics passing through A, B, C, and X(1). This ellipse is also the locus of crosssums of the intersections of the circumcircle and lines through X(1). Furthmore, this ellipse is the bicevian conic of X(1) and X(2) (i.e. the conic which passes through the vertices of the incentral and medial triangles). The ellipse passes through X(11), X(214), X(244), X(1015) and the midpoints of the sides of ABC. (Randy Hutson, 8/13/2011, Hyacinthos #20179; see also #20181, by Chris van Tienhoven.)

A construction of X(1125) is given by Antreas Hatipolakis and Angel Montesdeoca at 24185.

Let A'B'C' be the incentral triangle. Let A" be the reflection of A in A', and define B", C" cyclically. Let A* be the trilinear pole of line B"C", and define B*, C* cyclically. The lines AA*, BB*, CC* concur in X(1125). (Randy Hutson, July 21, 2017)

Let A'B'C' be the 2nd circumperp triangle. Let A"B"C" be the triangle bounded by the Simson lines of A', B', C'. A"B"C" is homothetic to A'B'C' at X(1125). (Randy Hutson, July 21, 2017)

X(1125) lies on these lines:
1,2   3,142   5,515   11,214   21,36   33,475   34,406   35,404   37,39   40,631   55,474   56,226   58,86   65,392   72,354   105,831   114,116   140,517   165,962   171,595   274,350   409,759   443,497   749,984   758,942   958,999   1015,1107

X(1125) is the {X(1),X(2)}-harmonic conjugate of X(10). For a list of other harmonic conjugates of X(1125), click Tables at the top of this page.

X(1125) = midpoint of X(i) and X(j) for these (i,j):
(1,10), (2,551), (3,946), (5,1385), (8,3244), (11,214), (142,1001), (226,993), (942,960), (999,3452), (1100,3775), (4065,4647)
X(1125) = isogonal conjugate of X(1126)
X(1125) = isotomic conjugate of X(1268)
X(1125) = complement of X(10)
X(1125) = anticomplement of X(3634)
X(1125) = crosspoint of X(2) and X(86)
X(1125) = crosssum of X(6) and X(42)

X(1125) = perspector of circumconic centered at X(1213)
X(1125) = center of circumconic that is locus of trilinear poles of lines passing through X(1213)
x(1125) = center of bicevian conic of X(1) and X(2)
X(1125) = Kosnita(X(1),X(2)) point
X(1125) = X(1)-Ceva conjugate of X(4065)
X(1125) = X(2)-Ceva conjugate of X(1213)
X(1125) = X(214)-of-X(1)-Brocard triangle
X(1125) = complement of X(4065) wrt incentral triangle
X(1125) = trilinear product of vertices of anti-Aquila triangle
X(1125) = X(10110)-of-excentral-triangle
X(1125) = {X(1),X(8)}-harmonic conjugate of X(3244)
X(1125) = {X(2),X(10)}-harmonic conjugate of X(3634)
X(1125) = {X(8),X(10)}-harmonic conjugate of X(4691)
X(1125) = perspector of Gemini triangle 11 and cross-triangle of ABC and Gemini triangle 11
X(1125) = homothetic center of anticomplementary triangle and cross-triangle of Aquila and anti-Aquila triangles

### X(1126) = ISOGONAL CONJUGATE OF X(1125)

Trilinears       a/(2a+b+c) : b/(a+2b+c) : c/(a+b+2c)
Barycentrics  a2/(2a+b+c) : b2/(a+2b+c) : c2/(a+b+2c)

X(1126) lies on these lines:
1,748   6,595   10,86   35,42   56,181   145,996   830,1027

X(1126) = isogonal conjugate of X(1125)
X(1126) = isotomic conjugate of X(1269)
X(1126) = cevapoint of X(6) and X(42)
X(1126) = X(512)-cross conjugate of X(101)
X(1126) = perspector of ABC and unary cofactor triangle of Gemini triangle 12

### X(1127) = 1st DE VILLIERS POINT

Trilinears    (sin A/4)/sin(3A/4) : :
Trilinears    [1 - 2 cos(A/2)]/(1 + 2 cos A) : :      (M. Iliev, 5/13/07)
Barycentrics    1/(cot A - cot(A/4)) : : (Nikolaos Dergiades, ADGEOM #1511, 8/19/2014)

Let A', B', C' be the incenters of triangles XBC, XCA, XAB, respectively, where X is the incenter, X(1). The triangle A'B'C' is perspective to ABC, and the perspector is X(1127). Coordinates found by Darij Grinberg, 8/22/02. (The triangle A'B'C' is the BCI triangle.)

Michael de Villiers,A dual to Kosnita's theorem, reprinted from Mathematics & Informatics Quarterly 6 (1996) 1996.

X(1127) lies on this line: 174,481

X(1127) = isogonal conjugate of X(1129)

X(1127) = trilinear product of vertices of BCI triangle

### X(1128) = 2nd DE VILLIERS POINT

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = (sin((π - A)/4))/sin((π + 3A)/4)
Trilinears       gA,B,C) : g(B,C,A) : g(C,A,B),
where g(A,B,C) = [1 - 2 sin(A/2)]/(1 - 2 cos A)     (M. Iliev, 5/13/07)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Let A",B",C" be the excenters of ABC, and let A', B', C' be the respective incenters of triangles A"BC, B"CA, C"AB, respectively. The triangle A'B'C' is perspective to ABC, and the perspector is X(1128). (Darij Grinberg, 8/22/02). See references at X(1127).

X(1128) lies on these lines:
164,173   188,519   258,505

X(1128) = isogonal conjugate of X(1130)

### X(1129) = ISOGONAL CONJUGATE OF X(1127)

Trilinears    (sin 3A/4)/sin(A/4) : :
Trilinears    1 + 2 cos(A/2)      (M. Iliev, 5/13/07)

Let A', B', C' be as at X(1127). Let A" = BC' ∩ CB', B" = CA' ∩ AC', C" = AB' ∩ BA'. The lines AA", BB", CC" concur in X(1129). Note: A'B'C' and A"B"C" are analogous to the 1st Morley triangle and adjunct Morley triangle, substituting angle quadrisectors for angle trisectors. (Randy Hutson, January 29, 2018)

X(1129) lies on this line: 1,168

X(1129) = isogonal conjugate of X(1127)
X(1129) = perspector of ABC and cross-triangle of ABC and BCI triangle

### X(1130) = 1st STEVANOVIC POINT

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = sin[(π + 3A)/4]/sin[(π - A)/4]

Trilinears       g(A,B,C) : g(B,C,A) : g(C,A,B),
where g(A,B,C) = 1 + 2 sin(A/2)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Let U be the A-excenter of triangle ABC; let A' be the incenter of triangle UBC, and define B', C' cyclically. Let A" = BC'∩CB', and define B", C" cyclically. The lines AA", BB", CC" concur in X(1130). (Milorad R. Stevanovic, Hyacinthos #7185, 5/21/03. See also X(1488) and X(1489).)

X(1130) lies on these lines: 1,164   173,505

X(1130) = isogonal conjugate of X(1128)

### X(1131) = ARCTAN(2) KIEPERT POINT

Trilinears    1/(sin A + 2 cos A) : :
Trilinears    csc(A + t) : : , where t = arctan(2)
Barycentrics    1/(2 SA + S) : :

To construct the Vecten point, X(485), squares are erected outward on the sides of ABC. If A', B', C' are the centers of these squares, then triangle A'B'C' is perspective to ABC with perspector X(485). Now let A" be the midpoint of the side of the A-square that does not touch line BC, and define B" and C" cyclically. Then triangle A"B"C" is perspective to ABC with perspector X(1131). Angle(A"BC) = angle(B"CA) = angle(C"AB), so that X(1131) lies on the Kiepert hyperbola. Here, the common angle is arctan(2). (Darij Grinberg 9/22/02)

If you have The Geometer's Sketchpad, you can view X(1131).

X(1131) lies on these lines: 2,490   6,1132   4,3311   20,485   175,226

X(1131) = isogonal conjugate of X(1151)
X(1131) = isotomic conjugate of X(1270)
X(1131) = polar conjugate of X(3535)

### X(1132) = ARCTAN(-2) KIEPERT POINT

Trilinears    1/(sin A - 2 cos A) : :
Trilinears    csc(A - t) : : , where t = arctan(2)

Barycentrics    1/(2 SA - S) : :

X(1132) is constructed in the manner for X(1131), using squares erected inward, so that the three equal angles have common measure arctan(-2), and X(1132) lies on the Kiepert hyperbola. (Darij Grinberg 9/22/02)

If you have The Geometer's Sketchpad, you can view X(1132).

X(1132) lies on these lines: 2,489   4,3312   6,1131   20,486   176,226

X(1132) = isogonal conjugate of X(1152)
X(1132) = isotomic conjugate of X(1271)
X(1132) = polar conjugate of X(3536)

### X(1133) = BURGESS POINT

Trilinears    sin(π/3 - A/3)/sin(π/3 + A/3) : :

Rotate line BC about B away from A through angle B/3, and rotate line BC about C away from A through angle C/3; let A' be the point in which the two rotated lines meet. Define B' and C' cyclically. Let A" be the point of intersection of lines BC' and B'C, and define B" and C" cyclically. The lines AA', BB', CC' concur in X(357), and AA", BB", CC" concur in X(358). The first of these with reference to triangle A'B'C' is X(1133); i.e., X(1133) = X(357)-of-A'B'C'.

A. G. Burgess, "Concurrency of lines joining vertices of a triangle to opposite vertices of triangles on its sides,"Proceedings of the Edinburgh Mathematical Society 33 (1913-14) 58-64; page 63.

X(1133) = X(3273)-isoconjugate of X(3602)

### X(1134) = 3rd MORLEY-TAYLOR-MARR CENTER

Trilinears    1/cos(A/3 + 2π/3) : :
Trilinears       sec(A/3 - π/3) : :

F. Glanville Taylor and W. L. Marr, "The six trisectors of each of the angles of a triangle," Proceedings of the Edinburgh Mathematical Society 33 (1913-14) 119-131; especially item 9, p. 127.

If you have The Geometer's Sketchpad, you can view X(1134).

X(1134) lies on these lines: 356,1135   357,3275

X(1134) = isogonal conjugate of X(1135)

### X(1135) = 4th MORLEY-TAYLOR-MARR CENTER

Trilinears    cos(A/3 + 2π/3) : :
Trilinears    cos(A/3 - π/3) : :
Trilinears    cos(B/3) cos(C/3) - sin(B/3) sin(C/3) : :

See the reference at X(1134).

If you have The Geometer's Sketchpad, you can view X(1135).

X(1135) lies on these lines: 16,358   356,1134

X(1135) = isogonal conjugate of X(1134)
X(1135) = perspector of ABC and 3rd Morley adjunct triangle
X(1135) = trilinear product of vertices of 3rd Morley adjunct triangle
X(1135) = {X(357),X(3603)}-harmonic conjugate of X(3272)

### X(1136) = 5th MORLEY-TAYLOR-MARR CENTER

Trilinears    1/cos(A/3 + 4π/3) : :
Trilinears      sec(A/3 + π/3) :
Trilinears    cos(A/3) - cos(B/3 - C/3) : :

See the reference at X(1134).

Hyacinthos 21423 (January 16, 2013), posted by Chris van Tienhoven.

If you have The Geometer's Sketchpad, you can view X(1136).

X(1136) = isogonal conjugate of X(1137)
X(1136) = perspector of ABC and 2nd Morley triangle
X(1136) = trilinear product of vertices of 2nd Morley triangle

### X(1137) = 6th MORLEY-TAYLOR-MARR CENTER

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = cos(A/3 + 4π/3)

Trilinears       cos(A/3 + π/3) : cos(B/3 + π/3) : cos(C/3 + π/3)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

See the reference at X(1134).

If you have The Geometer's Sketchpad, you can view X(1137).

X(1137) lies on this line: 16,358

X(1137) = isogonal conjugate of X(1136)
X(1137) = perspector of ABC and 2nd Morley adjunct triangle
X(1137) = trilinear product of vertices of 2nd Morley adjunct triangle

### X(1138) = ISOGONAL CONJUGATE OF X(399)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = 1/(5 cos A - 4 cos B cos C - 8 cos2A sin B sin C)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

There are only two points X such that the pedal triangle of X is similar to the cevian triangle of X. They are X(4) and X(1138). (Jean-Pierre Ehrmann, January 4, 2003)

Let A'B'C' be the anticomplementary triangle of a triangle ABC, and let EA be the line through A parallel to the Euler line. Let A" be the point of intersection, other than A, of EA and the circumcircle. Define EB and EC cyclically. The locus of a point P such that the Euler line of PBC is parallel to the Euler line of ABC is a conic ABCA'A'' having the midpoint of segment BC as center. The conics, ABCA'A'', ABCB'B'', ABCC'C'' pass through X(1138). (Francisco Javier García Capitán, April 3, 2015: ADGEOM 2458)

X(1138) lies on the following curves and lines: K001 (Neuberg Cubic), K279, K449, K490, K515, K528, K614, Q066, Q105, {1,5677}, {3,3471}, {4,2132}, {15,5624}, {16,5623}, {30,146}, {74,5670}, {186,1990}, {484,3464}, {616,5675}, {617,5674}, {1157,5667}, {1272,3260}, {3258,5627}, {3465,5685}, {3479,5679}, {3480,5678}, {3484,5684}, {5672,5673}

X(1138) = reflection of X(5627) in X(3258)
X(1138) = isogonal conjugate of X(399)
X(1138) = isotomic conjugate of X(1272)
X(1138) = X(30)-Ceva conjugate of X(5670)
X(1138) = X(523)-cevapoint of X(3258)
X(1138) = X(i)-cross conjugate of X(j) for these (i,j): (74,4), (1989,2)
X(1138) = X(i)-vertex conjugate of X(j) for these (i,j): (4,3447), (30,186)
X(1138) = trilinear pole of the line X(526)X(1637)

### X(1139) = OUTER PENTAGON POINT

Trilinears       (csc A)/(cot A + cot 2π/5) : (csc B)/(cot B + cot 2π/5) : (csc C)/(cot C + cot 2π/5)

Trilinears        csc(A + 2π/5) : csc(B + 2π/5) : csc(C + 2π/5)   (Joe Goggins, Oct. 19, 2005)

Barycentrics  1/(cot A + cot 2π/5) : 1/(cot B + cot 2π/5) : 1/(cot C + cot 2π/5)

Let A' be the outermost vertex of the regular pentagon erected outward on side BC of ABC. Define B' and C' cyclically. Then triangle A'B'C' is perspective to ABC, and the perspector is X(1139). (Steve Sigur and Antreas Hatzipolakis, #5246, 12/31/02; generalizations to n-gons, for odd n, by Milorad R. Stevanovic, #5253, 5256)

If you have The Geometer's Sketchpad, you can view Outer Pentagon Point.

X(1139) lies on this line: (6,1140)

X(1139) = X(3394)-cross conjugate of X(3397)

### X(1140) = INNER PENTAGON POINT

Trilinears       (csc A)/(cot A - cot 2π/5) : (csc B)/(cot B - cot 2π/5) : (csc C)/(cot C - cot 2π/5)

Trilinears        csc(A + 3π/5) : csc(B + 3π/5) : csc(C + 3π/5)   (Joe Goggins, Oct. 19, 2005)

Barycentrics  1/(cot A - cot 2π/5) : 1/(cot B - cot 2π/5) : 1/(cot C - cot 2π/5)

Let A' be the innermost vertex of the regular pentagon erected inward on side BC of ABC. Define B' and C' cyclically. Then triangle A'B'C' is perspective to ABC, and the perspector is X(1140). See references at X(1140).

If you have The Geometer's Sketchpad, you can view Inner Pentagon Point.

X(1140) lies on this line: (6,1139)

### X(1141) = GIBERT POINT

Trilinears    bc/[16D2 + (a2+c2-b2)(a2+b2-c2)][16D2 - 3(b2+c2-a2)2] : : , where D = area(ABC) = sec(B - C)/(1 - 4 cos2A) : sec(C - A)/(1 - 4 cos2B) : sec(A - B)/(1 - 4 cos2C) (Eric Weisstein, Nov. 17, 2005)
X(1141) = 3X(549) - 2X(6592)

X(1141) was first noted (Hyacinthos #1498, September 25, 2000) by Bernard Gibert as a point of intersection of the circumcircle and certain cubic, denoted Kn. To define Kn, note first that the Neuberg cubic is the locus of a point M such that the reflections of M in the sidelines of triangle ABC are the vertices of a triangle perspective to ABC. The locus of the perspector is the cubic Kn, and X(1141) is the point, other than A,B,C, in which Kn meets the circumcircle. Also, X(1141) is the perspector when M = X(1157).

In Jean-Pierre Ehrmann and Bernard Gibert, "Special Isocubics," downloadable from Cubics in the Triangle Plane, the point X(1141) is labeled E, barycentrics are given, and it is established that this point also lies on the line X(5)-to-X(110) [listed below as 5,49], two other cubics, and the hyperbola that passes through the points A, B, C, X(4), X(5).

Let A' be the reflection of A in line BC, and define B' and C' cyclically. Let AB be the reflection of A' in AB, and define AC, BC, BA, CA, CB cyclically. Let

A1 = BAB∩CAC, and define B1 and C1 cyclically,
A2 = BAC∩CAB, and define B2 and C2 cyclically,
A3 = BBA∩CCA, and define B3 and C3 cyclically,
A4 = BBC∩CCB, and define B4 and C4 cyclically,
A5 = BCA∩CBA, and define B5 and C5 cyclically,
A6 = BCB∩CBC, and define B6 and C6 cyclically.

Then triangle AnBnCn is perspective to ABC, for n = 1,2,3,4,5,6. The six perspectors are X(1141), X(186), X(4), X(54), X(265), X(5), respectively. (Keith Dean, #4953, 3/12/02; coordinates by Paul Yiu, #4963; summary by Dean, #4971)

X(1141) lies on the conic of {A, B, C, X(3), X(49)}, the conic of {A, B, C, X(6), X(567)}, and the conic of {A, B, C, X(70), X(253), X(254)}.

X(1141) is the antipode of X(930) on the circumcircle, and X(1141) lies on the line of the nine-point center, X(5), and its isogonal conjugate, X(54).

X(1141) lies on the circumcircle, and cubics K060, K112, K466, K467, K491, the circumconic {{A,B,C,X(4),X(5)}}, and these lines:
{2,128}, {3,252}, {4,137}, {5,49}, {20,11671}, {30,1157}, {53,112}, {55,7159}, {56,3327}, {79,109}, {94,96}, {95,99}, {101,7110}, {107,3518}, {140,11016}, {476,2070}, {549,6592}, {621,10409}, {622,10410}, {1303,5890}, {1304,5627}, {1487,10619}, {2166,2222}, {2413,5966}, {3153,10420}, {3459,12254}, {5994,11582}, {5995,11581}, {6069,11464}, {6240,6799}, {7418,9076}, {7731,9512}, {8800,12225}

X(1141) = midpoint of X(20) and X(11671)
X(1141) = reflection of X(i) in X(j) for these (i,j): (4,137), (5,12026), (930,3)
X(1141) = isogonal conjugate of X(1154)
X(1141) = isotomic conjugate of X(1273)
X(1141) = anticomplement of X(128)
X(1141) = X(231)-cross conjugate of X(2)
X(1141) = antipode of X(4) in hyperbola {{A,B,C,X(4),X(5)}}
X(1141) = the point of intersection, other than A, B, C, of the circumcircle and hyperbola {{A,B,C,X(4),X(5)}}
X(1141) = Collings transform of X(137)
X(1141) = cevapoint of X(i) and X(j) for these {i,j}: {3,539}, {13,6104}, {14,6105}, {54,1157}, {265,5961}
X(1141) = X(110)-of-Lucas-triangle (defined at X(95))
X(1141) = X(i)-cross conjugate of X(j) for these (i,j): (4,5627), (231,2), (2070,1166), (10412,476), (11063,288)
X(1141) = isoconjugate of X(j) and X(j) for these (i,j): {1,1154}, {2,2290}, {5,6149}, {31,1273}, {63,11062}, {323,1953}, {526,2617}, {662,2081}, {2179,7799}
X(1141) = trilinear pole of line {6,2623}
X(1141) = inverse of X(54) in the circle having diameter OH
X(1141) = X(110)-of-circumorthic-triangle
X(1141) = barycentric product X(i)*X(j) for these {i,j}: {54,94}, {95,1989}, {97,6344}, {264,11077}, {265,275}, {328,8882}, {930,2413}, {2166,2167}
X(1141) = barycentric quotient X(i)/X(j) for these (i,j): (i,j}: (2,1273), (6,1154), (25,11062), (31,2290), (54,323), (94,311), (95,7799), (231,128), (265,343), (275,340), (512,2081), (1989,5), (2148,6149), (2623,526), (6344,324), (8737,6116), (8738,6117), (8882,186), (11060,51), (11071,1263), (11077,3)

### X(1142) = 1st MALFATTI-RABINOWITZ POINT

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = 1 - 2(1 + cos B/2)(1 + cos C/2)/(1 + cos A/2)      (M. Iliev, 5/13/07)
Trilinears       g(A,B,C) : g(B,C,A) : g(C,A,B),
where g(A,B,C) = 1 - 4 sec2(A/4)cos2(B/4)cos2(C/4)      (M. Iliev, 5/13/07)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Let A', B', C' be the respective centers of the three Malfatti circles of ABC. Let A" be the point of intersection of lines BC' and CB', and define B" and C" cyclically. Then triangle A"B"C" is perspective to ABC, and the perspector is X(1142). (Stanley Rabinowitz, #4610, 12/29/01; coordinates by Paul Yiu, #4614, 12/30/01)

If you have The Geometer's Sketchpad, you can view X(1142) and X(1142) External.

X(1142) lies on this line: 1,179

### X(1143) = 2nd MALFATTI-RABINOWITZ POINT

Trilinears       csc A tan A/4 : csc B tan B/4 : csc C tan C/4
Barycentrics  tan A/4 : tan B/4 : tan C/4
= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)1/2[2b1/2c1/2 - (a + b + c)1/2(b + c - a)1/2]

Along each side of ABC there is a segment that is a common tangent to two of the three Malfatti circles of ABC. Let A', B', C' be the midpoints of these respective segments. Then triangle A'B'C' isperspective to ABC, and the perspector is X(1143). (Stanley Rabinowitz, #4611, 12/29/01) For coordinates, see Paul Yiu, #4615, 12/30/01, and

Milorad R. Stevanovic, "Triangle Centers Associated with the Malfatti Circles," Forum Geometricorum 3 (2003) 83-93.

Let A' be the touchpoint of the line BC and the incircle of the triangle BCI, where I = incenter of ABC. Define B' and C' cyclically. The line AA', BB', CC' concur in X(1143). (Randy Hutson, 9/23/2011)

If you have The Geometer's Sketchpad, you can view X(1143) and X(1143) External.

X(1143) lies on the cubic K200 and these lines: 8,177   174,175   558,1488

X(1143) = isotomic conjugate of X(1274)
X(1143) = X(1489)-cross conjugate of X(2)
X(1143) = {X(8),X(556)}-harmonic conjugate of X(1274)

### X(1144) = EHRMANN CONGRUENT SQUARES POINT

Trilinears       a/(a - L) : b/(b - L) : c/(c - L),
where L = L(a,b,c) is the smallest root of a2/(a - L) + b2/(b - L) + c2/(c - L) = 2D/L
where D = area(ABC).
Barycentrics  a2/(a - L) : b2/(b - L) : c2/(c - L)

Suppose P is a point inside triangle ABC. Let SA be the square inscribed in triangle PBC, having two vertices on segment BC, one on PB, and one on PC. Define SB and SC cyclically. Then X(1144) is the unique choice of P for which the three squares are congruent. The function L(a,b,c) is symmetric, homogeneous of degree 1, and satisfies 0 < L(a,b,c) < min{a,b,c}. Also, X(1144) lies on the hyperbola {A,B,C,X(1),X(6)}; indeed, X(1144) lies on the open arc from X(1) to the vertex of ABC opposite the shortest side. L(a,b,c) is the common length of the sides of the three squares. (Jean-Pierre Ehrmann, 12/16/01)

If you have The Geometer's Sketchpad, you can view X(1144).

### X(1145) = 3rd EHRMANN POINT

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = bc(b + c - 2a)[2abc - (b + c)(a2 - (b - c)2)]
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Let A',B',C' be the respective excenters of ABC, and let AB be the projection of A on A'B', let AC be the projection of A on A'C', and define BC, BA, CA, CB cyclically. The Euler lines of the three triangles A'ABAC, B'BCBA, C'CACB concur in X(1145). Also, X(1145) is X(974) of the excentral triangle. (Analogously, X(442) is X(973) of the excentral triangle; see the note at X(442).) Jean-Pierre Ehrmann (#4200, 10/24/01)

X(1145) lies on these lines:
2,1000   3,8   9,80   10,11   119,517   144,153   214,519   484,529

X(1145) = midpoint of X(8) and X(100)
X(1145) = reflection of X(i) in X(j) for these (i,j): (11,10), (1317,214), (1320,1387), (1537,119)
X(1145) = anticomplement of X(1387)
X(1145) = outer-Garcia-to-ABC similarity image of X(11)

### X(1146) = CENTER OF HYPERBOLA {{A,B,C,X(2),X(8)}}

Trilinears    bc[(b - c)(b + c - a)]2 : :
Let La, Lb, Lc be the lines through A, B, C, resp. parallel to the Soddy line. Let Ma, Mb, Mc be the reflections of BC, CA, AB in La, Lb, Lc, resp. Let A' = Mb∩Mc, and define B' and C' cyclically. Triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A"B"C" be the reflection of A'B'C' in the Soddy line. The triangle A"B"C" is homothetic to ABC, with center of homothety X(1146); see Hyacinthos #16741/16782, Sep 2008. (Randy Hutson, March 25, 2016)

X(1146) lies on the Steiner inellipse, the inconic having perspector X(2052), and these lines:
2,664   6,281   8,220   9,80   101,952   115,124   116,514   169,355   515,910   918,1086

X(1146) = midpoint of X(2) and X(1121)
X(1146) = reflection of X(1565) in X(116)
X(1146) = isogonal conjugate of X(1262)
X(1146) = isotomic conjugate of X(1275)
X(1146) = complement of X(664)
X(1146) = crosspoint of X(i) and X(j) for these (i,j): (2,522), (4,514), (9,1021)
X(1146) = crosssum of X(i) and X(j) for these (i,j): (3,101), (6,109), (56,1415), (57,1020), (1407,1461)
X(1146) = crossdifference of every pair of points on line X(109)X(692)
X(1146) = projection from Steiner circumellipse to Steiner inellipse of X(1121)
X(1146) = perspector of circumparabola centered at X(522)
X(1146) = center of circumconic that is locus of trilinear poles of lines passing through X(522)
X(1146) = X(2)-Ceva conjugate of X(522)
X(1146) = trilinear pole wrt medial triangle of line X(2)X(7)
X(1146) = barycentric square of X(522)

### X(1147) = ISOGONAL CONJUGATE OF X(847)

Trilinears       (sin 4A)/sin A : (sin 4B)/sin B : (sin 4C)/sin C
Trilinears       cos A cos 2A : cos B cos 2B : cos C cos 2C
Barycentrics   sin 4A : sin 4B : sin 4C
Barycentrics    a^4 (b^2 + c^2 - a^2) (a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2) : :

V. Thebault, "Sine-triple-angle-circle," Mathesis 65 (1956) 282-284. (Contributed by Edward Brisse, 3/4/02)

Let A'B'C' be the circumcevian triangle of X(4). Let RA be the radical axis of the circles (B', |B'C|) and (C',|C'B|), and define RB and RC cyclically. The lines RA, RB, RC concur in X(1147). For figures, see Concurrent Radical Axes.    (Antreas Hatzipolakis and Peter Moses, April 10, 2013)

Let A'B'C' be the orthic triangle. Let A'' be the orthogonal projection of A onto line B'C', and define B'' and C'' cyclically; then X(1147) is the circumcenter of A''B''C''. Let L be the reflection of line B'C' in the perpendicular bisector of segment BC, and define M and N cyclically. Let A* = M∩N, and define B* and C* cyclically; then X(1147) is the incenter of A*B*C*. (Randy Hutson, August 26, 2014)

Let A'B'C' be the Kosnita triangle. Let L be the line through A' parallel to the Euler line, and define M and N dyclically. Let L' be the reflection of L in sideline BC, and define M' and N' cyclically. The lines L',M',N' concur in X(1147). (Randy Hutson, August 26, 2014)

Let DEF be the anticevian triangle of the circumcircle, O; then X(1147) is the centroid of the quadrilateral DEFO. (Randy Hutson, August 26, 2014)

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). X(1147) = X(3)-of-A'B'C'. (Randy Hutson, October 15, 2018)

X(1147) is the insimilicenter of the circumcircle and the sine-triple-angle circle. (Randy Hutson, December 14, 2014)

X(1147) lies on these lines:
2,54   3,49   4,110   5,578   24,52   26,206   30,156   55,1069   56,215   140,141   143,576   195,568   912,960

X(1147) = midpoint of X(3) and X(155)
X(1147) = isogonal conjugate of X(847)
X(1147) = complement of X(68)
X(1147) = anticomplement of X(5449)
X(1147) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,577), (54,3)
X(1147) = crosspoint of X(i) and X(j) for these (i,j): (2,317), (371,372)
X(1147) = crosssum of X(485) and X(486)
X(1147) = perspector of the circumconic centered at X(577)
X(1147) = X(92)-isoconjugate of X(2165)
X(1147) = {X(3),X(49)}-harmonic conjugate of X(184)
X(1147) = X(4)-of-Kosnita-triangle
X(1147) = X(91)-isoconjugate of X(4)
X(1147) = perspector of 1st Hyacinth triangle and 1st Brocard triangle of 2nd Hyacinth triangle
X(1147) = Dao image of X(3)

### X(1148) = YIU-HATZIPOLAKIS POINT

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = (sec B + sec C - sec A) sec A
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Suppose LA, LB, LC are lines through a point P, respectively perpendicular to sidelines BC, CA, AB. Let AB be the point where LA meets AB, and let AC be the point where LA meets AC. Define BC, BA, CA, CB cyclically. Then X(1148) is the point P, which satisfies

|PAB| + |PAC| = |PBC| + |PBA| = |PCA| + |PCB|.

See Hyacinthos messages #4204-4206, 10/01.

X(1148) lies on these lines:
1,1075   3,653   4,65   46,243   92,942

X(1148) = X(1)-Ceva conjugate of X(4)
X(1148) = X(46)-Hirst inverse of X(243)

### X(1149) = ISOGONAL CONJUGATE OF X(1120)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = a(b2 + c2 + ba + ca - 4bc)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1149) lies on these lines:
1,2   31,999   36,106   38,392   244,517   513,663   672,1015   748,956

X(1149) = isogonal conjugate of X(1120)
X(1149) = crosspoint of X(1) and X(106)
X(1149) = crosssum of X(1) and X(519)
X(1149) = crossdifference of every pair of points on line X(9)X(649)
X(1149) = bicentric sum of PU(98)
X(1149) = PU(98)-harmonic conjugate of X(649)

### X(1150) = INTERSECTION OF LINES X(2)X(6) AND X(3)X(8)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = (b2 + c2 - a2) + bc(b + c)/a
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1150) lies on these lines:
2,6   3,8   10,750   58,964   63,321   76,799   88,330   239,980

### X(1151) = ISOGONAL CONJUGATE OF X(1131)

Trilinears    sin A + 2 cos A : :
Trilinears     sin(A + arctan(2)) : :
Trilinears     a(2 SA + S) : :
Trilinears     a(b^2 + c^2 - a^2 + S) : :
X(1151) = La/Ra + Lb/Rb + Lc/Rc + X(3)/R, where La, Lb, Lc are the centers of the Lucas circles, and Ra, Rb, Rc their radii
X(1151) = La/Ra + Lb/Rb + Lc/Rc - Li/Ri, where Li, Ri are the center and radius of the Lucas inner circle
X(1151) = (Rb+Rc)*La + (Rc+Ra)*Lb + (Ra+Rb)*Lc
(Combos by Randy Hutson, September 5, 2015)

X(1151) is the radical center of the Lucas circles, the incenter of the Lucas central triangle, and the perspector of triangle ABC and the Lucas inner triangle.

Fourteen constructions for X(1151) received from Randy Hutson, September 5, 2015:

(1)-(10): Each pair of the following triangles are perspective, and their perspector is X(1151): tangential triangle, Lucas tangents triangle, Lucas inner tangential triangle, 1st Lucas secondary tangents triangle, Lucas Brocard triangle.

(11) Let A', B', C' be the centers of the Kenmotu squares. Let A" be the reflection of A' in X(371), and define B" and C" cyclically. The triangle A"B"C" is homothetic to ABC at X(1151).

(12) Let A'B'C' be the Lucas tangents triangle. Let A" be the cevapoint of B' and C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1151).

(13) Let A'B'C' be the Lucas central triangle. Let A" be the pole, wrt the A-Lucas circle, of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1151). The point A" is also the intersection of the polars of B' and C' wrt the A-Lucas circle, and likewise for B" and C".

(14) Let A'B'C' be the Lucas central triangle. Let A" be the pole, wrt the A-Lucas circle, of line BC, and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1151).

Let A'B'C' be the triangle whose trilinear vertex matrix is the sum of the matrices for the Lucas central and Lucas tangents triangles, so that A' = 2a(SA + S) : b(2SB + S) : c(2SC + S). The lines AA', BB', CC' concur in X(1151). (Randy Hutson, September 14, 2016)

X(1151) lies on these lines:
2,489   3,6   4,590   30,485   35,1335   36,1124   140,486   141,487   488,524   615,631

X(1151) is the {X(3),X(6)}-harmonic conjugate of X(1152). For a list of other harmonic conjugates of X(1151), click Tables at the top of this page.

X(1151) = reflection of X(485) in X(8981)
X(1151) = isogonal conjugate of X(1131)
X(1151) = inverse-in-Brocard circle of X(1152)
X(1151) = X(493)-Ceva conjugate of X(6)
X(1151) = crosspoint of X(249) and X(1306)
X(1151) = perspector of Lucas(8) central triangle and circumsymmedial triangle
X(1151) = insimilicenter of circumcircle and Lucas inner circle
X(1151) = inner Soddy center (X(176)) of tangential triangle, if ABC is acute
X(1151) = X(3)-of-Lucas-tangents-triangle
X(1151) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,371,6), (3,3311,372), (371,372,3311), (372,3311,6)

### X(1152) = ISOGONAL CONJUGATE OF X(1132)

Trilinears    sin A - 2 cos A : :
Trilinears    sin(A - arctan(2)) : :
Trilinears    a(2SA - S) : :
Trilinears    a(b2 + c2 - a2 - S) : :

X(1152) is the radical center of the Lucas(-1:1) circles and the perspector of triangle ABC and the Lucas(-1,1) inner triangle.

X(1152) is the perpsector of each pairs of the following five triangles: tangential triangle, Lucas(-1) tangents triangle, Lucas(-1) inner tangential triangle, 1st Lucas(-1) secondary tangents triangle, Lucas(-1) Brocard triangle. Also, X(1152) perspector of Lucas(-8) central triangle and circumsymmedial triangle. (Randy Hutson, October 13, 2015)

Let A'B'C' be the Lucas(-1) tangents triangle. Let A" be the cevapoint of B' and C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1152). (Randy Hutson, October 13, 2015)

Let A'B'C' be the Lucas(-1) central triangle. Let A" be the pole, wrt the A-Lucas(-1) circle, of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1152). The pole A" can also be constructed as the intersection of the polars of B' and C' wrt the A-Lucas(-1) circle, and similarly for B" and C". (Randy Hutson, October 13, 2015)

Let A'B'C' be the Lucas(-1) central triangle. Let A" be the pole, wrt the A-Lucas(-1) circle, of line BC, and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1152). (Randy Hutson, October 13, 2015)

Let A'B'C' be the triangle whose trilinear vertex matrix is the sum of the matrices for the Lucas(-1) central and Lucas(-1) tangents triangles, so that A' = 2a(SA - S) : b(2SB - S) : c(2SC - S). The lines AA', BB', CC' concur in X(1152). (Randy Hutson, September 14,

2016) X(1152) lies on these lines:
2,490   3,6   4,615   30,486   35,1124   36,1335  140,485   141,488   487,524   590,631

X(1152) = isogonal conjugate of X(1132)
X(1152) = inverse-in-Brocard circle of X(1151)
X(1152) = X(494)-Ceva conjugate of X(6)
X(1152) = crosspoint of X(249) and X(1307)
X(1152) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (3,372,6), (3,3312,371), (371,372,3312), (371,3312,6)
X(1152) = exsimilicenter of circumcircle and Lucas(-1) inner circle; the insimilicenter is X(6398)
X(1152) = outer Soddy center (X(175)) of tangential triangle, if ABC is acute
X(1152) = X(1)-of-Lucas(-1)-central-triangle, if the Lucas(-1) circles are all externally tangent; otherwise, X(1152) is an excenter of the Lucas(-1) central triangle X(1152) = X(3)-of-Lucas(-1)-tangents-triangle

### X(1153) = CENTER OF THE VAN LAMOEN CIRCLE

Trilinears       g(a,b,c) : g(b,c,a) : g(c,a,b),
where g(a,b,c) = bc[13a2(b2 + c2) + 10b2c2 - 10a4 - 4b4 - 4c4]

Barycentrics  ag(a,b,c) : bg(b,c,a) : cg(c,a,b)

A triangle is divided by its three medians into 6 smaller triangles. The circumcenters of these smaller triangles are concyclic. Their circle, the Van Lamoen circle, is introduced in

Floor van Lamoen Problem 10830, American Mathematical Monthly 107 (2000) 863; solution by the editors, 109 (2002) 396-397.

Numerous messages about this circle and its center can be accessed from the Hyacinthos archive using "Floor's Monthly problem" as search words. M. Stevanovic's message (#5599, 5/28/02) gives coordinates.

If you have The Geometer's Sketchpad, you can view X(1153).

X(1153) lies on these lines: 2,187   140,524   543,549

### X(1154) = ISOGONAL CONJUGATE OF X(1141)

Trilinears    (1 - 4 cos2A)cos(B - C) : (1 - 4 cos2B)cos(C - A) : (1 - 4 cos2C)cos(A - B)
Barycentrics    a(1 - 4 cos2A)cos(B - C) : b(1 - 4 cos2B)cos(C - A) : c(1 - 4 cos2C)cos(A - B)
Barycentrics    a^2[(a^2 - b^2 - c^2)^2 - b^2c^2](b^4 + c^4 - a^2b^2 - a^2c^2 - 2b^2c^2) : :

As the isogonal conjugate of a point on the circumcircle, X(1154) lies on the line at infinity; X(1154) is, in fact, the point where the Euler line of the orthic triangle meets the line at infinity (Bernard Gibert, Hyacinthos 1498, September 25, 2000).

X(1154) lies on these (parallel) lines:
2,568   3,54   4,93   5,51   26,154   30,511   35,500   140,389   185,550   186,323   403,1112

X(1154) = isogonal conjugate of X(1141)
X(1154) = complementary conjugate of X(128)
X(1154) = X(i)-Ceva conjugate of X(j) for these (i,j): (3,1511), (4,128)
X(1154) = crosspoint of X(i) and X(j) for these (i,j): (5,1263), (323,340)
X(1154) = crosssum of X(i) and X(j) for these (i,j): (3,539), (54,1157)
X(1154) = X(30)-of-orthic-triangle
X(1154) = X(30)-of-tangential-triangle
X(1154) = infinite point of tangent to hyperbola {{A,B,C,X(4),X(15)}} at X(15) and tangent to hyperbola {{A,B,C,X(4),X(16)}} at X(16)

### X(1155) = SCHRÖDER POINT

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos B + cos C - 2 cos A;
= g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b - c)2 + a(b + c - 2a)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
= ag(a,b,c) : bg(b,c,a) : cg(c,a,b)

Let XYZ be the intouch triangle of ABC; i.e., the pedal triangle of the incenter, I. The circles AIX, BIY, CIZ concur in two points. One of them is I; the other is X(1155). This result is obtain by inversion in

Heinz Schröder, "Die Inversion und ihre Anwendung im Unterricht der Oberstufe," Der Mathematikunterricht 1 (1957) 59-80.

Each vertex of the tangential triangle of any triangle T is the inverse-in-the-circumcircle-of-T of the midpoints of the sides of T. Applying this to triangle XYZ shows that X(1155) is the inverse-in-the-incircle of the centroid of XYZ; i.e., X(1155) is X(23)-of-the-intouch-triangle. (Darij Grinberg, #6319, 1/11/03; coordinates by Jean-Pierre Ehrmann, #6320, 1/11/03)

X(1155) lies on the Darboux quintic and these lines:
1,3   10,535   11,516   37,750   44,513   47,582   63,210   88,105   89,1002   100,518   227,603   238,1054   243,653   244,902   377,667   404,960

X(1155) = midpoint of X(i) and X(j) for these (i,j): (1,3245), (36,484)
X(1155) = reflection of X(1319) in X(36)
X(1155) = isogonal conjugate of X(1156)
X(1155) = inverse-in-circumcircle of X(55)
X(1155) = inverse-in-incircle of X(354)
X(1155) = inverse-in-Bevan-circle of X(57)
X(1155) = crosspoint of X(i) and X(j) for these (i,j): (1,1156), (527,1323)
X(1155) = crosssum of X(1) and X(1155)
X(1155) = crossdifference of every pair of points on line X(1)X(650)
X(1155) = complement of X(5057)
X(1155) = anticomplement of X(5087)
X(1155) = orthogonal projection of X(1) on its trilinear polar
X(1155) = inverse-in-{circumcircle, incircle}-inverter of X(1)
X(1155) = homothetic center of intouch triangle and medial triangle of 1st circumperp triangle
X(1155) = endo-homothetic center of X(2)- and X(4)-Ehrmann triangles; the homothetic center is X(858)
X(1155) = X(468)-of-excentral-triangle

### X(1156) = ISOGONAL CONJUGATE OF X(1155)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = 1/[(b - c)2 + a(b + c - 2a)]
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1156) lies on the Darboux septic and these lines: 1,651   4,653   7,11   8,190   9,100   21,662   44,294   80,516   90,411   104,971   144,149   314,799   390,952   673,885

X(1156) = midpoint of X(144) and X(149)
X(1156) = reflection of X(i) in X(j) for these (i,j): (7,11), (100,9)
X(1156) = isogonal conjugate of X(1155)
X(1156) = antigonal conjugate of X(7)
X(1156) = symgonal of X(9)
X(1156) = trilinear pole of line X(1)X(650)

### X(1157) = INVERSE-IN-CIRCUMCIRCLE OF X(54)

Trilinears    a[(a2 - b2)2 - c2(a2 + b2)][(a2 - c2)2 - b2(c2 + a2)]U(a,b,c),
where U(a,b,c) = a6 - b6 - c6 + 3a2(b4 + c4 - a2b2 - a2c2) + b2c2(b2 + c2) - a2b2c2

Trilinears    4 cos A + cos 3A sec A sec(B - C) : :

For any point X, let XA be the reflection of X in sideline BC, and define XB and XC cyclically. Then X(1157) is the unique point X for which the lines AXA, BXB, CXC concur on the circumcircle; the point of concurrence is X(1141).

X(1157) is the tangential of X(3) on the Neuberg cubic.

Let A'B'C' be the reflection triangle. The circumcircles of AB'C', BC'A', CA'B' (i.e., the Yiu circles) concur in X(1157). (Randy Hutson, July 20, 2016)

X(1157) lies on the Neuberg cubic and these lines: 1,3483   3,54   4,3482   5,252   30,1141   74,3484   186,933   1337,1338   3065,3465

X(1157) = isogonal conjugate of X(1263)
X(1157) = inverse-in-circumcircle of X(54)
X(1157) = X(30)-Ceva conjugate of X(3484)
X(1157) = Yiu-isogonal conjugate of X(195)
X(1157) = Cundy-Parry Phi transform of X(195)

### X(1158) = CIRCUMCENTER OF EXTOUCH TRIANGLE

Trilinears    a6 - b6 - c6 + b2c2(b2 + c2) + 3a2(b4 + c4 - a2b2 - a2c2) + 2abc(a3 - b3 - c3 - abc + (a2 + bc)(b + c) - ab2 - ac2)

Trilinears    sin2B/2 cos B + sin2C/2 cos C - sin2A/2 cos A (D. Grinberg, 2/25/04)

X(1158) lies on these lines:
1,104   3,960   4,46   8,20   57,946   65,1012   117,208   165,191

X(1158) = X(318)-Ceva conjugate of X(1)

X(1158) = midpoint of X(40) and X(84)
X(1158) = X(3)-of-extouch triangle, so that X(210)X(1158) = Euler line of the extouch triangle

### X(1159) = GREENHILL POINT

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B),
where f(A,B,C) = cos A + 4 cos B + 4 cos C
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(1159) lies on these lines: 1,3   7,952

See Hyacinthos #6535 and

William Gallatly, The Modern Geometry of the Triangle, 2nd edition, Hodgson, London, 1913, page 23.

### X(1160) = CIRCUMCENTER OF OUTER GREBE TRIANGLE

Trilinears    (1 + 2 cot A + 2 cot B + 2 cot C) cos A - 2 sin A : :
Trilinears s   (1 + 2 cot ω) cos A - 2 sin A : :

See Hyacinthos #6537.

X(1160) lies on these lines: 3,6   4,1162

X(1160) = reflection of X(1161) in X(3)

### X(1161) = CIRCUMCENTER OF INNER GREBE TRIANGLE

Trilinears    (1 - 2 cot A - 2 cot B - 2 cot C) cos A + 2 sin A : :
Trilinears    (1 - 2 cot ω) cos A - 2 sin A : :

X(1161) lies on these lines: 3,6   4,1163

See Hyacinthos #6537.

X(1161) = reflection of X(1160) in X(3)

### X(1162) = OUTER GREBE-ORTHIC PERSPECTOR

Trilinears    (area + a2)(2 area + b2 + c2)/[a(b2 + c2 - a2)]

See Hyacinthos #6537.

X(1162) lies on these lines: 4,1160   428,1163

X(1162) = X(4)-Ceva conjugate of X(3127)

### X(1163) = INNER GREBE-ORTHIC PERSPECTOR

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = (area - a2)(2 area - b2 - c2)/[a(b2 + c2 - a2)]
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1163) lies on these lines: 4,1161   428,1162

See Hyacinthos #6537.

X(1163) = X(4)-Ceva conjugate of X(3128)

### X(1164) = POINT ALTAIR

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = (area + a2)(2 area - b2 - c2)/[a(b2 + c2 - a2)]
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1164) lies on this line: 468,1165

### X(1165) = POINT ALTAIS

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = (area - a2)(2 area + b2 + c2)/[a(b2 + c2 - a2)]
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1165) lies on this line: 468,1164

Saragossa Points 1166- 1208
Let A'B'C' be the cevian triangle of a point P, and let A", B", C" be the respective intersections of lines PA, PB, PC with the circumcircle of triangle ABC. Let

U = B'C"∩B"C'       V = C'A"∩C"A'       W = A'B"∩A"B'.

Lines AU, BV, CW concur in the 1st Saragossa point of P;
lines A'U, B'V, C'W concur in the 2nd Saragossa point of P;
lines A"U, B"V, C"W concur in the 3rd Saragossa point of P.

These concurrences were presented by Darij Grinberg (Hyacinthos #6531, February 14, 2003),
with coordinates as follows. Let P = x : y : z (trilinears), and abbreviate the 1st, 2nd, and 3rd Saragossa points
as Q, Q', Q", respectively; then first trilinears are

for Q:    f(a,b,c) = a/[x(bz + cy)],
for Q':     f(a,b,c) = ax[(b2z2 + c2y2)x + xyzbc + ayz(bz + cy)],
for Q":    f(a,b,c) = ax[(b2z2 + c2y2)x + ayz(bz + cy)].

The name Saragossa refers to the king who proved Ceva's theorem before Ceva did. See

J. B. Hogendijk, "Al-Mu'taman ibn Hud [bar over u], 11th century king of Saragossa and brilliant mathematician," Historia Mathematica, 22 (1995) 1-18.

The points P, Q', Q" are collinear.

The 1st Saragossa point of X(i) is X(j) for these (i,j):
(1,58)   (2,251)   (3,4)   (4,54)   (6,6)   (19,284)   (21,961)   (24,847)   (25,2)   (28,943)   (31,81)
(32,83)   (51,288)   (55,57)   (56,1)   (58,1126)   (64,3)   (84,947)   (154,1073)   (184,275)
(198,282)   (512,249)   (513,59)   (667,1016)   (939,937)   (1036,959).

The 2nd Saragossa point of X(i) is X(j) for these (i,j): (1,386)   (6,6)
The 3rd Saragossa point of X(i) is X(j) for these (i,j): (3,185)   (4,389)   (6,6)

### X(1166) = 1st SARAGOSSA POINT OF X(5)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1166) lies on these lines: 2,252   5,96   52,54

X(1166) = isogonal conjugate of X(1209)
X(1166) = isotomic conjugate of X(1225)

### X(1167) = 1st SARAGOSSA POINT OF X(34)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1167) lies on these lines: 31,937   34,40   56,580   255,269   271,936   595,998

X(1167) = isogonal conjugate of X(1210)
X(1167) = isotomic conjugate of X(1226)
X(1167) = cevapoint of X(i) and X(j) for these (i,j): (6,212), (31,198)

### X(1168) = 1st SARAGOSSA POINT OF X(36)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1168) lies on these lines: 36,88   44,517   80,519   484,759   535,903

X(1168) = isogonal conjugate of X(214)
X(1168) = isotomic conjugate of X(1227)
X(1168) = X(6)-cross conjugate of X(88)

### X(1169) = 1st SARAGOSSA POINT OF X(37)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1169) lies on these lines:
2,261   6,60   21,172   28,961   32,941   42,284   572,849   604,1178

X(1169) = isogonal conjugate of X(1211)
X(1169) = isotomic conjugate of X(1228)

X(1169) = cevapoint of X(6) and X(1333)

### X(1170) = 1st SARAGOSSA POINT OF X(41)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1170) lies on these lines:
2,220   3,955   6,279   7,218   41,57   56,1002   65,105   81,241   278,607

X(1170) = isogonal conjugate of X(1212)
X(1170) = isotomic conjugate of X(1229)
X(1170) = cevapoint of X(i) and X(j) for these (i,j): (1,218), (6,57)

### X(1171) = 1st SARAGOSSA POINT OF X(42)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1171) lies on these lines: 6,593   35,58   37,81

X(1171) = isogonal conjugate of X(1213)
X(1171) = isotomic conjugate of X(1230)
X(1171) = cevapoint of X(6) and X(58)
X(1171) = trilinear pole of line X(512)X(1326)

### X(1172) = 1st SARAGOSSA POINT OF X(48)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1172) lies on these lines:
1,19   4,6   7,27   8,29   9,33   21,270   25,941   37,943   58,84   104,112   162,1156   186,1030   286,648   406,966

X(1172) = isogonal conjugate of X(1214)
X(1172) = isotomic conjugate of X(1231)
X(1172) = X(27)-Ceva conjugate of X(28)
X(1172) = cevapoint of X(6) and X(19)
X(1172) = X(i)-cross conjugate of X(j) for these (i,j): (6,284), (33,29)
X(1172) = crosspoint of X(i) and X(j) for these (i,j): (27,29), (81,285)
X(1172) = crosssum of X(i) and X(j) for these (i,j): (37,227), (71,73)
X(1172) = crossdifference of every pair of points on line X(520)X(656)
X(1172) = trilinear pole of line X(650)X(1946)
X(1172) = polar conjugate of X(1441)

### X(1173) = 1st SARAGOSSA POINT OF X(54)

Trilinears    a/[2a^4 - 3a^2(b^2 + c^2) + (b^2 - c^2)^2] : :

Let P and Q be the intersections of line BC and the 2nd Lemoine circle. Let X = X(6). Let A' be the circumcenter of triangle PQX, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1173); c.f. X(592), where the circle is the 1st Lemoine circle. X(571): Let A'B'C' be the Kosnita triangle. Let A" be the barycentric product B'*C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(571). (Randy Hutson, December 2, 2017)

Let Ha be the foot of the A-altitude. Let Ba, Ca be the feet of perpendiculars from Ha to CA, AB, resp. Let Na be the nine-point center of HaBaCa. Define Nb and Nc cyclically. The lines ANa, BNb, CNc concur in X(1173). (Randy Hutson, December 2, 2017)

X(1173) lies on the the conics {{A, B, C, X(13), X(62)}} and {A, B, C, X(14), X(61)}} and on these lines:
3,143   51,54   69,576   74,389   265,546   575,1176

X(1173) = isogonal conjugate of X(140)
X(1173) = isotomic conjugate of X(1232)
X(1173) = cevapoint of X(i) and X(j) for these (i,j): (6,51), (61,62)
X(1173) = X(6)-cross conjugate of X(288)

X(1173) = X(5506)-of-orthic-triangle if ABC is acute

### X(1174) = 1st SARAGOSSA POINT OF X(57)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1174) lies on these lines: 41,57   55,218   101,354   226,673   284,672   661,1024

X(1174) = isogonal conjugate of X(142)
X(1174) = isotomic conjugate of X(1233)
X(1174) = cevapoint of X(6) and X(41)
X(1174) = X(513)-cross conjugate of X(101)
X(1174) = crosssum of X(354) and X(1212)

### X(1175) = 1st SARAGOSSA POINT OF X(65)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1175) lies on these lines:
3,60   21,72   28,65   35,71   58,73   69,261   110,942

X(1175) = isogonal conjugate of X(442)
X(1175) = isotomic conjugate of X(1234)
X(1175) = X(513)-cross conjugate of X(110)
X(1175) = trilinear pole of line X(647)X(2605)

### X(1176) = 1st SARAGOSSA POINT OF X(66)

Trilinears    a(b2 + c2 - a2)/(b2 + c2) : :      (M. Iliev, 5/13/07)
Trilinears    sin 2A csc(A + ω) : :

X(1176) lies on these lines:
2,66   4,83   6,22   54,511   65,82   67,110   69,184   74,827   216,248   290,308   575,1173

X(1176) = isogonal conjugate of X(427)
X(1176) = isotomic conjugate of X(1235)
X(1176) = X(83)-Ceva conjugate of X(251)
X(1176) = cevapoint of X(i) and X(j) for these (i,j): (3,184), (6,206)
X(1176) = X(39)-isoconjugate of X(92)

### X(1177) = 1st SARAGOSSA POINT OF X(67)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(b6 + c6 - a4b2 - a4c2 + 2a2b2c2 - b4c2 - b2c4)     (M. Iliev, 5/25/07)

See Angel Montesdeoca, Hyacinthos #21528, 2/12/2013

X(1177) lies on these lines:
6,1112   23,895   66,125   67,468   68,542   69,110   72,692   290,685

X(1177) = reflection of X(i) in X(j) for these (i,j): (66,125), (110,206)
X(1177) = isogonal conjugate of X(858)
X(1177) = isotomic conjugate of X(1236)
X(1177) = cevapoint of X(i) and X(j) for these (i,j): (3,101), (6,109)
X(1177) = trilinear pole of line X(32)X(647)
X(1177) = Jerabek-hyperbola antipode of X(66)
X(1177) = antigonal conjugate of X(66)

### X(1178) = 1st SARAGOSSA POINT OF X(82)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[(b + c)(a2 + bc)]      (M. Iliev, 5/13/07)

X(1178) lies on these lines:
6,694   21,238   82,662   284,893   409,1201   604,1169   741,985   759,995   765,872   869,983

X(1178) = isogonal conjugate of X(1215)
X(1178) = isotomic conjugate of X(1237)

### X(1179) = 1st SARAGOSSA POINT OF X(96)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1179) lies on these lines: 4,569   24,264   25,847   93,324

X(1179) = isogonal conjugate of X(1216)
X(1179) = isotomic conjugate of X(1238)
X(1179) = cevapoint of X(25) and X(53)
X(1179) = trilinear pole of line X(2501)X(3050)

### X(1180) = 2nd SARAGOSSA POINT OF X(2)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b4 + c4 + b2c2 + c2a2 + a2b2)      (M. Iliev, 5/13/07)

X(1180) lies on these lines: 2,39,   6,22   111,907

X(1180) = isotomic conjugate of X(1239)
X(1180) = anticomplement of X(8891)
X(1180) = crossdifference of every pair of points on line X(669)X(826)

### X(1181) = 2nd SARAGOSSA POINT OF X(3)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (1 - cos A sin B sin C) cos A

X(1181) lies on these lines:
3,49   4,6   5,1899   24,154   25,389   54,64   110,974   125,399   186,1192   1060,1069

X(1181) = reflection of X(1593) in X(578)
X(1181) = isogonal conjugate of X(1217)
X(1181) = crosssum of X(4) and X(631)

### X(1182) = 2nd SARAGOSSA POINT OF X(19)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1182) lies on these lines: 3,6   9,498   19,208

### X(1183) = 2nd SARAGOSSA POINT OF X(21)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1183) lies on these lines: 1,41   6,959   8,1036   391,958

### X(1184) = 2nd SARAGOSSA POINT OF X(25)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1184) lies on these lines: 2,6   3,1194   25,32

### X(1185) = 2nd SARAGOSSA POINT OF X(31)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1185) lies on these lines: 1,1206   2,6   31,32

X(1185) = isogonal conjugate of X(1218)
X(1185) = crossdifference of every pair of points on line X(512)X(693)

### X(1186) = 2nd SARAGOSSA POINT OF X(32)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1186) lies on these lines: 2,1207   6,76   32,184

X(1186) = crossdifference of every pair of points on line X(688)X(850)

### X(1187) = 2nd SARAGOSSA POINT OF X(37)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1187) lies on these lines: 6,60   10,37

### X(1188) = 2nd SARAGOSSA POINT OF X(41)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1188) lies on these lines: 6,279   31,32

### X(1189) = 2nd SARAGOSSA POINT OF X(43)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1189) lies on this line: 1,2

### X(1190) = 2nd SARAGOSSA POINT OF X(55)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1190) lies on these lines: 6,57   41,55   56,1202   165,218   294,940

### X(1191) = 2nd SARAGOSSA POINT OF X(56)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1191) lies on these lines:
1,6   3,595   28,957   31,56   55,1193   58,999   65,614   105,959   387,1058

X(1191) = isogonal conjugate of X(1219)
X(1191) = crosspoint of X(1016) and X(1310)

### X(1192) = 2nd SARAGOSSA POINT OF X(64)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1192) lies on these lines: 3,6   25,64

### X(1193) = 3rd SARAGOSSA POINT OF X(1)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2 + ab + ac)      (M. Iliev, 5/13/2007)
Trilinears       a(as + SA) : b(bs + SB) : c(cs + SC)      (C. Lozada, 9/07/2013)

X(1193) lies on these lines:
1,2   3,31   6,41   21,238   35,595   36,58   37,992   38,72   39,213   57,959   63,988   106,1126   171,404   222,1106   244,942   405,748   474,750   518,872   999,1066   1045,1050

X(1193) is the {X(1),X(43)}-harmonic conjugate of X(8). For a list of other harmonic conjugates of X(1193), click Tables at the top of this page.

X(1193) = midpoint of X(1) and X(3293)
X(1193) = isogonal conjugate of X(1220)
X(1193) = isotomic conjugate of X(1240)
X(1193) = crosspoint of X(i) and X(j) for these (i,j): (1,58), (6,893), (57,86)
X(1193) = crosssum of X(i) and X(j) for these (i,j): (1,10), (2,894), (9,42)
X(1193) = crossdifference of every pair of points on line X(522)X(649)

### X(1194) = 3rd SARAGOSSA POINT OF X(2)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b4 + c4 + a2b2 + a2c2)      (M. Iliev, 5/13/07)

X(1194) lies on these lines:
2,39   6,25   22,32   23,251   230,570

X(1194) = isotomic conjugate of X(1241)
X(1194) = crosspoint of X(i) and X(j) for these (i,j): (2,251), (4,308), (6,893)
X(1194) = crosssum of X(i) and X(j) for these (i,j): (2,384), (6,141)
X(1194) = crossdifference of every pair of points on line X(525)X(669)

### X(1195) = 3rd SARAGOSSA POINT OF X(19)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1195) lies on these lines: 19,208   41,71   43,165   60,283

X(1195) = crosspoint of X(19) and X(284)
X(1195) = crosssum of X(63) and X(226)

### X(1196) = 3rd SARAGOSSA POINT OF X(25)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b4 + c4 + a2b2 + a2c2 - 2b2c2)      (M. Iliev, 5/13/07)

X(1196) lies on these lines:
{2,39}, {3,1611}, {6,2056}, {22,187}, {23,1627}, {25,32}, {51,3051}, {111,251}, {115,427}, {184,1692}, {216,230}, {232,800}, {233,3815}, {394,5028}, {511,1613}, {612,1500}, {614,1015}, {682,3080}, {1084,2493}, {1368,5254}, {1495,1501}, {1570,1993}, {1915,3506}, {1995,5007}, {2092,5275}, {2275,5272}, {2276,5268}, {2670,4263}, {3094,3819}, {3231,3917}, {3796,5033}

X(1196) = complement of X(305)
X(1196) = crosspoint of X(2) and X(25)
X(1196) = crosssum of X(6) and X(96)

### X(1197) = 3rd SARAGOSSA POINT OF X(31)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(ab2 + ac2 + b2c + bc2)      (M. Iliev, 5/13/07)

X(1197) lies on these lines: 6,43   31,32   81,239   284,893

X(1197) = isogonal conjugate of X(1221)
X(1197) = crosspoint of X(i) and X(j) for these (i,j): (6,904), (31,81)
X(1197) = crosssum of X(37) and X(75)

### X(1198) = 3rd SARAGOSSA POINT OF X(43)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1198) lies on this line: 1,2

### X(1199) = 3rd SARAGOSSA POINT OF X(54)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1199) lies on these lines: 4,6   23,143   54,186   140,195   288,1157   578,1204

X(1199) = crosspoint of X(54) and X(1173)
X(1199) = crosssum of X(5) and X(140)

### X(1200) = 3rd SARAGOSSA POINT OF X(55)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1200) lies on these lines: 41,55   43,165   57,279   171,294

X(1200) is the {X(55),X(1190)}-harmonic conjugate of X(41). For a list of other harmonic conjugates, click Tables at the top of this page.

X(1200) = crosspoint of X(55) and X(57)
X(1200) = crosssum of X(7) and X(9)

### X(1201) = 3rd SARAGOSSA POINT OF X(56)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2 - 2bc + ab + ac)      (M. Iliev, 5/13/07)
Trilinears       csc2(B/2) + csc2(C/2) : csc2(C/2) + csc2(A/2) : csc2(A/2) + csc2(B/2)     (Randy Hutson, 9/23/2011)

X(1201) lies on these lines:
1,2   3,902   31,56   32,1055   36,595   38,960   58,106   65,244   73,1104   105,904   205,604   213,1015   409,1178   500,1064   748,958   651, 1476   1279,2293

X(1201) is the {X(56),X(1191)}-harmonic conjugate of X(31). For a list of other harmonic conjugates of X(1201), click Tables at the top of this page.

X(1201) = isogonal conjugate of X(1222)
X(1201) = crosspoint of X(1) and X(56)
X(1201) = crosssum of X(1) and X(8)
X(1201) = bicentric sum of PU(92)
X(1201) = PU(92)-harmonic conjugate of X(649)

### X(1202) = 3rd SARAGOSSA POINT OF X(57)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1202) lies on these lines: 6,31   57,279

X(1202) = isogonal conjugate of X(1223)
X(1202) = crosspoint of X(57) and X(1174)
X(1202) = crosssum of X(9) and X(142)

### X(1203) = 3rd SARAGOSSA POINT OF X(58)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[(a + b + c)2 - bc]      (M. Iliev, 5/13/2007)
Trilinears       as - rR : bs - rR : cs - rR     (C. Lozada, 9/07/2013)

X(1203) lies on these lines: 1,6   31,35   36,58   42,595   81,1125   580,1064   581,602

X(1203) = isogonal conjugate of X(1224)
X(1203) = crosspoint of X(58) and X(1126)
X(1203) = crosssum of X(10) and X(1125)

### X(1204) = 3rd SARAGOSSA POINT OF X(64)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1204) lies on these lines: 3,49   4,74   25,64   217,574   378,389   578,1199

X(1204) = reflection of X(1092) in X(3)
X(1204) = crosspoint of X(3) and X(64)
X(1204) = crosssum of X(i) and X(j) for these (i,j): (4,20), (489,490)

### X(1205) = 3rd SARAGOSSA POINT OF X(67)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1205) lies on this line: 74,511

X(1205) = crosspoint of X(67) and X(1177)
X(1205) = crosssum of X(23) and X(858)

### X(1206) = 3rd SARAGOSSA POINT OF X(81)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1206) lies on these lines: 6,31   81,239

### X(1207) = 3rd SARAGOSSA POINT OF X(83)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1207) lies on this line: 3,6

### X(1208) = 3rd SARAGOSSA POINT OF X(84)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1166))

X(1208) lies on these lines: 6,963   56,64

X(1208) = crosspoint of X(84) and X(947)
X(1208) = crosssum of X(40) and X(946)

### X(1209) = ISOGONAL CONJUGATE OF X(1166)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1166)

X(1209) lies on these lines:
2,54   3,161   5,51   6,17   12,942   125,128   127,129

X(1209) is the {X(17),X(18)}-harmonic conjugate of X(231). For a list of other harmonic conjugates of X(1209), click Tables at the top of this page.

X(1209) = reflection of X(52) in X(973)
X(1209) = isogonal conjugate of X(1166)
X(1209) = complement of X(54)
X(1209) = complementary conjugate of X(140)
X(1209) = crosspoint of X(2) and X(311)

### X(1210) = ISOGONAL CONJUGATE OF X(1167)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1167)

X(1210) lies on these lines:
1,2   3,950   4,57   5,226   11,65   12,354   29,58   36,411   40,497   46,516   56,515   79,1156   142,442   158,273   189,937   355,999   381,553   496,517   1089,1229

X(1210) is the {X(2),X(8)}-harmonic conjugate of X(936). For a list of other harmonic conjugates of X(1210), click Tables at the top of this page.

X(1210) = isogonal conjugate of X(1167)
X(1210) = crosspoint of X(i) and X(j) for these (i,j): (2,273), (75,189)
X(1210) = crosssum of X(i) and X(j) for these (i,j): (6,212), (31,198)

### X(1211) = ISOGONAL CONJUGATE OF X(1169)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c)(b2 + c2 + ab + ac)      (M. Iliev, 5/13/07)

X(1211) lies on these lines:
2,6   9,440   10,12   37,306   120,125   223,936   257,312   278,860   313,321   429,960   1086,1227

X(1211) = isogonal conjugate of X(1169)
X(1211) = complement of X(81)
X(1211) = crosspoint of X(i) and X(j) for these (i,j): (2,321), (76,1441)
X(1211) = crosssum of X(6) and X(1333)

### X(1212) = ISOGONAL CONJUGATE OF X(1170)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1170)

X(1212) lies on these lines:
1,6   2,85   3,169   10,1146   21,294   65,672   281,475

X(1212) = isogonal conjugate of X(1170)
X(1212) = complement of X(85)
X(1212) = X(i)-Ceva conjugate of X(j) for these (i,j) : (2,142), (142,354)
X(1212) = crosspoint of X(i) and X(j) for these (i,j): (1,277), (2,9)
X(1212) = crosssum of X(i) and X(j) for these (i,j): (1,218), (6,57)

### X(1213) = ISOGONAL CONJUGATE OF X(1171)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c)(2a + b + c)      (M. Iliev, 5/13/07)

X(1213) lies on these lines:
2,6   5,573   9,46   10,37   19,429   21,1030   115,121   140,572   190,1268   281,860   440,910   451,1172   1100,1125   1230,1269

X(1213) = isogonal conjugate of X(1171)
X(1213) = complement of X(86)
X(1213) = crosspoint of X(2) and X(10)
X(1213) = crosssum of X(6) and X(58)
X(1213) = crossdifference of every pair of points on line X(512)X(1326)

### X(1214) = ISOGONAL CONJUGATE OF X(1172)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1172)
= cot A (cos B + cos C) : cot B (cos C + cos A) : cot C (cos A + cos B) (Darij Grinberg, 4/11/03)

X(1214) lies on these lines:
1,3   2,92   7,464   9,223   10,227   34,405   37,226   63,77   72,73   216,1108   225,442   304,345   306,307   333,664   343,914

X(1214) = isogonal conjugate of X(1172)
X(1214) = complement of X(92)
X(1214) = X(i)-Ceva conjugate of X(j) for these (i,j) : (2,226), (77,73), (307,72), (348, 307)
X(1214) = cevapoint of X(i) and X(j) for these (i,j): (37,227), (71,73)
X(1214) = X(i)-cross conjugate of X(j) for these (i,j): (71,72), (201,307)
X(1214) = crosspoint of X(i) and X(j) for these (i,j): (2,63), (77,348), (1231,1441)
X(1214) = crosssum of X(i) and X(j) for these (i,j): (6,19), (33,607)

### X(1215) = ISOGONAL CONJUGATE OF X(1178)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b + c)(a2 + bc)      (M. Iliev, 5/13/07)

X(1215) lies on these lines:
1,312   2,38   10,12   37,714   42,321   43,75   171,385   190,846   964,976

X(1215) = midpoint of X(42) and X(321)
X(1215) = isogonal conjugate of X(1178)
X(1215) = complement of X(38)
X(1215) = crosssum of X(893) and X(904)

### X(1216) = ISOGONAL CONJUGATE OF X(1179)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1179)

X(1216) lies on these lines:
2,52   3,49   5,141   54,323   68,69   140,389

X(1216) = midpoint of X(1352) and X(3313)
X(1216) = reflection of X(389) in X(140)
X(1216) = isogonal conjugate of X(1179)
X(1216) = complement of X(52)
X(1216) = crosspoint of X(69) and X(97)
X(1216) = crosssum of X(25) and X(53)

### X(1217) = CRETAN PERSPECTOR

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sec A)/(1 - cos A sin B sin C)
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (tan A)/(1 - cos A sin B sin C)
Barycentrics  h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = 1/(a2 - 3b2 - 3c2)      (Peter Moses, Oct. 22, 2012)

Let A'B'C' be the medial triangle of the orthic triangle of triangle ABC. Let A" be the reflection of X(4) in A', and define B" and C" cyclically. Let Kab and Kac be the circumcenters of triangles A"BA and A"CA, respectively. Let A''' = BKac∩CKab, and define B''' and C''' cyclically. The lines AA''', BB''', CC''' concur in X(1217). (Antreas Hatzipolakis, Anopolis #39, 3/19/2002)

Using barycentric coordinates, let P = p : q : r be a point not on a sideline of triangle ABC, and let P' be the isogonal conjugate of P. Let DEF and D'E'F' be the pedal triangles of P and P', respectively. Let

X = PD'∩P'D,     Y = PE'∩P'E,     Z = PF'∩P'F.

Then the triangle XYZ is perspective to ABC (Dominik Burek, June 8, 2012) and the perspector is given by barycentric coordinates k(a,b,c) : k(b,c,a) : k(c,a,b), where k = 1/(pqr(b2 + c2 - a2) + 2p(b2r2 + c2q2))      (Peter Moses, Oct. 22, 2012). The point X(1217) results from taking P = X(3).

X(1217) lies on these lines:
2,1093   3,393   4,394   5,1073   20,97   254,378

X(1217) = isogonal conjugate of X(1181)

### X(1218) = ISOGONAL CONJUGATE OF X(1185)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1185)

X(1218) lies on these lines: 6,274   25,286   37,76   42,75   767,785

X(1218) = isogonal conjugate of X(1185)

### X(1219) = ISOGONAL CONJUGATE OF X(1191)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1191)

X(1219) lies on these lines:
1,346   2,341   8,57   28,956   72,957   75,279   81,145   105,958   278,318   518,959

X(1219) = isogonal conjugate of X(1191)
X(1219) = isotomic conjugate of X(3672)
X(1219) = trilinear pole of orthic axis of 2nd extouch triangle

### X(1220) = ISOGONAL CONJUGATE OF X(1193)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1193)

X(1220) lies on these lines:
1,312   2,12   6,8   10,58   34,92   42,1043   65,257   85,269   86,313   106,1125   292,1107   341,612   519,1126

X(1220) = isogonal conjugate of X(1193)
X(1220) = cevapoint of X(i) and X(j) for these (i,j): (1,10), (9,42)
X(1220) = crosspoint of X(1) and X(10) wrt the excentral triangle
X(1220) = trilinear pole of line X(522)X(649)
X(1220) = pole wrt polar circle of trilinear polar of X(1848)
X(1220) = X(48)-isoconjugate (polar conjugate) of X(1848)
X(1220) = intersection of tangents at X(1) and X(10) to hyperbola passing through X(1), X(10) and the excenters

### X(1221) = ISOGONAL CONJUGATE OF X(1197)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1197)

X(1221) lies on these lines: 43,75   76,192   213,274

X(1221) = isogonal conjugate of X(1197)
X(1221) = isotomic conjugate of X(1107)
X(1221) = cevapoint of X(37) and X(75)

### X(1222) = ISOGONAL CONJUGATE OF X(1201)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1201)

X(1222) lies on these lines:
1,341   6,145   8,56   10,106   34,318   58,519   75,269

X(1222) = isogonal conjugate of X(1201)
X(1222) = cevapoint of X(1) and X(8)
X(1222) = crosssum of X(1) and X(1050)

### X(1223) = ISOGONAL CONJUGATE OF X(1202)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1202)

X(1223) lies on these lines: 2,480   7,220   9,1038   75,728

X(1223) = isogonal conjugate of X(1202)
X(1223) = cevapoint of X(9) and X(142)

### X(1224) = ISOGONAL CONJUGATE OF X(1203)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1203)

X(1224) lies on these lines:
1,594   2,1089   10,81   12,57   274,313   1125,1255

X(1224) = isogonal conjugate of X(1203)

### X(1225) = ISOTOMIC CONJUGATE OF X(1166)

Trilinears       1/(a2f(a,b,c)) : 1/(b2f(b,c,a)) : 1/(c2f(c,a,b)), where f(a,b,c) is as in X(1166)

X(1225) lies on these lines: 5,311   76,95   339,1232

X(1225) = isotomic conjugate of X(1166)

### X(1226) = ISOTOMIC CONJUGATE OF X(1167)

Trilinears       1/(a2f(a,b,c)) : 1/(b2f(b,c,a)) : 1/(c2f(c,a,b)), where f(a,b,c) is as in X(1167)

X(1226) lies on these lines: 10,75   85,264   86,811   311,349   1233,1234

X(1226) = isotomic conjugate of X(1167)

### X(1227) = ISOTOMIC CONJUGATE OF X(1168)

Trilinears       1/(a2f(a,b,c)) : 1/(b2f(b,c,a)) : 1/(c2f(c,a,b)), where f(a,b,c) is as in X(1168)

X(1227) lies on these lines:
63,190   75,537   313,1232   320,758   321,545   1086,1211   1234,1269

X(1227) = isotomic conjugate of X(1168)

### X(1228) = ISOTOMIC CONJUGATE OF X(1169)

Trilinears       1/(a2f(a,b,c)) : 1/(b2f(b,c,a)) : 1/(c2f(c,a,b)), where f(a,b,c) is as in X(1169)

X(1228) lies on these lines: 2,39   12,313   312,857

X(1228) = isotomic conjugate of X(1169)

### X(1229) = ISOTOMIC CONJUGATE OF X(1170)

Trilinears       1/(a2f(a,b,c)) : 1/(b2f(b,c,a)) : 1/(c2f(c,a,b)), where f(a,b,c) is as in X(1170)

X(1229) lies on these lines: 2,37   294,314   1089,1210

X(1229) = isotomic conjugate of X(1170)
X(1229) = crosspoint of X(76) and X(312)
X(1229) = crosssum of X(32) and X(604)

### X(1230) = ISOTOMIC CONJUGATE OF X(1171)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b3c3(b + c)(2a + b + c)      (M. Iliev, 5/13/07)

X(1230) lies on these lines:
2,39   312,1234   313,321   339,440   469,1235   1213,1269

X(1230) = isotomic conjugate of X(1171)
X(1230) = crosspoint of X(76) and X(313)

### X(1231) = ISOTOMIC CONJUGATE OF X(1172)

Trilinears       1/(a2f(a,b,c)) : 1/(b2f(b,c,a)) : 1/(c2f(c,a,b)), where f(a,b,c) is as in X(1172)

X(1231) lies on these lines:
7,8   76,331   201,307   304,345   321,349   664,1043

X(1231) = isogonal conjugate of X(2204)
X(1231) = isotomic conjugate of X(1172)
X(1231) = X(76)-Ceva conjugate of X(349)
X(1231) = cevapoint of X(306) and X(307)
X(1231) = crosspoint of X(76) and X(304)

### X(1232) = ISOTOMIC CONJUGATE OF X(1173)

Trilinears       1/(a2f(a,b,c)) : 1/(b2f(b,c,a)) : 1/(c2f(c,a,b)), where f(a,b,c) is as in X(1173)

X(1232) lies on these lines: 4,69   95,252   313,1227   339,1225   1238,1273

X(1232) = isotomic conjugate of X(1173)
X(1232) = anticomplement of X(5421)

### X(1233) = ISOTOMIC CONJUGATE OF X(1174)

Trilinears       1/(a2f(a,b,c)) : 1/(b2f(b,c,a)) : 1/(c2f(c,a,b)), where f(a,b,c) is as in X(1174)

X(1233) lies on these lines: 69,674   76,85   310,333   1226,1234

X(1233) = isotomic conjugate of X(1174)

### X(1234) = ISOTOMIC CONJUGATE OF X(1175)

Trilinears       1/(a2f(a,b,c)) : 1/(b2f(b,c,a)) : 1/(c2f(c,a,b)), where f(a,b,c) is as in X(1175)

X(1234) lies on these lines: 4,69   12,313   312,1230   1226,1233   1227,1269

X(1234) = isotomic conjugate of X(1175)

### X(1235) = ISOTOMIC CONJUGATE OF X(1176)

Trilinears       1/(a2f(a,b,c)) : 1/(b2f(b,c,a)) : 1/(c2f(c,a,b)), where f(a,b,c) is as in X(1176)

X(1235) lies on these lines:
4,69   5,339   24,183   25,1239   54,276   83,648   112,384   297,324   469,1230

X(1235) = isotomic conjugate of X(1176)
X(1235) = X(264)-Ceva conjugate of X(427)
X(1235) = cevapoint of X(141) and X(427)

### X(1236) = ISOTOMIC CONJUGATE OF X(1177)

Trilinears       1/(a2f(a,b,c)) : 1/(b2f(b,c,a)) : 1/(c2f(c,a,b)), where f(a,b,c) is as in X(1177)

X(1236) lies on these lines: 4,69   325,339   826,850

X(1236) = isotomic conjugate of X(1177)

### X(1237) = ISOTOMIC CONJUGATE OF X(1178)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b3c3(b + c)(a2 + bc)      (M. Iliev, 5/13/07)

X(1237) lies on these lines: 12,313   75,1240   76,334   561,756

X(1237) = isotomic conjugate of X(1178)

### X(1238) = ISOTOMIC CONJUGATE OF X(1179)

Trilinears       1/(a2f(a,b,c)) : 1/(b2f(b,c,a)) : 1/(c2f(c,a,b)), where f(a,b,c) is as in X(1179)

X(1238) lies on these lines: 3,69   311,325   1232,1273

X(1238) = isotomic conjugate of X(1179)

### X(1239) = ISOTOMIC CONJUGATE OF X(1180)

Trilinears       1/(a2f(a,b,c)) : 1/(b2f(b,c,a)) : 1/(c2f(c,a,b)), where f(a,b,c) is as in X(1180)

X(1239) lies on these lines: 25,1235   76,251

X(1239) = isotomic conjugate of X(1180)

### X(1240) = ISOTOMIC CONJUGATE OF X(1193)

Trilinears       1/(a2f(a,b,c)) : 1/(b2f(b,c,a)) : 1/(c2f(c,a,b)), where f(a,b,c) is as in X(1193)

X(1240) lies on these lines: 7,76   75,1237   86,313   903,1269

X(1240) = isotomic conjugate of X(1193)

X(1240) = cevapoint of X(i) and X(j) for these (i,j): (10,312), (75,313)

### X(1241) = ISOTOMIC CONJUGATE OF X(1194)

Trilinears       1/(a2f(a,b,c) : 1/(b2f(b,c,a)) : 1/(c2f(c,a,b)), where f(a,b,c) is as in X(1194)

X(1241) lies on these lines: 6,305   25,76   251,384

X(1241) = isotomic conjugate of X(1194)
X(1241) = cevapoint of X(39) and X(69)

### X(1242) = ISOGONAL CONJUGATE OF X(1005)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1005)

X(1242) lies on these lines: 71,1155   72,527

X(1242) = isogonal conjugate of X(1005)

### X(1243) = ISOGONAL CONJUGATE OF X(1006)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1006)

X(1243) lies on these lines: 5,72   28,54   71,517   73,942   270,1175

X(1243) = isogonal conjugate of X(1006)

### X(1244) = ISOGONAL CONJUGATE OF X(1009)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1009)

X(1244) lies on these lines: 71,238   72,239

X(1244) = isogonal conjugate of X(1009)

### X(1245) = ISOGONAL CONJUGATE OF X(1010)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1010)

X(1245) lies on these lines:
1,69   3,31   4,1039   42,72   71,213   895,923   1176,1203

X(1245) = isogonal conjugate of X(1010)
X(1245) = crosspoint of X(1036) and X(1039)
X(1245) = crosssum of X(388) and X(1038)

### X(1246) = ISOGONAL CONJUGATE OF X(1011)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1011)

X(1246) lies on these lines:
2,71   3,86   6,27   7,73   65,273   69,310   72,75

X(1246) = isogonal conjugate of X(1011)
X(1246) = X(386)-cross conjugate of X(2)

### X(1247) = ISOGONAL CONJUGATE OF X(1046)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1046))

X(1247) lies on these lines:
1,409   10,846   65,1046   158,415

X(1247) = isogonal conjugate of X(1046)
X(1247) = X(21)-cross conjugate of X(1)

### X(1248) = ISOGONAL CONJUGATE OF X(1047)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1047)

X(1248) lies on these lines: 1,410   73,1047   255,416

X(1248) = isogonal conjugate of X(1047)
X(1248) = X(29)-cross conjugate of X(1)

### X(1249) = ISOGONAL CONJUGATE OF X(1073)

Trilinears    (tan A)(cos A - cos B cos C) : :
Barycentrics    [3a^4 - 2a^2b^2 - 2a^2c^2 - (b^2 - c^2)^2]/(b^2 + c^2 - a^2) : :

X(1249) is the perspector of triangle ABC and the tangential triangle of the circumconic centered at X(4). X(1249) is also the perspector of the medial triangle and the triangle formed by the trilinear poles of the sidelines of the orthic triangle. (Randy Hutson, 9/23/2011)

X(1249) is the unique point whose anticomplement is also its polar conjugate, namely X(253). (Randy Hutson, March 14, 2018)

Let Ea be the ellipse with B and C as foci and passing through X(4), and define Eb and Ec cyclically.
Let La be the line tangent to Ea at X(4), and define Lb and Lc cyclically.
Let A' be the trilinear pole of line La, and define B' and C' cyclically.
Then A', B', C' lie on the circumconic centered at X(1249). (Randy Hutson, March 14, 2018)

Let A'B'C' be the medial triangle. Let A" be the pole, wrt the polar circle, of line B'C', and define B", C" cyclically. Also, A"B"C" is the triangle formed by trilinear poles of sides of the orthic triangle. The lines A'A", B'B", C'C" concur in X(1249). (Randy Hutson, March 14, 2018)

X(1249) lies on hyperbola {{X(2),X(6),X(216),X(233),X(1249),X(1560),X(3162)}}. This hyperbola is a circumconic of the medial triangle, and the locus of perspectors of circumconics centered at a point on the Euler line. The hyperbola is tangent to Euler line at X(2).

X(1249) lies on the Thomson cubic and these lines:
{1,281}, {2,253}, {3,1033}, {4,6}, {9,1712}, {19,57}, {20,3172}, {25,5304}, {69,648}, {108,198}, {112,376}, {154,3079}, {186,1609}, {193,297}, {208,2270}, {216,631}, {219,1783}, {223,3352}, {232,800}, {233,3090}, {264,3618}, {273,5222}, {317,1992}, {346,1897}, {347,653}, {461,3192}, {579,1715}, {604,2202}, {608,2122}, {610,3213}, {920,1720}, {1108,1148}, {1118,2264}, {1743,1785}, {1838,2956}, {1870,3554}, {1941,2138}, {1968,5065}, {2165,3018}, {3003,3147}, {3068,3535}, {3069,3536}, {3089,5305}, {3199,5319}, {3284,3529}, {3343,3356}, {5200,5411}}

X(1249) = isogonal conjugate of X(1073)
X(1249) = X(2)-Ceva conjugate of X(4)
X(1249) = X(154)-cross conjugate of X(20)
X(1249) = X(2)-crosspoint of X(20)
X(1249) = crosssum of X(6) and X(64)
X(1249) = X(4)-Hirst inverse of X(1503)

X(1249) = complement of X(253)
X(1249) = polar conjugate of X(253)
X(1249) = perspector of ABC and antipedal triangle of X(3346)
X(1249) = perspector of pedal and anticevian triangles of X(3183)
X(1249) = perspector of ABC and medial triangle of pedal triangle of X(1498)
X(1249) = center of circumconic that is locus of trilinear poles of lines passing through X(4)
X(1249) = trilinear product X(6)*X(661)
X(1249) = perspector, wrt medial triangle, of polar circle

### X(1250) = ISOGONAL CONJUGATE OF X(1081)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1081)
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = cos(A/2) cos(A/2 + π/6)

X(1250) lies on these lines: 1,16   6,31   15,35   37,1251

X(1250) = isogonal conjugate of X(1081)
X(1250) = crosssum of X(1) and X(1277)
X(1250) = {X(6),X(55)}-harmonic conjugate of X(10638)

### X(1251) = ISOGONAL CONJUGATE OF X(1082)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1082)
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = sin(A) csc(A/2) sec(A/2 + π/6)
Trilinears    (cos A/2) csc(A/2 - π/3) : :

X(1251) lies on these lines: 1,15   7,559   13,80   37,1250   55,199

X(1251) = isogonal conjugate of X(1082)

### X(1252) = ISOGONAL CONJUGATE OF X(1086)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1086)

X(1252) is the vertex conjugate of the foci of the inellipse that is the barycentric square of the Nagel line. (Randy Hutson, October 15, 2018)

X(1252) lies on these lines:
44,765   59,672   100,650   101,649   110,813   241,1262   644,906   902,1110   1018,1021

X(1252) = isogonal conjugate of X(1086)
X(1252) = cevapoint of X(6) and X(101)
X(1252) = X(i)-cross conjugate of X(j) for these (i,j): (6,101), (31,110), (55,100)
X(1252) = trilinear pole of line X(101)X(692) (the tangent to circumcircle at X(101))
X(1252) = polar conjugate of X(2973)
X(1252) = X(92)-isoconjugate of X(3937)
X(1252) = crossdifference of every pair of points on line X(764)X(1647)

### X(1253) = ISOGONAL CONJUGATE OF X(1088)

Trilinears       a2(b + c - a)2 : b2(c + a - b)2 : c2(a + b - c)2                         = cos4A/2 : cos4B/2 : cos4C/2

X(1253) lies on these lines:
1,1170   3,1037   6,31   9,294   33,756   35,255   38,1040   40,1254   48,692   165,269   219,949   220,480   238,390   497,748

X(1253) = isogonal conjugate of X(1088)
X(1253) = X(55)-Ceva conjugate of X(41)
X(1253) = crosspoint of X(55) and X(220)
X(1253) = trilinear square of X(55)
X(1253) = crosssum of X(i) and X(j) for these (i,j): (1,1445), (7,279)

### X(1254) = ISOGONAL CONJUGATE OF X(1098)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1098)

X(1254) lies on these lines:
1,411   4,774   7,986   10,307   12,201   31,34   38,388   40,1253   42,65   46,255   56,244   57,961   200,1257   208,1096   269,1126   279,291   651,1046   750,1038

X(1254) = isogonal conjugate of X(1098)
X(1254) = crosspoint of X(65) and X(225)
X(1254) = crosssum of X(i) and X(j) for these (i,j): (1,411), (21,283)
X(1254) = trilinear square of X(65)

### X(1255) = ISOGONAL CONJUGATE OF X(1100)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1100))

Let A12B12C12 be Gemini triangle 12. Let A' be the perspector of conic {{A,B,C,B12,C12}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1255). (Randy Hutson, January 15, 2019)

X(1255) lies on these lines:
1,748   2,594   37,81   89,940   274,321   278,469   1125,1224

X(1255) = isogonal conjugate of X(1100)
X(1255) = isotomic conjugate of X(4359)
X(1255) = cevapoint of X(i) and X(j) for these (i,j): (1,37), (6,35)
X(1255) = crosssum of X(1) and X(1051)
X(1255) = trilinear pole of line X(484)X(513)

### X(1256) = ISOGONAL CONJUGATE OF X(1103)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1103))

X(1256) lies on these lines:
1,280   6,282   56,84   58,285   189,937   271,936

X(1256) = isogonal conjugate of X(1103)
X(1256) = trilinear square of X(84)

### X(1257) = ISOGONAL CONJUGATE OF X(1104)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1104))

X(1257) lies on these lines:
2,1265   8,278   28,72   57,78   69,279   105,960   200,1254   518,961

X(1257) = isogonal conjugate of X(1104)
X(1257) = cevapoint of X(i) and X(j) for these (i,j): (1,72), (37,200)

### X(1258) = ISOGONAL CONJUGATE OF X(1107)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1107)

X(1258) lies on these lines:
6,330   81,172   171,904   213,274   291,1193

X(1258) = isogonal conjugate of X(1107)
X(1258) = cevapoint of X(i) and X(j) for these (i,j): (1,213), (6,171), (37,43)

### X(1259) = ISOGONAL CONJUGATE OF X(1118)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1118))
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = (cos2A)/(1 - cos A)

X(1259) lies on these lines:
3,63   7,404   8,21   12,377   20,100   35,200   219,283   255,394   268,271   318,1013   329,411   355,1012   651,1035

X(1259) = isogonal conjugate of X(1118)
X(1259) = crosspoint of X(345) and X(1264)
X(1259) = X(92)-isoconjugate of X(608)

### X(1260) = ISOGONAL CONJUGATE OF X(1119)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1119))
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = (1 + cos A)/(1 - sec A)

X(1260) lies on these lines:
1,939   2,954   3,63   8,405   9,55   31,218   100,329   101,154   212,219   956,1006

X(1260) = isogonal conjugate of X(1119)
X(1260) = crosspoint of X(i) and X(j) for these (i,j): (346,1265), (1252,1331)
X(1260) = crosssum of X(i) and X(j) for these (i,j): (34,1435), (1398,1407)
X(1260) = X(92)-isoconjugate of X(1407)

### X(1261) = ISOGONAL CONJUGATE OF X(1122)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1122))

X(1261) lies on these lines: 8,56   31,200   41,728

X(1261) = isogonal conjugate of X(1122)
X(1261) = cevapoint of X(55) and X(200)

### X(1262) = ISOGONAL CONJUGATE OF X(1146)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1146))

X(1262) lies on these lines: 36,59   109,663   241,1252   651,905   919,934   1019,1020

X(1262) = isogonal conjugate of X(1146)
X(1262) = X(i)-cross conjugate of X(j) for these (i,j): (6,109), (48,110), (198,100)
X(1262) = cevapoint of X(i) and X(j) for these (i,j): (3,101), (6,109)
X(1262) = trilinear pole of line X(109)X(692) (the tangent to the circumcircle at X(109))
X(1262) = X(92)-isoconjugate of X(3270)

### X(1263) = ISOGONAL CONJUGATE OF X(1157)

Trilinears       1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b), where f(a,b,c) is as in X(1157))

Let A' be the reflection in BC of the A-vertex of the antipedal triangle of X(5), and define B' and C' cyclically. The circumcircles of A'BC, B'CA, and C'AB concur at X(1263). (Randy Hutson, December 2, 2017)

X(1263) lies on the Neuberg cubic and these lines: 4,195   5,128   30,1141   140,930

X(1263) = reflection of X(i) in X(j) for these (i,j): (5,137), (930,140)
X(1263) = isogonal conjugate of X(1157)
X(1263) = X(399)-of-orthic-triangle
X(1263) = tangential of X(484) on the Neuberg cubic
X(1263) = antigonal conjugate of X(5)
X(1263) = Cundy-Parry Psi transform of X(195)

### X(1264) = ISOTOMIC CONJUGATE OF X(1118)

Trilinears       1/(a2f(a,b,c)) : 1/(b2f(b,c,a)) : 1/(c2f(c,a,b)), where f(a,b,c) is as in X(1118))
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = (cot2A)/(1 - cos A)

X(1264) lies on these lines: 8,314   69,72   219,332   319,341

X(1264) = isotomic conjugate of X(1118)

### X(1265) = ISOTOMIC CONJUGATE OF X(1119)

Trilinears       1/(a2f(a,b,c)) : 1/(b2f(b,c,a)) : 1/(c2f(c,a,b)), where f(a,b,c) is as in X(1119))
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = (cos A)(csc4A/2)

X(1265) lies on these lines:
1,344   2,1257   8,210   20,190   69,72   78,345   145,1191   220,346

X(1265) = isogonal conjugate of X(1398)
X(1265) = isotomic conjugate of X(1119)

### X(1266) = ISOTOMIC CONJUGATE OF X(1120)

Trilinears       1/(a2f(a,b,c)) : 1/(b2f(b,c,a)) : 1/(c2f(c,a,b)), where f(a,b,c) is as in X(1120))

X(1266) lies on these lines:
7,145   10,75   44,545   142,192   239,527   320,519   522,693   536,1086

X(1266) = isotomic conjugate of X(1120)
X(1266) = anticomplement of X(2325)
X(1266) = crosspoint of X(75) and X(903)
X(1266) = crosssum of X(31) and X(902)

### X(1267) = ISOTOMIC CONJUGATE OF X(1123)

Trilinears       1/(a2f(a,b,c)) : 1/(b2f(b,c,a)) : 1/(c2f(c,a,b)), where f(a,b,c) is as in X(1123))
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = (csc A)(1 + csc A)
Trilinears        (bc + S)/a : (ca + S)/b : (ab + S)/c

X(1267) lies on these lines:
2,37   7,492   8,491   319,1271   320,1270

X(1267) = isotomic conjugate of X(1123)
X(1267) = {X(2),X(75)}-harmonic conjugate of X(5391)

### X(1268) = ISOTOMIC CONJUGATE OF X(1125)

Trilinears       1/(a2f(a,b,c)) : 1/(b2f(b,c,a)) : 1/(c2f(c,a,b)), where f(a,b,c) is as in X(1125))

Let A22B22C22 be Gemini triangle 22. Let A' be the perspector of conic {{A,B,C,B22,C22}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1268). (Randy Hutson, January 15, 2019)

X(1268) lies on these lines:
2,594   7,12   10,86   75,1089   190,1213   310,313   333,1171

X(1268) = isogonal conjugate of X(2308)
X(1268) = isotomic conjugate of X(1125)
X(1268) = X(523)-cross conjugate of X(190)
X(1268) = cevapoint of X(2) and X(10)
X(1268) = polar conjugate of X(1839)

### X(1269) = ISOTOMIC CONJUGATE OF X(1126)

Trilinears       1/(a2f(a,b,c)) : 1/(b2f(b,c,a)) : 1/(c2f(c,a,b)), where f(a,b,c) is as in X(1126))

X(1269) lies on these lines:
7,349   10,75   69,674   79,314   86,310   141,321   903,1240   1213,1230   1227,1234

X(1269) is the {X(75),X(76)}-harmonic conjugate of X(313). For a list of other harmonic conjugates of X(1269), click Tables at the top of this page.

X(1269) = isotomic conjugate of X(1126)
X(1269) = crosspoint of X(76) and X(310)

### X(1270) = ISOTOMIC CONJUGATE OF X(1131)

Trilinears       1/(a2f(a,b,c)) : 1/(b2f(b,c,a)) : 1/(c2f(c,a,b)), where f(a,b,c) is as in X(1131))
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = (csc A)(1 + 2 cot A)

X(1270) lies on these lines:
2,6   4,1160   8,175   20,488   76,1132   320,1267

X(1270) is the {X(2),X(69)}-harmonic conjugate of X(1271). For a list of other harmonic conjugates of X(1270), click Tables at the top of this page.

X(1270) = isotomic conjugate of X(1131)
X(1270) = anticomplement of X(3068)
X(1270) = X(1151)-cross conjugate of X(3535)
X(1270) = homothetic center of anticomplementary triangle and cross-triangle of ABC and outer Grebe triangle
X(1270) = homothetic center of outer Grebe triangle and cross-triangle of ABC and outer Grebe triangle

### X(1271) = ISOTOMIC CONJUGATE OF X(1132)

Trilinears       1/(a2f(a,b,c)) : 1/(b2f(b,c,a)) : 1/(c2f(c,a,b)), where f(a,b,c) is as in X(1132))
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = (csc A)(1 - 2 cot A)

X(1271) lies on these lines:
2,6   4,1161   8,176   20,487   76,1131   319,1267

X(1271) = isotomic conjugate of X(1132)
X(1271) = anticomplement of X(3069)
X(1271) = X(1152)-cross conjugate of X(3536)
X(1271) = homothetic center of anticomplementary triangle and cross-triangle of ABC and inner Grebe triangle
X(1271) = homothetic center of inner Grebe triangle and cross-triangle of ABC and inner Grebe triangle

### X(1272) = ISOTOMIC CONJUGATE OF X(1138)

Trilinears       1/(a2f(a,b,c)) : 1/(b2f(b,c,a)) : 1/(c2f(c,a,b)), where f(a,b,c) is as in X(1138))
Barycentrics    a^8 - 4 a^6 (b^2 + c^2) + a^4 (6 b^4 + b^2 c^2 + 6 c^4) - a^2 (4 b^6 - b^4 c^2 - b^2 c^4 + 4 c^6) + (b^2 - c^2)^2 (b^4 + 4 b^2 c^2 + c^4) : :

X(1272) lies on these lines: 2,94   69,74

X(1272) = isotomic conjugate of X(1138)
X(1272) = anticomplement of X(1989)

### X(1273) = ISOTOMIC CONJUGATE OF X(1141)

Trilinears       1/(a2f(a,b,c)) : 1/(b2f(b,c,a)) : 1/(c2f(c,a,b)), where f(a,b,c) is as in X(1141))

X(1273) lies on these lines:
2,231   5,311   54,69   93,264   186,340   325,523   1232,1238

X(1273) = isotomic conjugate of X(1141)
X(1273) = anticomplement of X(231)
X(1273) = X(128)-cross conjugate of X(2)

### X(1274) = ISOTOMIC CONJUGATE OFX(1143)

Trilinears       1/(a2f(a,b,c)) : 1/(b2f(b,c,a)) : 1/(c2f(c,a,b)), where f(a,b,c) is as in X(1143))
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = (csc A)(cot A/4)

Let I be the incenter of ABC and EA the excircle of triangle BCI that touches segment BC, and let A' be the touchpoint. Define B' and C' cyclically. The lines AA', BB', CC' concur in X(1274). (Randy Hutson, 9/23/2011)

X(1274) lies on the cubic K200 and these lines: 2,1489   8,177   174,176   557,1488

X(1274) = isotomic conjugate of X(1143)
X(1274) = {X(8),X(556)}-harmonic conjugate of X(11143)

### X(1275) = ISOTOMIC CONJUGATE OF X(1146)

Trilinears       1/(a2f(a,b,c)) : 1/(b2f(b,c,a)) : 1/(c2f(c,a,b)), where f(a,b,c) is as in X(1146))

X(1275) lies on these lines: 7,59   320,765   513,927   522,664   651,666   898,934

X(1275) = isotomic conjugate of X(1146)
X(1275) = cevapoint of X(i) and X(j) for these (i,j): (69,190), (100,220)
X(1275) = X(i)-cross conjugate of X(j) for these (i,j): (63,99), (144,190), (220,100)
X(1275) = trilinear pole of line X(100)X(658) (the tangent to Steiner circumellipse at X(664))
X(1275) = X(2)-cross conjugate of X(664)
X(1275) = barycentric square of X(664)

### X(1276) = 2nd EVANS PERSPECTOR

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sqrt(3/4) + sin(A + π/3) - sin(B + π/3) - sin(C + π/3)
= sqrt(3/4) + cos(A - π/6) - cos(B - π/6) - cos(C - π/6)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Let T be the excentral triangle, whose vertices are the A-, B-, C- excenters of triangle ABC. Let U be the equilateral triangle having segment BC as base with vertex A' on the side of BC that does not contain vertex A. Define B' and C' cyclically, and let T' be the triangle A'B'C'. Let V be the equilateral triangle having BC as base with vertex A" on the side of BC that contains A. Define B" and C" cyclically, and let T" = A"B"C". Then T and T' are perspective, and X(1276) is their perspector. (Lawrence Evans, 2/4/2003)

Evans conjectured that X(1), X(484), X(1276), X(1277) are concyclic, and he reported that Paul Yiu confirmed this conjecture and noted that the center of this circle is X(1019). (Lawrence Evans, 2/24/2003)

X(1276) is the perspector of the excentral triangle and the apices of equilateral triangles constructed outward from the sides, as in the construction of X(13). More generally, the excentral triangle is perspective to every Kiepert triangle. The locus of the perspector Kθ is the line X(4)X(9). Specifically, Kθ divides the segment from X(75) to X(9) in the ratio - ((4R+r)/s) cot θ. (Paul Yiu, 2/27/04).

X(1276) lies on the Neuberg cubic and these lines: 1,15   4,9   14,484   63,616

X(1276) = inverse-in-Bevan-circle of X(1277) (noted by Peter J. C. Moses, Sept. 8, 2004)
X(1276) = X(16)-of-excentral-triangle
X(1276) = reflection of X(1277) in X(5011)

### X(1277) = 3rd EVANS PERSPECTOR

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sqrt(3/4) - sin(A - π/3) + sin(B - π/3) + sin(C - π/3)
= - sqrt(3/4) - cos(A + π/6) + cos(B + π/6) + cos(C + π/6)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Continuing from X(1276), the triangles T and T" are perspective with perspector X(1277). (Lawrence Evans, 2/4/2003).

X(1277) is the perspector of the excentral triangle and the apices of equilateral triangles constructed inward from the sides, as in the construction of X(14); for a generalization, see X(1276). (Paul Yiu, 2/27/04).

X(1277) lies on the Neuberg cubic and these lines: 1,16   4,9   13,484   63,617

X(1277) = reflection of X(1276) in X(5011)
X(1277) = X(15)-of-excentral-triangle
X(1277) = inverse-in-Bevan-circle of X(1276) (noted by Peter J. C. Moses, Sept. 8, 2004)

### X(1278) = CONGRUENT MIDWAY-PARALLELIANS POINT

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(ab +ac - 3bc)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Paul Yiu, Introduction to the Geometry of the Triangle, 2002, Section 3.3.1, exercise 2.

X(1278) lies on these lines: 2,37   8,726   145,740   193,742

X(1278) is the {X(75),X(192)}-harmonic conjugate of X(2). For a list of harmonic conjugates, click Tables at the top of this page.

X(1278) = reflection of X(192) in X(75)
X(1278) = anticomplement of X(192)

X(1278) = complement of X(4788)
X(1278) = polar conjugate of isogonal conjugate of X(22149)

### X(1279) = MIDPOINT OF X(1) AND X(238)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b - c)2 - a(b + c - 2a)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1279) lies on these lines:
1,6   31,354   55,614   105,910   145,344   210,748   244,902   513,663   516,1086   551,752   595,942   56,1418   1201,2293

X(1279) is the {X(1),X(1001)}-harmonic conjugate of X(37). For a list of other harmonic conjugates of X(1279), click Tables at the top of this page.

X(1279) = midpoint of X(1) and X(238)
X(1279) = reflection of X(44) in X(238)
X(1279) = isogonal conjugate of X(1280)
X(1279) = crosspoint of X(i) and X(j) for these (i,j): (1,105), (927,1016)
X(1279) = crosssum of X(i) and X(j) for these (i,j): (1,518), (926,1015)
X(1279) = crossdifference of every pair of points on line X(9)X(513)
X(1279) = {X(1),X(9)}-harmonic conjugate of X(3242)
X(1279) = crossdifference of PU(56)
X(1279) = midpoint of PU(96)
X(1279) = bicentric sum of PU(96)

### X(1280) = ISOGONAL CONJUGATE OF X(1279)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[(b-c)2 - a(b + c - 2a)]

X(1280) lies on these lines:
1,644   8,277   57,100   81,643   105,518   145,279   200,244

X(1280) = reflection of X(644) in X(1)
X(1280) = isogonal conjugate of X(1279)
X(1280) = trilinear pole of line X(9)X(513)

### X(1281) = 3rd SHARYGIN POINT

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(a2 - bc)(b3 + c3 - a3 - abc)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

The Sharygin points are described in

Darij Grinberg, Sharygin Points Report, Hyacinthos #6293 (1/8/03) and #6315 (1/10/03)

The first of ten sections is an Introduction quoted, in part, here:

We will treat two remarkable triangles: the triangle bounded by the perpendicular bisectors of the internal angle bisectors of a triangle ABC, and the triangle bounded by the perpendicular bisectors of the external angle bisectors of triangle ABC. These two triangles and the triangle ABC are three perspective triangles, having a common perspectrix: the Lemoine axis of ABC. The mutual perspectors of the three triangles will be called the first, second and third Sharygin points of ABC (after a problem of Igor Sharygin - see Section 10).
The report introduces fifteen Sharygin points, of which the 1st, 2nd, 4th, and 6th are X(256), X(291), X(846),
and X(1054), respectively. X(1281) is the 3rd Sharygin point. See also Hyacinthos #6293 and #6315.

Let A' be the point where the internal angle bisector of angle CAB meets line BC, and let A" be the point where the external angle bisector of angle CAB meets line BC. Let x be the perpendicular bisector of segment AA', and let x' be the perpendicular bisector of segment AA". Define y, z, y', z' cyclically. Let D be the point where lines y and z meet, and let D' be the point where lines y' and z' meet. Define E, F, E', F' cyclically. Then

X(1281) = points of concurrence of lines DD', EE', FF'
X(846) = homothetic center of the excentral triangle and triangle DEF
X(1054) = center of similitude of the excentral triangle and triangle D'E'F'.

X(1281) lies on the Yff contact circle and these lines:
2,846   21,99   63,147   98,100   256,291   350,1284   385,740   659,804

X(1281) = isogonal conjugate of X(30648)
X(1281) = perspector of unary cofactor triangles of Gemini triangles 33 and 34
X(1281) = crossdifference of every pair of points on line X(5029)X(30654)
X(1281) = X(1)-of-1st-anti-Brocard-triangle
X(1281) = perspector of (cross-triangle of ABC and 1st Sharygin triangle) and (cross-triangle of ABC and 2nd Sharygin triangle)

### X(1282) = 5th SHARYGIN POINT

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4 + a3(b + c) - a2(2b2 + 3bc + 2c2)
+ a(b + c)(b2 + c2) - (b - c)2(b2 + bc + c2)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1282) is the perspector of the excentral triangle and the triangle D'E'F' constructed at X(1281). Coordinates were found by Jean-Pierre Ehrmann. See Hyacinthos #6293 and #6315.

Let IaIbIc be the excentral triangle. Let La be the line parallel to the Brocard axis of BCIa and passing through Ia. Define Lb and Lc cyclically. The lines La, Lb, Lc concur in X(1282). (Randy Hutson, December 2, 2017)

X(1282) lies on the Bevan circle and these lines:
1,41   8,1281   10,150   40,170   43,57   55,846   63,100   152,516   354,1051   518,910   (659,926)

X(1282) = X(98)-of-excentral-triangle

### X(1283) = 7th SHARYGIN POINT

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[a4 - (b + c)a3 + (b3 + c3)a + b2c2 - b4 - c4]
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1283) is the center of similitude of the triangles DEF and D'E'F' constructed at X(1281). Coordinates were found by Jean-Pierre Ehrmann. See Hyacinthos #6293 and #6315.

X(1283) lies on these lines: 3,1054   10,21   36,244   55,846   242,243

X(1283) = inverse-in-circumcircle of X(1054)

### X(1284) = 8th SHARYGIN POINT

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)(bc - a2)/(b + c - a)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1284) is the homothetic center of the intouch intriangle and the triangle DEF constructed at X(1281). Coordinates were found by Jean-Pierre Ehrmann. See Hyacinthos #6293 and #6315.

X(1284) lies on these lines:
1,256   7,21   37,65   57,846   350,1281   513,663

X(1284) = crosspoint of X(i) and X(j) for these (i,j): (1,98), (238,242), (1429,1447)
X(1284) = crosssum of X(i) and X(j) for these (i,j): (1,511), (291,295)
X(1284) = X(65)-Hirst inverse of X(1400)
X(1284) = bicentric sum of PU(88)
X(1284) = PU(88)-harmonic conjugate of X(3287)

### X(1285) = LEMOINE HOMOTHETIC CENTER

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(3a2 + b2 - c2)(3a2 - b2 + c2)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1285) is the homothetic center of the antipedal triangle of X(2) and the pedal triangle of X(6). (Darij Grinberg, Hyacinthos #6577, 2/21/03). See also X(3066).

Let T denote the antipedal triangle of X(2), and let T(m) denote X(m)-of-T. Then T(m) is a triangle center of the reference triangle ABC. The appearance of (m,n) in the following list means that T(m) = X(n). Note the special case m = 1285.

(2,376), (6,2), (115,99), (125,1296), (597,3), (599,20), (1285,1285), (1992,4), (2030,187), (2549,69)

Let U denote the pedal triangle of X(6), and let U(m) denote X(m)-of-U. Then U(m) is a triangle center of the reference triangle ABC. The appearance of (m,n) in the following list means that U(m) = X(n). (This list and the one just above were contributed by Peter Moses, 11/15/2007.)

(2,6), (3,597), (4,1992), (20,599), (69,2549), (99,115), (187,2030), (376,2), (1285, 1285), (1296, 125)

If you have The Geometer's Sketchpad, you can view X(1285) and X(1285) Intersection.

X(1285) lies on these lines: 2,1384   4,32   6,376   99,1992   172,1058   193,1003   497,609   631,3053   1056,1914

Collings Transforms: 1286 - 1311

If ABC is a triangle, P is a point, and A', B', C' are the reflections of A, B, C in P, then the circumcircles of triangles AB'C', A'BC' and A'B'C concur at a point Q on the circumcircle of triangle ABC. The transformation T given by Q = T(P) was described by

S. N. Collings, "Reflections on reflections 2", Mathematical Gazette 1974, page 264.

It was further discussed by Floor van Lamoen and Darij Grinberg, and coordinates were found by Barry Wolk; see Hyacinthos #4547, #4548, #6469, #6538, #6546, #6560. Paul Yiu noted that T(P) is the point, other than A, B, C, in which the circumconic centered at P meets the circumcircle of triangle ABC (#4548).

If P = x : y : z (trilinears), then Q = f(a,b,c) : f(b,c,a) : f(c,a,b), where

f(a,b,c) = 1/[bz(ax + by - cz) - cy(ax + cz - by)].

For given Q, the set of points P satisfying T(P) = Q is a conic. Examples follow:

if Q = X(74), the conic passes through X(i) for I = 125;
if Q = X(98), the conic passes through X(i) for I = 115, 868;
if Q = X(99), the conic passes through X(i) for I = 2, 39, 114, 618, 619, 629, 630, 641, 642, 1125;
if Q = X(100), the conic passes through X(i) for I = 1, 9, 10, 119, 142, 214, 442, 1145;
if Q = X(107), the conic passes through X(i) for I = 4, 133, 800, 1249;
if Q = X(110), the conic passes through X(i) for I = 5, 6, 113, 141, 206, 942, 960, 1147, 1209;
if Q = X(476), the conic passes through X(i) for I = 30.

Bernard Gibert (4/02/03) identified T(P) as the trilinear pole of the line of X(6) and the X(2)-Ceva conjugate of P. He identified the locus of P as the rectangular hyperbola that circumscribes the medial triangle and has center W given by the vector equation 4X(2)W = X(2)Q. The anticomplement of this hyperbola is the rectangular ABC-circumhyperbola whose center is the complement of Q. Thus, referring to examples given above:

if Q = X(99), the conic is the Kiepert hyperbola of the medial triangle;
if Q = X(100), the conic is the Feuerbach hyperbola of the medial triangle;
if Q = X(110), the conic is the Jerabek hyperbola of the medial triangle.

### X(1286) = COLLINGS TRANSFORM OF X(22)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1286))

X(1286) lies on the circumcircle and this line: 26,98

### X(1287) = COLLINGS TRANSFORM OF X(23)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1286))

X(1287) lies on the circumcircle and these lines: 5,842   110,826   523,827

### X(1288) = COLLINGS TRANSFORM OF X(24)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1286))

X(1288) lies on the circumcircle and this line: 70,74

### X(1289) = COLLINGS TRANSFORM OF X(25)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1286))

X(1289) lies on the circumcircle and these lines: 4,127   24,98   25,339   66,74   111,459   378,1294   403,842   648,827

### X(1290) = COLLINGS TRANSFORM OF X(36)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1286))

X(1290) lies on the circumcircle and these lines:
23,105   30,104   36,759   74,517   99,693   100,523   101,661   110,513   186,915   354,840

X(1290) = reflection of X(1325) in X(36)
X(1290) = isogonal conjugate of X(8674)
X(1290) = cevapoint of X(36) and X(513)
X(1290) = trilinear pole of line X(6)X(1718)
X(1290) = Ψ(X(6), X(1718))
X(1290) = reflection of X(100) in the Euler line
X(1290) = reflection of X(110) in line X(1)X(3)

### X(1291) = COLLINGS TRANSFORM OF X(50)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1286))
Barycentrics    a^2/((b^2 - c^2) (a^12 - 4 a^10 (b^2 + c^2) + 5 a^8 (b^4 + b^2 c^2 + c^4) + a^6 b^2 c^2 (b^2 + c^2) - a^4 (5 b^8 - b^6 c^2 + 2 b^4 c^4 - b^2 c^6 + 5 c^8) + a^2 (b^2 - c^2)^2 (4 b^6 + 3 b^4 c^2 + 3 b^2 c^4 + 4 c^6) - (b^2 - c^2)^4 (b^2 + c^2)^2)) : :

X(1291) lies on the circumcircle and these lines: 30,1141   74,1154   477,550   523,930

X(1291) = cevapoint of X(50) and X(512)

### X(1292) = COLLINGS TRANSFORM OF X(55)

Trilinears    1/((b - c) (a^2 + b^2 + c^2 - 2 a b - 2 a c)) : :

Let LA be the reflection of the line X(1)X(6) in the line BC, and define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1292). Let MA be the reflection of the line X(7)X(8) in the line BC, and define MB and MC cyclically. Let A'' = MB∩MC, and define B'' and C'' cyclically. The lines AA'', BB'', CC'' concur in X(1292). (Randy Hutson, 9/23/2011)

Let A'B'C' be the excentral triangle. The van Aubel lines of triangles A'BC, B'CA, C'AB bound a triangle perspective to ABC at X(1292). (Randy Hutson, June 27, 2018)

X(1292) lies on the circumcircle and these lines:
3,105   4,120   40,103   104,376   378,915   411,1311   517,840   601,727   906,919

X(1292) = reflection of X(i) in X(j) for these (i,j): (4,120), (105,3)
X(1292) = isogonal conjugate of X(3309)
X(1292) = trilinear pole of line X(6)X(354)
X(1292) = cevapoint of X(55) and X(513)
X(1292) = Ψ(X(6), X(354))
X(1292) = Λ(X(1), X(3309))
X(1292) = Λ(X(4), X(885))
X(1292) = X(127)-of-excentral-triangle
X(1292) = X(132)-of-hexyl-triangle
X(1292) = Cundy-Parry Phi transform of X(14267)
X(1292) = Thomson-isogonal conjugate of X(518)
X(1292) = Lucas-isogonal conjugate of X(518)

### X(1293) = COLLINGS TRANSFORM OF X(121)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/[(b - c)(3a - b - c)]      (M. Iliev, 5/13/07)

Let LA be the reflection of the line X(1)X(2) in the line BC, and define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1293). (Randy Hutson, 9/23/2011)

X(1293) lies on the circumcircle and these lines:
3,106   4,121   40,104   105,165   182,727   572,739

X(1293) = reflection of X(i) in X(j) for these (i,j): (4,121), (106,3)
X(1293) = isogonal conjugate of X(3667)
X(1293) = trilinear pole of line X(6)X(1201)
X(1293) = cevapoint of X(55) and X(649)
X(1293) = Ψ(X(6), X(1201))
X(1293) = trilinear pole wrt 1st circumperp triangle of line X(40)X(518)
X(1293) = X(107)-of-1st-circumperp-triangle
X(1293) = X(122)-of-excentral-triangle
X(1293) = X(133)-of-hexyl-triangle
X(1293) = Λ(X(649), X(4949))
X(1293) = Thomson-isogonal conjugate of X(519)
X(1293) = Lucas-isogonal conjugate of X(519)

### X(1294) = COLLINGS TRANSFORM OF X(122)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1286))

X(1294) lies on the circumcircle and these lines:
2,133   3,107   4,122   20,110   22,1302   30,1304   112,376   378,1289   550,933

X(1294) = reflection of X(i) in X(j) for these (i,j): (4,122), (107,3)
X(1294) = cevapoint of X(3) and X(30)
X(1294) = X(193)-Hirst inverse of X(297)
X(1294) = isogonal conjugate of X(6000)
X(1294) = anticomplement of X(133)
X(1294) = Λ(X(74), X(186))
X(1294) = Λ(X(5), X(2883))
X(1294) = X(134)-of-hexyl-triangle
X(1294) = eigencenter of circumanticevian triangle of X(4)
X(1294) = Cundy-Parry Phi transform of X(14249)
X(1294) = Cundy-Parry Psi transform of X(14379)
X(1294) = Thomson-isogonal conjugate of X(520)
X(1294) = Lucas-isogonal conjugate of X(520)
X(1294) = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {A,B,C,X(4),X(20)} (circumconic centered at X(122))

### X(1295) = COLLINGS TRANSFORM OF X(123)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1286))

X(1295) lies on the circumcircle and these lines:
3,108   4,123   20,100   21,107   28,1301   40,109   101,610   268,281   347,934

X(1295) = reflection of X(i) in X(j) for these (i,j): (4,123), (108,3)
X(1295) = cevapoint of X(3) and X(517)
X(1295) = isogonal conjugate of X(6001)
X(1295) = Λ(X(3), X(960))
X(1295) = Λ(X(104), X(1319))
X(1295) = X(135)-of-hexyl-triangle
X(1295) = trilinear pole of line X(6)X(2431)
X(1295) = Ψ(X(6), X(2431))
X(1295) = Cundy-Parry Phi transform of X(14257)
X(1295) = Thomson-isogonal conjugate of X(521)
X(1295) = Lucas-isogonal conjugate of X(521)

### X(1296) = COLLINGS TRANSFORM OF X(126)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1286))

Let LA be the reflection of the line X(2)X(6) in the line BC, and define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1296). (Randy Hutson, 9/23/2011)

X(1296) lies on the circumcircle and these lines:
3,111   4,126   98,376   182,729   511,843

X(1296) = reflection of X(i) in X(j) for these (i,j): (4,126), (111,3)
X(1296) = isogonal conjugate of X(1499)
X(1296) = cevapoint of X(512) and X(574)
X(1296) = trilinear pole of line X(6)X(373)
X(1296) = Ψ(X(6), X(373))
X(1296) = reflection of X(2696) in the Euler line
X(1296) = reflection of X(2709) in the Brocard axis
X(1296) = reflection of X(2746) in line X(1)X(3)
X(1296) = reflection of X(74) in line X(3)X(351)
X(1296) = X(138)-of-hexyl-triangle
X(1296) = X(74)-of-circumsymmedial-triangle
X(1296) = Λ(trilinear polar of X(1992))
X(1296) = 1st-Parry-to-ABC similarity image of X(111)
X(1296) = X(4) of 4th anti-Brocard triangle
X(1296) = perspector of 4th anti-Brocard and 1st Ehrmann triangles
X(1296) = Cundy-Parry Phi transform of X(14263)
X(1296) = Thomson-isogonal conjugate of X(524)
X(1296) = Lucas-isogonal conjugate of X(524)

### X(1297) = COLLINGS TRANSFORM OF X(127)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(b6 + c6 - 2a6 + a4b2 + a4c2 - b4c2 - b2c4)      (M. Iliev, 5/13/07)

X(1297) lies on the circumcircle, the hyperbolas {{A,B,C,X(4),X(22)}} and {{A,B,C,X(2),X(3)}}, and these lines:
2,107   3,112   4,127   20,99   22,110   23,1304   25,1073   30,935   97,933   108,1214   476,858

X(1297) = reflection of X(i) in X(j) for these (i,j): (4,127), (112,3)
X(1297) = X(232)-cross conjugate of X(2)
X(1297) = cevapoint of X(3) and X(511)
X(1297) = crosssum of X(20) and X(147)
X(1297) = isogonal conjugate of X(1503)
X(1297) = anticomplement of X(132)
X(1297) = trilinear pole of line X(6)X(520)
X(1297) = Ψ(X(6), X(520))
X(1297) = Λ(X(4), X(6))
X(1297) = Λ(X(98), X(230))
X(1297) = X(139)-of-hexyl triangle
X(1297) = inverse-in-{circumcircle, nine-point circle}-inverter of X(122)
X(1297) = Cundy-Parry Phi transform of X(8743)
X(1297) = Cundy-Parry Psi transform of X(14376)
X(1297) = Thomson-isogonal conjugate of X(525)
X(1297) = Lucas-isogonal conjugate of X(525)
X(1297) = SR(P,U), where P and U are the circumcircle intercepts of the van Aubel line

### X(1298) = COLLINGS TRANSFORM OF X(130)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1286))

X(1298) lies on the circumcircle and these lines:
2,129   3,1303   4,130   51,107   54,112   97,110   184,933

X(1298) = reflection of X(i) in X(j) for these (i,j): (4,130), (1303,3)
X(1298) = anticomplement of X(129)
X(1298) = X(107)-of-Lucas-triangle (defined at X(95))
X(1298) = X(99)-of-circumorthic-triangle
X(1298) = point of intersection, other than A, B, C, of circumcircle and hyperbola {{A,B,C,X(4),X(51)}} (circumconic centered at X(130))

### X(1299) = COLLINGS TRANSFORM OF X(135)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1286))

X(1299) lies on the circumcircle and these lines:
4,131   24,110   99,317   403,476   459,1302

X(1299) = reflection of X(4) in X(135)

### X(1300) = COLLINGS TRANSFORM OF X(136)

Trilinears    (sec A)/(1 + cos 2B + cos 2C) : :
Trilinears    (sec A)/(-1 + 2cos^2 B + 2 cos^2 C) : :

X(1300) lies on the circumcircle and these lines:
2,131   3,847   4,110   20,254   24,107   25,1302   93,930   99,264   109,225   112,393   186,476   403,1304   687,691

X(1300) = reflection of X(i) in X(j) for these (i,j): (4,136), (925,3)
X(1300) = cevapoint of X(4) and X(186)
X(1300) = X(i)-cross conjugate of X(j) for these (i,j): (30,4), (50,275)
X(1300) = Λ(X(5), X(389))
X(1300) = anticomplement of X(131)
X(1300) = inverse-in-polar-circle of X(113)
X(1300) = pole wrt polar circle of trilinear polar of X(3580)
X(1300) = X(48)-isoconjugate (polar conjugate) of X(3580)
X(1300) = eigencenter of circumanticevian triangle of X(3)
X(1300) = the point of intersection, other than A, B, C, of the circumcircle and hyperbola {{A,B,C,X(4),X(93)}} (circumconic centered at X(136))

### X(1301) = COLLINGS TRANSFORM OF X(235)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1286))

X(1301) lies on the circumcircle and these lines:
4,122   24,64   25,1073   28,1295   98,459   162,934   403,477

X(1301) = X(520)-cross conjugate of X(4)
X(1301) = cevapoint of X(i) and X(j) for these (i,j): (25,647), (235,523)
X(1301) = isogonal conjguate of X(8057)
X(1301) = trilinear pole of line X(6)X(64)
X(1301) = concurrence of reflections in sides of ABC of line X(4)X(64)
X(1301) = Ψ(X(3), X(64))
X(1301) = Ψ(X(4), X(64))
X(1301) = Ψ(X(6), X(64))
X(1301) = Ψ(X(69), X(20))
X(1301) = ?(X(6587), X(8057)) (line X(6587)X(8057) is the trilinear polar of X(20), which is also perspectrix of ABC and half-altitude triangle)
X(1301) = inverse-in-polar-circle of X(122)
X(1301) = X(63)-isoconjugate of X(6587)

### X(1302) = COLLINGS TRANSFORM OF X(381)

Trilinears    1/((cos A) (sin 2B - sin 2C) + cos B sin C - sin B cos C) : :
Barycentrics    1/((b^2 - c^2) (a^4 + b^4 + c^4 - 2 a^2 b^2 - 2 a^2 c^2 + 4 b^2 c^2)) : :

X(1302) lies on the circumcircle and these lines:
2,74   22,1294   23,477   25,1300   30,841   459,1299   648,1304

X(1302) = cevapoint of X(381) and X(523)
X(1302) = isogonal conjugate of X(8675)
X(1302) = isotomic conjugate of X(30474)
X(1302) = trilinear pole of line X(6)X(30)
X(1302) = Ψ(X(6), X(30))
X(1302) = Λ(X(6), X(647))
X(1302) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(113)

### X(1303) = COLLINGS TRANSFORM OF X(389)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1286))

X(1303) lies on the circumcircle and these lines:
2,130   3,1298   4,129   98,185

X(1303) = reflection of X(i) in X(j) for these (i,j): (4,129), (1298,3)
X(1303) = cevapoint of X(389) and X(512)
X(1303) = anticomplement of X(130)
X(1303) = X(1294)-of-Lucas-triangle (defined at X(95))
X(1303) = X(98)-of-circumorthic-triangle

### X(1304) = COLLINGS TRANSFORM OF X(403)

Trilinears    (sin A)/[(tan B - tan C)(tan B + tan C - 2 tan A)] : :

Let A', B', C' be the intersections of the Euler line and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(1304). (Randy Hutson, February 10, 2016)

X(1304) lies on the circumcircle and these lines:
{2, 2697}, {3, 2693}, {4, 477}, {21, 2694}, {23, 1297}, {25, 842}, {27, 2688}, {28, 2687}, {29, 2695}, {30, 1294}, {74, 186}, {98, 468}, {99, 3233}, {100, 5379}, {102, 2075}, {103, 2073}, {104, 2074}, {107, 523}, {110, 250}, {111, 232}, {112, 647}, {378, 841}, {403, 1300}, {476, 4240}, {648, 1302}, {691, 4230}, {877, 2855}, {925, 7471}, {933, 1624}, {935, 2394}, {1141, 5627}, {1290, 4246}, {1292, 7476}, {1295, 1325}, {1296, 7482}, {1305, 7479}, {1494, 2373}, {2071, 5897}, {2159, 2249}, {2328, 2738}, {2360, 2732}, {2370, 7478}, {2433, 2715}, {2689, 7452}, {2690, 4241}, {2691, 4238}, {2696, 4235}, {2734, 3109}, {2752, 4233}, {2758, 4248}, {2766, 7435}, {2770, 4232}, {3470, 3518}, {3565, 7468}

X(1304) = reflection of X(2693) in X(3)
X(1304) = isogonal conjugate of X(9033)
X(1304) = X(186)-cross conjugate of X(250)
X(1304) = cevapoint of X(403) and X(523)
X(1304) = crosssum of X(30) and X(402)
X(1304) = cevapoint of {403, 523}, {647, 1495}
X(1304) = X(i)-cross conjugate of X(j) for these (i,j): (186,250), (526,4), (686,2052), (3003,249),. (5502,110)
X(1304) = trilinear pole of {6, 74}
X(1304) = trilinear product X(i)*X(j) for these {i,j}: {74, 162}, {112, 2349}, {648, 2159}, {662, 8749}
X(1304) = barycentric product X(i)*X(j) for these {i,j}: {74, 648}, {99, 8749}, {112, 1494}, {162, 2349}, {250, 2394}, {811, 2159}
X(1304) = polar-circle-inverse of X(3258)
X(1304) = isogonal conjugate of X(9033)
X(1304) = trilinear pole of line X(6)X(74)
X(1304) = Ψ(X(i), X(j)) for these (i,j): (3,74), (6,74), (69,74)
X(1304) = Λ(X(i), X(j)) for these (i,j): (74,1294), (107,110), (113,133), (122,125), (1636,1637), (1494,3268)
X(1304) = reflection of X(107) in the Euler line
X(1304) = inverse-in-polar-circle of X(3258)
X(1304) = perspector of circumorthic triangle and cross-triangle of ABC and circumcevian triangle of X(186)
X(1304) = X(i)-isoconjugate of X(j) for these {i,j}: {1,9033}, {2,2631}, {30,656}, {63,1637}, {92,1636}, {162,1650}, {520,1784}, {525,2173}, {810,3260}, {1568,2616}, {1577,3284}, {2407,3708}, {2632,4240}, {6357,8611}

### X(1305) = COLLINGS TRANSFORM OF X(440)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1286))

X(1305) lies on the circumcircle and these lines:
3,917   20,103   22,675   106,347   110,664   112,653   272,759   915,1006

X(1305) = reflection of X(917) in X(3)
X(1305) = cevapoint of X(i) and X(j) for these (i,j): (3,514), (440,523), (513,1214)

### X(1306) = COLLINGS TRANSFORM OF X(639)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1286))

X(1306) lies on the circumcircle and these lines: 98,637   111,493

### X(1307) = COLLINGS TRANSFORM OF X(640)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1286))

X(1307) lies on the circumcircle and these lines: 98,638   111,494

### X(1308) = COLLINGS TRANSFORM OF X(661)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1286))

Let A'B'C' be the excentral triangle. The Fermat axes of triangles A'BC, B'CA, C'AB bound a triangle perspective to ABC at X(1308). (Randy Hutson, June 27, 2018)

X(1308) lies on the circumcircle and these lines:
1,840   36,105   100,514   101,513   103,517   104,516   106,1279   110,1019   813,876   901,1022   919,1027

X(1308) = reflection of X(101) in line X(1)X(3)
X(1308) = isogonal conjugate of X(3887)
X(1308) = cevapoint of X(513) and X(1155)

### X(1309) = COLLINGS TRANSFORM OF X(860)

Trilinears    (csc 2A)/[4(sin^2 B - sin^2 C) cos B cos C + sin^2 B cos B - sin^2 C cos C] : :
Trilinears    bc/[(b - c)(a^2 - b^2 - c^2)(a^2b + a^2c - 2abc + b^2c + bc^2 - b^3 - c^3)] : :

Let A', B', C' be the intersections of line X(4)X(8) and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X(1309). (Randy Hutson, January 29, 2018)

X(1309) lies on the circumcircle and these lines:
4,953   102,515   105,243   109,522   693,934

X(1309) = isogonal conjugate of X(8677)
X(1309) = cevapoint of X(i) and X(j) for these (i,j): (515,522), (523,860)
X(1309) = Λ(X(1459), X(1946))
X(1309) = Λ(trilinear polar of X(905))
X(1309) = trilinear pole of line X(6)X(281)
X(1309) = circumcircle-antipode of X(2734)
X(1309) = Ψ(X(3), X(8))
X(1309) = Ψ(X(6), X(281))
X(1309) = inverse-in-polar-circle of X(3259)
X(1309) = pole wrt polar circle of Sherman line (line X(3259)X(3326)) (see http://forumgeom.fau.edu/FG2012volume12/FG201220.pdf)
X(1309) = intersection of antipedal lines of X(102) and X(109)

### X(1310) = COLLINGS TRANSFORM OF X(940)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1286))

X(1310) lies on the circumcircle and these lines:
98,336   105,1036   107,811   108,664   112,662   741,1245

X(1310) = cevapoint of X(513) and X(940)

### X(1311) = COLLINGS TRANSFORM OF X(1146)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b) (see note above X(1286))

X(1311) lies on the circumcircle and these lines:
2,109   8,101   29,112   85,934   92,108   100,312   110,333   411,1292

### X(1312) = 1st MOSES INTERSECTION

Trilinears       bcf(a,b,c) : caf(b,c,a): abf(c,a,b), where f(a,b,c) is the 1st barycentric given below

Trilinears       g(A,B,C) : g(B,C,A) : g(C,A,B),
where g(A,B,C) = (J - 1)cos A + 2(J + 1)cos B cos C, where J = |OH|/R; see X(1113)

Barycentrics  f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (J - 1)a2SA + 2(J + 1) SB SC, where
SA = (b2 + c2 - a2)/2, and SB and SC are defined cyclically.
X(1312) = 3X(2) - (3 + |OH|/R)*X(5) = 3(1 + |OH|/R)*X(2) - (3 + |OH|/R)*X(3)

As a point on the Euler line, X(1312) has Shinagawa coefficients (J - 1, J + 3).

X(1312) is a point of intersection of the Euler line and the nine-point circle. Its antipode on the nine-point circle
is X(1313). Of the two points, X(1312) is the one closer to X(4). The asymptotes of the Jerabek hyperbola meet at X(125) on the nine-point circle. One of the asymptotes meets the circle again at X(1312), and the other, at X(1313). Thus, the points X(125), X(1312), X(1313) form a right triangle of which the midpoint of the hypotenuse X(1312)-to-X(1313) is X(5). (Peter J. C. Moses, 3/14/2003)

Of the points other than X(125) in which the nine-point circle meets the asymptotes of the Jerabek hyperbola, X(1312) is the one farther from X(3). (Randy Hutson, December 2, 2017)

X(1312) lies on the nine-point circle, the MacBeath inconic and this line: 2,3

X(1312) = midpoint of X(4) and X(1113)
X(1312) = reflection of X(1313) in X(5)
X(1312) = complement of X(1114)
X(1312) = X(1113)-Ceva conjugate of X(523)
X(1312) = inverse-in-polar-circle of X(1114)
X(1312) = X(1381)-of-orthic-triangle if ABC is acute
X(1312) = {X(2),X(1113)}-harmonic conjugate of X(468)
X(1312) = {X(2),X(858)}-harmonic conjugate of X(1313)
X(1312) = {X(4),X(403)}-harmonic conjugate of X(1313)
X(1312) = {X(427),X(468)}-harmonic conjugate of X(1313)

For a list of harmonic conjugates, click Tables at the top of this page.

### X(1313) = 2nd MOSES INTERSECTION

Trilinears    (J + 1)cos A + 2(J - 1)cos B cos C, where J = |OH|/R; see X(1113)
Barycentrics    (J + 1)a2SA + 2(J - 1) SB SC; cf. X(1312)
X(1313) = 3X(2) + (-3 + |OH|/R)*X(5) = 3(1 - |OH|/R)*X(2) + (-3 + |OH|/R)*X(3)

As a point on the Euler line, X(1313) has Shinagawa coefficients (J + 1, J - 3).

Of the points other than X(125) in which the nine-point circle meets the asymptotes of the Jerabek hyperbola, X(1313) is the one nearer to X(3). (Randy Hutson, December 2, 2017)

X(1313) lies on the nine-point circle, the MacBeath inconic, and this line: 2,3

X(1313) = midpoint of X(4) and X(1114)
X(1313) = reflection of X(1312) in X(5)
X(1313) = complement of X(1113)
X(1313) = X(1114)-Ceva conjugate of X(523)
X(1313) = inverse-in-polar-circle of X(1113)
X(1313) = X(1382)-of-orthic-triangle if ABC is acute
X(1313) = {X(2),X(1114)}-harmonic conjugate of X(468)

### X(1314) = 3rd MOSES INTERSECTION

Trilinears        bcf(a,b,c) : caf(b,c,a): abf(c,a,b), where f(a,b,c) is the 1st barycentric given below

Barycentrics  f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [d2 + (4r - R)R + sqrt(Q)]SBSC + (d2 + 2r2 - R2)a2SA,
where Q = 4d2R(4r - R) + [d2 - 3R2 + 4r(r + R)]2,
d = distance between X(3) and X(4),
SA = (b2 + c2 - a2)/2, and SB and SC are defined cyclically (Peter J. C. Moses, 3/2003)

As a point on the Euler line, X(1314) has Shinagawa coefficients (d2 + 2r2 - R2,4rR - 2r2 + Q1/2).

X(1314) is a point of intersection of the Euler line and the incircle. For some obtuse triangles, this point is not in the real plane (specifically, for those a,b,c such that Q < 0).

X(1314) lies on the incircle and this line: 2,3

### X(1315) = 4th MOSES INTERSECTION

Trilinears        bcf(a,b,c) : caf(b,c,a): abf(c,a,b), where f(a,b,c) is the 1st barycentric given below

Barycentrics  f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [d2 + (4r - R)R - sqrt(Q)]SBSC + (d2 + 2r2 - R2)a2SA,
where Q = 4d2R(4r - R) + [d2 - 3R2 + 4r(r + R)]2,
d = distance between X(3) and X(4),
SA = (b2 + c2 - a2)/2, and SB and SC are defined cyclically (Peter J. C. Moses, 3/2003)

As a point on the Euler line, X(1315) has Shinagawa coefficients (d2 + 2r2 - R2, 4rR - 2r2 - Q1/2).

X(1315) is a point of intersection of the Euler line and the incircle. For some obtuse triangles, this point is not in the real plane (specifically, for those a,b,c such that Q < 0).

X(1315) lies on the incircle and this line: 2,3

### X(1316) = 5th MOSES INTERSECTION

Trilinears    bc sin^2(B - C) + a^2 sin(A - B) sin(A - C) : :

Barycentrics    a8 + a4b2c2 - a6(b2 + c2) + b2c2(b2 - c2)2 : :
Barycentrics    b2c2(b2 - c2)2 + a4(a2 - b2)(a2 - c2)

As a point on the Euler line, X(1316) has Shinagawa coefficients (3(E + F)F - S2, -(E + F)2 + 3S2).

X(1316) is the point of intersection, other than X(3), of the Euler line and Brocard circle.
X(1316) is the point of intersection, other than X(6), of the Brocard circle and orthosymmedial circle.
X(1316) is the point of intersection of the Euler line and the trilinear polar of X(98). (P.J.C. Moses, 6/22/04)
X(1316) is the orthogonal projection of X(6) on the Euler line.

Let A2B2C2, A3B3C3, A4B4C4 be the 2nd, 3rd and 4th Brocard triangles, respectively. Let A' = B3B4∩C3C4, and define B' and C' cyclically. The lines A2A', B2B', C2C' concur in X(1316). (Randy Hutson, January 29, 2015)

If you have The Geometer's Sketchpad, you can view X(1316).

X(1316) lies on these lines: 2,3   6,523   250,264   262,842   338,1576

X(1316) = {X(1113),X(1114)}-harmonic conjugate of X(237)
X(1316) = {X(2),X(4)}-harmonic conjugate of X(868).

For a longer list of harmonic conjugates of X(1316), click Tables at the top of this page.

X(1316) = inverse-in-circumcircle of X(237)
X(1316) = inverse-in-orthocentroidal-circle of X(868)
X(1316) = inverse-in-2nd-Lemoine-circle of X(2451)
X(1316) = crossdifference of every pair of points on line X(511)X(647)
X(1316) = X(6)-Hirst inverse of X(523)
X(1316) = X(110) of 1st Brocard triangle
X(1316) = orthocentroidal-to-1st-Brocard similarity image of X(2)
X(1316) = inverse-in-polar-circle of X(297)
X(1316) = crosssum of S1 and S2 on the Brocard (third) cubic, K019, these being the Brocard-circle intercepts of the line X(2)X(98)

### X(1317) = REFLECTION OF X(11) IN X(1)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(c + a - b)(a + b - c)(b + c - 2a)2
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (c + a - b)(a + b - c)(b + c - 2a)2

X(1317) is the antipode of X(11) on the incircle.

X(1317) lies on these lines:
1,5   7,528   55,104   56,100   149,388   153,497   214,519

X(1317) = midpoint of X(100) and X(145)
X(1317) = reflection of X(i) in X(j) for these (i,j): (11,1), (80,1387), (1145,214)
X(1317) = isogonal conjugate of X(1318)
X(1317) = anticomplement of X(3036)

### X(1318) = ISOGONAL CONJUGATE OF X(1317)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)/(b + c - 2a)2
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1318) lies on these lines:
1,1168   36,106   88,517   679,1319

### X(1319) = BEVAN-SCHRÖDER POINT

Trilinears    (b + c - 2a)/ (b + c - a) : :
Trilinears    2 - 2 cos A - cos B - cos C
X(1319) = (R-r)*X(1) - r*X(3)

Let X'Y'Z' be the pedal triangle of the Bevan point, W = X(40); then X(1319) is the point, other than W, in which the circles AWX', BWY', CWZ' concur. (Floor van Lamoen, Hyacinthos #6321, 6352).

X(1319) lies on the Darboux quintic and these lines:
1,3   11,515   12,1125   37,604   44,1317   48,1108   59,518   73,1104   77,1122   106,1168   108,953   210,956   214,519   226,535   355,499   392,993   513,663   529,908   840,934   910,1055   961,1255

X(1319) is the {X(1),X(56)}-harmonic conjugate of X(65). For a list of other harmonic conjugates of X(1319), click Tables at the top of this page.

X(1319) = midpoint of X(1) and X(36)
X(1319) = reflection of X(1155) in X(36)
X(1319) = isogonal conjugate of X(1320)
X(1319) = inverse-in-circumcircle of X(56)
X(1319) = inverse-in-incircle of X(65)
X(1319) = cevapoint of X(902) and X(1404)
X(1319) = crosspoint of X(1) and X(104)
X(1319) = crosssum of X(1) and X(517)
X(1319) = crossdifference of every pair of points on line X(9)X(650)
X(1319) = complement of X(5176)
X(1319) = anticomplement of X(5123)
X(1319) = {X(1),X(3)}-harmonic conjugate of X(3057)
X(1319) = {X(1),X(40)}-harmonic conjugate of X(2098)
X(1319) = {X(55),X(56)}-harmonic conjugate of X(1470)
X(1319) = X(186)-of-intouch- triangle
X(1319) = X(2077)-of-Mandart-incircle- triangle
X(1319) = homothetic center of intangents triangle and reflection of extangents triangle in X(2077)
X(1319) = orthocenter of pedal triangle of X(36)
X(1319) = perspector of ABC and reflection of extangents triangle in antiorthic axis

### X(1320) = ISOGONAL CONJUGATE OF X(1319)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)/ (b + c - 2a)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1320) lies on the Darboux septic and on these lines:
1,88   2,1000   4,145   7,528   8,11   9,644   21,643   80,519   104,517   518,1156   900,1120   1022,1280

X(1320) = midpoint of X(145) and X(149)
X(1320) = reflection of X(i) in X(j) for these (i,j): (8,11), (100,1), (1145,1387)
X(1320) = isogonal conjugate of X(1319)
X(1320) = anticomplement of X(1145)
X(1320) = cevapoint of X(1) and X(517)
X(1320) = crosssum of X(902) and X(1404)
X(1320) = antigonal conjugate of X(8)
X(1320) = symgonal of X(1)
X(1320) = trilinear pole of line X(9)X(650)
X(1320) = Kirikami concurrent circles image of X(1)

### X(1321) = 1st YIU SQUARES PERSPECTOR

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (2 cos A + sin A)/(cos2A + cos A sin A)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Let BBACAC be the external square on side BC, and define CCBABA and AACBCB cyclically. Let X = BCB∩CBC and X' = BAB∩CAC, and define Y, Z and Y', Z' cyclically. The lines AX, BY, CZ concur in X(4), and the lines AX', BY', CZ' concur in X(485). The lines XX', YY', ZZ' concur in X(1321), as shown in

X(1321) lies on this line: 4,371

### X(1322) = 2nd YIU SQUARES PERSPECTOR

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (2 cos A - sin A)/(cos2A - cos A sin A)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

The construction of X(1322) is like that of X(1321), but using internally positioned squares. See the reference
at X(1321).

X(1322) lies on this line: 4,372

### X(1323) = FLETCHER POINT

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sec2A/2)(2 cos2A/2 - cos2B/2 - cos2C/2)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
= g(a,b,c) : g(b,c,a) : g(c,a,b),
where g(a,b,c) = (2a2 - b2 - c2 - ab - ac + 2bc)/(b + c - a) [P. J. C. Moses, 6/25/04]

X(1323) is the point of intersection of the line X(1)X(7) and the trilinear polar of X(7). These two lines are orthogonal.
X(1323) is named in honor of T. J. Fletcher in

Adrian Oldknow, "The Euler-Gergonne-Soddy Triangle of a Triangle," American Mathematical Monthly 103 (1996) 319-329.

X(1323) is the radical trace of the inner and outer Soddy circles.

X(1323) lies on these lines:
1,7   10,348   36,934   40,738   85,1125   106,927   165,479   241,514   519,664   1319,1355

X(1323) = midpoint of X(1155) and X(3328)
X(1323) = inverse-in-incircle of X(7)
X(1323) = X(1260)-cross conjugate of X(527)
X(1323) = crossdifference of every pair of points on line X(55)X(657)

### X(1324) = INVERSE-IN-CIRCUMCIRCLE OF X(10)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = a[a5 - b5 - c5 + a3bc - b3ca - c3ab + a2b2(b - a) + a2c2(c - a) + b2c2(b + c)]
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1324) lies on thes lines:
3,10   35,228   36,1054   58,181   98,929   759,859

X(1324) = inverse-in-circumcircle of X(10)

### X(1325) = INVERSE-IN-CIRCUMCIRCLE OF X(21)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [b4 + c4 - a4 + abc(b + c - a) - 2b2c2]/(b + c)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

As a point on the Euler line, X(1325) has Shinagawa coefficients (\$a\$(E + 4F) + 2abc, -4\$a\$(E + F) - 6abc).

X(1325) lies on thes lines:
1,229   2,3   36,759   60,65   104,476   105,691   110,517   1295,1304

X(1325) = reflection of X(1290) in X(36)
X(1325) = inverse-in-circumcircle of X(21)

### X(1326) = INVERSE-IN-CIRCUMCIRCLE OF X(58)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2 - a2 + bc - ab - ac)/(b + c)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1326) lies on thes lines:
3,6   10,261   35,849   42,593   99,726   106,691   110,902   238,662   249,1101   727,805

X(1326) = inverse-in-circumcircle of X(58)
X(1326) = X(741)-Ceva conjugate of X(58)
X(1326) = crossdifference of every pair of points on line X(523)X(1213)
X(1326) = X(6)-Hirst inverse of X(58)

### X(1327) = ARCTAN(3) KIEPERT POINT

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(sin A + 3 cos A)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Let BBACAC be the external square on side BC, and define CCBABA and AACBCB cyclically. The lines ABBA, BCCB, CAAC form a triangle perspective to triangle ABC, and the perspector is X(1327).

If you have The Geometer's Sketchpad, you can view X(1327).

X(1327) lies on thes lines: 6,1328   30,485   371,1131   381,486   547,1152

### X(1328) = ARCTAN(-3) KIEPERT POINT

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(sin A - 3 cos A)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

The construction of X(1328) is like that of X(1327), but using internally positioned squares. See the reference
at X(1321).

X(1328) lies on thes lines: 6,1327   30,486   372,1132   381,485   547,1151

### X(1329) = COMPLEMENTARY CONJUGATE OF X(1)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[b2/(a - b + c) + c2/(a + b - c)]
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1329) lies on thes lines:
2,12   3,119   5,10   8,11   9,46   65,908   121,124   140,993   355,997   405,498   495,1125   496,519   499,956   518,1210   975,998

X(1329) = isogonal conjugate of X(3450)
X(1329) = complement of X(56)
X(1329) = complementary conjugate of X(1)
X(1329) = crosssum of X(6) and X(1397)

### X(1330) = ANTICOMPLEMENTARY CONJUGATE OF X(1)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[b2/(a + c) + c2/(a + b) - a2/(b + c)]
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1330) lies on thes lines:
2,58   4,69   8,79   10,894   30,1043   193,387   320,942   333,442   1010,1211

X(1330) = reflection of X(1046) in X(10)
X(1330) = isogonal conjugate of X(3437)
X(1330) = anticomplement of X(58)
X(1330) = anticomplementary conjugate of X(1)
X(1330) = X(313)-Ceva conjugate of X(2)

### X(1331) = ORTHOCORRESPONDENT OF X(101)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2 - a2)/(b - c)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

For the definition of orthocorrespondent, see the notes just before X(1992).

X(1331) lies on thes lines:
63,212   71,895   72,283   78,255   100,109   101,110   145,595   162,190   228,295   287,293   394,1260   677,1252   901,1293

X(1331) = X(i)-Ceva conjugate of X(j) for these (i,j): (190,101), (643,100)
X(1331) = X(i)-cross conjugate of X(j) for these (i,j): (521,283), (652,63), (1260,1252)
X(1331) = cevapoint of X(i) and X(j) for these (i,j): (3,1459), (72,521), (212,652)

### X(1332) = ORTHOCORRESPONDENT OF X(100)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 + c2 - a2)/(b - c)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

For the definition of orthocorrespondent, see the notes just before X(1992).

X(1332) lies on these lines:
6,344   69,219   72,895   100,110   101,1310   190,644   287,336   345,394   645,648   646,1016   677,765   815,932

X(1332) = X(i)-Ceva conjugate of X(j) for these (i,j): (645, 190), (668,100), (1016,345)
X(1332) = X(i)-cross conjugate of X(j) for these (i,j): (521,69), (905,63), (906,100)
X(1332) = cevapoint of X(i) and X(j) for these (i,j): (63,905), (71,1459), (219,521)

### X(1333) = POINT ALULA

Trilinears    a2/(b + c) : :

X(1333) lies on these lines: {2,18744}, {3,6}, {9,609}, {19,2217}, {21,37}, {27,3772}, {28,1104}, {31,48}, {36,16470}, {44,1778}, {45,4877}, {47,22134}, {53,7511}, {56,608}, {65,1950}, {81,593}, {86,3662}, {99,713}, {100,21858}, {104,112}, {110,739}, {141,5337}, {163,9456}, {213,2174}, {219,1780}, {261,27164}, {272,379}, {286,16732}, {292,4628}, {314,19623}, {321,17587}, {333,4386}, {346,17539}, {385,3770}, {536,16046}, {594,5291}, {595,16685}, {603,604}, {662,18274}, {692,1911}, {741,825}, {759,5341}, {849,1437}, {859,2178}, {872,18266}, {896,3958}, {910,5324}, {940,16368}, {963,2332}, {992,27660}, {1010,4426}, {1014,1418}, {1043,17299}, {1086,17189}, {1171,28625}, {1178,3863}, {1191,2255}, {1193,22054}, {1213,5277}, {1396,1427}, {1399,1409}, {1400,1415}, {1407,1412}, {1436,2299}, {1449,7031}, {1575,13588}, {1627,5276}, {1766,15952}, {1790,2221}, {1801,2327}, {1811,5546}, {1817,3752}, {1901,13442}, {1931,4469}, {1951,2264}, {1977,17961}, {2160,3125}, {2162,21769}, {2197,5172}, {2241,4658}, {2242,4653}, {2256,2328}, {2262,18191}, {2268,10457}, {2269,22361}, {2276,4184}, {2277,4225}, {2311,3862}, {2345,11115}, {2361,22074}, {2423,7252}, {2699,2715}, {2991,4558}, {3290,4228}, {3330,28381}, {3454,24935}, {3739,26643}, {3998,4641}, {4000,14953}, {4234,17281}, {4567,5381}, {4749,23868}, {5358,16583}, {5563,16488}, {5839,16704}, {7735,26118}, {8069,22132}, {8822,17276}, {9341,19297}, {10315,21866}, {10789,18194}, {11102,16974}, {11320,18147}, {12194,18170}, {12610,17197}, {13728,17398}, {16047,17263}, {16050,17279}, {16054,17278}, {16350,19701}, {16580,17171}, {16706,21997}, {16917,25457}, {16973,18206}, {17052,24890}, {17053,21773}, {17187,21764}, {18697,24335}, {19308,24530}, {19785,26830}, {21353,21833}

X(1333) = isogonal conjugate of X(321)
X(1333) = X(i)-Ceva conjugate of X(j) for these (i,j): (593,58), (1169,6), (1175,184)
X(1333) = X(31)-cross conjugate of X(58)
X(1333) = cevapoint of X(31) and X(32)
X(1333) = crosspoint of X(i) and X(j) for these (i,j): (28,81), (58,1412), (593,849)
X(1333) = crosssum of X(i) and X(j) for (i,j) = (37,72), (594,1089)
X(1333) = trilinear pole of line X(667)X(838)
X(1333) = X(92)-isoconjugate of X(72)
X(1333) = X(100)-isoconjugate of X(1577)
X(1333) = perspector of ABC and unary cofactor triangle of Gemini triangle 21
X(1333) = perspector of ABC and unary cofactor triangle of Gemini triangle 27

### X(1334) = POINT ALYA

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)(b + c - a)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1334) lies on thes lines:
1,672   3,1055   8,9   10,1018   21,644   32,902   35,101   37,65   39,1201   41,55   42,213   607,1253   756,862

X(1334) = X(i)-Ceva conjugate of X(j) for these (i,j): (9, 210), (37,42), (644,663)
X(1334) = crosspoint of X(i) and X(j) for these (i,j): (9,55), (37,210)
X(1334) = crosssum of X(i) and X(j) for (i,j) = (7,57), (81,1014)
X(1334) = crossdifference of every pair of points on line X(1443)X(1447)>br> X(1334) = trilinear pole of line X(1443)X(1447)

### X(1335) = {X(1),X(6)}-HARMONIC CONJUGATE OF X(1124)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1 - sin A
Trilinears       a(S - bc) : b(S - ca) : c(S - ab)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(1335) lies on thes lines:
1,6   11,486   12,485   35,1151   36,1152   42,493   55,371   56,372   81,1123   175,651   255,606   498,590   499,615

X(1335) = isogonal conjugate of X(1336)
X(1335) = exsimilicenter of incircle and 2nd Lemoine circle; the insimilicenter is X(1124)

### X(1336) = ISOGONAL CONJUGATE OF X(1335)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(1 - sin A)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

The parabola with focus A and directrix BC meets line AB in two points; let AB be the one further from B, and define AC similarly. Let LA be the line ABAC, and define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1336). (Randy Hutson, 9/23/2011)

In the configuration for the Paasche point X(1123), there are 4 circles tangent to the circle with diameter BC and also tangent to the lines AB and AC. Of the 4 circles, there are two pairs, one having X(1123) as perspector, and the other having X(1336). (Peter Moses, 21 January 2013)

If you have The Geometer's Sketchpad, you can view X(1336)

X(1336) lies on thes lines: 1,3068   2,585   4,2362   37,158   57,481   81,1124   274,1267   498,3300   499,3302   9203069

X(1336) = isogonal conjugate of X(1335)
X(1336) = isotomic conjugate of X(5391)

### X(1337) = 1st WERNAU POINT

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[x1 + 2Dx2/sqrt(3)], where
D = area(ABC),
x1 = 4a2U3 + (a2 + V)(a2 + W)U2 - 7V2W2 - 5a2UVW,
x2 = 3U3 + 7a2U2 - 6a2VW - 5 UVW,
U = (b2 + c2 - a2)/2, V = (c2 + a2 - b2)/2, W = (a2 + b2 - c2)/2

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2(2DU - 31/2VW)/(4D + 31/2a2),
D, U, V, W as above; see Hyacinthos #8874

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2(2DU - 31/2VW)/(4D + 31/2a2),

Let A'BC be the external equilateral triangle on side BC, and define CB'A and AC'B cyclically. Let (AB'C') be the circle passing through the points A, B', C', and define (BC'A') and (CA'B') cyclically. The three circles concur in X(1337). Wernau is a town near Stuttgart, the site of a mathematics olympiad seminar in Spring 2003. (Darij Grinberg; Hyacinthos, April, 2003: #6874, 6881, 6882; coordinates by Jean-Pierre Ehrmann)

X(1337) is the tangential of X(15) on the Neuberg cubic.

X(1337) lies on the Neuberg cubic and these lines: 4,616   15,2981   399,3441   1157,1338

X(1337) = antigonal conjugate of X(662)
X(1337) = symgonal of X(16)

### X(1338) = 2nd WERNAU POINT

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[x1 - 2Dx2/sqrt(3)], where
D = area(ABC), x1, x2 are as at X(1337).

Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a2(2DU + 31/2VW)/(4D - 31/2a2),
D, U, V, W as at X(1337); see Hyacinthos #8874

Let A'BC be the internal equilateral triangle on side BC, and define CB'A and AC'B cyclically. Let (AB'C') be the circle passing through the points A, B', C', and define (BC'A') and (CA'B') cyclically. The three circles concur in X(1338). For details, see X(1337).

X(1338) is the tangential of X(16) on the Neuberg cubic.

X(1338) lies on the Neuberg cubic and these lines: 4,617   16,3458   399,3440   1157,1337

X(1338) = antigonal conjugate of X(661)
X(1338) = symgonal of X(15)

### X(1339) = NAGEL-SCHRÖDER POINT

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b+c-2a)[4bc(b+c-a) - (a+b+c)(b2 + c2 - a2)]/(b+c-3a)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Let X'Y'Z' be the extouch triangle of ABC; viz., X' is where the A-excircle meets line BC, and X'Y'Z' is the pedal triangle of X(40). Let I = incenter of ABC. The circles (AIX'), (BIY'), (CIZ') concur in two points: I and X(1339). (Jean-Pierre Ehrmann, Hyacinthos #6545)

X(1339) lies on this line: 1,474

### X(1340) = INSIMILICENTER(CIRCUMCIRCLE, BROCARD CIRCLE)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a + 2(|OK| + R) cot ω cos A, where
|OK| = distance between X(3) and X(6),
ω = Brocard angle of ABC, and R = circumradius of ABC

Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = e cos A + cos(A - ω), e = (1 - 4 sin2ω)1/2

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1340) = (1 + |OK|/R)*X(3) + X(6)

If you have The Geometer's Sketchpad, you can view X(1340).

X(1340) = internal center of similitude of circumcircle and Brocard circle (Peter J. C. Moses, 4/2003)

X(1340) lies on these lines: 2,1349   3,6   4,1348

X(1340) = inverse-in-Brocard-circle of X(1380)

X(1340) = {X(3),X(182)}-harmonic conjugate of X(1341)
X(1340) = radical center of Lucas(t) circles, for t where circles are tangent to Brocard circle
X(1340) = intersection of Brocard axis and minor axis of Steiner circumellipse
X(1340) = homothetic center of 1st Brocard triangle and circumcevian triangle of X(3414)

### X(1341) = EXSIMILICENTER(CIRCUMCIRCLE, BROCARD CIRCLE)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a - 2(|OK| - R) cot ω cos A

Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = e cos A - cos(A - ω), e = (1 - 4 sin2ω)1/2

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1341) = (-1 + |OK|/R)*X(3) - X(6)

X(1341) = external center of similitude of circumcircle and Brocard circle (Peter J. C. Moses, 4/2003)

X(1341) lies on these lines: 2,1348   3,6   4,1349

X(1341) = inverse-in-Brocard-circle of X(1379)
X(1341) = {X(3),X(182)}-harmonic conjugate of X(1340)
X(1341) = intersection of Brocard axis and major axis of Steiner circumellipse
X(1341) = homothetic center of 1st Brocard triangle and circumcevian triangle of X(3413)

### X(1342) = INSIMILICENTER(CIRCUMCIRCLE, 1stLEMOINE CIRCLE)

Trilinears    sin A + cos A cot(ω/2) : :
Trilinears    sin A - sin(A - ω) : :
Trilinears    cos A + cos(A - ω) : :
Trilinears    cos(A - ω/2) : :
Trilinears    sin A + (csc ω + cot ω) cos A : :
Trilinears    cos A + (csc ω - cot ω) sin A : :

X(1342) =(sec ω)*X(3) + 2*X(182) = (1 + sec ω)*X(3) + X(6)

X(1342) = internal center of similitude of circumcircle and 1st Lemoine circle (Peter J. C. Moses, 4/2003; cf. X(1343), X(1670), X(1671))

X(1342) lies on this line: 3,6

X(1342) = reflection of X(1343) in X(3398)
X(1342) = isogonal conjugate of X(5403)
X(1342) = inverse-in-Brocard-circle of X(1670)
X(1342) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (371,372,1671), (1687,1688,1343)
X(1342) = insimilicenter of 1st and 2nd Brocard circles; the insimilicenter is X(1343)

### X(1343) = EXSIMILICENTER(CIRCUMCIRCLE, 1stLEMOINE CIRCLE)

Trilinears    sin A - cos A tan(ω/2) : :
Trilinears    sin A + sin(A - ω) : :
Trilinears    cos A - cos(A - ω) : :
Trilinears    sin(A - ω/2) : :
Trilinears    sin A - (sec ω + tan ω) cos A: :
Trilinears    sin A + (cot ω - csc ω) cos A: :
Trilinears    cos A - (csc ω + cot ω) sin A : :
X(1343) =(sec ω)*X(3) - 2*X(182) = (1 - sec ω)*X(3) + X(6)

X(1343) = external center of similitude of circumcircle and 1st Lemoine circle (Peter J. C. Moses, 4/2003)

Let Lbc be the line obtained by rotating line CA through C by an angle of ω/2 toward B. Let Lcb be the line obtained by rotating line AB through B by an angle of ω/2 toward C. Let A' =Lbc∩\Lcb. Define B' and C' cyclically. The lines AA', BB', CC' concur in X(1343). (Randy Hutson, October 13, 2015)

X(1343) lies on this line: 3,6

X(1343) = reflection of X(1342) in X(3398)
X(1343) = isogonal conjugate of X(5404)
X(1343) = inverse-in-Brocard-circle of X(1671)
X(1343) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (371,372,1670), (1687,1688,1342)
X(1343) = exsimilicenter of 1st and 2nd Brocard circles (the insimilicenter is X(1342))

### X(1344) = INSIMILICENTER(CIRCUMCIRCLE, ORTHOCENTROIDAL CIRCLE)

Trilinears     (|OH| + R) cos A + 4R cos B cos C : :
Barycentrics    a^4+a^2 b^2-2 b^4+a^2 c^2+4 b^2 c^2-2 c^4-a^2 (a^2-b^2-c^2) J : :
X(1344) = 6X(2) + (-3 + |OH|/R)*X(3)

As a point on the Euler line, X(1344) has Shinagawa coefficients (R + |OH|,3R - |OH|).

X(1344) = internal center of similitude of circumcircle and orthocentroidal circle (Peter J. C. Moses, 4/2003)

X(1344) lies on the cubics K281, K297, K708, K843, and on these lines: {2,3}, {6,2574}, {55,2464}, {56,2463}, {64,14375}, {111,8427}, {112,8426}, {183,15165}, {371,2466}, {372,2465}, {958,2468}, {1340,2470}, {1341,2469}, {1342,2472}, {1343,2471}, {1351,8116}, {1376,2467}, {1689,2015}, {1690,2016}, {2575,11472}, {5640,24651}, {5968,16070}, {6090,8115}

X(1344) = orthocentroidal circle inverse of X(1313)
X(1344) = X(11472)-line conjugate of X(2575)
X(1344) = X(1345)-vertex conjugate of X(11181)
X(1344) = crossdifference of every pair of points on line {647, 2575}
X(1344) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 4, 1313), (2, 1113, 3), (2, 1346, 5094), (2, 1995, 1345), (2, 2553, 1347), (3, 381, 1345), (3, 3830, 15155), (4, 378, 1345), (4, 1113, 25), (4, 1346, 381), (4, 10737, 3830), (4, 14709, 3), (5, 6644, 1345), (5, 20478, 3), (22, 5169, 1345), (23, 10719, 15155), (24, 7577, 1345), (25, 5094, 1345), (458, 4230, 1345), (868, 3148, 1345), (1113, 10719, 23), (1346, 14709, 378), (1347, 2553, 381), (2070, 7579, 1345), (3091, 14709, 3516), (3091, 15078, 1345), (3526, 21308, 1345), (7418, 13860, 1345), (7485, 7533, 1345), (14807, 15158, 20063)

### X(1345) = EXSIMILICENTER(CIRCUMCIRCLE, ORTHOCENTROIDAL CIRCLE)

Trilinears    (|OH| - R) cos A - 4R cos B cosC : :
Barycentrics    a^4+a^2 b^2-2 b^4+a^2 c^2+4 b^2 c^2-2 c^4+a^2 (a^2-b^2-c^2) J : :
X(1345) = 6X(2) - (3 + |OH|/R)*X(3)

As a point on the Euler line, X(1345) has Shinagawa coefficients (R - |OH|,3R + |OH|).

X(1345) = external center of similitude of circumcircle and orthocentroidal circle (Peter J. C. Moses, 4/2003)

X(1345) lies on the cubics K281, K297, K708, K843, and on these lines: {2,3}, {6,2575}, {55,2463}, {56,2464}, {64,14374}, {111,8426}, {112,8427}, {183,15164}, {371,2465}, {372,2466}, {958,2467}, {1340,2469}, {1341,2470}, {1342,2471}, {1343,2472}, {1351,8115}, {1376,2468}, {1689,2016}, {1690,2015}, {2574,11472}, {5640,24650}, {5968,16071}, {6090,8116}

X(1345) = orthocentroidal circle inverse of X(1312)
X(1345) = X(11472)-line conjugate of X(2574)
X(1345) = X(1344)-vertex conjugate of X(11181)
X(1345) = crossdifference of every pair of points on line {647, 2574}
X(1345) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (2, 4, 1312), (2, 1114, 3), (2, 1347, 5094), (2, 1995, 1344), (2, 2552, 1346), (3, 381, 1344), (3, 3830, 15154), (4, 378, 1344), (4, 1114, 25), (4, 1347, 381), (4, 10736, 3830), (4, 14710, 3), (5, 6644, 1344), (5, 20479, 3), (22, 5169, 1344), (23, 10720, 15154), (24, 7577, 1344), (25, 5094, 1344), (458, 4230, 1344), (868, 3148, 1344), (1114, 10720, 23), (1346, 2552, 381), (1347, 14710, 378), (2070, 7579, 1344), (3091, 14710, 3516), (3091, 15078, 1344), (3526, 21308, 1344), (7418, 13860, 1344), (7485, 7533, 1344), (14808, 15159, 20063)

### X(1346) = INSIMILICENTER(NINE-POINT CIRCLE, ORTHOCENTROIDAL CIRCLE)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = R cos A + |OH|cos(B - C) + 4 R cos B cos C
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(1346) = 3(2 + |OH|/R)*X(2) - (3 + |OH|/R)*X(3)

As a point on the Euler line, X(1346) has Shinagawa coefficients (R + |OH|,3R + |OH|).

X(1346) = internal center of similitude of nine-point circle and orthocentroidal circle (Peter J. C. Moses, 4/2003)

X(1346) lies on this line: 2,3

### X(1347) = EXSIMILICENTER(NINE-POINT CIRCLE, ORTHOCENTROIDAL CIRCLE)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = R cos A - |OH|cos(B - C) + 4 R cos B cos C
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(1347) = 3(-2 + |OH|/R)*X(2) + (3 - |OH|/R)*X(3)

As a point on the Euler line, X(1347) has Shinagawa coefficients (R - |OH|,3R - |OH|).

X(1347) = external center of similitude of nine-point circle and orthocentroidal circle (Peter J. C. Moses, 4/2003)

X(1347) lies on this line: 2,3

### X(1348) = INSIMILICENTER(NINE-POINT CIRCLE, BROCARD CIRCLE)

Trilinears    e cos(B - C) + cos(A - ω), where e = (1 - 4 sin2ω)1/2
X(1348) = |OK|*X(5) + R*X(182) = (e sec ω)*X(5) + X(182) = X(3) + 2(e sec ω)*X(5) + X(6)

X(1348) = internal center of similitude of nine-point circle and Brocard circle (Peter J. C. Moses, 4/2003)

X(1348) lies on these lines: 4,1340   2,1341   5,182   316,1341

### X(1349) = EXSIMILICENTER(NINE-POINT CIRCLE, BROCARD CIRCLE)

Trilinears    e cos(B - C) - cos(A - ω), where e = (1 - 4 sin2ω)1/2
X(1348) = |OK|*X(5) - R*X(182) = (e sec ω)*X(5) - X(182) = X(3) - 2(e sec ω)*X(5) + X(6)

X(1349) = external center of similitude of nine-point circle and Brocard circle (Peter J. C. Moses, 4/2003)

X(1349) lies on these lines: 2,1340   4,1341   5,182

### X(1350) = REFLECTION OF X(6) IN X(3)

Trilinears    a - 4R cot ω cos A : :
Trilinears    sin A - 2 cos A cot ω : :
Trilinears        h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = 2 cos A - sin A tan ω   (Peter J. C. Moses, 8/22/03)

Barycentrics    a^2(a^4 - 3b^4 - 3c^4 + 2a^2b^2 + 2a^2c^2 - 2b^2c^2) : :
X(1350) = 2X(3) - X(6)

Let A'B'C' be the reflection of ABC in X(3). Let AB = BC∩C'A', and define BC and CA cyclically. AC = BC∩A'B', and define BA and CB cyclically. The 6 points AB, BC, CA, AC, BA, CB lie on a conic. Let A" be the intersection of the tangents to the conic at AB and AC, and define B", C" cyclically. The lines A'A", B'B", C'C" concur in X(1350). (Randy Hutson, January 29, 2015)

X(1350) lies on these lines:
2,3066  3,6   4,141   20,64   22,110   30,599   35,611   36,613   40,518   74,1296   103,1293   206,1092   343,1370   376,524   517,990

X(1350) = midpoint of X(20) and X(69)
X(1350) = reflection of X(i) in X(j) for these (i,j): (4,141), (6,3), (1351,182), (1498,159)
X(1350) = isogonal conjugate of X(3424)
X(1350) = X(6) of circumcevian triangle of X(511)
X(1350) = radical center of Lucas(-tan ω) circles
X(1350) = {X(182),X(1351)}-harmonic conjugate of X(6)
X(1350) = inverse-in-2nd-Brocard-circle of X(5188)
X(1350) = antipedal-isogonal conjugate of X(6)
X(1350) = X(53)-of-the-hexyl-triangle
X(1350) = exsimilicenter of circle centered at X(1151) through X(372) and circle centered at X(1152) through X(371); the insimilicenter is X(3053)

### X(1351) = REFLECTION OF X(3) IN X(6)

Trilinears         a - R cot ω cos A : :
Trilinears       2 sin A - cos A cot ω : :
Trilinears        cos A - 2 sin A tan ω : :   (Peter J. C. Moses, 8/22/03)
Barycentrics    a^2[a^4 - 4a^2(b^2 + c^2) + 3b^4 - 2b^2c^2 + 3c^4] : :
Barycentrics    a^2(S^2 - SA^2 + SB SC) : :
X(1351) = X(3) - 2X(6)

Let T be a triangle inscribed in the circumcircle and circumscribing the orthic inconic. As T varies, its orthocenter traces a circle centered at X(1351) with segment X(4)X(193) as diameter. (Randy Hutson, August 29, 2018)

X(1351) lies on these lines:
3,6   4,193   5,69   25,110   30,1353   49,206   51,394   159,195   183,262   381,524   613,999

X(1351) = midpoint of X(4) and X(193)
X(1351) = reflection of X(i) in X(j) for these (i,j): (3,6), (6,576), (69,5), (1350,182)
X(1351) = isogonal conjugate of X(7612)
X(1351) = inverse-in-2nd-Lemoine-circle of X(1692)
X(1351) = radical center of Lucas(-4 tan ω) circles
X(1351) = intersection of tangents to 2nd Lemoine circle at intersections with circumcircle
X(1351) = inverse-in-{circle centered at X(3) with radius |OK|} of X(182)
X(1351) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (6,1350,182), (1687,1688,5033)
X(1351) = exsimilicenter of circle centered at X(371) through X(1152) and circle centered at X(372) through X(1151); the insimilicenter is X(3053)

### X(1352) = REFLECTION OF X(6) IN X(5)

Trilinears    a - 2R cos(B-C) cot ω : :
Trilinears    sin A - cos(B - C) cot ω : :
Barycentrics    S^2 SA + SB SC SW : :
Barycentrics    (cot A + cot B + cot C) tan A + (tan A + tan B + tan C) cot A : :
Barycentrics    a^6 - a^4 (b^2 + c^2) + a^2 (b^2 + c^2)^2 - (b^2 + c^2) (b^2 - c^2)^2 : :

X(1352) lies on these lines:
2,98   3,66   4,69   5,6   11,613   12,611   25,343   30,599   70,1176   193,576   206,1209   298,383   299,1080   355,518   381,524   394,426

X(1352) = midpoint of X(4) and X(69)
X(1352) = reflection of X(i) in X(j) for these (i,j): (3,141), (6,5), (193,576)
X(1352) = isogonal conjugate of X(3425)
X(1352) = anticomplement of X(182)
X(1352) = X(327)-Ceva conjugate of X(2)
X(1352) = complement of X(6776)
X(1352) = X(4)-of-1st-Brocard-triangle
X(1352) = X(3)-of-X(2)-Fuhrmann-triangle
X(1352) = center of the perspeconic of these triangles: Ehrmann side and Johnson
X(1352) = insimilicenter of X(13)- and X(14)-Fuhrmann circles (aka -Hagge circles

### X(1353) = REFLECTION OF X(5) IN X(6)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2a - R cos(B-C) cot ω

Trilinears       g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 4 sin A - cos(B - C) cot ω

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1353) lies on these lines:
3,193   5,6   30,1351   69,140   141,575   182,524   511,550

X(1353) = midpoint of X(3) and X(193)
X(1353) = reflection of X(i) in X(j) for these (i,j): (5,6), (69,140), (141,575)

Brisse Transforms 1354- 1367

Suppose that P is a point on the circumcircle Γ of triangle ABC. Let U and V be the lines through P tangent to the incircle. Line U meets Γ in a point U' other than P, and line V meets Γ in a point V' other than P. The line U'V' is tangent to the incircle. The touchpoint, denoted by T(P), is the Brisse transform of P. Suppose P is given by barycentrics u : v : w. Barycentrics for T(P) are found in

Edward Brisse, Perspective Poristic Triangles: a4/[(b + c - a)u2] : b4/[(c + a - b)v2] : c4/[(a + b - c)w2].

If X is given by trilinears x : y : z, then T(X) has trilinears a/[(b + c - a)x2] : b/[(c + a - b)y2] : c/[(a + b - c)z2].

Examples: X(11) = Feuerbach point = T(X(109))
X(1317) = incircle-antipode of X(11) = T(X(106))

Still open is the question posed in Hyacinthos #6832: to list all polynomial centers on the incircle having low degree and to prove that there are no others. Here, "degree" of X = p(a,b,c) : p(b,c,a) : p(c,a,b) [barycentrics] refers to the degree of homogeneity of p(a,b,c), and "low" means less than 6. (The Feuerbach point, X(11), has degree 3.)

In Hyacinthos #6835, Paul Yiu gives two methods for constructing polynomial centers on the incircle:

(1) If X is a polynomial center on the incircle and W is any other polynomial center, then the line XW meets the incircle in another point that is a polynomial center.

(2) If W is on the line at infinity, then the barycentric square W2 is on the Steiner inscribed ellipse, and the barycentric product X(7)*W2 is on the incircle.

### X(1354) = BRISSE TRANSFORM OF X(74)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is as above X(1354), using X = X(74)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

The following ten points lie on a circle: X(i) for i = 11, 36, 65, 80, 108, 759, 1354, 1845, 2588, 2589. (Chris Van Tienhoven, Hyacinthos, January 4, 2011)

X(1354) lies on the incircle and these lines: 7,1367   56,759   942,1364

### X(1355) = BRISSE TRANSFORM OF X(98)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is as above X(1354), using X = X(98)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1355) lies on the incircle and these lines: 56,741   222,1363

### X(1356) = BRISSE TRANSFORM OF X(99)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is as above X(1354), using X = X(99)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1356) = anticomplement of X(3037)

X(1356) lies on the incircle and this line: 56,741

### X(1357) = BRISSE TRANSFORM OF X(100)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is as above X(1354), using X = X(100)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1357) lies on the incircle and these lines:
12,121   43,57   55,1293   56,106   65,1317   1086,1365

X(1357) = anticomplement of X(3038) X(1357) = crosssum of X(i) and X(j) for these (i,j): (55,644), (100,145), (190,344)

### X(1358) = BRISSE TRANSFORM OF X(101)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is as above X(1354), using X = X(101)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1358) lies on the incircle and these lines:
7,528   11,1111   12,85   55,1292   56,105   65,1362   269,1359   553,1366   1120,1125   1122,1361   1319,1323

X(1358) = anticomplement of X(3039)
X(1358) = X(244)-cross conjugate of X(1086)
X(1358) = crosspoint of X(277) and X(514)
X(1358) = crosssum of X(101) and X(218)

### X(1359) = BRISSE TRANSFORM OF X(102)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is as above X(1354), using X = X(102)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1359) lies on the incircle and these lines:
4,11   12,123   55,1295   65,1364   269,1358

### X(1360) = BRISSE TRANSFORM OF X(103)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) where f(a,b,c) is as above X(1354), using X = X(103)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1360) lies on the incircle and these lines:
11,57   12,208   55,108   56,105   354,1364

### X(1361) = BRISSE TRANSFORM OF X(104)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is as above X(1354), using X = X(104)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1361) lies on the incircle and these lines:
1,1364   11,65   12,124   55,102   56,106   151,497   181,994   928,1362   962,1118   1122,1358

X(1361) = reflection of X(1364) in X(3)
X(1361) = anticomplement of X(3040)
X(1361) = crosspoint of X(7) and X(1465)

### X(1362) = BRISSE TRANSFORM OF X(105)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is as above X(1354), using X = X(105)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1362) lies on the incircle and these lines:
7,1002   11,118   12,116   43,57   55,103   56,101   59,840   65,1358   105,651   150,388   152,497   928,1361

X(1362) = anticomplement of X(3041)
X(1362) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,241), (651,665)
X(1362) = crosspoint of X(7) and X(241)
X(1362) = crosssum of X(i) and X(j) for (i,j) = (11,885), (55,294)
X(1362) = crossdifference of every pair of points on line X(294)X(885)
X(1362) = X(672)-Hirst inverse of X(1458)

### X(1363) = BRISSE TRANSFORM OF X(107)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is as above X(1354), using X = X(107)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1363) lies on the incircle and this line:
222,1355

### X(1364) = BRISSE TRANSFORM OF X(108)

Trilinears    csc A tan(A/2) / (sec B - sec C) : :
Trilinears    a(b + c - a)[(b - c)(b^2 + c^2 - a^2)]^2 : :

X(1364) lies on the incircle and these lines:
1,1361   11,124   12,117   55,103   56,102   65,1359   77,296   151,388   185,603   354,1360   942,1354

X(1364) = reflection of X(1361) in X(1)
X(1364) = anticomplement of X(3042)
X(1364) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,905), (189,650), (222,652), (255,520)
X(1364) = crosspoint of X(i) and X(j) for these (i,j): (3,521), (7,905)
X(1364) = crosssum of X(4) and X(108)
X(1364) = X(925)-of-intouch-triangle
X(1364) = X(109)-of-Mandart-incircle-triangle
X(1364) = homothetic center of intangents triangle and reflection of extangents triangle in X(109)
X(1364) = trilinear pole wrt intouch triangle of line X(4)X(7)
X(1364) = intersection, other than X(11), of incircle and Mandart circle
X(1364) = extouch isogonal conjugate of X(522)
X(1364) = crossdifference of every pair of points on line X(1783)X(4559)

### X(1365) = BRISSE TRANSFORM OF X(110)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is as above X(1354), using X = X(110)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1365) lies on the incircle and these lines:
7,1366   56,759   125,1109   1086,1357   1283,1284

X(1365) = crosssum of X(643) and X(1098)

### X(1366) = BRISSE TRANSFORM OF X(111)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is as above X(1354), using X = X(111)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1366) lies on the incircle and these lines: 7,1365   222,1367   553,1358

### X(1367) = BRISSE TRANSFORM OF X(112)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is as above X(1354), using X = X(112)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1367) lies on the incircle and these lines: 7,1354   222,1366

### X(1368) = COMPLEMENTARY CONJUGATE OF X(6)

Trilinears    bc[b2/(a2 - b2 + c2) + c2/(a2 + b2 - c2)] : :
Barycentrics    b2/(a2 - b2 + c2) + c2/(a2 + b2 - c2) : :
Barycentrics    (b^2 + c^2 - a^2)[a^2(b^2 + c^2) + (b^2 - c^2)^2] : :

As a point on the Euler line, X(1368) has Shinagawa coefficients (E - F, -E - F).

X(1368) lies on these lines: 2,3   11,1040   12,1038   98,801   114,122   120,123   125,343   126,127   230,577   495,612   496,614

X(1368) = midpoint of X(25) and X(1370)
X(1368) = reflection of X(1596) in X(5)
X(1368) = complement of X(25)
X(1368) = complementary conjugate of X(6)
X(1368) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,1196), (670,525), (222,652), (255,520)
X(1368) = crosspoint of X(2) and X(305)
X(1368) = perspector of circumconic centered at X(1196)
X(1368) = center of circumconic that is locus of trilinear poles of lines passing through X(1196)
X(1368) = X(2)-Ceva conjugate of X(1196)
X(1368) = homothetic center of the medial triangle and the 3rd pedal triangle of X(3)
X(1368) = X(6244)-of-orthic-triangle if ABC is acute

### X(1369) = ANTICOMPLEMENTARY CONJUGATE OF X(6)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[b2/(a2 + c2) + c2/(a2 + b2) - a2/(b2 + c2)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = b2/(a2 + c2) + c2/(a2 + b2) - a2/(b2 + c2)

X(1369) lies on this line: 2,32

X(1369) = anticomplement of X(251)
X(1369) = anticomplementary conjugate of X(6)

### X(1370) = ANTICOMPLEMENT OF X(25)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[b2/(a2 - b2 + c2) + c2/(a2 + b2 - c2) - a2/(- a2 + b2 + c2)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = b2/(a2 - b2 + c2) + c2/(a2 + b2 - c2) - a2/(- a2 + b2 + c2)

As a point on the Euler line, X(1370) has Shinagawa coefficients (E, -2E - 2F).

X(1370) lies on these lines: 2,3   66,69   305,315   343,1350   925,1297

X(1370) = reflection of X(25) in X(1368)
X(1370) = anticomplement of X(25)
X(1370) = anticomplementary conjugate of X(193)
X(1370) = X(i)-Ceva conjugate of X(j) for these (i,j): (305,2), (315,69)

### X(1371) = 1st RIGBY POINT

Trilinears    1 + 8(area ABC)/[3a(b + c - a)] : :
X(1371) = 3s*X(1) + (2r + 8R)*X(7)

Adrian Oldknow, "The Euler-Gergonne-Soddy Triangle of a Triangle," American Mathematical Monthly 103 (1996) 319-329. See page 326.

X(1371) lies on this line: 1,7

X(1371) = {X(1),X(7)}-harmonic conjugate of X(1372)
X(1371) = {X(176),X(482)}-harmonic conjugate of X(1)

### X(1372) = 2nd RIGBY POINT

Trilinears     1 - 8(area ABC)/[3a(b + c - a)] : :
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(1372) = 3s*X(1) - (2r + 8R)*X(7)

Adrian Oldknow, "The Euler-Gergonne-Soddy Triangle of a Triangle," American Mathematical Monthly 103 (1996) 319-329. See page 326.

X(1372) lies on this line: 1,7

X(1372) = {X(1),X(7)}-harmonic conjugate of X(1371)
X(1372) = {X(175),X(481)}-harmonic conjugate of X(1)

### X(1373) = 1st GRIFFITHS POINT

Trilinears    1 + 8(area ABC)/[a(b + c - a)] : :
Trilinears    1 + 4 sec A/2 cos B/2 cos C/2 : :
X(1373) = 2s*X(1) + (2r + 8R)*X(7)

Adrian Oldknow, "The Euler-Gergonne-Soddy Triangle of a Triangle," American Mathematical Monthly 103 (1996) 319-329. See page 327.

X(1373) lies on these lines: {1,7}, {226,3317}

X(1373) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,7,1374), (7,176,481), (7,482,1), (175,482,1), (176,481,1), (481,482,176)
X(1373) = perspector of inner Soddy triangle and cross-triangle of inner and outer Soddy triangles
X(1373) = perspector of inner Soddy tangential triangle and cross-triangle of inner and outer Soddy tangential triangles

### X(1374) = 2nd GRIFFITHS POINT

Trilinears    1 - 8(area ABC)/[a(b + c - a)] : :
Trilinears    1 - 4 sec A/2 cos B/2 cos C/2 : :
X(1374) = 2s*X(1) - (2r + 8R)*X(7)

Adrian Oldknow, "The Euler-Gergonne-Soddy Triangle of a Triangle," American Mathematical Monthly 103 (1996) 319-329. See page 327.

X(1374) lies on these lines: {1,7}, {226,3316}

X(1374) = {X(1),X(7)}-harmonic conjugate of X(1373)
X(1374) = {X(7),X(175)}-harmonic conjugate of X(482)
X(1374) = {X(7),X(481)}-harmonic conjugate of X(1)
X(1374) = {X(481),X(482)}-harmonic conjugate of X(175)
X(1374) = perspector of outer Soddy triangle and cross-triangle of inner and outer Soddy triangles
X(1374) = perspector of outer Soddy tangential triangle and cross-triangle of inner and outer Soddy tangential triangles

### X(1375) = EVANS POINT

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[- 2a2cos A + b(a - b + c)cos B + c(a + b - c)cos C]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = - 2a2cos A + b(a - b + c)cos B + c(a + b - c)cos C

As a point on the Euler line, X(1375) has Shinagawa coefficients (2\$aSBSC\$ - \$a\$S2, \$a\$S2). Adrian Oldknow, "The Euler-Gergonne-Soddy Triangle of a Triangle," American Mathematical Monthly 103 (1996) 319-329. See page 328.

X(1375) lies on these lines: 2,3   241,514

X(1375) = complement of X(857)
X(1375) = crossdifference of every pair of points on line X(55)X(647)

### X(1376) = EXSIMILICENTER(CIRCUMCIRCLE, SPIEKER CIRCLE)

Trilinears    a2 - ab - ac + 2bc : :
Barycentrics    b + c - a cos A : :
X(1376) = X(1) - 3X(2) + (r/R)X(3) = r*X(3) - 2R*X(10) = R*X(1) - r*X(3) + R*X(8)

X(1376) lies on these lines:
1,474   2,11   3,10   4,1329   6,43   7,480   8,56   9,165   12,377   31,899   35,405   36,956   40,936   42,750   45,846   46,72   57,200   63,210   65,78   71,965   75,183   226,1260   227,1038   371,1377   372,1378   442,498   517,997   519,999   748,902   851,1211   978,1191   982,1054

X(1376) = midpoint of X(i) and X(j) for these (i,j): (8,3476), (57,200), (329,3474)
X(1376) = complement of X(497)
X(1376) = X(294)-Ceva conjugate of X(518)
X(1376) = cevapoint of X(43) and X(165)
X(1376) = anticomplement of X(3816)
X(1376) = crosssum of PU(46)
X(1376) = crosspoint of PU(112)
X(1376) = crossdifference of every pair of points on line X(665)X(4083)
X(1376) = homothetic center of ABC and cross-triangle of ABC and inner Johnson triangle

### X(1377) = INSIMILICENTER(2nd LEMOINE CIRCLE, SPIEKER CIRCLE)

Trilinears    bc(b + c + a sin A) : :
X(1377) = r*X(6) + (2R tan ω)*X(10) = R*X(1) + (r cot ω)*X(6) + R*X(8)

X(1377) lies on these lines:
2,1335   6,10   8,1124   371,1376   372,958   485,1329   993,1152

X(1377) = {X(6),X(10)}-harmonic conjugate of X(1378)

### X(1378) = EXSIMILICENTER(2nd LEMOINE CIRCLE, SPIEKER CIRCLE)

Trilinears    bc(b + c - a sin A) : :
X(1378) = r*X(6) - (2R tan ω)*X(10) = R*X(1) - (r cot ω)*X(6) + R*X(8)

X(1378) lies on these lines:
2,1124   6,10   8,1335   371,958   372,1376   486,1329   993,1151

X(1378) = {X(6),X(10)}-harmonic conjugate of X(1377)

### X(1379) = 1st BROCARD-AXIS INTERCEPT OF CIRCUMCIRCLE

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a + 2(|OK| - R)cot ω cos A
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = e cos A - cos(A + ω), e = (1 - 4 sin2ω)1/2
Trilinears        h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = a(a4 - b2c2 + a2T), where T = (a4 + b4 + c4 - b2c2 - c2a2 - a2 b2)1/2

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1379) = (-1 + |OK|/R)*X(3) + X(6)

The Brocard axis, OK, is the line of the circumcenter, O [= X(3)], and the symmedian point, K [= X(6)]. This line meets the circumcircle in two points, X(1379) and X(1380); the closer to X(6) is X(1379.

X(1379) lies on this line: 3,6

X(1379) = isogonal conjugate of X(3413)
X(1379) = reflection of X(i) in X(j) for these (i,j): (4,2040), (1380,3), (3557,2029)
X(1379) = anticomplement of X(2039)
X(1379) = inverse-in-Brocard-circle of X(1341)
X(1379) = X(249)-Ceva conjugate of X(1380)
X(1379) = Ψ(X(2), X(1340))
X(1379) = trilinear pole of line X(6)X(5639)
X(1379) = {X(371),X(372)}-harmonic conjugate of X(3558)
X(1379) = {X(1687),X(1688)}-harmonic conjugate of X(1380)
X(1379) = circumcircle intercept, other than A, B, C, of hyperbola {A,B,C,X(6),PU(118)}
X(1379) = perspector of triangles AiBiCi and (AaBbCc)*, and of triangles AaBbCc and (AiBiCi)*; see preamble before X(11752)
X(1379) = perspector of ABC and (degenerate) cross-triangle of 1st and 2nd anti-circummedial triangles

### X(1380) = 2nd BROCARD-AXIS INTERCEPT OF CIRCUMCIRCLE

Trilinears    a - 2(|OK| + R)cot ω cos A : :
Trilinears    e cos A + cos(A + ω) : : , where e = (1 - 4 sin2ω)1/2
Trilinears    a(a4 - b2c2 - a2T) : : , where T = (a4 + b4 + c4 - b2c2 - c2a2 - a2 b2)1/2

X(1380) = (1 + |OK|/R)*X(3) - X(6)

Let A' be the incenter of BCX(15), and define B' and C' cyclically. Let A" be the incenter of BCX(16), and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(1380). (Randy Hutson, February 10, 2016)

X(1380) lies on this line: 3,6

X(1380) = isogonal conjugate of X(3414)
X(1380) = reflection of X(i) in X(j) for these (i,j): (4,2039), (1379,3), (3558,2028)
X(1380) = anticomplement of X(2040)
X(1380) = inverse-in-Brocard-circle of X(1340)
X(1380) = X(249)-Ceva conjugate of X(1379)
X(1380) = Ψ(X(2), X(1341))
X(1380) = {X(371),X(372)}-harmonic conjugate of X(3557)
X(1380) = {X(1687),X(1688)}-harmonic conjugate of X(1379)
X(1380) = trilinear pole of line X(6)X(5638) (tangent to hyperbola {{A,B,C,X(6),PU(118)}} at X(6))

### X(1381) = 1st INTERCEPT OF LINE X(1)X(3) AND CIRCUMCIRCLE

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = r + (|OI| - R) cos A
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

The line IO, where I = X(1) = incenter and O = X(3) = circumcenter, meets the circumcircle in two points, X(1381) and X(1382), where X(1381) is the nearer of the two to X(1).

X(1381) lies on the circumcircle and this line: 1,3

X(1381) = reflection of X(1382) in X(3)
X(1381) = isogonal conjugate of X(3307)
X(1381) = X(59)-Ceva conjugate of X(1382)
X(1381) = X(513)-cross conjugate of X(1382)

### X(1382) = 2nd INTERCEPT OF LINE X(1)X(3) AND CIRCUMCIRCLE

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = r - (|OI| + R) cos A
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1382) lies on the circumcircle and this line: 1,3

X(1382) = reflection of X(1381) in X(3)
X(1382) = isogonal conjugate of X(3308)
X(1382) = X(59)-Ceva conjugate of X(1381)
X(1382) = X(513)-cross conjugate of X(1381)

### X(1383) = 1st GRINBERG HOMOTHETIC CENTER

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(2b2 + 2c2 - a2)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Let A'B'C' be the circumcevian triangle of X(2), and let P(A) be the line through A' parallel to line BC. Define P(B) and P(C) cyclically. Let A" = P(B)∩P(C), and define B" and C" cyclically. Triangle A"B"C" is homothetic to ABC, and X(1383) is the center of the homothety.

X(1383) lies on these lines: 2,187   6,23   32,111

X(1383) = isogonal conjugate of X(599)
X(1383) = cevapoint of X(6) and X(1384)

### X(1384) = 2nd GRINBERG HOMOTHETIC CENTER

Trilinears    a(b2 + c2 - 5a2) : :
Trilinears    sin(A + ω) - 5 sin(A - ω) : :

Let GA be the circumcenter of triangle BCG, where G - centroid(ABC). Define GB and GC cyclically. Triangle GAGBGC is homothetic to the pedal triangle of X(6), and X(1384) is the center of the homothety.

Let DEF be the circumsymmedial triangle and let (Oa) be the circle tangent at D to circumcircle and with center Oa=AH∩ OD. Define (Ob) and (Oc) cyclically. The radical center of the circles (Oa), (Ob), (Oc) is X(1384). (Angel Montesdeoca, October 2, 2018)

X(1384) lies on these lines: 3,6   25,111   55,609   230,381   385,1003

X(1384) = X(1383)-Ceva conjugate of X(6)
X(1384) = isogonal conjugate of X(5485)
X(1384) = intersection of tangents at PU(2) to hyperbola {X(6),PU(1),PU(2)}
X(1384) = {X(3),X(6)}-harmonic conjugate of X(5024)
X(1384) = inverse-in-1st-Brocard-circle of X(5024)

### X(1385) = MIDPOINT OF INCENTER AND CIRCUMCENTER

Trilinears    16(area ABC)2 + a(a + b + c)(b2 + c2 - a2) : :
Trilinears    2 cos A + cos B + cos C - 1 : :
Trilinears    r + R cos A : : : :
X(1385) = X(1) + X(3)

Let Na = X(5)-of-BCX(1), Nb = X(5)-of-CAX(1), Nc = X(5)-of-ABX(1). X(1385) is the antigonal image of X(5) wrt NaNbNc. Also, X(1385) = X(265)-of-NaNbNc. (Randy Hutson, December 10, 2016)

Let A'B'C' be the medial triangle. X(1385) is the radical center of the incircles of AB'C', BC'A', CA'B'. (Randy Hutson, December 10, 2016)

X(1385) lies on these lines:
1,3   2,355   5,515   8,631   10,140   21,104   30,551   37,572   54,72   77,945   78,956   101,1212   182,518   284,1108   376,962   496,950   500,1064   511,1386   516,550   519,549   573,1100   581,995   602,1468   912,960   943,1476   953,1290   958,997   971,1001   991,1279

X(1385) = midpoint of X(i) and X(j) for these (i,j): (1,3), (40,1482), (355,944)
X(1385) = reflection of X(i) in X(j) for these (i,j): (5,1125), (10,140)
X(1385) = isogonal conjugate of X(1389)
X(1385) = complement of X(355)
X(1385) = {X(1),X(40)}-harmonic conjugate of X(1483)
X(1385) = X(5)-of-2nd circumperp triangle
X(1385) = X(3)-of-X(1)-Brocard triangle
X(1385) = X(140)-of-hexyl triangle
X(1385) = X(26)-of-incircle-circles-triangle
X(1385) = endo-homothetic center of Ehrmann side-triangle and circumorthic triangle; the homothetic center is X(5)
X(1385) = X(546)-of-excentral-triangle

### X(1386) = MIDPOINT OF INCENTER AND SYMMEDIAN POINT

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a + b + c) + (a2 + b2 + c2)
Trilinears       as + Sω : bs + Sω : cs + Sω
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1386) lies on these lines:
1,6   7,1456   56,77   65,82   81,105   141,1125   171,1054   182,517   206,942   241,1471   511,1385   519,597   524,551   614,940   751,1319

X(1386) = midpoint of X(1) and X(6)
X(1386) = reflection of X(141) in X(1125)
X(1386) = isogonal conjugate of X(1390)
X(1386) = complement of X(3416)
X(1386) = crosspoint of X(1) and X(985)
X(1386) = crosssum of X(1) and X(984)

### X(1387) = MIDPOINT OF INCENTER AND FEUERBACH POINT

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = bc[2a4 - 2(b + c)a3 + (8bc - 3b2 - 3c2)a2
+ 2(b + c)(b - c)2a + (b2 - c2)2]
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1387) lies on these lines:
1,5   2,1000   7,104   30,1319   100,474   106,1086   142,214   149,377   153,1056

X(1387) = midpoint of X(i) and X(j) for these (i,j): (1,11), (80,1317), (1145,1320)
X(1387) = isogonal conjugate of X(1391)
X(1387) = inverse-in-incircle of X(80)
X(1387) = complement of X(1145)
X(1387) = crosssum of X(202) and X(203)

### X(1388) = POINT ANCHA

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (3a - 2b -2c)/(a - b - c)      (M. Iliev, 5/13/07)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1388) lies on these lines:
1,3   8,1317   11,944   45,1404   73,1149   499,952   603,1339

X(1388) = isogonal conjugate of X(1392)

### X(1389) = ISOGONAL CONJUGATE OF X(1385)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[16(area ABC)2 + a(a + b + c)(b2 + c2 - a2)]
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1389) lies on these lines:
1,1393   5,8   7,944   21,517   65,104   79,515   80,946   942,1476

X(1389) = isogonal conjugate of X(1385)

### X(1390) = ISOGONAL CONJUGATE OF X(1386)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[a(a + b + c) + (a2 + b2 + c2)]
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1390) lies on these lines:
37,105   38,57   81,518   278,427   279,388   984,985

X(1390) = isogonal conjugate of X(1386)
X(1390) = cevapoint of X(1) and X(984)

### X(1391) = ISOGONAL CONJUGATE OF X(1387)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = a/[2a4 - 2(b + c)a3 + (8bc - 3b2 - 3c2)a2 + 2(b + c)(b - c)2a + (b2 - c2)2]
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1391) lies on this line: 517,1443

X(1391) = isogonal conjugate of X(1387)
X(1391) = cevapoint of X(202) and X(203)

### X(1392) = ISOGONAL CONJUGATE OF X(1388)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a - b -c)/(3a - 2b - 2c)      (M. Iliev, 5/13/07)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1392) lies on this line: 80,145

X(1392) = isogonal conjugate of X(1388)

Beth Conjugates 1393- 1477

If X is a triangle center then X = P-beth conjugate of P for some triangle center P. Using the definition of beth conjugate (in the Glossary), it is easy to prove that the P-beth conjugate of P is the trilinear product P*X(56),
so that P = X*X(8). In trilinears,

if P = p : q : r, then (P-beth conjugate of P) = up : vq : wr, where

u : v : w = a/(b + c - a) : b/(c + a - b) : c/(a + b - c), or equivalently,
u : v : w = 1 - cos A : 1 - cos B : 1 - cos C.

Following is a list of pairs (i,j) for which X(i) = X(j)-beth conjugate of X(j):

 1,8 2,312 3,78 4,318 6,9 7,75 8,341 9,346 12,1089 19,281 21,1043 25,33 28,29 31,55 32,41 34,4 41,220 42,210 48,219 55,200 56,1 57,2 58,21 59,765 60,1098 63,345 65,10 73,72 77,69 78,1265 81,333 84,280 85,76 86,314 101,644 109,100 110,643 142,1229 174,556 181,756 184,212 190,646 212,1260 220,728 221,40 222,63 223,329 226,321 244,11 255,1259 266,188 269,7 273,264 278,92 279,85 326,1264 347,322 348,304 479,1088 513,522 552,873 603,3 604,6 604,6 608,19 614,497 649,650 651,190 662,645 664,668 667,663 738,279 757,261 849,60 934,664 951,1257 961,1220 1014,86 1027,885 1042,65 1104,950 1106,56 1118,158 1119,273 1193,960 1214,306 1253,480 1254,12 1284,740

### X(1393) = X(5)-BETH CONJUGATE OF X(5)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(5)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1393) lies on these lines:
1,1389   2,201   11,774   12,38   28,34   31,1454   46,602   56,244   65,1193   73,942   225,1210   227,354   278,1148   388,982   595,1421

X(1393) = crosspoint of X(57) and X(273)
X(1393) = crosssum of X(9) and X(212)

### X(1394) = X(20)-BETH CONJUGATE OF X(20)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(20)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1394) lies on these lines:
1,84   3,223   9,478   21,77   28,34   40,109   56,269   73,991   78,651   165,227   614,1106   1104,1407   1398,1473   1420,1457

X(1394) = X(i)-Ceva conjugate of X(j) for these (i,j): (21,56), (77,57)
X(1394) = X(154)-cross conjugate of X(610)
X(1394) = cevapoint of X(221) and X(1035)

### X(1395) = X(25)-BETH CONJUGATE OF X(25)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(25)
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = sin A tan A sin2(A/2)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1395) lies on these lines:
4,171   24,602   25,31   28,34   32,1402   56,1472   108,727   212,573   238,459   278,985   427,750   468,748   607,1200   1106,1398   1416,1435

X(1395) = X(34)-Ceva conjugate of X(604)
X(1395) = crosspoint of X(608) and X(1398)
X(1395) = crosssum of X(345) and X(1265)

### X(1396) = X(27)-BETH CONJUGATE OF X(27)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(27)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1396) lies on these lines:
4,940   7,27   21,1214   28,34   108,741   223,284   269,1412   593,1014   1119,1407   1333,1427

X(1396) = X(i)-cross conjugate of X(j) for these (i,j): (1407,1412), (1408,1414), (1474,28)
X(1396) = cevapoint of X(i) and X(j) for these (i,j): (34,608), (1407,1435)

### X(1397) = X(31)-BETH CONJUGATE OF X(31)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(31)
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = [sin A sin A/2]2
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1397) is the vertex conjugate of the foci of the inellipse that is the isogonal conjugate of the isotomic conjugate of the incircle. (Randy Hutson, October 15, 2018)

X(1397) lies on these lines:
1,987   6,181   31,184   42,1404   55,572   56,58   57,985   60,959   109,727   171,182   278,1365   392,993   602,1092   603,1472   1257,1407

X(1397) = X(i)-Ceva conjugate of X(j) for these (i,j): (59,1415), (604,32), (1408,604)
X(1397) = X(560)-cross conjugate of X(32)
X(1397) = crosspoint of X(i) and X(j) for these (i,j): (56,608), (59,1415), (604,1106)
X(1397) = crosssum of X(i) and X(j) for these (i,j): (8,345), (75,322), (312,341)
X(1397) = isogonal conjugate of X(3596)
X(1397) = X(8)-isoconjugate of X(75)
X(1397) = trilinear product of extraversions of X(55)

### X(1398) = X(34)-BETH CONJUGATE OF X(34)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(34)
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = sec A sin4(A/2)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1398) lies on these lines:
1,1037   4,496   6,1425   25,34   28,279   184,221   278,961   388,427   475,956   604,608   607,1475   1106,1395   1254,1460   1394,1473   1407,1408

X(1398) = isogonal conjugate of X(1265)
X(1398) = X(i)-Ceva conjugate of X(j) for these (i,j): (1119,1407), (1435,608)
X(1398) = X(1395)-cross conjugate of X(608)

### X(1399) = X(35)-BETH CONJUGATE OF X(35)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(35)
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = cos A - cos 2A
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1399) lies on these lines:
3,47   6,1195   12,171   31,56   34,1454   35,500   55,255   58,65   201,896   213,1415   580,1155   595,1319   920,1060   1333,1409   1402,1408

### X(1400) = X(37)-BETH CONJUGATE OF X(37)

Trilinears    (1 - cos A)(b + c) : :
Trilinears    a(cos B + cos C) : :

X(1400) lies on these lines:
1,573   2,7   6,41   12,1213   19,208   25,31   36,572   37,65   42,181   44,583   58,1169   85,1218   108,1172   109,111   171,256   213,1042   222,967   292,694   308,349   388,966   478,603   651,1014   910,1200   1100,1319   1122,1418   1171,1412   1254,1426   1258,1432   1333,1415   1420,1449

X(1400) = isogonal conjugate of X(333)
X(1400) = X(i)-Ceva conjugate of X(j) for these (i,j): (6,1409), (56,1402), (57,65), (65,42), (108,663), (226,73), (951,55), (1415,649), (1427,1042)
X(1400) = X(i)-cross conjugate of X(j) for these (i,j): (181,65), (213,42), (1402,1402)
X(1400) = cevapoint of X(213) and X(1402)
X(1400) = crosspoint of X(i) and X(j) for these (i,j): (6,19), (56,57), (65,1427), (225,226)
X(1400) = crosssum of X(i) and X(j) for these (i,j): (1,573), (2,63), (8,9), (283,284)
X(1400) = crossdifference of every pair of points on line X(522)X(663)
X(1400) = X(65)-Hirst inverse of X(1284)
X(1400) = bicentric sum of PU(18)
X(1400) = PU(18)-harmonic conjugate of X(663)
X(1400) = barycentric product of PU(81)
X(1400) = trilinear pole of line X(512)X(810)
X(1400) = X(92)-isoconjugate of X(283)
X(1400) = perspector of ABC and unary cofactor triangle of Gemini triangle 1

### X(1401) = X(38)-BETH CONJUGATE OF X(38)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(38)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1401) lies on these lines:
7,310   43,57   51,244   56,58   65,519   226,1463   354,1122   511,982   1106,1425   1355,1365   1356,1366   1402,1458   1407,1460

X(1401) = crosspoint of X(7) and X(56)
X(1401) = crosssum of X(8) and X(55)

### X(1402) = X(42)-BETH CONJUGATE OF X(42)

Trilinears    (1 - cos A)u(a,b,c) : : , where u : v : w = X(42)

Trilinears    (sin^2 A) (cos B + cos C) : :
Trilinears    a^2 (b + c)/(a - b - c) : :

X(1402) lies on these lines:
1,3   21,961   25,1096   31,184   32,1395   42,181   73,1245   98,108   109,741   172,893   226,1284   923,1415   968,1011   1042,1410   1399,1408   1401,1458   1441,1447

X(1402) = isogonal conjugate of X(314)
X(1402) = X(i)-Ceva conjugate of X(j) for these (i,j): (56,1400), (65,1409), (961,6), (1037,73), (1400,213)
X(1402) = crosspoint of X(i) and X(j) for these (i,j): (25,31), (56,604), (1042, 1400)
X(1402) = crosssum of X(i) and X(j) for these (i,j): (8,312),. (69,75), (333,1043)
X(1402) = X(21)-isoconjugate of X(75)
X(1402) = X(92)-isoconjugate of X(1812)

### X(1403) = X(43)-BETH CONJUGATE OF X(43)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(43)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1403) lies on these lines:
1,3   2,1284   6,893   31,1428   42,1469   43,1423   75,183   109,727   1326,1412

X(1403) = X(604)-Ceva conjugate of X(56)
X(1403) = X(1423)-cross conjugate of X(56)
X(1403) = X(31)-Hirst inverse of X(1428)

### X(1404) = X(44)-BETH CONJUGATE OF X(44)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(44)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1404) lies on these lines:
6,41   35,572   42,1397   44,1319   57,89   59,672   217,1409   649,854   651,1429

X(1404) = X(1319)-Ceva conjugate of X(902)
X(1404) = crosspoint of X(i) and X(j) for these (i,j): (6,909), (57,1411)
X(1404) = crosssum of X(2) and X(908)
X(1404) = crossdifference of every pair of points on line X(8)X(522)

### X(1405) = X(45)-BETH CONJUGATE OF X(45)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(45)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1405) lies on these lines:
6,41   31,51   35,573   44,65   57,88   169,1046

X(1405) = isogonal conjugate of X(30608)

### X(1406) = X(46)-BETH CONJUGATE OF X(46)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(46)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1406) lies on these lines:
3,1464   31,56   55,1066   57,1203   65,222   1411,1413   1427,1454

X(1406) = X(34)-Ceva conjugate of X(56)

### X(1407) = X(57)-BETH CONJUGATE OF X(57)

Trilinears    (1 - cos A)(1 - cos A + cos B + cos C) : :
Trilinears    a/(b + c - a)^2 : :

X(1407) is the vertex conjugate of the foci of the inellipse that is the barycentric square of the Gergonne line (with center X(4000) and perspector X(279)). (Randy Hutson, October 15, 2018)

X(1407) lies on these lines:
3,951   6,57   7,940   31,56   34,1413   55,1458   63,220   73,1466   81,279   109,1477   189,1146   278,1086   478,1122   479,1462   534,553   608,1435   614,1456   739,934   942,1448   1104,1394   1119,1396   1333,1412   1357,1397   1398,1408   1401,1460   1464,1470

X(1407) = isogonal conjugate of X(346)
X(1407) = X(i)-Ceva conjugate of X(j) for these (i,j): (269,56), (1119,1398), (1262,1461), (1275,934), (1396,1435)
X(1407) = X(i)-cross conjugate of X(j) for these (i,j): (604,56), (608,1413), (1042,269)
X(1407) = cevapoint of X(604) and X(1106)
X(1407) = crosspoint of X(i) and X(j) for these (i,j): (57,1422), (269,738), (279,1119), (934,1275), (1262, 1461), (1396,1412)
X(1407) = crosssum of X(i) and X(j) for these (i,j): (200,728), (220,1260)
X(1407) = trilinear product X(56)*X(57)
X(1407) = X(92)-isoconjugate of X(1260)
X(1407) = perspector of ABC and unary cofactor triangle of Ayme triangle
X(1407) = barycentric square of X(57)

### X(1408) = X(58)-BETH CONJUGATE OF X(58)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(58)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1408) lies on these lines:
21,1319   56,58   60,757   65,81   283,1037   284,1466   603,604   1398,1407   1399,1402   1413,1474

X(1408) = X(1412)-Ceva conjugate of X(1333)
X(1408) = X(604)-cross conjugate of X(1412)
X(1408) = cevapoint of X(604) and X(1397)
X(1408) = crosspoint of X(1014) and X(1396)

### X(1409) = X(71)-BETH CONJUGATE OF X(71)

Trilinears    a(sec B + sec C) : :
Trilinears    (sin 2A)(cos B + cos C) : :

Let V = isotomic conjugate of polar conjugate of line X(1)X(3) and W = polar conjugate of isotomic conjugate of line X(1)X(3); then X(1409) = V∩W. (Randy Hutson, December 26, 2015)

X(1409) lies on these lines:
6,19   31,184   48,577   63,77   71,73   109,284   213,1042   217,1404   287,651   800,1195   1333,1399

X(1409) = X(i)-Ceva conjugate of X(j) for these (i,j): (6,1400), (65,1402), (73,228), (222,73)
X(1409) = crosspoint of X(i) and X(j) for these (i,j): (6,48), (65,1214), (222,603)
X(1409) = crosssum of X(i) and X(j) for these (i,j): (2,92), (21,1172), (281,318)
X(1409) = X(92)-isoconjugate of X(21)
X(1409) = bicentric sum of PU(19)
X(1409) = PU(19)-harmonic conjugate of X(1946)
X(1409) = barycentric product of PU(83)
X(1409) = polar conjugate of isotomic conjugate of X(22341)

### X(1410) = X(73)-BETH CONJUGATE OF X(73)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(73)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1410) lies on these lines:
3,77   25,34   32,604   58,1461   73,228   98,934   184,603   1042,1402

X(1410) = X(56)-Ceva conjugate of X(1042)
X(1410) = crosspoint of X(i) and X(j) for these (i,j): (56,603), (1427,1439)
X(1410) = crosssum of X(8) and X(318)

### X(1411) = X(80)-BETH CONJUGATE OF X(80)

Trilinears    (1 - cos A)u(a,b,c) : : , where u : v : w = X(80)
Trilinears    (1 - cos A)/(1 - 2 cos A) : :

X(1411) lies on these lines:
1,5   56,244   58,65   86,664   106,1168   269,1358   388,977   996,1215   1406,1413

X(1411) = X(i)-cross conjugate of X(j) for these (i,j): (1404,57), (1457,56)
X(1411) = cevapoint of X(65) and X(1319)
X(1411) = isogonal conjugate of X(4511)
X(1411) = inverse-in-Feuerbach-hyperbola of X(1807)
X(1411) = {X(1),X(80)}-harmonic conjugate of X(1807)

### X(1412) = X(81)-BETH CONJUGATE OF X(81)

Trilinears    (1 - cos A)u(a,b,c), where u : v : w = X(81)
Trilinears    cot(?BAACA) : :, where BA and CA are the touchpoints of the B- and C-excircles with line BC, resp.

X(1412) lies on these lines:
21,1420   28,1422   56,58   57,77   86,226   109,741   110,1477   269,1396   283,951   394,579   552,553   572,940   580,1092   1171,1400   1326,1402   1333,1407   1427,1461

X(1412) = isogonal conjugate of X(2321)
X(1412) = isotomic conjugate of X(30713)
X(1412) = X(1014)-Ceva conjugate of X(58)
X(1412) = X(i)-cross conjugate of X(j) for these (i,j): (56,1014), (604,1408), (1333,58), (1407,1396)
X(1412) = cevapoint of X(i) and X(j) for these (i,j): (56,604), (1333,1408)

### X(1413) = X(84)-BETH CONJUGATE OF X(84)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(84)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1413) lies on these lines:
1,84   3,1167   6,603   34,1407   57,937   64,1364   73,939   86,285   189,1220   280,1222   998,1448   1406,1411   1408,1474

X(1413) = X(i)-Ceva conjugate of X(j) for these (i,j): (84,56), (1422,1436)
X(1413) = X(i)-cross conjugate of X(j) for these (i,j): (608,1407), (1106,56)
X(1413) = crosspoint of X(84) and X(1256)
X(1413) = crosssum of X(40) and X(1103)

### X(1414) = X(99)-BETH CONJUGATE OF X(99)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(99)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1414) lies on these lines:
7,757   58,1416   85,603   99,109   110,934   162,658   163,1019   348,1098   651,662   1014,1122

X(1414) = X(i)-cross conjugate of X(j) for these (i,j): (110,662), (1019,1434)
X(1414) = cevapoint of X(i) and X(j) for these (i,j): (58,1019), (109,651)

X(1414) = isogonal conjugate of X(4041)
X(1414) = isotomic conjugate of X(4086)
X(1414) = trilinear pole of line X(57)X(77)

### X(1415) = X(101)-BETH CONJUGATE OF X(101)

Trilinears    a^2/(cos B - cos C) : :
Trilinears    a^2/((a - b - c) (b - c)) : :

X(1415) lies on these lines:
6,909   32,56   41,609   57,609   65,172   101,109   108,112   198,478   213,1399   571,608   604,1417   651,662   910,1455   919,934   923,1402   1055,1457   1333,1400

X(1415) = X(i)-Ceva conjugate of X(j) for these (i,j): (59,1397), (109,692), (1262,56)
X(1415) = X(i)-cross conjugate of X(j) for these (i,j): (649,1333), (667,56), (1397,59)
X(1415) = cevapoint of X(i) and X(j) for these (i,j): (32,667), (649,1400)
X(1415) = crosspoint of X(i) and X(j) for these (i,j): (108,651), (109,1461), (112,163)
X(1415) = crosssum of X(521) and X(650)
X(1415) = crossdifference of every pair of points on line X(11)X(123)
X(1415) = barycentric product of X(1381) and X(1382)

X(1415) = isogonal conjugate of X(4391)
X(1415) = trilinear pole of line X(31)X(184)
X(1415) = X(92)-isoconjugate of X(521)
X(1415) = barycentric product of PU(102)
X(1415) = perspector of unary cofactor triangles of outer and inner Garcia triangles

### X(1416) = X(105)-BETH CONJUGATE OF X(105)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(105)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1416) lies on these lines:
6,1362   31,57   32,56   58,1414   294,1468   727,927   738,1106   919,1458   951,1193   1357,1397   1395,1435

X(1416) = X(1462)-Ceva conjugate of X(1438)
X(1416) = cevapoint of X(56) and X(1428)
X(1416) = X(56)-Hirst inverse of X(1438)

### X(1417) = X(106)-BETH CONJUGATE OF X(106)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(106)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1417) lies on these lines:
56,106   65,1320   88,961   604,1415   901,1319   1014,1122   1037,1470

### X(1418) = X(142)-BETH CONJUGATE OF X(142)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(142)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1418) lies on these lines:
6,57   7,37   44,1445   56,1279   65,1458   77,1100   142,1212   354,2293   553,1214   603,1456   942,991   1014,1333   1086,1108   1104,1448   1122,1400   1155,1253

X(1418) = X(57)-Ceva conjugate of X(1475)
X(1418) = X(1475)-cross conjugate of X(354)
X(1418) = crosspoint of X(57) and X(279)
X(1418) = crosssum of X(9) and X(220)

### X(1419) = X(144)-BETH CONJUGATE OF X(144)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(144)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1419) lies on these lines:
1,971   6,57   7,1449   9,77   41,738   48,1461   73,991   109,1253   347,527   1201,1420   1443,1445

X(1419) = X(1)-Ceva conjugate of X(57)
X(1419) = crosspoint of X(1) and X(165)
X(1419) = X(57)-Hirst inverse of X(910)

### X(1420) = X(145)-BETH CONJUGATE OF X(145)

Trilinears    (1 - cos A) u(a,b,c), where u :v : w = X(145)
Trilinears    3 cos A + cos B + cos C - 3 : cos A + 3 cos B + cos C - 3 : cos A + cos B + 3 cos C - 3 (Peter Moses, March 12, 2011)
Trilinears    -1 + sin(A/2)csc(B/2)csc(C/2) (Randy Hutson, August 23, 2011)
Trilinears    (3a - b - c)/(a - b - c) : : (Randy Hutson, July 31, 2018)

Let Ja, Jb, Jc be the excenters and I the incenter. Let A' be the centroid of JbJcI, and define B' and C' cyclically; A'B'C' is also the cross-triangle of the excentral and 2nd circumperp triangles. The triangle A'B'C' is homothetic to the Hutson intouch triangle at X(1420). (Randy Hutson, July 31 2018)

X(1420) lies on these lines:
1,3   9,604   21,1412   34,106   73,995   84,104   109,1106   222,1191   223,1104   226,452   269,1279   386,1450   388,1125   595,603   610,1108   738,934   936,956   944,1210   1042,1149   1201,1419   1394,1457   1400,1449

X(1420) = isogonal conjugate of X(3680)
X(1420) = X(i)-Ceva conjugate of X(j) for these (i,j): (269,57), (765,109)
X(1420) = {X(1),X(56)}-harmonic conjugate of X(57)
X(1420) = {X(3513),X(3514)}-harmonic conjugate of X(165)
X(1420) = homothetic center of 1st Johnson-Yff triangle and cross-triangle of Aquila and anti-Aquila triangles

### X(1421) = X(149)-BETH CONJUGATE OF X(149)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(149)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1421) lies on these lines:
1,5   31,57   34,106   595,1393   1279,1465

### X(1422) = X(189)-BETH CONJUGATE OF X(189)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(189)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1422) lies on these lines:
1,84   2,77   28,1412   34,1256   35,1079   57,1436   268,1214   269,278   271,1257   280,1219

X(1422) = isogonal conjugate of X(2324)
X(1422) = X(i)-Ceva conjugate of X(j) for these (i,j): (189,57), (1440,84)
X(1422) = X(i)-cross conjugate of X(j) for these (i,j): (34,269), (1407,57), (1436,84)
X(1422) = cevapoint of X(1413) and X(1436)

### X(1423) = X(192)-BETH CONJUGATE OF X(192)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (ab + ac - bc)/(b + c - a)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1423) lies on these lines:
1,256   2,7   6,1429   43,1403   56,87   65,984   85,1221   241,1122   269,292   604,651   1201,1419

X(1423) = isogonal conjugate of X(2319)
X(1423) = X(56)-Ceva conjugate of X(57)
X(1423) = crosspoint of X(56) and X(1403)
X(1423) = X(6)-Hirst inverse of X(1429)

### X(1424) = X(194)-BETH CONJUGATE OF X(194)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(194)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1424) lies on these lines: 56,87   57,85   222,1429

X(1424) = X(604)-Ceva conjugate of X(57)

### X(1425) = X(201)-BETH CONJUGATE OF X(201)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(201)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1425) lies on these lines:
1,185   6,1398   12,125   25,221   34,51   55,1204   56,184   65,225   72,307   73,228   181,1254   213,1042   217,1015   999,1181   1093,1148   1106,1401

X(1425) = X(i)-Ceva conjugate of X(j) for these (i,j): (65,1254), (1020,647)
X(1425) = crosspoint of X(65) and X(73)
X(1425) = crosssum of X(21) and X(29)

### X(1426) = X(225)-BETH CONJUGATE OF X(225)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(225)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1426) lies on these lines:
4,7   25,34   65,225   72,860   226,429   227,228   278,959   517,1068   1254,1400

X(1426) = isogonal conjugate of X(1792)
X(1426) = X(34)-Ceva conjugate of X(1042)
X(1426) = crosspoint of X(i) and X(j) for these (i,j): (34,1118), (1119,1435)
X(1426) = crosssum of X(78) and X(1259)

### X(1427) = X(226)-BETH CONJUGATE OF X(226)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(226)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1427) lies on these lines:
2,85   3,1448   6,57   7,941   25,34   31,1456   37,226   42,65   77,940   111,934   212,1155   278,393   307,1211   354,1458   581,942   1014,1169   1106,1451   1333,1396   1406,1454   1412,1461

X(1427) = isogonal conjugate of X(2287)
X(1427) = X(i)-Ceva conjugate of X(j) for these (i,j): (269,1042), (1446,1439)
X(1427) = X(i)-cross conjugate of X(j) for these (i,j): (1400,65), (1410,1439)
X(1427) = cevapoint of X(1042) and X(1400)
X(1427) = crosspoint of X(i) and X(j) for these (i,j): (57,278), (269, 279)
X(1427) = crosssum of X(i) and X(j) for these (i,j): (6,610), (9,219), (200,220)

### X(1428) = X(238)-BETH CONJUGATE OF X(238)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(238)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1428) lies on these lines:
1,182   3,613   6,41   31,1403   36,511   57,985   58,1178   59,518   60,757   65,82   184,614   238,1284   499,1352   611,999   651,1463   692,1279   961,1258   1456,1462

X(1428) = X(1416)-Ceva conjugate of X(56)
X(1428) = crosspoint of X(1014) and X(1462)
X(1428) = X(i)-Hirst inverse of X(j) for these (i,j): (31,1403), (56,604)

### X(1429) = X(239)-BETH CONJUGATE OF X(239)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(239)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1429) lies on these lines:
1,3   6,1423   7,604   73,1244   81,1432   83,226   222,1424   238,1284   239,385   552,553   651,1404   1458,1462

X(1429) = X(i)-Ceva conjugate of X(j) for these (i,j): (1447,238), (1462,57)
X(1429) = X(1284)-cross conjugate of X(1447)
X(1429) = crossdifference of every pair of points on line X(210)X(650)
X(1429) = X(i)-Hirst inverse of X(j) for these (i,j): (6,1423), (56,57)

### X(1430) = X(243)-BETH CONJUGATE OF X(243)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(243)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1430) lies on these lines:
1,1013   4,1468   25,34   27,58   31,278   57,1096   92,171   108,1458   162,238   281,750   603,1118

X(1430) = X(34)-Hirst inverse of X(56)

### X(1431) = X(256)-BETH CONJUGATE OF X(256)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(256)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1431) lies on these lines:
1,256   6,893   7,870   56,904   57,87   58,1178   65,257   86,1447   292,694   518,1222   758,996   979,1046

X(1431) = X(904)-cross conjugate of X(893)

### X(1432) = X(257)-BETH CONJUGATE OF X(257)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(257)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1432) lies on these lines:
1,256   2,257   7,330   28,1178   56,985   57,893   65,291   81,1429   105,904   961,1042   1258,1400

X(1432) = isogonal conjugate of X(2329)
X(1432) = X(893)-cross conjugate of X(256)
X(1432) = cevapoint of X(893) and X(1431)

### X(1433) = X(271)-BETH CONJUGATE OF X(271)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(271)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1433) lies on these lines:
1,84   6,282   29,81   55,947   56,102   78,271   145,280   219,255   284,1436   945,999

X(1433) = X(i)-Ceva conjugate of X(j) for these (i,j): (189,1436), (271,3), (285,84)
X(1433) = X(i)-cross conjugate of X(j) for these (i,j): (6,222), (603,3)
X(1433) = cevapoint of X(1364) and X(1459)

### X(1434) = X(274)-BETH CONJUGATE OF X(274)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(274)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1434) lies on these lines:
7,21   27,1088   57,85   58,1414   65,664   81,279   99,1477   270,757   310,349   332,951   552,553   658,1446   576,1475

X(1434) = isogonal conjugate of X(1334)
X(1434) = isotomic conjugate of X(2321)
X(1434) = X(552)-Ceva conjugate of X(1014)
X(1434) = X(i)-cross conjugate of X(j) for these (i,j): (57,1014), (81,86), (553,7), (1019,1414)
X(1434) = cevapoint of X(i) and X(j) for these (i,j): (7,57), (81,1014)

### X(1435) = X(278)-BETH CONJUGATE OF X(278)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(278)
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = tan A tan2(A/2)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1435) lies on these lines:
1,951   19,57   25,34   27,1088   33,354   48,223   108,1477   154,1456   244,1096   269,1396   608,1407   913,1461   1395,1416

X(1435) = X(i)-Ceva conjugate of X(j) for these (i,j): (1119,34), (1396,1407)
X(1435) = X(i)-cross conjugate of X(j) for these (i,j): (608,34), (1106,269), (1426,1119)
X(1435) = cevapoint of X(608) and X(1398)

### X(1436) = X(282)-BETH CONJUGATE OF X(282)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(282)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1436) lies on these lines:
3,9   6,603   19,56   48,55   57,1422   189,333   284,1433   673,1440

X(1436) = isogonal conjugate of X(329)
X(1436) = X(i)-Ceva conjugate of X(j) for these (i,j): (189,1433), (282,6), (1422,1413)
X(1436) = X(i)-cross conjugate of X(j) for these (i,j): (25,56), (604,6)
X(1436) = crosspoint of X(84) and X(1422)
X(1436) = crosssum of X(9) and X(1490)

### X(1437) = X(283)-BETH CONJUGATE OF X(283)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(283)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1437) lies on these lines:
3,49   21,104   28,60   35,692   48,255   56,58   163,911   182,474   215,1364   284,1433   849,1333   1014,1175

X(1437) = X(i)-Ceva conjugate of X(j) for these (i,j): (60,58), (81,1333)
X(1437) = X(603)-cross conjugate of X(58)
X(1437) = cevapoint of X(48) and X(184)
X(1437) = crosspoint of X(81) and X(1444)
X(1437) = crosssum of X(4) and X(451)

### X(1438) = X(294)-BETH CONJUGATE OF X(294)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(294)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1438) lies on these lines:
1,41   6,692   32,56   58,163   86,142   87,572   106,919   269,604   665,911   909,1024   950,1220

X(1438) = X(1462)-Ceva conjugate of X(1416)
X(1438) = crosspoint of X(105) and X(1462)
X(1438) = X(i)-Hirst inverse of X(j) for these (i,j): (56,1416)

X(1438) = isogonal conjugate of X(3912)
X(1438) = trilinear pole of PU(48) (line X(31)X(649))
X(1438) = barycentric product of PU(96)
X(1438) = crossdifference of every pair of points on line X(918)X(2254)

### X(1439) = X(307)-BETH CONJUGATE OF X(307)

Trilinears   (1 - cos A)u(a,b,c) : : , where u : v : w = X(307)
Trilinears    (b + c)(b^2 + c^2 - a^2)/(b + c - a)^2 : :

Let A' be the homothetic center of the orthic triangles of the intouch and A-extouch triangles, and define B' and C' cyclically. The triangle A'B'C' is perspective to the intouch triangle at X(1439). (Randy Hutson, September 14, 2016)

X(1439) lies on these lines:
1,64   3,77   4,7   6,57   37,1020   54,1443   71,1214   72,307   74,934   86,658   241,579   284,1461   347,517   1014,1175   1042,1245   1088,1246

X(1439) = X(i)-Ceva conjugate of X(j) for these (i,j): (658,905), (1446,1427)
X(1439) = X(i)-cross conjugate of X(j) for these (i,j): (73,1214), (656,1020), (1410,1427)
X(1439) = crosspoint of X(7) and X(77)
X(1439) = crosssum of X(i) and X(j) for these (i,j): (24,204), (33,55)
X(1439) = isogonal conjugate of X(4183)
X(1439) = perspector of intouch triangle and 3rd extouch triangle
X(1439) = perspector of ABC and cross-triangle of ABC and 3rd extouch triangle

### X(1440) = X(309)-BETH CONJUGATE OF X(309)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(309)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1440) lies on these lines:
2,77   7,84   27,1014   75,280   86,285   269,1256   271,307   273,279   673,1436

X(1440) = X(i)-cross conjugate of X(j) for these (i,j): (84,189), (269,7), (278,279)
X(1440) = cevapoint of X(84) and X(1422)

### X(1441) = X(313)-BETH CONJUGATE OF X(313)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(313)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1441) lies on these lines:
2,92   7,8   10,307   12,313   19,379   21,286   34,964   57,1150   86,664   95,404   226,306   253,318   269,996   274,961   287,651   305,561   443,1119   1074,1111   1402,1447

X(1441) = isogonal conjugate of X(2194)
X(1441) = isotomic conjugate of X(21)
X(1441) = X(i)-Ceva conjugate of X(j) for these (i,j): (75,307), (85,226), (349,321), (664,693)
X(1441) = X(i)-cross conjugate of X(j) for these (i,j): (10,321), (12,226), (226,1446), (442,2), (121,76), (1214,1231)
X(1441) = cevapoint of X(i) and X(j) for these (i,j): (10,226), (65,1214)
X(1441) = crosspoint of X(75) and X(264)
X(1441) = crosssum of X(31) and X(184)

### X(1442) = X(319)-BETH CONJUGATE OF X(319)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(319)
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = (1 + 2 cos A)/(1 + cos A)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1442) lies on these lines:
1,7   2,914   8,326   37,651   65,1014   74,934   81,1214   86,664   226,1029   241,1100   319,1273   1082,1250   1445,1449

### X(1443) = X(320)-BETH CONJUGATE OF X(320)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(320)
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = (1 - 2 cos A)/(1 + cos A)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1443) lies on these lines:
1,7   44,241   54,1439   57,89   59,1155   60,757   88,1465   320,1464   679,1318   934,953   1419,1445

X(1443) = anticomplement of X(1489)
X(1443) = cevapoint of X(1319) and X(1465)
X(1443) = crossdifference of every pair of points on line X(657)X(1334)

### X(1444) = X(332)-BETH CONJUGATE OF X(332)

Trilinears       (cot A)/(b + c) : (cot B)/(c + a) : (cot C)/(a + b)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1444) lies on these lines:
1,969   3,69   7,21   28,242   48,63   58,988   71,1332   77,283   81,593   99,104   100,319   189,333   524,1030   662,911   963,1043

X(1444) = isogonal conjugate of X(1824)
X(1444) = X(i)-Ceva conjugate of X(j) for these (i,j): (261,86), (274,81)
X(1444) = X(i)-cross conjugate of X(j) for these (i,j): (77,86), (1437,58), (1459,1332), (1473,58)
X(1444) = cevapoint of X(3) and X(63)

### X(1445) = X(344)-BETH CONJUGATE OF X(344)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(344)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1445) lies on these lines:
1,1170   2,7   6,77   19,273   40,390   44,1418   46,516   56,78   65,1001   169,1446   269,651   942,954   1038,1451   1419,1443   1442,1449

X(1445) = X(i)-Ceva conjugate of X(j) for these (i,j): (765,651), (1088,1)

### X(1446) = X(349)-BETH CONJUGATE OF X(349)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(349)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1446) lies on these lines:
2,85   4,7   10,307   57,379   76,1229   98,934   169,1445   226,857   294,1170   321,349   658,1434   1111,1210

X(1426) = isotomic conjugate of X(2287)
X(1446) = X(226)-cross conjugate of X(1441)
X(1446) = cevapoint of X(1427) and X(1439)
X(1446) = crosspoint of X(85) and X(331)

### X(1447) = X(350)-BETH CONJUGATE OF X(350)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(350)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1447) lies on these lines:
2,7   25,273   36,1111   56,85   75,183   77,614   86,1431   87,269   105,927   230,1086   239,385   241,292   261,552   320,325   350,1281   459,1119   664,1319   673,910   1402,1441

X(1447) = X(i)-cross conjugate of X(j) for these (i,j): (238,239), (1284,1429)
X(1447) = cevapoint of X(i) and X(j) for these (i,j): (238,1429), (241,1463)
X(1447) = crossdifference of every pair of points on line X(663)X(1334)
X(1447) = X(7)-Hirst inverse of X(57)

### X(1448) = X(377)-BETH CONJUGATE OF X(377)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(377)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1448) lies on these lines:
1,7   3,1427   28,34   46,255   65,222   85,1010   223,386   226,975   241,405   443,948   942,1407   998,1413   1104,1418   1465,1466

### X(1449) = {X(1),X(6)}-HARMONIC CONJUGATE OF X(9)

Trilinears       3a + b + c : 3b + c + a : 3c + a + b
Trilinears       a + s : b + s : c + s
Barycentrics  a(3a + b + c) : b(3b + c + a) : c(3c + a + b)

X(1449) lies on these lines:
1,6   7,1419   32,988   34,1172   40,572   43,1051   57,77   65,380   87,1045   198,999   579,1475   610,942   894,1278   966,1125   1400,1420   1442,1445

X(1449) = {X(1),X(6)}-harmonic conjugate of X(9)
X(1449) = X(391)-beth conjugate of X(391)

### X(1450) = X(392)-BETH CONJUGATE OF X(392)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(392)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1450) lies on these lines:
1,631   3,1057   6,41   31,1470   36,1064   42,1319   57,957   65,244   386,1420   388,978   1191,1466

### X(1451) = X(405)-BETH CONJUGATE OF X(405)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(405)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1451) lies on these lines:
1,201   6,41   28,34   31,65   36,581   46,601   255,942   270,273   1038,1445   1106,1427

### X(1452) = X(406)-BETH CONJUGATE OF X(406)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(406)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1452) lies on these lines:
1,24   4,46   19,208   25,65   28,34   33,40   227,607   1038,1039

### X(1453) = X(452)-BETH CONJUGATE OF X(452)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 3a3 + b3 + c3 + 3a2b + 3a2c + ab2 + ac2 - b2c - bc2 + 2abc
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1453) lies on these lines:
1,6   4,204   28,34   31,40   43,1009   56,223   73,995   84,1039   212,595   222,1467   387,950   581,1193   614,1468

### X(1454) = X(498)-BETH CONJUGATE OF X(498)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(498)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1454) lies on these lines:
1,3   5,920   12,63   31,1393   34,1399   90,381   201,750   208,407   453,1014   603,1254   1406,1427

### X(1455) = X(515)-BETH CONJUGATE OF X(515)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(515)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1455) lies on these lines:
1,84   3,227   25,34   36,1465   37,478   65,603   73,820   109,517   117,515   513,663   608,1108   910,1415   958,1038   993,1214

X(1455) = X(104)-Ceva conjugate of X(56)
X(1455) = crosspoint of X(1) and X(1295)

### X(1456) = X(516)-BETH CONJUGATE OF X(516)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(516)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1456) lies on these lines:
1,971   6,19   7,1386   31,1427   55,223   56,269   77,1001   109,1155   154,1435   222,354   227,1253   238,241   513,663   518,651   603,1418   614,1407   1042,1104   1428,1462

X(1456) = X(105)-Ceva conjugate of X(56)
X(1456) = crosspoint of X(i) and X(j) for these (i,j): (1,972), (269,1462)
X(1456) = crosssum of X(1) and X(971)
X(1456) = crossdifference of every pair of points on line X(9)X(521)

### X(1457) = X(517)-BETH CONJUGATE OF X(517)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(517)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1457) lies on these lines:
1,4   3,945   31,56   36,109   48,608   57,957   65,1193   201,960   222,999   350,664   392,1214   478,604   513,663   517,1465   1055,1415   1394,1420

X(1457) = X(106)-Ceva conjugate of X(56)
X(1457) = crosspoint of X(i) and X(j) for these (i,j): (1,102), (56,1411)
X(1457) = crosssum of X(1) and X(515)
X(1457) = crossdifference of every pair of points on line X(9)X(652)

### X(1458) = X(518)-BETH CONJUGATE OF X(518)

Trilinears    (1 - cos A)u(a,b,c) : : , where u : v : w = X(518)

X(1458) lies on these lines:
1,7   3,1037   6,41   31,222   36,59   38,1214   42,57   55,1407   64,963   65,1418   108,1430   109,840   185,1208   223,614   238,651   241,518   244,1465   256,1476   354,1427   513,663   672,1362   919,1416   942,1254   976,1038   999,1064   1201,1419   1401,1402   1429,1462

X(1458) = X(i)-Ceva conjugate of X(j) for these (i,j): (241,672), (1477,56)
X(1458) = crosspoint of X(i) and X(j) for these (i,j): (1,103)
X(1458) = crosssum of X(1) and X(516)
X(1458) = crossdifference of every pair of points on line X(9)X(522)
X(1458) = X(i)-Hirst inverse of X(j) for these (i,j): (6,56), (6,72)
X(1458) = perspector of conic {A,B,C,X(109),PU(48)}
X(1458) = crossdifference of the isogonal conjugates of PU(48)

### X(1459) = X(521)-BETH CONJUGATE OF X(521)

Trilinears        (b - c)cos A: (c - a)cos B: (a - b)cos C
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1459) lies on these lines:
1,522   6,657   106,953   242,514   513,663   520,647   521,656   649,834

X(1459) = reflection of X(656) in X(905)
X(1459) = isogonal conjugate of X(1897)
X(1459) = X(i)-Ceva conjugate of X(j) for these (i,j):
(101,1473), (109,603), (514,649), (905,652), (1331,3), (1332,71), (1433,1364), (1461,6)

X(1459) = X(647)-cross conjugate of X(905)
X(1459) = cevapoint of X(647) and X(810)
X(1459) = crosspoint of X(i) and X(j) for these (i,j): (1,109), (3,1331), (81,934), (1332,1444)
X(1459) = crosssum of X(1) and X(522)
X(1459) = crossdifference of every pair of points on line X(4)X(9)
X(1459) = orthojoin of X(1146)

### X(1460) = X(612)-BETH CONJUGATE OF X(612)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(612)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1460) lies on these lines:
1,3   6,181   8,961   25,31   42,604   222,1469   332,1014   388,1010   959,1036   1254,1398   1401,1407

X(1460) = isogonal conjugate of X(30479)
X(1460) = crossdifference of every pair of points on line X(650)X(3910)

### X(1461) = X(651)-BETH CONJUGATE OF X(651)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(651)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1461) lies on these lines:
6,911   48,1419   57,909   58,1410   77,572   101,651   109,692   269,604   284,1439   514,653   658,662   913,1435   923,1042   1025,1332   1412,1427

X(1461) = isogonal conjugate of X(3239)
X(1461) = X(i)-Ceva conjugate of X(j) for these (i,j): (934,109), (1262,1407)
X(1461) = X(i)-cross conjugate of X(j) for these (i,j): (649,56), (1407,1262), (1415,109)
X(1461) = cevapoint of X(i) and X(j) for these (i,j): (6,1459), (56,649)
X(1461) = trilinear pole of line X(31)X(56) (the isogonal conjugate of the isotomic conjugate of the Soddy line)

### X(1462) = X(673)-BETH CONJUGATE OF X(673)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(673)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1462) lies on these lines:
6,7   31,57   213,1170   241,1279   269,604   479,1407   608,1119   739,927   919,1465   934,1015   1014,1333   1428,1456   1429,1458

X(1462) = X(i)-cross conjugate of X(j) for these (i,j): (1428,1014), (1438,105), (1456,269)
X(1462) = cevapoint of X(i) and X(j) for these (i,j): (6,1279), (57,1429), (1416,1438)
X(1462) = X(i)-Hirst inverse of X(j) for these (i,j): (57,105)
X(1462) = trilinear pole of line X(56)X(667)

### X(1463) = X(726)-BETH CONJUGATE OF X(726)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(726)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1463) lies on these lines:
7,8   44,583   56,87   181,553   226,1401   513,663   651,1428

X(1463) = isogonal conjugate of X(8851)
X(1463) = X(1447)-Ceva conjugate of X(241)
X(1463) = {P,Q}-harmonic conjugate of X(65), where P and Q are the incircle intercepts of line X(7)X(8)

### X(1464) = X(758)-BETH CONJUGATE OF X(758)

Trilinears    (1 - cos A)(b + c)(b^2 + c^2 - a^2 - bc) : :

X(1464) lies on these lines:
1,30   3,1406   42,65   56,58   320,1443   354,1064   513,663   1407,1470

X(1464) = crosspoint of X(1) and X(74)
X(1464) = crosssum of X(1) and X(30)
X(1464) = crossdifference of every pair of points on line X(9)X(1021)
X(1464) = PU(86)-harmonic conjugate of X(9404)

### X(1465) = X(908)-BETH CONJUGATE OF X(908)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(908)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1465) lies on these lines:
1,227   2,92   3,34   5,225   6,57   36,1455   46,221   56,998   65,386   73,942   88,1443   106,1168   109,1155   241,514   244,1458   474,1038   517,1457   919,1462   1193,1254   1279,1421   1448,1466

X(1465) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,1361), (88,57), (1443,1319)
X(1465) = X(1361)-cross conjugate of X(7)

### X(1466) = X(936)-BETH CONJUGATE OF X(936)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(936)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1466) lies on these lines:
1,3   6,603   7,404   12,443   42,1106   73,1407   221,1193   222,386   226,474   284,1408   388,1376   1012,1210   1191,1450   1448,1465

### X(1467) = X(938)-BETH CONJUGATE OF X(938)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(938)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1467) lies on these lines:
1,3   7,452   28,1412   34,269   142,388   207,278   222,1453   604,610   614,1042   1066,1103   1104,1394

### X(1468) = X(958)-BETH CONJUGATE OF X(958)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(958)
Trilinears       a(as + bc) : b(bs + ca) : c(cs + ab)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Let Oa be the A-extraversion of the Conway circle (the circle centered at the A-excenter and passing through A, with radius sqrt(r_a^2 + s^2), where r_a is the A-exradius). Let Pa be the perspector of Oa, and La the polar of Pa wrt Oa. Define Lb and Lc cyclically. Let A' = Lb ∩ Lc, B' = Lc ∩ La, C' = La ∩ Lb. The lines AA', BB', CC' concur in X(1468). (Randy Hutson, April 9, 2016)

X(1468) lies on these lines:
1,21   3,42   4,1430   6,41   8,171   10,750   36,386   43,404   57,961   65,603   75,757   222,1042   294,1416   330,985   354,1104   474,899   517,601   518,976   602,1385   614,1453   748,1125   756,975   940,958   995,1203   999,1201   1149,1191

X(1468) = crosssum of X(9) and X(612)
X(1468) = crossdifference of every pair of points on line X(522)X(661)

### X(1469) = X(984)-BETH CONJUGATE OF X(984)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(984)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1469) lies on these lines:
1,256   3,611   6,41   7,8   12,141   36,182   42,1403   43,57   51,614   55,1350   109,753   193,330   222,1460   613,999   751,1319   970,888

### X(1470) = X(997)-BETH CONJUGATE OF X(997)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(997)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1470) lies on these lines:
1,3   6,909   11,1012   12,474   31,1450   73,1106   108,378   109,995   388,404   603,1193   1037,1417   1407,1464

### X(1471) = X(1001)-BETH CONJUGATE OF X(1001)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(1001)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1471) lies on these lines:
1,1170   6,41   7,238   31,57   36,991   58,269   65,1279   212,354   226,748   241,1386   307,1125   602,942   603,1418

### X(1472) = X(1036)-BETH CONJUGATE OF X(1036)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(1036)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1472) lies on these lines:
3,31   28,614   32,48   56,1395   58,988   104,1039   238,987   595,997   603,1397   727,1310

### X(1473) = X(1040)-BETH CONJUGATE OF X(1040)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(1040)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1473) lies on these lines:
3,63   25,57   31,56   38,55   184,222   197,1155   198,672   988,1036   1394,1398

X(1473) = X(101)-Ceva conjugate of X(1459)
X(1473) = crosspoint of X(58) and X(1444)

### X(1474) = X(1172)-BETH CONJUGATE OF X(1172)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(1172)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1474) lies on these lines:
1,19   4,572   6,25   24,573   27,86   29,1220   34,604   56,608   106,112   163,913   198,939   269,1396   281,996   286,870   459,966   468,1213   1408,1413

X(1474) = isogonal conjugate of X(306)
X(1474) = X(27)-Ceva conjugate of X(58)
X(1474) = X(604)-cross conjugate of X(608)
X(1474) = cevapoint of X(604) and X(608)
X(1474) = crosspoint of X(28) and X(1396)
X(1474) = crossdifference of every pair of points on line X(525)X(656)

### X(1475) = X(1212)-BETH CONJUGATE OF X(1212)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(1212)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1475) lies on these lines:
1,672   6,41   39,42   57,279   58,163   71,583   213,1015   218,999   354,1212   579,1449   607,1398   649,764   673,1434   934,1170

X(1475) = X(i)-Ceva conjugate of X(j) for these (i,j): (57,1418), (692,649), (934,663)
X(1475) = crosspoint of X(i) and X(j) for these (i,j): (6,57), (354,1418)
X(1475) = crosssum of X(2) and X(9)

### X(1476) = X(1222)-BETH CONJUGATE OF X(1222)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(1222)
Trilinears       g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 1/[(sin B) tan(B/2) + (sin C) tan(C/2)]
Trilinears        h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = 1/[(b + c - a)(b2 + c2 - 2bc + ab + ac)]
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1476) lies on these lines:
1,1106   3,1000   4,496   8,56   9,604   21,1319   65,1320   80,1210   172,294   256,1458   314,1014   651,1201   942,1389   943,1385

X(1476) = isogonal conjugate of X(3057)
X(1476) = X(i)-cross conjugate of X(j) for these (i,j): (15,1222), (649,651)
X(1476) = cevapoint of X(1) and X(56)

### X(1477) = X(1280)-BETH CONJUGATE OF X(1280)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (1 - cos A)u(a,b,c), where u : v : w = X(1280)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1477) lies on the circumcircle and these lines:
1,1292   55,1293   56,101   57,100   99,1434   108,1435   109,1407   110,1412   738,934   919,1416   1308,1319

X(1477) = X(672)-cross conjugate of X(57)
X(1477) = cevapoint of X(56) and X(1458)

### X(1478) = CENTER OF JOHNSON-YFF CIRCLE

Trilinears    1 + 2 cos B cos C : :
Barycentrics    a^4 + 2 a^2 b c - (b^2 - c^2)^2 : :
X(1478) = (R/r)*X(1) + 3X(2) - 2X(3)
X(1478) = R*X(1) - r*X(3) + 2r*X(5)

The Johnson Circle Theorem is the fact that if three congruent circles intersect in a point, then the circle passing through the other three intersections is congruent to them. This fourth circle is the Johnson circle of the three given circles. There are three congruent circles each tangent to two sides of triangle ABC. Peter Yff proved that their Johnson circle has center X(1478). The circle is here named the Johnson-Yff circle of the triangle.

Roger A. Johnson, Advanced Euclidean Geometry, Dover, New York, 1960, page 75.

Peter Yff, "Three concurrent congruent circles 'inscribed' in a triangle," manuscript, 1998; X(1478) is the point C on page 5. See also X(495)-X(499).

X(1478) lies on these lines:
1,4   2,36   3,12   5,56   7,80   8,79   10,46   11,381   13,203   14,202   30,55   65,68   119,1470   148,192   442,958   474,1329   496,546   529,956   612,1370   908,997   975,1076   990,1074   1352,1469

X(1478) = reflection of X(i) in X(j) for these (i,j): (1,226), (55,495), (63,10)
X(1478) = isogonal conjugate of X(3422)
X(1478) = anticomplement of X(993)
X(1478) = X(1065)-Ceva conjugate of X(1)
X(1478) = homothetic center of 2nd isogonal triangle of X(1) and anticomplementary triangle; see X(36)
X(1478) = homothetic center of anticomplementary triangle and cross-triangle of ABC and 2nd isogonal triangle of X(1)
X(1478) = homothetic center of 2nd isogonal triangle of X(1) and cross-triangle of ABC and 2nd isogonal triangle of X(1)
X(1478) = homothetic center of Johnson triangle and cross-triangle of ABC and 1st Johnson-Yff triangle
X(1478) = homothetic center of Ehrmann mid-triangle and 2nd Johnson-Yff triangle

### X(1479) = {X(1),X(4)}-HARMONIC CONJUGATE OF X(1478)

Trilinears    1 - 2 cos B cos C : :
Barycentrics    a^4 - 2a^2bc - (b^2 - c^2)^2 : :
X(1479) = (R/r)*X(1) - 3X(2) + 2X(3)
X(1479) = R*X(1) + r*X(3) - 2r*X(5)

Peter Yff, "Three concurrent congruent circles 'inscribed' in a triangle," manuscript, 1998; X(1479) is the point C' on page 5. See also X(495)-X(499).

X(1479) lies on these lines:
1,4   2,35   3,11   5,55   7,79   8,80   12,381   20,36   30,56   46,516   63,90   148,330   156,215   315,350   377,1125   382,999   387,1203   442,1001   495,546   528,1329   614,1370   1387,1388

X(1479) = reflection of X(i) in X(j) for these (i,j): (46,1210), (56,496)
X(1479) = X(1067)-Ceva conjugate of X(1)
X(1479) = homothetic center of intangents triangle and reflection of tangential triangle in X(5)
X(1479) = homothetic center of Johnson triangle and cross-triangle of ABC and 2nd Johnson-Yff triangle
X(1479) = homothetic center of Ehrmann mid-triangle and 1st Johnson-Yff triangle
X(1479) = Ursa-major-to-Ursa-minor similarity image of X(3)
X(1479) = homothetic center of 2nd isogonal triangle of X(1) and the reflection of the anticomplementary triangle in X(4); see X(36)

### X(1480) = 1st SHADOW POINT

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (2 - cos A)(2 + cos A - cos B - cos C)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Antreas P. Hatzipolakis and Paul Yiu, Pedal Triangles and Their Shadows, Forum Geometricorum 1 (2001) 81-90. (X(1480) is point M on page 88. See X(6580) for the 2nd Shadow Point.

X(1480) lies on these lines: 1,1406   3,902   6,517   651,1000

X(1480) = X(1)-Ceva conjugate of X(999)

### X(1481) = POINT ASTEROPE

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (2 - cos A)/(2 + cos A - cos B - cos C)

(For the 2nd Shadow Point, see X(6580).

X(1481) lies on these lines: {1,1406}, {2316,4253}

### X(1482) = REFLECTION OF CIRCUMCENTER IN INCENTER

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 2r - R cos A (where r = inradius, R = circumradius)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1482) lies on these lines:
1,3   4,145   5,8   30,944   104,1392   355,381   382,515   518,1351

X(1482) = midpoint of X(i) and X(j) for these (i,j): (4,145), (944,962)
X(1482) = reflection of X(i) in X(j) for these (i,j): (3,1), (8,5), (40,1385), (355,946), (944,1483)

### X(1483) = REFLECTION OF X(5) IN X(1)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 4r - R cos(B - C)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1483) lies on these lines:
1,5   3,145   8,140   10,632   30,944   40,548   517,550   518,1353   519,549

X(1483) = midpoint of X(i) and X(j) for these (i,j): (3,145), (944,1482)
X(1483) = reflection of X(i) in X(j) for these (i,j): (5,1), (8,140)

### X(1484) = REFLECTION OF X(5) IN X(11)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 4r - (2r + R)cos(B - C)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1484) lies on these lines:
1,5   3,149   30,104   100,140   153,381   528,549

X(1484) = midpoint of X(3) and X(149)
X(1484) = reflection of X(i) in X(j) for these (i,j): (5,11), (100,140)

### X(1485) = PERSPECTOR OF ABC AND TANGENTIAL-OF-TANGENTIAL TRIANGLE

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(b3cos B + c3cos C - a3cos A)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1485) = X(1486)-of-tangential triangle if ABC is acute. (Darij Grinberg, 5/24/03)

Let A'B'C' be the orthic triangle. Let La be the reflection of line B'C' in the perpedicular bisector of BC, and define Lb and Lc cyclically. Let A" = Lb∩Lc, B" = Lc∩La, C" = La∩Lb. The lines AA", BB", CC" concur in X(1485). (Randy Hutson, July 31 2018)

X(1485) lies on these lines: 22,160   26,206   157,264   232,571

### X(1486) = PERSPECTOR OF TANGENTIAL AND INTOUCH TRIANGLES

Trilinears    a[ - a2(b + c - a) + b2(c + a - b) + c2(a + b - c)] : :

Let A'B'C' be the intouch triangle. Let A" be the crosspoint of the circumcircle intercepts of line B'C', and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(1486).

X(1486) lies on these lines:
1,159   3,142   6,692   19,25   56,1279   100,344   219,674   354,1473   513,1037

X(1486) = X(7)-Ceva conjugate of X(6)
X(1486) = crosssum of X(116) and X(522)
X(1486) = crossdifference of every pair of points on line X(905)X(918)
X(1486) = X(173)-of-the-tangential-triangle if ABC is acute; see note at X(1485)
X(1486) = pole, with respect to circumcircle, of the Gergonne line

### X(1487) = NAPOLEON CEVAPOINT

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc A)/[(3 - cot B cot C)(3 - cot2A)]
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Let N denote the nine-point center, X(5). Let NA = N-of-triangle NBC, and define NB and NC cyclically. Triangle NANBNC is perspective to ABC, and X(1487) is the perspector. X(1487) is the cevapoint of the Napoleon points, X(17) and X(18). (Coordinates found by Paul Yiu.)

The construction just given for X(1487) shows that it is a solution X of the "four-triangle problem" posed in

C. Kimberling, "Triangle centers as functions," Rocky Mountain Journal of Mathematics 23 (1993) 1269-1286. See Section 5; a complete solution to the problem remains to be found.

X(1487) lies on these lines: 4,252   5,1173   140,930

X(1487) = isogonal conjugate of X(1493)
X(1487) = cevapoint of X(17) and X(18)
X(1487) = X(523)-cross conjugate of X(930)

### X(1488) = 2nd STEVANOVIC POINT

Trilinears    1/[1 + sin(A/2)] : :
Trilinears    sec^2(A'/2) : : , where A'B'C' is the excentral triangle

Let U be the A-excenter of triangle ABC; let A' be the incenter of triangle UBC, and define B', C' cyclically. Let A" = IA'∩BC, and define B", C" cyclically. The lines AA", BB", CC" concur in X(1488). (Milorad R. Stevanovic, Hyacinthos #7185, 5/21/03. See also X(1130) and X(1489).)

X(1488) lies on these lines: 1,166   7,2089   57,173   145,188   557,1274   558,1143

X(1488) = X(1)-cross conjugate of X(174)

X(1488) = SS(A->A') of X(7), where A'B'C' is the excentral triangle
X(1488) = trilinear pole of Monge line of incircles of BCI, CAI, ABI
X(1488) = trilinear pole of Monge line of I-excircles of BCI, CAI, ABI

### X(1489) = 3rd STEVANOVIC POINT

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (csc A)[1 - cos(A/2) - sin(A/2)]
Barycentrics  1 - cos(A/2) - sin(A/2) : 1 - cos(B/2) - sin(B/2) : 1 - cos(C/2) - sin(C/2)

Let U be the A-excenter of triangle ABC; let A' be the incenter of triangle UBC, and define B', C' cyclically. Let I be the incenter of ABC, let A'' be the incenter of triangle IBC, and define B'', C'' cyclically. Let X = A'A''∩BC, and define Y, Z cyclically. Then AX, BY, CZ concur in X(1489). (Milorad R. Stevanovic, Hyacinthos #7185, 5/21/03. See also X(1130) and X(1489).)

X(1489) lies on these lines: 1,188   2,1143   174,558   258,483

X(1489) = complement of X(1143)
X(1489) = cevapoint of X(1) and X(258)
X(1489) = crosspoint of X(2) and X(1274)

### X(1490) = X(1)X(4)∩X(3)X(9)

Trilinears    1 + sec A - sec B - sec C : :

X(1490) is X(68)-of-the-excentral triangle and the reflection of X(84) in X(3). These and other properties itemized here were reported by Darij Grinberg, May 19, 2003.

Let U(A) be the circle with center A having the radius of the A-excircle, and define U(B) and U(C) cyclically. Then X(1490) is the radical center of the three circles. (Hauke Reddmann, Hyacinthos, Jan. 8, 2009)

Let Oa be the circle centered at A and passing through the A-excenter, and define Ob and Oc cyclically. The radical center of Oa, Ob, Oc = X(1490). (Randy Hutson, September 14, 2016)

Let A'B'C' be the cevian triangle of X(1034). Let A" be the orthocenter of AB'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1490). (Randy Hutson, September 14, 2016)

X(1490) lies on the Darboux cubic and these lines:
1,4   3,9   20,78   40,64   63,411   165,191   224,908   386,990   910,1192   975,991   1045,1047   1210,1467   1498,3347   2130,3473   2131,3353   3182,3348   3183,3354

X(1490) = reflection of X(84) in X(3)
X(1490) = X(i)-Ceva conjugate of X(j) for these (i,j): (20,40), (78,1), (329,9)
X(1490) = X(207)-cross conjugate of X(1)
X(1490) = homothetic center of hexyl and 2nd extouch triangles
X(1490) = X(155)-of-hexyl-triangle
X(1490) = X(155)-of-2nd-extouch-triangle
X(1490) = perspector of hexyl triangle and cevian triangle of X(20)
X(1490) = perspector of hexyl triangle and anticevian triangle of X(40)
X(1490) = perspector of ABC and the reflection in X(282) of the antipedal triangle of X(282)
X(1490) = {X(1),X(1745)}-harmonic conjugate of X(223)
X(1490) = excentral isogonal conjugate of X(46)
X(1490) = excentral isotomic conjugate of X(1721)

### X(1491) = CROSSDIFFERENCE OF X(1) AND X(32)

Trilinears    b3 - c3 : :

Let L be the line PU(10) = X(10)X(514); let M be the trilinear polar of the cevapoint of PU(10), so that M = X(522)X(1491). Let V = P(10)-Ceva conjugate of U(10) and let W = U(10)-Ceva conjugate of P(10). The lines L, M, and VW concur in X(1491). (Randy Hutson, December 26, 2015)

X(1491) lies on these lines: 10,514   44,513   325,523   663,1193   667,830

X(1491) = reflection of X(659) in X(650)
X(1491) = isogonal conjugate of X(1492)
X(1491) = isotomic conjugate of X(789)
X(1491) = X(i)-Ceva conjugate of X(j) for these (i,j): (262,11), (1492,1)
X(1491) = cevapoint of X(1) and X(1492)
X(1491) = crosspoint of X(i) and X(j) for these (i,j): (1,1492)
X(1491) = crosssum of X(i) and X(j) for these (i,j): (1,1491), (100,1390)
X(1491) = crossdifference of every pair of points on line X(1)X(32)

### X(1492) = COLUMBUS POINT

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/(b3 - c3)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1492) lies on these lines:
88,985   100,825   101,660   110,789   190,692

X(1492) = isogonal conjugate of X(1491)
X(1492) = cevapoint of X(513) and X(1386)
X(1492) = X(i)-cross conjugate of X(j) for these (i,j): (182,59), (1491,1)

### X(1493) = NAPOLEON CROSSSUM

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = csc(B + π/6)csc(C - π/6) + csc(C + π/6)csc(B - π/6)
= g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (3 sin2A - cos2A)(3 sin B sin C - cos B cos C)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

A construction of X(1493) is given by Antreas Hatipolakis and Angel Montesdeoca at 24179.

X(1493) lies on these lines:
3,54   5,539   49,143   110,1173   113,137   141,575   206,576

X(1493) = midpoint of X(54) and X(195)
X(1493) = isogonal conjugate of X(1487)
X(1493) = complement of X(3519)
X(1493) = X(110)-Ceva conjugate of X(1510)
X(1493) = crosspoint of X(61) and X(62)
X(1493) = crosssum of X(17) and X(18)

### X(1494) = ISOTOMIC CONJUGATE OF X(30)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc/[(b2 - c2)2 + a2(b2 + c2 - 2a2)]
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1494) lies on the Steiner circumellipse and these lines:
2,648   30,340   69,74   95,549   190,306   253,317   264,339   287,524   305,670   307,319   325,892

X(1494) = reflection of X(648) in X(2)
X(1494) = isogonal conjugate of X(1495)
X(1494) = isotomic conjugate of X(30)
X(1494) = anticomplement of X(3163)
X(1494) = cevapoint of X(i) and X(j) for these (i,j): (2,30), (3,323), (298,299)
X(1494) = X(i)-cross conjugate of X(j) for these (i,j): (30,2), (340,95)
X(1494) = antipode of X(2) in hyperbola {A,B,C,X(2),X(69)}
X(1494) = trilinear pole of line X(2)X(525)
X(1494) = crossdifference of PU(87)
X(1494) = pole wrt polar circle of trilinear polar of x(1990)
X(1494) = X(48)-isoconjugate (polar conjugate) of X(1990)

### X(1495) = CROSSSUM OF X(2) AND X(30)

Trilinears    a[(b2 - c2)2 + a2(b2 + c2 - 2a2)] : :

Let La be the line through A parallel to the de Longchamps line, and define Lb and Lc cyclically. Let Ma be the reflection of BC in La, and define Mb and Mc cyclically. Let A' = Mb∩Mc, and define B', C' cyclically. The triangle A'B'C' is inversely similar to, and 3 times the size of, ABC. Let A"B"C" be the reflection of A'B'C' in the de Longchamps line. The triangle A"B"C" is homothetic to ABC, with center of homothety X(1495). (See Hyacinthos #16741/16782, Sep 2008.) (Randy Hutson, December 26, 2015)

X(1495) lies on these lines:
3,3426   6,25   23,110   24,185   30,113   52,156   74,186   125,468   182,373   187,237   263,1383   1204,1498

X(1495) = midpoint of X(23) and X(110)
X(1495) = reflection of X(i) in X(j) for these (i,j): (125,468), (1531,113)
X(1495) = isogonal conjugate of X(1494)
X(1495) = X(i)-Ceva conjugate of X(j) for these (i,j): (78,1), (329,9)
X(1495) = crosspoint of X(6) and X(74)
X(1495) = crosssum of X(i) and X(j) for these (i,j): (74,6), (1304,647)
X(1495) = crossdifference of every pair of points on line X(2)X(525)
X(1495) = pole of Brocard axis wrt Moses radical circle
X(1495) = {X(3),X(5651)}-harmonic conjugate of X(5650)
X(1495) = trilinear pole of PU(87)
X(1495) = inverse-in-Parry-isodynamic-circle of X(3569); see X(2)
X(1495) = X(3218)-of-orthic-triangle if ABC is acute

### X(1496) = POINT BETELGEUSE I

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1 + cos2A
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(1496) lies on these lines:
1,21   3,1037   48,820   55,603   56,212   75,775   354,1451   580,1471   602,999

### X(1497) = POINT BETELGEUSE II

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1 + sin2A
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(1497) lies on these lines:
1,21   3,1057   46,1471   55,602   56,601   82,158   498,748   499,750   517,1451   603,999   605,1335   606,1124

### X(1498) = REFLECTION OF X(64) IN X(3)

Trilinears    (sin A)(tan2B + tan2C - tan2A) : :
Trilinears    a*[S^2 - 2*(4*R^2 - SA)*SA] : :

Let A'B'C' be the cevian triangle of X(1032). Let A" be the orthocenter of AB'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1498). (Randy Hutson, December 10, 2016)

X(1498) is the perspector of the tangential triangle and the reflection of triangle ABC in X(3); also, X(1498) is X(8)-of tangential triangle if ABC is acute. (Darij Grinberg, 6/2/03)

X(1498) lies on the Darboux cubic and these lines:
1,84   3,64   4,6   20,394   24,1192   25,185   30,155   40,219   159,1350   195,382   1158,1214   1490,3347   2131,3183   3182,3354   3353,3473

X(1498) = reflection of X(i) in X(j) for these (i,j): (64,3), (1350,159)
X(1498) = isogonal conjugate of X(3346)
X(1498) = X(i)-Ceva conjugate of X(j) for these (i,j): (20,3), (394,6)
X(1498) = crosssum of X(122) and X(523)
X(1498) = tangential isogonal conjugate of X(24)
X(1498) = tangential isotomic conjugate of X(6
X(1498) = antipedal isogonal conjugate of X(64
X(1498) = Nagel point of tangential triangle if ABC is acute
X(1498) = perspector of antipedal triangle of X(64) and cevian triangle of X(20)
X(1498) = isogonal conjugate, wrt tangential triangle of MacBeath circumconic (or anticevian triangle of X(3)), of X(64)
X(1498) = perspector of ABC and the reflection in X(1073) of the antipedal triangle of X(1073)
X(1498) = perspector of excentral triangle and cross-triangle of ABC and hexyl triangle

### X(1499) = POINT BIHAM

Trilinears    bc(b2 - c2)(b2 + c2 - 5a2) : :

X(1499) lies on these (parallel) lines: 3,669   4,1550   30,511   74,2770   98,843

X(1499) = isogonal conjugate of X(1296)
X(1499) = crosspoint of X(99) and X(598)
X(1499) = crosssum of X(512) and X(574)
X(1499) = crossdifference of every pair of points on line X(6)X(373)
X(1499) = bicentric difference of PU(7)
X(1499) = ideal point of PU(7)
X(1499) = X(2780) of 4th Brocard triangle
X(1499) = X(2780) of orthocentroidal triangle
X(1499) = X(2780) of X(4)-Brocard triangle
X(1499) = Thomson-isogonal conjugate of X(111)
X(1499) = Lucas-isogonal conjugate of X(111)
X(1499) = Cundy-Parry Psi transform of X(14263)

### X(1500) = INSIMILICENTER OF MOSES CIRCLE AND INCIRCLE

Trilinears    a(b + c)2 : b(c + a)2 : c(a + b)2
Barycentrics    a2(b + c)2 : b2(c + a)2 : c2(a + b)2
X(1500) = (tan ω sin 2ω)R*X(1) + r*X(39)

The circle having center X(39) and radius R tan ω sin 2ω, where R denotes the circumradius of triangle ABC, is here introduced as the Moses circle. It is tangent to the nine-point circle at X(115), and its internal and external centers of similitude with the incircle are X(1500) and X(1015), respectively. (Peter J. C. Moses, 5/29/03)

X(1500) lies on these lines:
1,39   6,595   10,37   11,1508   12,115   32,55   35,172   41,1017   42,213   56,574   76,192   216,1062   346,941   519,1107   612,1196   756,762   1124,1505   1335,1504

X(1500) = isogonal conjugate of X(1509)
X(1500) = X(i)-Ceva conjugate of X(j) for these (i,j): (37,756), (42,872), (1018,512)
X(1500) = X(872)-cross conjugate of X(181)
X(1500) = crosspoint of X(37) and X(42)
X(1500) = crosssum of X(81) and X(86)
X(1500) = barycentric square of X(37)

### X(1501) = TRILINEAR 5th POWER POINT

Trilinears    a5: b5 : c5
Barycentrics    a6 : b6 : c6

X(1501) is the vertex conjugate of the foci of the inellipse that is the barycentric square of the Lemoine axis. The center of this inellipse is X(8265) and its perspector is X(32). (Randy Hutson, October 15, 2018)

X(1501) lies on these lines:
6,22   32,184   101,697   110,699   154,1184   701,825   703,827   1196,1495

X(1501) = isogonal conjugate of X(1502)
X(1501) = crosssum of X(i) and X(j) for these (i,j): (2,315), (76,305), (115,850)
X(1501) = crossdifference of every pair of points on line X(826)X(850)
X(1501) = trilinear product of PU(12)
X(1501) = X(92)-isoconjugate of X(305)
X(1501) = barycentric product of vertices of circumsymmedial triangle

### X(1502) = ISOGONAL CONJUGATE OF X(1501)

Trilinears    a-5 : b- 5 : c- 5
Barycentrics  a- 4 : b- 4 : c- 4

X(1502) is the Brianchon point (perspector) of the inellipse that is the barycentric square of the de Longchamps line. The center of this inellipse is X(626). (Randy Hutson, October 15, 2018)

X(1502) lies on these lines:
1,704   2,308   6,706   31,708   32,710   66,315   69,290   75,700   76,141   99,160   264,305   311,327   313,561

X(1502) = isogonal conjugate of X(1501)
X(1502) = isotomic conjugate of X(32)
X(1502) = cevapoint of X(i) and X(j) for these (i,j): (2,315), (76,305), (115,850)
X(1502) = X(i)-cross conjugate of X(j) for these (i,j): (115,850), (626,2)
X(1502) = barycentric product of PU(14)
X(1502) = trilinear pole of line X(826)X(850)
X(1502) = pole wrt polar circle of trilinear polar of X(1974)
X(1502) = X(48)-isoconjugate (polar conjugate) of X(1974)
X(1502) = barycentric square of X(76)

### X(1503) = POINT ARKAB

Trilinears    sin A tan ω - 2 cos B cos C : :
Trilinears    bc(b6 + c6 - 2a6 + a4b2 +a4c2 - b4c2 - c4b2) : :
Barycentrics    S^2 a^2 - 2 SB SC SW : :
X(1503) = X(4) - X(6)

As the isogonal conjugate of a point on the circumcircle, X(1503) lies on the line at infinity.

X(1503) lies on these (parallel) lines:
2,154   3,66   4,6   5,182   11,1428   20,64   22,161   30,511   51,428   67,74   98,230   110,858   125,468   147,325   184,427   221,388   242,1146   265,1177   287,297   376,599   381,597   382,1351   383,395   394,1370   396,1080   546,575   576,1353   611,1478   613,1479   946,1386

X(1503) = isogonal conjugate of X(1297)
X(1503) = complementary conjugate of X(132)
X(1503) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,132), (287,6), (297,230), (685,523)
X(1503) = cevapoint of X(20) and X(147)
X(1503) = crosspoint of X(4) and X(98)
X(1503) = crosssum of X(3) and X(511)
X(1503) = crossdifference of every pair of points on line X(6)X(520)
X(1503) = X(4)-Hirst inverse of X(1249)
X(1503) = intercept of the van Aubel line and the line at infinity
X(1503) = Cundy-Parry Phi transform of X(14376)
X(1503) = Cundy-Parry Psi transform of X(8743)
X(1503) = crosspoint of X(20) and X(147) wrt both the excentral and anticomplementary triangles
X(1503) = infinite point of tangents at X(4) and X(20) to Darboux cubic K004

### X(1504) = INSIMILICENTER OF MOSES AND 2nd LEMOINE CIRCLES

Trilinears    a[b2 + c2 + 4(area ABC)]
Trilinears    2 sin A sin ω + sin(A + ω) : : (Peter J. C. Moses, 9/12/03)
Trilinears    cos A + (sin A)(2 + cot ω) : : (Peter J. C. Moses, 9/12/03)
X(1504) = sin(2ω)*X(6) + X(39)

X(1504) = internal center of similitude of the Moses circle and the 2nd Lemoine circle. [See X(1500); Peter J. C. Moses, 6/2/03]

X(1504) lies on these lines:
2,588   3,6   115,485   394,493   486,1508   491,626   590,639   1015,1124   1335,1500

X(1504) = X(1306)-Ceva conjugate of X(512)
X(1504) = crosspoint of X(6) and X(493)
X(1504) = {X(6),X(39)}-harmonic conjugate of X(1505)
X(1504) = intersection of tangents to hyperbola {A,B,C,X(2),X(6)} at X(6) and X(493)

### X(1505) = EXSIMILICENTER OF MOSES AND 2nd LEMOINE CIRCLES

Trilinears    a[b2 + c2 - 4(area ABC)]
Trilinears    2 sin A sin ω - sin(A + ω) (Peter J. C. Moses, 9/12/03)
Trilinears    cos A + (sin A)(- 2 + cot ω) (Peter J. C. Moses, 9/12/03)
X(1505) = sin(2ω)*X(6) - X(39)

X(1505) = external center of similitude of the Moses circle and the 2nd Lemoine circle. [See X(1500); Peter J. C. Moses, 6/2/03]

X(1505) lies on these lines:
2,589   3,6   115,486   394,494   485,1508   492,626   615,640   1015,1335   1124,1500

X(1505) = X(1307)-Ceva conjugate of X(512)
X(1505) = crosspoint of X(6) and X(494)

X(1505) = {X(6),X(39)}-harmonic conjugate of X(1504)
X(1505) = intersection of tangents to hyperbola {{A,B,C,X(2),X(6)}} at X(6) and X(494)

### X(1506) = INSIMILICENTER OF MOSES AND NINE-POINT CIRCLES

Trilinears    bc[(b2 - c2)2 - 2a2(b2 + c2)] : :
X(1506) = 2(tan ω sin 2ω)*X(5) + X(39)

X(1506) = internal center of similitude of the Moses circle and the nine-point circle. [The external center is X(115); see X(1500); Peter J. C. Moses, 6/2/03.]

X(1506) lies on these lines:
2,32   4,574   5,39   6,17   11,1500   12,1015   51,211   125,217   140,187   384,620   485,1505   486,1504

X(1506) = complement of X(1078)
X(1506) = {X(6),X(1656)}-harmonic conjugate of X(7746)
X(1506) = {X(5),X(39)}-harmonic conjugate of X(115)

### X(1507) = 1st MORLEY-GIBERT POINT

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1 + 2(cos B/3 + cos C/3 - cos A/3)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(1507) and X(1508) are centers on Morley cubic, which also passes through X(356), X(357), X(358); see Bernard Gibert's site.

X(1507) lies on this line: 1,358

X(1507) = SS(A->A/3) of X(3336)
X(1507) = perspector of excentral and 1st Morley triangles

### X(1508) = 2nd MORLEY-GIBERT POINT

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 2 - sec A/3 + sec B/3 + sec C/3
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

See the notes at X(1507) and X(356).

X(1508) is the perspector of the excentral tiangle and the 1st Morley adjunct triangle. (César Lozada, January 18, 2015)

X(1508) lies on this line: 1,357

X(1508) = SS(A->A/3) of X(3468)
X(1508) = perspector of excentral and 1st Morley adjunct triangles

### X(1509) = ISOGONAL CONJUGATE OF X(1500)

Trilinears       1/[a(b + c)2] : 1/[b(c + a)2] : 1/[c(a + b)2]
Barycentrics  1/(b + c)2 : 1/(c + a)2 : 1/(a + b)2

Let A'B'C' be the anticomplement of the Feuerbach triangle. Let A" be the trilinear pole of the tangent to the circumcircle at A', and define B" and C" cyclically. The lines AA", BB", CC" concur at X(1509). (Randy Hutson, June 27, 2018)

X(1509) lies on these lines:
1,99   2,1171   58,86   76,940   81,239   552,553   593,763

X(1509) = isogonal conjugate of X(1500)
X(1509) = isotomic conjugate of X(594)
X(1509) = X(873)-Ceva conjugate of X(261)
X(1509) = cevapoint of X(81) and X(86)
X(1509) = X(i)-cross conjugate of X(j) for these (i,j): (81,757), (86,873), (757,552), (1019,99)
X(1509) = barycentric square of X(86)

### X(1510) = NAPOLEON CROSSDIFFERENCE

Trilinears    csc(B + π/6)csc(C - π/6) - csc(C + π/6)csc(B - π/6) : :
Trilinears    (3 sin2A - cos2A)(cos B sin C - sin B cos C) : :

X(1510) lies on these lines: 30,511   110,1291

X(1510) = isogonal conjugate of X(930)
X(1510) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,137), (110,1493)
X(1510) = X(137)-cross conjugate of X(143)
X(1510) = crosspoint of X(i) and X(j) for these (i,j): (4,933), (110,1173)
X(1510) = crosssum of X(i) and X(j) for these (i,j): (140,523), (512,570)
X(1510) = crossdifference of every pair of points on line X(6)X(17)
X(1510) = X(523)-of-orthic-triangle
X(1510) = infinite point of normal to hyperbola {{A,B,C,X(4),X(15)}} at X(15) and normal to hyperbola {{A,B,C,X(4),X(16)}} at X(16)

### X(1511) = FERMAT CROSSSUM

Trilinears    csc(B + π/3)csc(C - π/3) + csc(C + π/3)csc(B - π/3)
Trilinears    (sin2A - 3 cos2A)(sin B sin C - 3 cos B cos C)
Barycentrics    a^2[(a^2 - b^2 - c^2)^2 - b^2c^2][2a^4 - a^2(b^2 + c^2) - (b^2 - c^2)^2] : :

Let A'B'C' be the cevian triangle of X(30). Let A", B", C" be the inverse-in-circumcircle of A', B', C'. The lines AA", BB", CC" concur in X(1511). Let NA be the reflection of X(5) in the perpendicular bisector of segment BC, and define NB and NC cyclically; then X(1511) = X(186)-of-NANBNC. (Randy Hutson, August 26, 2014)

Let A'B'C' be the medial triangle. Let L be the line through A' parallel to the Euler line, and define M and N cyclically. Let L' be the reflection of L in sideline BC, and define M' and N' cyclically. The lines L',M',N' concur in X(1511). Let (O') be the circle passing through the points X(3), X(4), X(399), X(6069), let Q be the radical axis (O') and the circumcircle; then X(1511) = Q∩X(3)X(74). (Randy Hutson, August 26, 2014)

Let U be the Simson line of X(110), which is the line X(30)X(113). Let V be the line normal to the circumcircle at X(110), which is the line X(3)X(74). Then X(1511) = U∩V. (Randy Hutson, January 29, 2015)

X(1511) lies on (Johnson circumconic of medial triangle), the bicevian conic of X(2) and X(110), and on these lines:
2,265   3,74   24,1112   30,113   36,1464   125,128   141,542   146,376   184,974   186,323   214,960   249,842   389,1493

X(1511) = midpoint of X(i) and X(j) for these (i,j): (3,110), (74,399)
X(1511) = reflection of X(i) in X(j) for these (i,j): (125,140), (1539,113)
X(1511) = complement of X(265)
X(1511) = complementary conjugate of X(2072)
X(1511) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,3284), (3,1154), (110,526)
X(1511) = crosspoint of X(i) and X(j) for these (i,j): (2,340), (15,16)
X(1511) = crosssum of X(13) and X(14)
X(1511) = crossdifference of every pair of points on the line X(1637)X(1989)
X(1511) = inverse-in-circumcircle of X(399)
X(1511) = perspector of the circumconic centered at X(3284)
X(1511) = {X(3),X(399)}-harmonic conjugate of X(74)
X(1511) = orthocentroidal-to-ABC similarity image of X(5)
X(1511) = 4th-Brocard-to-circumsymmedial similarity image of X(5)
X(1511) = reflection of X(5) in X(5972)
X(1511) = antipode of X(5) in the bicevian conic of X(2) and X(110)
X(1511) = X(12515)-of-orthic-triangle if ABC is acute

Orthojoins: X(1512)- X(1568)

The orthojoin of a point X = x : y : z other than X(6) is defined in the Glossary algebraically in terms of variables a,b,c. When these are sidelengths of a triangle, orthojoin(X) is the orthopole of the trilinear polar of the isogonal conjugate of X. (Added to ETC 6/18/03.) Let

D(a,b,c) = bc[2abcx + c3y + b3z - bc(by + cz) - a2(cy + bz)],
E(a,b,c) = [(a4 - (b2 - c2)2]x - 4a2bc(y cos B + z cos C),
f(a,b,c) = D(a,b,c)E(a,b,c).

Then orthojoin(X) = f(a,b,c) : f(b,c,a) : f(c,a,b). Below, orthojoin(X) is written as H(X).

Suppose X is not X(2) and does not lie on a sideline of triangle ABC. The crossdifference of X and X(2) has first trilinear
a(b-c)x', where x' = (by - cz)/(b-c).

Let X -1 denote the isogonal conjugate of X. Then
H(X -1) = (by - cz) cos A : (cz - ax) cos B : (ax - by) cos C.

In other words, if X lies on a line PG through the centroid G, then H(X -1) lies on the line HQ, where H denotes the orthocenter and Q is a point that can be determined from the above formula. Examples:

If X is on the Euler line, L(2,3), then H(X -1) is on the line L(4,6);
If X is on L(2,6,), then H(X -1) is on the Euler line;
If X is on L(1,2), then H(X -1) is on L(4,9);
If X is on L(2,7), then H(X -1) is on L(1,4).

Suppose P is not X(6), and let

 S = crossdifference(P, X(6)) (S lies on the line at infinity) S' = orthopoint(S) (S' lies on the line at infinity) S" = complementary conjugate of S' (S" lies on the nine-point circle)

Let X be a point on line PX(6) and not on a sideline of ABC. Then H(X -1) is on line S"S'. Examples:

If X is a center on L(1,6) and X is not X(6), then H(X -1) is on L(119,517).
If X is a center on L(2,6) and X is not X(6), then H(X -1) is on L(114,511).
If X is a center on L(3,6) and X is not X(6), then H(X -1) is on L(113,30).
If X is a center on L(6,31) and X is not X(6), then H(X -1) is on L(118,516).
If X is a center on L(6,44) and X is not X(6), then H(X -1) is on L(117,515).

Further,

 H(X(11)) = L(117,515)∩L(118,516) H(X(37)) = L(117,515)∩L(119,517) H(X(244)) = L(118,516)∩L(119,517)

If X lies on the line L(230,231), then H(X) lies on the nine-point circle. Examples:

 H(X(230)) = X(114) H(X(231)) = X(128) H(X(232)) = X(132) H(X(468)) = X(1560) H(X(523)) = X(115) H(X(647)) = X(125) H(X(650)) = X(11)

### X(1512) = ORTHOJOIN OF X(1)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1512) lies on these lines: 4,9   5,392   6,80   119,517   355,956   944,1210

X(1512) = reflection of X(i) in X(j) for these (i,j): (908,119), (1519,1532), (1537,1538)
X(1512) = orthpole of antiorthic axis

### X(1513) = ORTHOJOIN OF X(2)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = bc(b4 + c4 - a2b2 - a2c2)[3a4 + (b2 - c2)2]

As a point on the Euler line, X(1513) has Shinagawa coefficients (S2, -(E + F)2).

X(1513) lies on these lines: 2,3   98,230   114,325   132,232   147,385   183,1352   1181,1184

X(1513) = reflection of X(i) in X(j) for these (i,j): (98,230), (325,114)
X(1513) = orthopole of Lemoine axis
X(1513) = X(114)-of-1st-anti-Brocard-triangle

### X(1514) = ORTHOJOIN OF X(3)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1514) lies on these lines: 4,6   30,113   74,403   187,1516

X(1514) = orthopole of orthic axis

### X(1515) = ORTHOJOIN OF X(4)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1515) lies on the Simson quartic (Q101) and these lines: 4,6   30,1294   133,1559   297,1533

X(1515) = intersection of Simson line of X(107) (line X(133)X(1515)) and trilinear polar of X(107) (line X(4)X(6))

### X(1516) = ORTHOJOIN OF X(5)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1516) lies on these lines: 4,96   187,1514

### X(1517) = ORTHOJOIN OF X(7)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1517) lies on this line: 4,218

### X(1518) = ORTHOJOIN OF X(8)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1518) lies on this line: 4,608

### X(1519) = ORTHOJOIN OF X(9)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1519) lies on these lines: 1,4   119,517   499,1158

X(1519) = reflection of X(i) in X(j) for these (i,j): (1512,1532), (1532,1538)
X(1519) = orthopole of PU(96)

### X(1520) = ORTHOJOIN OF X(10)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1520) lies on this line: 4,572

### X(1521) = ORTHOJOIN OF X(11)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1521) lies on these lines: 117,515   118,516

### X(1522) = ORTHOJOIN OF X(13)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1522) lies on these lines: 4,14   1523,1553

### X(1523) = ORTHOJOIN OF X(14)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1523) lies on this line: 4,13   1522,1553

### X(1524) = ORTHOJOIN OF X(15)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1524) lies on these lines: 4,13   30,113

X(1524) = reflection of X(i) in X(j) for these (i,j): (1525,1514), (1546,1525)

### X(1525) = ORTHOJOIN OF X(16)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1525) lies on these lines: 4,14   30,113

X(1525) = reflection of X(i) in X(j) for these (i,j): (1524,1514), (1545,1524)

### X(1526) = ORTHOJOIN OF X(17)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1526) lies on these lines: 4,16   128,1154

### X(1527) = ORTHOJOIN OF X(18)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1527) lies on these lines: 4,15   128,1154

### X(1528) = ORTHOJOIN OF X(19)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1528) lies on this line: 1,4

### X(1529) = ORTHOJOIN OF X(25)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

As a point on the Euler line, X(1529) has Shinagawa coefficients (FS2, -(E + F)[4(E + F)F - S2]).

X(1529) lies on these lines: 2,3   132,1503

### X(1530) = ORTHOJOIN OFX(31)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1530) lies on these lines: 4,8   118,516   152,971   908, 1543

X(1530) = reflection of X(i) in X(j) for these (i,j): (910,118), (1536,1541)

### X(1531) = ORTHOJOIN OF X(32)

Trilinears    [2a^4 - a^2(b^2 + c^2) - b^4 - c^4 + 2b^2c^2][a^5 - a(b^2 - c^2)^2 - 4bc(b^3 cos B + c^3 cos C)] : :

Let MaMbMc be the Ehrmann mid-triangle. Let A' be the crosspoint of Mb and Mc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1531). (Randy Hutson, July 31 2018)

X(1531) lies on these lines: 4,69   30,113   382,1092

X(1531) = reflection of X(i) in X(j) for these (i,j): (1495,113), (1533,1514)
X(1531) = orthopole of de Longchamps line

### X(1532) = ORTHOJOIN OF X(37)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = bc[2abc - (b + c)(a - b + c)(a + b - c)][a(b + c)(a + b - c)(a - b + c) - (a + b + c)(a - b - c)(b - c)2]

As a point on the Euler line, X(1532) has Shinagawa coefficients (\$a\$S2, \$a\$S2 + 2abc(E + F) + 2\$a(SA)2\$).

X(1532) lies on these lines: 2,3   11,515   12,946   40,1329   119,517   496,944

X(1532) = reflection of X(i) in X(j) for these (i,j): (1519,1538), (1537,1519)

### X(1533) = ORTHOJOIN OF X(39)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1533) lies on these lines: 4,83   30,113   297,1515

X(1533) = reflection of X(1531) in X(1514)

### X(1534) = ORTHOJOIN OF X(40)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1534) lies on this line: 4,937

### X(1535) = ORTHOJOIN OF X(41)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1535) lies on these lines: 4,7   117,515   151,517

### X(1536) = ORTHOJOIN OF X(42)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

As a point on the Euler line, X(1536) has Shinagawa coefficients (S2, -2(E + F)2 + 2\$bcSA\$ - S2).

X(1536) lies on these lines: 2,3   118,516

X(1536) = reflection of X(1530) in X(1541)

### X(1537) = ORTHOJOIN OF X(44)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1537) lies on these lines: 4,145   7,104   11,65   100,962   119,517   214,516   515,1317

X(1537) = reflection of X(i) in X(j) for these (i,j): (11,946), (104,1387), (1145,119), (1512,1538), (1532,1519)
X(1537) = orthopole of PU(55)

### X(1538) = ORTHOJOIN OF X(45)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1538) lies on these lines: 4,1385   11,971   119,517   495,946   515,1387

### X(1539) = ORTHOJOIN OF X(50)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1539) lies on these lines: 4,94   5,2777   30,113   74,381   110,382   125,546

X(1539) = reflection of X(i) in X(j) for these (i,j): (125,546), (1511,113)
X(1539) = orthopole of PU(5)

### X(1540) = ORTHOJOIN OF X(54)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1540) lies on this line: 4,6

### X(1541) = ORTHOJOIN OF X(55)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1541) lies on these lines: 1,4   118,516

X(1541) = orthopole of Gergonne line

### X(1542) = ORTHOJOIN OF X(56)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1542) lies on these lines: 4,9   117,515

### X(1543) = ORTHOJOIN OF X(57)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1543) lies on these lines: 1,4   516,972   908,1530

### X(1544) = ORTHOJOIN OF X(58)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1544) lies on these lines: 4,9   30,113

### X(1545) = ORTHOJOIN OF X(61)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1545) lies on these lines: 4,15   30,113

X(1545) = reflection of X(i) in X(j) for these (i,j): (1525,1524), (1546,1514)

### X(1546) = ORTHOJOIN OF X(62)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1546) lies on these lines: 4,16   30,113

X(1546) = reflection of X(i) in X(j) for these (i,j): (1524,1525), (1545,1514)

### X(1547) = ORTHOJOIN OF X(71)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1547) lies on this line: 4,6   118,516

### X(1548) = ORTHOJOIN OF X(72)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1548) lies on these lines: 4,6   119,517

### X(1549) = ORTHOJOIN OF X(73)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1549) lies on these lines: 4,6   117,515

X(1539) = orthopole of PU(173)

### X(1550) = ORTHOJOIN OF X(110)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1550) lies on these lines: 4,1499   30,74   98,230   542,1551

X(1550) = orthopole of line X(115)X(125)

### X(1551) = ORTHOJOIN OF X(111)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

As a point on the Euler line, X(1551) has Shinagawa coefficients (2(E + F)3 - 9(E - 2F)S2, -3(E + F)[(E +10F)(E + F) - 6S2]).

X(1551) lies on this line: 2,3   542,1550

### X(1552) = ORTHOJOIN OF X(112)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1552) lies on these lines: 4,523   30,1294   74,403

### X(1553) = ORTHOJOIN OF X(115)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1553) lies on these lines: 4,250   30,113   146,476   1522,1523

### X(1554) = ORTHOJOIN OF X(125)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1554) lies on these lines: 30,113   132,1503

### X(1555) = ORTHOJOIN OF X(182)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1555) lies on these lines: 4,39   30,113

X(1555) = reflection of X(1561) in X(1514)

### X(1556) = ORTHOJOIN OF X(251)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

As a point on the Euler line, X(1556) has Shinagawa coefficients (2(E + F)3 + (E + 2F)S2, -(E + F)[5(E + 6F)(E + F) - 2S2]).

X(1556) lies on this line: 2,3

### X(1557) = ORTHOJOIN OF X(263)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

As a point on the Euler line, X(1557) has Shinagawa coefficients ([(E + F)3F + 2(E + F)2S2 + S4] S2, -4(E + F)3F + (E - 11F)(E + F)3S2 - (E + 10F)(E + F)S4 - 3S6).

X(1557) lies on this line: 2,3

### X(1558) = ORTHOJOIN OF X(284)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1558) lies on these lines: 1,4   30,113

### X(1559) = ORTHOJOIN OF X(393)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

As a point on the Euler line, X(1559) has Shinagawa coefficients (F2, -2(2E - F)F + S2).

X(1559) lies on these lines: 2,3   133,1515

### X(1560) = ORTHOJOIN OF X(468)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1560) lies on the nine-point circle and these lines:
2,112   4,111   6,67   53,136   115,427   122,216   135,571   187,468

X(1560) = complement of X(2373)
X(1560) = X(2)-Ceva conjugate of X(468)
X(1560) = crosspoint of X(2) and X(858)
X(1560) = crosssum of X(6) and X(1177)

### X(1561) = ORTHOJOIN OF X(511)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1561) lies on these lines: 4,32   30,113

X(1561) = reflection of X(1555) in X(1514)
X(1561) = orthopole of PU(45)

### X(1562) = ORTHOJOIN OF X(520)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1562) lies on these lines: 115,125   127,525   148,287

X(1562) = X(393)-Ceva conjugate of X(647)
X(1562) = crosspoint of X(4) and X(525)
X(1562) = crosssum of X(393) and X(647)

X(1562) = crossdifference of every pair of points on line X(110)X(1301)

### X(1563) = ORTHOJOIN OF X(588)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

As a point on the Euler line, X(1563) has Shinagawa coefficients ([2(E + F)2 + (9E + 8F)S+10S2]S, -(E + F)2E - 2(5E - F)(E + F)S - 4(7E - 2F)S2 - 22S3).

X(1563) lies on this line: 2,3

### X(1564) = ORTHOJOIN OF X(589)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

As a point on the Euler line, X(1564) has Shinagawa coefficients ([2(E + F)2 - (9E + 8F)S+10S2]S, (E + F)2E - 2(5E - F)(E + F)S + 4(7E - 2F)S2 - 22S3).

X(1564) lies on this line: 2,3

### X(1565) = ORTHOJOIN OF X(657)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b - c)2(b2 + c2 - a2)      (M. Iliev, 5/13/07)

X(1565) lies on these lines:
3,348   4,279   5,85   7,104   11,1111   77,1060   84,738   116,514   150,664   304,337   515,1323   812,1015   1119,1440   1364,1367

X(1565) = midpoint of X(150) and X(664)
X(1565) = reflection of X(1146) in X(116)
X(1565) = X(i)-Ceva conjugate of X(j) for these (i,j): (279,514), (304,525), (348,905)
X(1565) = crosspoint of X(7) and X(693)
X(1565) = crosssum of X(55) and X(692)
X(1565) = orthopole of Soddy line

### X(1566) = ORTHOJOIN OF X(676)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1566) lies on the nine-point circle and these lines:
2,927   11,650   116,514   118,516   125,661   132,242

X(1566) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,676), (4,926)
X(1566) = crosspoint of X(514) and X(516)
X(1566) = crosssum of X(i) and X(j) for these (i,j): (101,103), (651,1814)

### X(1567) = ORTHOJOIN OF X(694)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

As a point on the Euler line, X(1567) has Shinagawa coefficients ((E + F)6 - (5E - 3F)(E + F)3S2 + 3(E + F)2S4 + S6, -(E + 3F)(E + F)5 + 3(E - 3F)(E + F)3S2 + 9(E - F)(E + F)S4 - 3S6).

X(1567) lies on the nine-point circle and this line: 2,3

### X(1568) = ORTHOJOIN OF X(800)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1512))

X(1568) lies on these lines: 4,801   5,51   30,113   265,539   381,394   403,511

X(1568) = reflection of X(125) in X(2072)

### X(1569) = MOSES-CIRCLE ANTIPODE OF X(115)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[2a2(b2 + c2)(1 - 4 sin2ω) - (b2 - c2)2]
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1569) lies on this line: 5,39   32,99   76,620   98,574   543,598

X(1569) = midpoint of X(99) and X(194)
X(1569) = reflection of X(i) in X(j) for these (i,j): (76,620), (115,39)
X(1569) = inverse-in-circle-{{X(3102),X(3103),PU(1)}} of X(5)

### X(1570) = MOSES-CIRCLE INVERSE OF X(3)

Trilinears     a(cot A + cot ω - 4 tan ω) : :
Trilinears    a(2a4 + 3b4 + 3c4 - 2b2c2 -3a2b2 - 3a2c2) : :

X(1570) lies on these lines: 3,6   193,625

X(1570) = reflection of X(i) in X(j) for these (i,j): (39,2025), (187,1692), (1692,6)
X(1570) = inverse-in-2nd-Lemoine-circle of X(3)
X(1570) = crosssum of X(2) and X(230)
X(1570) = 2nd-Lemoine-circle inverse of X(3)
X(1570) = radical trace of the Brocard and 2nd Lemoine circles
X(1570) = radical trace of 1st Lemoine circle and Ehrmann circle

### X(1571) = INSIMILICENTER OF EXCENTRAL AND MOSES CIRCLES

Trilinears    a[bcr + (a2 - 2b2 - 2c2)R] : :, where R = circumradius, r = inradius

X(1571) = 2*X(39) + (tan ω sin 2ω)*X(40) = X(1) - 2*X(3) - (2 cot ω csc 2ω)*X(39)

X(1571) lies on these lines: 1,574   9,1574   32,165   39,40   57,1500

### X(1572) = EXSIMILICENTER OF EXCENTRAL AND MOSES CIRCLES

Trilinears    a(bcr + a2R) : :
Trilinears    a3 + 4(area)r : :, where r =inradius
Trilinears    sa3 + S2 : sb3 + S2 : sc3 + S2
X(1572) = 2*X(39) - (tan ω sin 2ω)*X(40) = X(1) - 2*X(3) + (2 cot ω csc 2ω)*X(39)

X(1572) lies on these lines: 1,32   6,517   9,1573   39,40   57,1015   165,574

### X(1573) = INSIMILICENTER OF SPIEKER AND MOSES CIRCLES

Trilinears    a(b2 + c2) + 2bc(b + c) : :
X(1573) = (2R tan ω sin 2ω)*X(10) + r*X(39) = R*X(1) + R*X(8) + (r cot ω csc 2ω)*X(39)

X(1573) lies on these lines:
2,668   8,1500   10,39   32,958   37,519   75,538   187,993   574,1376   1329,1508   1377,1505   1378, 1504

### X(1574) = EXSIMILICENTER OF SPIEKER AND MOSES CIRCLES

Trilinears    a(b2 + c2) - 2bc(b + c) : :
X(1574) = (2R tan ω sin 2ω)*X(10) - r*X(39) = R*X(1) + R*X(8) - (r cot ω csc 2ω)*X(39)

X(1574) lies on these lines:
2,1500   8,1015   10,39   32,1376   38,762   115,1329   213,899   574,993   1377,1504   1378,1505

### X(1575) = EXSIMILICENTER OF SPIEKER AND (1/2)-MOSES CIRCLES

Trilinears    a(b2 + c2) - bc(b + c)
X(1575) = (R tan ω sin 2ω)*X(10) - r*X(39) = R*X(1) + R*X(8) - (2r cot ω csc 2ω)*X(39)

The Moses circle, M, is introduced at X(1015); the (1/2)-Moses circle is concentric to M with half the radius of M. The insimilicenter of the Spieker and (1/2)-Moses circles is X(1107).

X(1575) lies on these lines:
2,37   6,43   10,39   42,1100   44,513   71,992   172,404   239,292   291,518   519,1015   574,993   1009,1104   1125,1500

X(1575) = complement of X(350)
X(1575) = X(i)-Ceva conjugate of X(j) for these (i,j): (239,518), (292,37)
X(1575) = cevapoint of X(43) and X(2108)
X(1575) = crosspoint of X(2) and X(291)
X(1575) = crosssum of X(i) and X(j) for these (i,j): (1,1575), (6,238)
X(1575) = crossdifference of every pair of points on line X(1)X(667)
X(1575) = {X(10),X(39)}-harmonic conjugate of X(1107)

### X(1576) = ISOGONAL CONJUGATE OF X(850)

Trilinears    a3/(b2 - c2) : b3/(c2 - a2) : c3/(a2 - b2)
Barycentrics  a4/(b2 - c2) : b4/(c2 - a2) : c4/(a2 - b2)

X(1576) is the center of the conic transform of the Stammler quartic (Q066 in Bernard Gibert' catalogue) by X(31)-isoconjugation. This conic is given by the barycentric equation b^4c^4(b^2-c^2)x^2+c^4a^4(c^2-a^2)y^2+a^4b^4(a^2-b^2)z^2 = 0, and it passes through the following triangle centers: X(6), X(31), X(48), X(154), X(1613), X(2578), X(2579), X(5638), X(5639). (Angel Montesdeoca, May 7, 2016)

X(1576) lies on these lines:
3,1177   6,157   32,1084   50,237   99,827   107,933   110,351   160,206   163,692   250,523   338,1316   662,1492

X(1576) = midpoint of X(648) and X(1632)
X(1576) = isogonal conjugate of X(850)
X(1576) = X(i)-Ceva conjugate of X(j) for these (i,j): (249,1501), (250,6), (827,110), (933,112)
X(1576) = cevapoint of X(i) and X(j) for these (i,j): (32,669), (39,647), (51,512)
X(1576) = X(i)-cross conjugate of X(j) for these (i,j): (669,32), (1501,249)
X(1576) = crosspoint of X(110) and X(112)
X(1576) = crosssum of X(523) and X(525)
X(1576) = crossdifference of every pair of points on line X(115)X(127)
X(1576) = barycentric product of PU(2)
X(1576) = barycentric product of vertices of circumtangential triangle
X(1576) = trilinear pole of line X(32)X(184)
X(1576) = X(92)-isoconjugate of X(525)
X(1576) = X(1577)-isoconjugate of X(2)
X(1576) = barycentric product X(1379)*X(1380)

### X(1577) = ISOGONAL CONJUGATE OF X(163)

Trilinears    b2c2(b2 - c2) : c2a2(c2 - a2) : a2b2(a2 - b2)
Trilinears    (csc 2A)(sin 2B - sin 2C) : :
Trilinears    d(a,b,c) : : , where d(a,b,c) = directed distance from A to Brocard axis
Trilinears    |AP(1)|^2 - |AU(1)|^2 : :
Barycentrics    bc(b2 - c2) : ca(c2 - a2) : ab(a2 - b2)
Barycentrics    sin(B - C) : sin(C - A) : sin(A - B)

Let A'B'C' be the excentral triangle, and let U be the bianticevian conic of X(1) and X(4). Let T be the tangential triangle, wrt the anticevian triangle of X(19), of U. Then A'B'C' and T are perspective, and their perspector is X(1577). (Randy Hutson, December 26, 2015)

X(1577) lies on these lines:
1,810   115,1111   163,811   240,522   514,661   667,814   784,149   798,812   826,1089

X(1577) = isogonal conjugate of X(163)
X(1577) = isotomic conjugate of X(662)
X(1577) = X(i)-Ceva conjugate of X(j) for these (i,j): (75,1109), (76,1111), (693,523), (799,75), (811,1), (823,92)
X(1577) = cevapoint of X(656) and X(661)
X(1577) = X(i)-cross conjugate of X(j) for these (i,j): (115,1089), (1109,75)
X(1577) = crosspoint of X(i) and X(j) for these (i,j): (75,799), (82,162), (92,823), (662,2167), (811,1969), (1240,1978)
X(1577) = crosssum of X(i) and X(j) for these (i,j): (31,798), (38,656), (48,822), (649,2260), (652,2269), (661,1953), (1923,1924)
X(1577) = crossdifference of every pair of points on line X(31)X(48)
X(1577) = X(i)-aleph conjugate of X(j) for these (i,j): (648,656), (811,1969)
X(1577) = complement of X(4560)
X(1577) = bicentric difference of PU(14)
X(1577) = PU(14)-harmonic conjugate of X(1930)
X(1577) = trilinear product of PU(40)
X(1577) = perspector of hyperbola {{A,B,C,X(75),X(92)}} (centered at X(4858))
X(1577) = center of circumconic that is locus of trilinear poles of lines passing through X(4858)
X(1577) = X(2)-Ceva conjugate of X(4858)
X(1577) = X(6)-isoconjugate of X(110)
X(1577) = pole wrt polar circle of trilinear polar of X(162) (line X(1)X(19))
X(1577) = X(48)-isoconjugate (polar conjugate) of X(162)
X(1577) = trilinear pole of line X(1109)X(2632)

### X(1578) = POINT ALTERF I

Trilinears       sin A + sec B sec C : sin B + sec C sec B : sin C + sec A sec B
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)(sin A + sec B sec C)

X(1578) lies on these lines: 3,6   394,488   485,1368   1038,1335   1040,1124

X(1578) = inverse-in-Brocard-circle of X(1579)

### X(1579) = POINT ALTERF II

Trilinears       sin A - sec B sec C : sin B - sec C sec B : sin C - sec A sec B
Barycentrics  g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin A)(sin A - sec B sec C)

X(1579) lies on these lines: 3,6   394,487   486,1368   1038,1124   1040,1335

X(1579) = inverse-in-Brocard-circle of X(1578)

### X(1580) = POINT ALUDRA

Trilinears    a4 - b2c2 : :

X(1580) lies on these lines:
1,21   6,256   41,43   75,560   87,604   171,172   238,1284   239,1281   284,1045   661,830   662,922

X(1580) = isogonal conjugate of X(1581)
X(1580) = isotomic conjugate of X(1934)
X(1580) = X(i)-Ceva conjugate of X(j) for these (i,j): (1755,1955), (1910,1), (1967,1582)
X(1580) = X(2236)-cross conjugate of X(1966)
X(1580) = crosssum of X(i) and X(j) for these (i,j): (1755,1964), (1965,1966)
X(1580) = crossdifference of every pair of points on line X(38)X(661)
X(1580) = X(98)-aleph conjugate of X(1755)
X(1580) = perspector of conic {A,B,C,PU(36)}
X(1580) = trilinear product X(171)*X(238)
X(1580) = perspector of unary cofactor triangles of Gemini triangles 32 and 34

### X(1581) = ISOGONAL CONJUGATE OF X(1580)

Trilinears    1/(a4 - b2c2) : :

X(1581) lies on these lines:
10,257   37,256   65,291   82,662   171,292   733,831   759,805   876,882

Let DEF and D'E'F' be the 1st and 2nd Sharygin triangles. Let A' be the trilinear product D*D', and define B' and C' cyclically. The lines AA', BB', CC' concur in X(1581). (Randy Hutson, December 26, 2015)

X(1581) = isogonal conjugate of X(1580)
X(1581) = isotomic conjugate of X(1966)
X(1581) = cevapoint of X(i) and X(j) for these (i,j): (1755,1964), (1965,1966)
X(1581) = X(i)-cross conjugate of X(j) for these (i,j): (1821,1956), (1959,1), (2227,75)
X(1581) = trilinear pole of PU(35) (line X(38)X(661))
X(1581) = trilinear product of circumcircle intercepts of line PU(11)
X(1581) = trilinear product X(256)*X(291)

### X(1582) = POINT CANOPUS

Trilinears    a4 + b2c2 : :

X(1582) lies on these lines:
1,19   6,291   75,560   82,662   238,992   978,1472

X(1582) = X(1967)-Ceva conjugate of X(1580)
X(1582) = crosssum of PU(35)
X(1582) = crosspoint of PU(36)
X(1582) = intersection of tangents at PU(36) to conic {{A,B,C,PU(36)}}

### X(1583) = POINT CAPELLA I

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = csc A + cos A
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(1583) = 6R2X(2) + S*X(3)

As a point on the Euler line, X(1583) has Shinagawa coefficients (E + S, -S).

X(1583) lies on these lines: 2,3   6,493   371,394

X(1583) = inverse-in-orthocentroidal-circle of X(1591)

### X(1584) = POINT CAPELLA II

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = csc A - cos A
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
X(1584) = 6R2X(2) - S*X(3)

As a point on the Euler line, X(1584) has Shinagawa coefficients (E - S, S).

X(1584) lies on these lines: 2,3   6,494   372,394

X(1584) = inverse-in-orthocentroidal-circle of X(1592)

### X(1585) = POINT CAPH I

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = csc A + sec A
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(1585) has Shinagawa coefficients (F, S).

X(1585) lies on these lines:
2,3   53,590   264,491   275,486   317,492   343,638   393,493   394,637   490,1321

X(1585) = inverse-in-orthocentroidal-circle of X(1586)
X(1585) = cevapoint of X(1599) and X(1993)
X(1585) = X(i)-cross conjugate of X(j) for these (i,j): (371,492), (1993,1586)

### X(1586) = POINT CAPH II

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = csc A - sec A
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(1586) has Shinagawa coefficients (F, -S).

X(1586) lies on these lines:
2,3   53,615   264,494   317,491   343,637   393,494   394,638   489,1322

X(1586) = inverse-in-orthocentroidal-circle of X(1585)
X(1586) = cevapoint of X(1600) and X(1993)
X(1586) = X(i)-cross conjugate of X(j) for these (i,j): (372,491), (1993,1585)

### X(1587) = POINT CASTOR I

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + cos B cos C
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(1587) lies on these lines:
2,372   4,6   20,371   193,637   194,487   376,1151   388,1335   394,1587   486,1131   497,1124   590,631   639,1270   1132,1327

### X(1588) = POINT CASTOR II

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A - cos B cos C
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(1588) lies on these lines:
2,371   4,6   20,372   193,638   194,488   376,1152   388,1124   394,1588   485,1132   497,1335   615,631   640,1271   1131,1328

### X(1589) = POINT CHARA I

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = csc A + sec B sec C
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(1589) has Shinagawa coefficients (F + S, -S).

X(1589) lies on these lines: 2,3   343,487   394,488

### X(1590) = POINT CHARA II

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = csc A - sec B sec C
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(1590) has Shinagawa coefficients (F - S, S).

X(1590) lies on these lines: 2,3   343,488   394,487

### X(1591) = POINT CHARA III

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = csc A + cos(B - C)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(1591) has Shinagawa coefficients (E + S, S).

X(1591) lies on these lines: 2,3   343,639   394,485

X(1591) = complement of X(1599)
X(1591) = inverse-in-orthocentroidal-circle of X(1583)
X(1591) = X(1306)-Ceva conjugate of X(523)

### X(1592) = POINT CHARA IV

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = csc A - cos(B - C)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(1592) has Shinagawa coefficients (E - S, -S).

X(1592) lies on these lines: 2,3   343,640   394,486

X(1592) = complement of X(1600)
X(1592) = inverse-in-orthocentroidal-circle of X(1584)
X(1592) = X(1307)-Ceva conjugate of X(523)

### X(1593) = POINT CEBALRAI

Trilinears    = cos A + sec A : :

As a point on the Euler line, X(1593) has Shinagawa coefficients (F, E - F).

X(1593) lies on these lines:
1,1037   2,3   6,64   19,1212   33,56   34,55   51,1204   74,1112   84,1473   184,1498   208,1466   264,1105   578,1181   607,672   1155,1452   1208, 1471

X(1593) = reflection of X(i) in X(j) for these (i,j): (4,1595), (1181,578)
X(1593) = inverse-in-orthocentroidal-circle of X(235)
X(1593) = crosspoint of X(4) and X(1595)
X(1593) = crosssum of X(3) and X(1181)
X(1593) = homothetic center of orthic triangle and reflection of tangential triangle in X(3)
X(1593) = homothetic center of tangential triangle and reflection of orthic triangle in X(4)
X(1593) = exsimilicenter of circumcircle and incircle of orthic triangle if ABC is acute; the insimilicenter is X(25)
X(1593) = X(1697)-of-orthic-triangle if ABC is acute

### X(1594) = RIGBY-LALESCU ORTHOPOLE

Trilinears    sec A + 2 cos(B - C) : :
Trilinears    cos(2B-C) cos(A-C) + cos(2C-B) cos(A-B) : :

As a point on the Euler line, X(1594) has Shinagawa coefficients (2F, E + 2F).

Traian Lalescu (Trajan Lalesco) proved in "A Class of Remarcable Triangles," Gazeta Matematica 20 (1915) 213 [in Romanian], that if triangles DEF and D'E'F' are inscribed in a circle and directed arclengths satisfyarc DD' + arc EE' + arc FF' = 0 mod 2π, then the Simson lines of D,E,F with respect to D',E',F' and the Simson lines of D',E',F' with respect to D,E,F concur in the midpoint X of the segment of the orthocenters of DEF and D'E'F'. Daniel Vacaretu considered triangles DEF and D'E'F' associated with left and right isoscelizers and inscribed in the sine-triple-angle circle. He obtained the second set of trilinears shown above for the midpoint X. (See also the bicentric pair PU(61).)

In Episodes in Nineteenth and Twentieth Century Euclidean Geometry,, page 132, Ross Honsberger presents X(1594) as the orthopole of the six sides of two triangles and as the point common to six Simson lines. Honsberger calls this orthopole the Rigby Point. (Notes on Lalescu and Honsberger received from D. Vacaretu, 19/16/03)

X(1594) lies on these lines:
2,3   6,70   50,252   53,566   67,1173   96,275   125,389   128,136   232,1508   264,847   325,1235   933,1166   1209,1216   1225,1238

X(1594) = inverse-in-nine-point-circle of X(186)
X(1594) = inverse-in-orthocentroidal-circle of X(24)
X(1594) = X(933)-Ceva conjugate of X(523)
X(1594) = crosspoint of X(i) and X(j) for these (i,j): (4,93), (264,275)
X(1594) = crosssum of X(i) and X(j) for these (i,j): (3,49), (184,216)
X(1594) = X(35)-of-orthic-triangle if ABC is acute
X(1594) = {X(4),X(5)}-harmonic conjugate of X(403)

### X(1595) = POINT CHELEB I

Trilinears    2 sec A + cos(B - C) : :

As a point on the Euler line, X(1595) has Shinagawa coefficients (F, 2E + F).

X(1595) lies on these lines: 2,3   33,496   34,495   39,53   578,1503

X(1595) = midpoint of X(4) and X(1593)
X(1595) = inverse-in-orthocentroidal-circle of X(1598)
X(1595) = {X(4),X(5)}-harmonic conjugate of X(1596)
X(1595) = X(3295)-of-orthic-triangle if ABC is acute

### X(1596) = POINT CHELEB II

Trilinears    2 sec A - cos(B - C) : :

As a point on the Euler line, X(1596) has Shinagawa coefficients (F, -2E + F).

X(1596) lies on these lines: 2,3   33,495   34,496   53,115

X(1596) = midpoint of X(4) and X(25)
X(1596) = reflection of X(1368) in X(5)
X(1596) = inverse-in-orthocentroidal-circle of X(1597)
X(1596) = {X(4),X(5)}-harmonic conjugate of X(1595)
X(1596) = X(999)-of-orthic-triangle if ABC is acute
X(1596) = center of inverse-in-polar-circle-of-de-Longchamps-line
X(1596) = homothetic center of Ehrmann mid-triangle and 3rd pedal triangle of X(4)
X(1596) = Ehrmann-side-to-orthic similarity image of X(18531)

### X(1597) = POINT CHERTAN I

Trilinears    cos A + 2 sec A : :

As a point on the Euler line, X(1597) has Shinagawa coefficients (F, 2E - F).

X(1597) lies on these lines: 2,3   33,999   64,389   578,1498

X(1597) = inverse-in-orthocentroidal-circle of X(1596)
X(1597) = {X(3),X(4)}-harmonic conjugate of X(1598)
X(1597) = center of conic that is locus of {X(4),P}-harmonic conjugate of Q, where P and Q lie on the circumcircle and are collinear with X(4)

### X(1598) = POINT CHERTAN II

Trilinears    cos A - 2 sec A : :

As a point on the Euler line, X(1598) has Shinagawa coefficients (F, -2E - F).

X(1598) lies on these lines:
1,1057   2,3   34,999   51,1181   154,578   155,1351   389,1498   399,1112

X(1598) = inverse-in-orthocentroidal-circle of X(1595)
X(1598) = {X(3),X(4)}-harmonic conjugate of X(1597)
X(1598) = X(3333)-of-orthic-triangle if ABC is acute
X(1598) = homothetic center of 3rd pedal triangle of X(4) and 3rd antipedal triangle of X(3)

### X(1599) = POINT CHORT I

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = csc A + 2 cos A
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(1599) has Shinagawa coefficients (E + 2S, -2S).

X(1599) lies on these lines: 2,3   6,588   394,1151

X(1599) = anticomplement of X(1591)
X(1599) = X(1585)-Ceva conjugate of X(1993)

### X(1600) = POINT CHORT II

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = csc A - 2 cos A
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(1600) has Shinagawa coefficients (E - 2S, 2S).

X(1600) lies on these lines: 2,3   6,589   394,1152

X(1600) = anticomplement of X(1592)
X(1600) = X(i)-Ceva conjugate of X(j) for these (i,j): (346,6), (1586,1993)

TCC Perspectors 1601- 1634

Suppose P is a point. As noted in TCCT, p. 201, the tangential triangle is perspective to the circumcevian triangle of P. In August 2003, Jean-Pierre Ehrmann noted that if P = x : y : z (barycentrics), then the perspector is given by

a2(b4/y2 + c4/z2 - a4/x2) : b2(c4/z2 + a4/x2 - b4/y2) : c2(a4/x2 + b4/y2 - c4/z2) .

Denote this perspector by T(P), and call it the TCC-perspector of P. If P = x : y : z (trilinears), then

T(P) = a(b2/y2 + c2/z2 - a2/x2) : b(c2/z2 + a2/x2 - b2/y2) : c(a2/x2 + b2/y2 - c2/z2).

The transformation T carries triangle centers to triangle centers. For example, T(X(i)) = X(j) for each of these (i,j):

(1,3),   (2,22),   (3,1498),   (4,24),   (6,6),   (58,595),   (83,1498),   (259,198),   (266,56),   (365,55),   (509,1486)

Ehrmann suggested that interesting new points might be found using this transformation. These are typified by X(1601)-X(1634).

Beginning with a note that the TCC-perspector the centroid, X(2), is the Exeter point, X(22),

Oliver Funck, in Geometrische Untersuchungen mit Computerunterstützung, generalizes to include TCC-perspectors. As X(22) is named for Phillips Exeter Academy in Exeter, New Hampshire, USA, where X(22) was detected in 1986 using a computer, the generalized points are sometimes called Steinbart points for the school, Steinbart-Gymnasium, in Duisburg, Germany. See Darij Grinberg, Variations of the Steinbart Theorem and his Extended Steinbart Theorem in Hyacinthos #7984, 9/23/03.

### X(1601) = TCC-PERSPECTOR OF X(5)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1601))

X(1601) lies on these lines: 3,128   6,1166

X(1601) = X(60)-of-tangential-triangle if ABC is acute
X(1601) = tangential isogonal conjugate of X(1614)

### X(1602) = TCC-PERSPECTOR OF X(7)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1601))

X(1602) lies on these lines: 7,1486   22,1626   24,242

### X(1603) = TCC-PERSPECTOR OF X(8)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1601))

X(1603) lies on this line: 8,197   24,1324

### X(1604) = TCC-PERSPECTOR OF X(9)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1601))

X(1604) lies on these lines: 3,9   25,1863

X(1604) = X(346)-Ceva conjugate of X(6)

### X(1605) = TCC-PERSPECTOR OF X(13)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1601))

X(1605) lies on this line: 3,618   26,1607

### X(1606) = TCC-PERSPECTOR OF X(14)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1601))

X(1606) lies on this line: 3,619   26,1608

### X(1607) = TCC-PERSPECTOR OF X(17)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1601))

X(1607) lies on this line: 3,619   26,1605

### X(1608) = TCC-PERSPECTOR OF X(18)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1601))

X(1608) lies on this line: 3,618   26,1606

### X(1609) = TCC-PERSPECTOR OF X(19)

Trilinears    f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1601))
Trilinears    a(cos^2 B + cos^2 C - cos^2 A) : :

X(1609) lies on these lines: 3,6   24,254   25,53   48,1195   112,1299   159,237   186,1249   590,1583   615,1584

X(1609) = X(i)-Ceva conjugate of X(j) for these (i,j): (24,25), (393,6)
X(1609) = crosspoint of X(107) and X(249)
X(1609) = crosssum of X(115) and X(520)

### X(1610) = TCC-PERSPECTOR OF X(21)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1601))

X(1610) lies on these lines: 1,19   6,959   8,197   20,1633   24,944   172,910   198,958   315,1310

### X(1611) = TCC-PERSPECTOR OF X(25)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[(a2 + b2 + c2)2 - 8b2c2]      (M. Iliev, 5/13/07)

X(1611) lies on these lines: 2,6   3,1196   232,1033   1498,1513

X(1611) = X(i)-Ceva conjugate of X(j) for these (i,j): (459,25), (2207,6)
X(1611) = crosssum of X(520) and X(1084)
X(1611) = center of bicevian conic of PU(4)

### X(1612) = TCC-PERSPECTOR OF X(28)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1601))

X(1612) lies on these lines: 1,21   6,943   55,387   681,1924   1006,1104

X(1612) = crosspoint of X(107) and X(765)
X(1612) = crosssum of X(244) and X(520)

### X(1613) = TCC-PERSPECTOR OF X(31)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2c2 - c2a2 - a2b2)      (M. Iliev, 5/13/07)

X(1613) lies on these lines: 1,1197   2,6   3,695   25,694   110,699   154,237   305,732   511,1196

X(1613) = isogonal conjugate of X(2998)
X(1613) = X(i)-Ceva conjugate of X(j) for these (i,j): (32,6)
X(1613) = X(194)-cross conjugate of X(6)
X(1613) = crosspoint of X(i) and X(j) for these (i,j): (1424,1740)
X(1613) = crosssum of X(523) and X(1084)
X(1613) = crossdifference of every pair of points on line X(512)X(625) (complement of Lemoine axis)
X(1613) = X(92)-isoconjugate of X(3504)
X(1613) = trilinear pole of polar wrt 2nd Brocard circle of perspector of 2nd Brocard circle
X(1613) = vertex conjugate of PU(148)

### X(1614) = TCC-PERSPECTOR OF X(54)

Trilinears    a[b^2 cos^2(C - A) + c^2 cos^2(A - B) - a^2 cos^2(B - C)] : :
Barycentrics    a^2 (a^8 - 3 a^6 (b^2 + c^2) + a^4 (3 b^4 + b^2 c^2 + 3 c^4) - a^2 (b^2 - c^2)^2 (b^2 + c^2) + b^2 c^2 (b^2 - c^2)^2) : :

Let P1 and P2 be the two points on the circumcircle whose Steiner lines are tangent to the circumcircle. X(1614) is the crosspoint of P1 and P2. (Randy Hutson, August 29, 2018)

Let A'B'C' be the triangle described in ADGEOM #2697 (8/26/2015, Tran Quang Hung). A'B'C' is homothetic to the anti-Euler triangle at X(1614). Also, X(1614) is the exsimilicenter of the circumcircle of ABC and nine-point circle of A'B'C'. (Randy Hutson, August 29, 2018)

> X(1614) lies on these lines:
3,74   4,184   6,1173   20,1147   22,155   23,52   24,154   30,49   51,1199   70,1176   185,186   376,1092   378,1498   389,1495   546,567   1503,1594

X(1614) = X(12)-of-tangential-triangle if ABC is acute
X(1614) = tangential isogonal conjugate of X(1601)
X(1614) = {X(3),X(156)}-harmonic conjugate of X(110)
X(1614) = exsimilicenter of circumcircle and nine-point circle of tangential triangle

### X(1615) = TCC-PERSPECTOR OF X(55)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1601))

X(1615) lies on these lines: 6,57   56,1200   165,220

X(1615) = X(i)-Ceva conjugate of X(j) for these (i,j): (165,55), (200,6)

### X(1616) = TCC-PERSPECTOR OF X(56)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[(a + b + c)2 - 8bc]      (M. Iliev, 5/13/07)

X(1616) lies on these lines: 1,6   42,1293   55,1201   221,1319   595,999   614,3057   902,1392   962,1086   1035,1457   1407,1420

X(1616) = X(i)-Ceva conjugate of X(j) for these (i,j): (1407,6), (1420,56)
X(1616) = crosspoint of X(934) and X(1016)

### X(1617) = TCC-PERSPECTOR OF X(57)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1601))

X(1617) lies on these lines:
1,3   6,1174   7,1621   22,347   25,105   31,222   41,1202   42,1471   109,1407   219,604   221,595   226,1001   388,405   479,934   518,1260   602,1066   614,1465   910,1108   1006,1056   1055,1200   1279,1427   1362,1397   1384,1415

X(1617) = isogonal conjugate of X(6601)
X(1617) = X(i)-Ceva conjugate of X(j) for these (i,j): (279,6), (1252,109), (1445,218)
X(1617) = crosspoint of X(59) and X(934)
X(1617) = pole wrt circumcircle of line X(513)X(676) (the trilinear polar of X(279))
X(1617) = {X(55),X(56)}-harmonic conjugate of X(57)
X(1617) = {X(3513),X(3514)}-harmonic conjugate of X(3)
X(1617) = inverse-in-circumcircle of X(3660)

### X(1618) = TCC-PERSPECTOR OF X(59)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1601))

X(1618) lies on these lines: 109,1459   110,901   513,651

X(1618) = reflection of X(59) in X(692)

### X(1619) = TCC-PERSPECTOR OF X(63)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1601))

X(1619) lies on these lines: 3,64   22,69   25,1503   161,542

X(1619) = crosssum of (122,512)

### X(1620) = TCC-PERSPECTOR OF X(64)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1601))

X(1620) lies on these lines: 3,6   154,1204   186,1498

### X(1621) = TCC-PERSPECTOR OF X(81)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2 - bc - ca - ab      (M. Iliev, 5/13/07)
X(1621) = 2X(1) + 3X(2) + 2(r/R)X(3)

X(1621) lies on these lines:
1,21   2,11   3,962   7,1617   8,405   9,1174   22,1486   35,404   36,551   37,82   42,238   43,748   99,873   145,958   171,902   213,1206   226,1005   278,1013   329,954   411,946   517,1006   739,932   985,1255

X(1621) = X(i)-Ceva conjugate of X(j) for these (i,j): (1252,100), (1509,6)
X(1621) = crosspoint of X(99) and X(765)
X(1621) = crosssum of (244,512)
X(1621) = {X(1),X(31)}-harmonic conjugate of X(81)

### X(1622) = TCC-PERSPECTOR OF X(84)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1601))

X(1622) lies on these lines: 1,3   6,947   84,963

### X(1623) = TCC-PERSPECTOR OF X(88)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1601))

X(1623) lies on these lines: 3,8   23,105   36,1168

### X(1624) = TCC-PERSPECTOR OF X(162)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1601))

X(1624) lies on these lines: 3,113   25,132   110,351   112,1301   852,1503   925,1302   933,1304

### X(1625) = TCC-PERSPECTOR OF X(163)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1601))
Barycentrics    a^2 (a^2 (b^2 + c^2) - (b^2 - c^2)^2) / (b^2 - c^2) : :

X(1625) lies on these lines: 5,217   6,13   110,112

X(1625) = midpoint of X(3289) and X(3331)
X(1625) = X(75)-Ceva conjugate of X(6)
X(1625) = crosspoint of X(i) and X(j) for these (i,j): (110,648), (107,110)
X(1625) = crosssum of (523,647), (520,523)

X(1625) = crossdifference of every pair of points on line X(125)X(526)

### X(1626) = TCC-PERSPECTOR OF X(174)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1601))

X(1626) lies on these lines: 3,10   22,1602   38,55   982,1283

X(1626) = X(85)-Ceva conjugate of X(6)

### X(1627) = TCC-PERSPECTOR OF X(251)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1601))
Barycentrics    a^2 (a^4 + a^2 b^2 + a^2 c^2 - b^2 c^2) : :

X(1627) lies on these lines:
2,32   3,1180   22,1184   23,1196   25,111   110,699   187,1194   571,1370   609,612

### X(1628) = TCC-PERSPECTOR OF X(254)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1601))

As a point on the Euler line, X(1628) has Shinagawa coefficients (2(E + F)2E2F - 2(E + 2F)EFS2 + 4FS4, [(E2 + 6EF + 4F2)E - 4(E + F)S2]S2).

X(1628) lies on this line: 2,3

### X(1629) = TCC-PERSPECTOR OF X(275)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1601))

X(1629) lies on these lines: 4,54   22,264   23,324   25,98   251,393   436,1495   1093,1179

### X(1630) = TCC-PERSPECTOR OF X(284)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1601))

X(1630) lies on these lines: 1,19   101,102   109,577   1055,1195

### X(1631) = TCC-PERSPECTOR OF X(366)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a3 - b3 - c3)      (M. Iliev, 5/13/07)

X(1631) lies on these lines:
3,142   6,560   22,1602   25,1826   48,674   55,199   198,480   573,692   789,1502

X(1631) = X(75)-Ceva conjugate of X(6)
X(1631) = crossum of X(116) and X(513)

X(1631) = tangential isogonal conjugate of X(55)

### X(1632) = TCC-PERSPECTOR OF X(648)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1601))

X(1632) lies on these lines:
98,338   99,670   110,925   112,1289   157,264   250,523   476,1302   827,1286   933,1288

X(1632) = reflection of X(648) in X(1576)
X(1632) = cevapoint of X(157) and X(523)
X(1632) = crosspoint of X(99) and X(107)
X(1632) = crosssum of X(512) and X(520)
X(1632) = perspector of ABC and side triangle of circumanticevian triangles of X(2) and X(4)

### X(1633) = TCC-PERSPECTOR OF X(651)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a2 + b2 + c2 - 2bc)/(b - c)      (M. Iliev, 5/13/07)

X(1633) lies on these lines:
7,1486   19,1721   20,1610   28,1770   48,1742   59,1310   99,1310   100,190   101,1292   105,1086   108,109   497,1473

X(1633) = reflection of X(651) in X(692)
X(1633) = X(1275)-Ceva conjugate of X(6)
X(1633) = cevapoint of X(513) and X(1486)
X(1633) = crosspoint of X(i) and X(j) for these (i,j): (99,162), (100,934)
X(1633) = crosssum of X(512) and X(656)
X(1633) = X(i)-aleph conjugate of X(j) for these (i,j): (100,610), (1783,1707), (1897,19)

### X(1634) = TCC-PERSPECTOR OF X(662)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2)/(b2 - c2)      (M. Iliev, 5/13/07)

X(1634) lies on these lines:
3,67   6,694   69,160   99,670   110,351   112,907   237,524   660,765   1306,1307

X(1634) = X(688)-cross conjugate of X(6)
X(1634) = crosspoint of X(99) and X(110)
X(1634) = crosssum of X(512) and X(523)

Tripolar Centroids 1635- 1651

Suppose P, Q, R are collinear points not on the line at infinity. Let M = midpoint{Q, R}. The segment PM has two trisectors. The trisector closer to M here defines the centroid of {P, Q, R}.

Suppose X = x : y : z (trilinears) is a point other than X(2), and define the tripolar centroid of X as the point TG(X) given by

TG(X) = x(by - cz)(by + cz - 2ax) : y(cz - ax)(cz + ax - 2by) : z(ax - by)(ax + by - 2cz).

TG(X) is the centroid of the points BC∩B'C', CA∩C'A', AB∩A'B', where A'B'C' denotes the cevian triangle of X. The notions of centroid and tripolar centroid were contributed by Darij Grinberg, August 24, 2003.

The appearance of (i,j) in the following list means that TG(X(i)) = X(j):

(1,1635), (3,1636), (4,1637), (5,14391), (6,351), (7,1638), (8,1639), (9,14392), (10,4120), (11,14393), (12,14394), (13,9200), (14,9201), (17,14446), (18,14447), (21,14395), (22,14396), (24,14397), (25,14398), (27,11125), (28,14399), (29,14400), (30,14401), (31,14402), (32,14403), (37,14404), (38,14405), (39,14406), (42,14407), (43,14408), (44,14409), (45,14410), (55,14411), (56,14412), (57,14413), (63,14414), (65,14415), (66,14417), (69,14417), (75,4728), (76,9148), (78,14418), (81,14419), (83,14420), (86,4750), (88,14421), (89,14422), (98,1640), (99,1641}, {100,1642), (105,1643), (111,9171), (115,14423), (141,14424), (145,14425), (190,1644), (192,14426), (200,14427), (239,4448), (251,14428), (262,3569), (263,2491), (298,9204), (299,9205), (306,14429), (312,14430), (321,14431), (333,14432), (350,14433), (385,11183), (512,1645), (513,1646), (514,1647}, {519,6544), (523,1648), (524,1649), (525,1650), (536,14434), (551,14435), (648,1651), (671,8371), (869,14436), (899,14437), (957,3310), (985,14438), (1002,665), (1022,244), (1125,4984), (1026,14439), (1267,14440), (1646,14441), (1647,14442), (1648,14443), (1649,14444), (1698,4958), (1916,11182), (1976,6041}, {1992,9125), (2394,125), (2395,6784), (2396,6786), (2403,3756), (2407,5642), (2408,6791), (2409,6793), (2418,12036), (2419,12037), (2421,9155), (3413,13636}, {3414,13722), (3616,4773), (4049,3120), (4240,3163), (5391,14445), (5466,115), (5468,2482), (6548,1086), (9178,3124), (9213,2088), (9221,2081), (13582,1116), (14223,868)

### X(1635) = TRIPOLAR CENTROID OF X(1)

Trilinears    (b - c)(b + c - 2a) : :      (M. Iliev, 5/13/07)

X(1635) lies on these lines: 2,812   44,513   88,1022   100,101   105,1024   244,665   900,1644

X(1635) = reflection of X(1962) in X(351)
X(1635) = X(i)-Ceva conjugate of X(j) for these (i,j): (44,2087), (88,244), (104,2310), (662,214), (1022,513), (1023,44), (2161,2170)
X(1635) = X(2087)-cross conjugate of X(44)
X(1635) = crosspoint of X(i) and X(j) for these (i,j): (44,1023), (57,2222), (88,100), (101,909), (190,1120), (513,1022), (662,759)
X(1635) = crosssum of X(i) and X(j) for these (i,j): (1,1635), (44,513), (88,1022), (100,1023), (514,908), (649,1149), (661,758)
X(1635) = bicentric sum of PU(34)
X(1635) = PU(34)-harmonic conjugate of X(244)
X(1635) = trilinear pole of line X(2087)X(3251)
X(1635) = centroid of antiorthic axis intercepts with sidelines of ABC
X(1635) = isogonal conjugate of X(3257)
X(1635) = X(647)-of-2nd-extouch-triangle

### X(1636) = TRIPOLAR CENTROID OF X(3)

Trilinears    (cos A)(tan B - tan C)(tan B + tan C - 2 tan A) : :
Trilinears    cos A [sec B csc(A - C) + sec C csc(A - B)] : :
Barycentrics    (sin 2A)(sin 2B - sin 2C)(sin 2B + sin 2C - 2 sin 2A) : :

X(1636) lies on these lines: 110,112   520,647   525,3268   1637,1651

X(1636) = crosspoint of X(648) and X(1294)
X(1636) = crosssum of X(i) and X(j) for these (i,j): (4,1637), (523,1990)
X(1636) = intersection of perspectrix of ABC and orthocentroidal triangle (line X(1636)X(1637)) and perspectrix of ABC and anti-orthocentroidal triangle (line X(520)X(647))

### X(1637) = TRIPOLAR CENTROID OF X(4)

Barycentrics    tan A (tan B - tan C)(tan B + tan C - 2 tan A) : :
Barycentrics    (b^2 - c^2)(2a^4 - b^4 - c^4 - a^2b^2 - a^2c^2 + 2b^2c^2) : :
Barycentrics    (cot B - cot C)(tan B + tan C - 2 tan A) : :

X(1637) lies on these lines: 2,3268   98,111   107,112   115,125   132,1560   230,231   1499,1514   1636,1651

X(1637) = complement of X(3268)
X(1637) = X(1989)-Ceva conjugate of X(115)
X(1637) = crosspoint of X(2) and X(476)
X(1637) = crosssum of X(i) and X(j) for these (i,j): (3,1636), (6,526)
X(1637) = crossdifference of every pair of points on line X(3)X(74)
X(1637) = PU(4)-harmonic conjugate of X(6103)
X(1637) = midpoint of circumcenters of X(13)X(14)X(15) and X(13)X(14)X(16)
X(1637) = perspector of circumconic centered at X(3258)
X(1637) = center of circumconic that is locus of trilinear poles of lines passing through X(3258)
X(1637) = X(2)-Ceva conjugate of X(3258)
X(1637) = centroid of orthic axis intercepts with sidelines of ABC
X(1637) = tripolar centroid of X(4) wrt orthic triangle
X(1637) = center of Dao-Moses-Telv circle
X(1637) = X(115) of 2nd Parry triangle
X(1637) = insimilicenter of circles {{X(98),X(107),X(125),X(132)}} and {{X(6),X(111),X(112),X(115),X(187),X(1560)}}; the exsimilicenter is X(6103)
X(1637) = radical center of orthocentroidal circles of ABC, orthocentroidal triangle, anti-orthocentroidal triangle
X(1637) = inverse-in-Hutson-Parry-circle of X(1640)
X(1637) = {X(13636),X(13722)}-harmonic conjugate of X(1640)

### X(1638) = TRIPOLAR CENTROID OF X(7)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1635))
Barycentrics    (b - c) (2 a^2 - b^2 - c^2 - a b - a c + 2 b c) : :

Let Oa be the circle centered at A with radius k(b + c -a) (for some constant k>0), and define Ob and Oc cyclically. Let A' be the exsimilicenter of Ob and Oc, and define B' and C' cyclically. The centroid of (degenerate) triangle A'B'C' = X(1638). This is independent of the choice of k. (Randy Hutson, January 29, 2018)

X(1638) lies on these lines: 2,918   11,244   57,654   88,673   241,514   354,926   651,658

X(1638) = reflection of X(1639) in X(2)
X(1638) = centroid of Gergonne line intercepts with sidelines of ABC
X(1638) = X(351)-of-intouch-triangle
X(1638) = complement of X(30565)

### X(1639) = TRIPOLAR CENTROID OF X(8)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b - c)(b + c - a)(b + c - 2a)      (M. Iliev, 5/13/07)

X(1639) lies on these lines: 2,918   9,654   11,1146   210,926   522,650

X(1639) = reflection of X(1638) in X(2)
X(1639) = centroid of (degenerate) side-triangle of ABC and Gemini triangle 15

### X(1640) = TRIPOLAR CENTROID OF X(98)

Trilinears    bc(b2 - c2)(2a6 - b6 - c6 - 2a4b2 - 2a4c2 + a2b4 + a2c4 + b4c2 + b2c4) : :     (M. Iliev, 5/25/07)

X(1640) lies on these lines:
2,525   4,1499   6,523   39,647   51,512   115,125

X(1640) = radical center of Brocard circle, orthocentroidal circle, and orthosymmedial circle
X(1640) = radical center of orthocentroidal circle, orthosymmedial circle, and circle O(2,6)
X(1640) = X(3268)-of-1st-Brocard-triangle
X(1640) = orthocenter of X(2)X(4)X(6)
X(1640) = inverse-in-Hutson-Parry-circle of X(1637)
X(1640) = {X(13636),X(13722)}-harmonic conjugate of X(1637)

### X(1641) = TRIPOLAR CENTROID OF X(99)

Trilinears    f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1635))

X(1641) lies on these lines: 2,6   351,690

X(1641) = reflection of X(1648) in X(2)
X(1641) = centroid of line X(2)X(6) intercepts with sidelines of ABC
X(1641) = intersection of tangents to circle {X(2),X(110),X(2770),X(5463),X(5464)} at X(5463) and X(5464)
X(1641) = centroid of degenerate cross-triangle of anticomplementary and Schroeter triangles

### X(1642) = TRIPOLAR CENTROID OF X(100)

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

X(1642) lies on these lines: 1,6   241,1025

### X(1643) = TRIPOLAR CENTROID OF X(105)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1635))

X(1643) lies on these lines: 1,650   6,513   42,663   57,1022   244,665   649,764

### X(1644) = TRIPOLAR CENTROID OF X(190)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1635))

X(1644) lies on these lines: {1,2}, {900,1635}

X(1644) = reflection of X(1647) in X(2)
X(1644) = centroid of Nagel line intercepts with sidelines of ABC

### X(1645) = TRIPOLAR CENTROID OF X(512)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1635))

X(1645) lies on these lines: 2,39   351,865

X(1645) = X(i)-Ceva conjugate of X(j) for these (i,j): (538,888), (728,888)
X(1645) = crosspoint of X(i) and X(j) for these (i,j): (538,888), (728,888)

### X(1646) = TRIPOLAR CENTROID OF X(513)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b - c)2(2bc - ab - ac)      (M. Iliev, 5/13/07)

X(1646) lies on these lines: 2,37   88,292   244,665

X(1646) = X(i)-Ceva conjugate of X(j) for these (i,j): (536,891), (738,891)
X(1646) = crosspoint of X(i) and X(j) for these (i,j): (536,891), (738,891)
X(1646) = crosssum of X(739) and X(898)

### X(1647) = TRIPOLAR CENTROID OF X(514)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(b - c)2(2a - b - c)      (M. Iliev, 5/13/07)

X(1647) lies on these lines: 1,2   11,244   80,106   149,1054

X(1647) = reflection of X(1644) in X(2)
X(1647) = X(i)-Ceva conjugate of X(j) for these (i,j): (80,513), (519,900), (903,514), (1120,522)
X(1647) = crosspoint of X(i) and X(j) for these (i,j): (514,903), (519,900)
X(1647) = crosssum of X(i) and X(j) for these (i,j): (101,902), (106,901)

### X(1648) = TRIPOLAR CENTROID OF X(523)

Trilinears    bc(b2 - c2)2(b2 + c2 - 2a2) : :

X(1648) lies on these lines: 2,6   115,125   669,865

X(1648) = reflection of X(1641) in X(2)
X(1648) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,1649), (67,512), (468,351), (524,690), (671,523)
X(1648) = crosspoint of X(i) and X(j) for these (i,j): (523,671), (524,690)
X(1648) = crosssum of X(i) and X(j) for these (i,j): (110, 187), (111,691)
X(1648) = intersection of tangents to Hutson-Parry circle at X(13) and X(14)
X(1648) = pole wrt Hutson-Parry circle of Fermat axis
X(1648) = inverse-in-Hutson-Parry-circle of X(115)
X(1648) = perspector of circumconic centered at X(1649)
X(1648) = center of circumconic that is locus of trilinear poles of lines passing through X(1649)
X(1648) = intersection of line PU(40) (X(115)X(125)) and trilinear polar of cevapoint of PU(40)
X(1648) = X(2502)-of-4th-Brocard-triangle
X(1648) = X(2502)-of-orthocentroidal-triangle
X(1648) = centroid of (degenerate) cross-triangle of 4th Brocard and orthocentroidal triangle
X(1648) = trilinear pole of line X(2682)X(14443)
X(1648) = {X(13636),X(13722)}-harmonic conjugate of X(115)

### X(1649) = TRIPOLAR CENTROID OF X(524)

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

X(1649) lies on the Kiepert parabola and these lines: 2,523   3,669   39,647   114,126   351,690

X(1649) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,1648), (99,524), (523,690)
X(1649) = crosspoint of X(i) and X(j) for these (i,j): (99,524), (523,690)
X(1649) = crosssum of X(i) and X(j) for these (i,j): (110, 691), (111,512)
X(1649) = perspector of circumconic centered at X(1648)
X(1649) = center of circumconic that is locus of trilinear poles of lines passing through X(1648)
X(1649) = center of circle {{X(2),X(110),X(2770),X(5463),X(5464)}}
X(1649) = harmonic center of circles {{X(13),X(15),X(5463),X(5464)}} and {{X(14),X(16),X(5463),X(5464)}}

### X(1650) = TRIPOLAR CENTROID OF X(525)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1635))

As a point on the Euler line, X(1650) has Shinagawa coefficients ((4E - 5F)F - S2, -3(E + F)F + S2).

Let W be the circumconic with center X(1650). One of the asymptotes of W is the Euler line. The other is in the direction of X(9033). For a sketch, click X(9033). (Angel Montesdeoca, April 19, 2016)

X(1650) lies on these lines: 2,3   122,125

X(1650) = reflection of X(1651) in X(2)
X(1650) = anticomplement of X(402)
X(1650) = X(i)-Ceva conjugate of X(j) for these (i,j): (265,520), (1294,523), (1494,525)
X(1650) = crosspoint of X(525) and X(1494)
X(1650) = crosssum of X(i) and X(j) for these (i,j): (74,1304), (110,2071), (112,1495)
X(1650) = complement of X(4240)
X(1650) = homothetic center of Gossard and medial triangles

### X(1651) = TRIPOLAR CENTROID OF X(648)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b)  (see note above X(1635))

As a point on the Euler line, X(1651) has Shinagawa coefficients (3(2E - 7F)F - S2,9(E + F)F - 3S2).

X(1651) lies on these lines: 2,3   1636,1637

X(1651) = reflection of X(i) in X(j) for these (i,j): (2,402), (1650,2)
X(1651) = Euler line intercept of trilinear polar of X(30)
X(1651) = centroid of Euler line intercepts with sidelines of ABC
X(1651) = X(2)-of-Gossard triangle

### X(1652) = 4th EVANS PERSPECTOR

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sqrt(3/4) - sin(A + π/3) + sin(B + π/3) + sin(C + π/3)
= sqrt(3/4) - cos(A - π/6) + cos(B - π/6) + cos(C - π/6)
Trilinears        sqrt(3) tan(A/2) + tan(B/2) tan(C/2) : sqrt(3) tan(B/2) + tan(C/2) tan(A/2) : sqrt(3) tan(C/2) + tan(A/2) tan(B/2)
= sqrt(3) tan(A/2) + (b + c - a)/(a + b + c) : sqrt(3) tan(B/2) + (c + a - b)/(a + b + c) : sqrt(3) tan(C/2) + (a + b - c)/(a + b + c)     (M. Iliev, 5/13/07)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

See also X(1276) and X(1277) for the 2nd and 3rd Evans perspectors.

X(1652) lies on these lines:
{1,15}, {2,7}, {13,3464}, {17,3336}, {46,1277}, {56,5240}, {61,3468}, {65,5239}, {396,554}, {559,1100}, {3638,5011}, {4848,5245}

X(1652) = X(i)-Ceva conjugate of X(j) for these (i,j): (554,1), (2160,1653)
X(1652) = X(554)-aleph conjugate of X(1652)
X(1652) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,3179,1276), (9,57,1653), (1400,3218,1653), (2285,3306,1653)

### X(1653) = 5th EVANS PERSPECTOR

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sqrt(3/4) + sin(A - π/3) - sin(B - π/3) - sin(C - π/3)
= - sqrt(3/4) + cos(A + π/6) - cos(B + π/6) - cos(C + π/6)
Trilinears        sqrt(3) tan(A/2) - tan(B/2) tan(C/2) : sqrt(3) tan(B/2) - tan(C/2) tan(A/2) : sqrt(3) tan(C/2) - tan(A/2) tan(B/2)
= sqrt(3) tan(A/2) - (b + c - a)/(a + b + c) : sqrt(3) tan(B/2) - (c + a - b)/(a + b + c) : sqrt(3) tan(C/2) - (a + b - c)/(a + b + c)     (M. Iliev, 5/13/07)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(1653) lies on these lines: 1,16   2,7   46,1276   395,1081   1082,1100

X(1653) = X(i)-Ceva conjugate of X(j) for these (i,j): (1081,1), (2160,1652)
X(1653) = X(1081)-aleph conjugate of X(1653)

### X(1654) = 1st HATZIPOLAKIS PARALLELIAN POINT

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(-a2 + b2 + c2 + bc + ca + ab)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = -a2 + b2 + c2 + bc + ca + ab

Let P be a point in the plane of, but not on a sideline of, triangle ABC. Let BA be the point where the line through P parallel to line BC meets line BA, and let CA be the point where the line through P parallel to line BC meets line CA. Define CB, AB, AC, and BC cyclically. If P = X(1654), then

|ABA| + |ACA| = |BCB| + |BAB| = |CAC| + |CBC|

(Antreas Hatzipolakis, Anopolis #20, 1/20/02)

X(1654) lies on these lines: 2,6   8,192   10,894   37,319   71,1762   190,594

X(1654) = reflection of X(86) in X(1213)
X(1654) = isotomic conjugate of X(6625)
X(1654) = crosspoint of X(2) and X(8) wrt 2nd Conway triangle
X(1654) = anticomplement of X(86)
X(1654) = X(i)-Ceva conjugate of X(j) for these (i,j): (10,2), (894,192)
X(1654) = anticomplementary isotomic conjugate of X(1)
X(1654) = complement of X(20090)
X(1654) = polar conjugate of isogonal conjugate of X(22139)
X(1654) = {X(2),X(2895)}-harmonic conjugate of X(17778)
X(1654) = perspector of Gemini triangle 39 and cross-triangle of Gemini triangles 39 and 40

### X(1655) = 2nd HATZIPOLAKIS PARALLELIAN POINT

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[-1/a2 + 1/b2 + 1/c2 + 1/(bc) + 1/(ca) + 1/(ab)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = -1/a2 + 1/b2 + 1/c2 + 1/(bc) + 1/(ca) + 1/(ab)

Continuing from the description of X(1654), let h(B,A) be the distance from the point BA to the line CA, and define five other distances cyclically. If P = X(1655), then

h(B,A) + h(C,A) = h(C,B) + h(A,B) = h(A,C) + h(B,C)

(Antreas Hatzipolakis, Anopolis #20, 1/20/02)

X(1655) lies on these lines: 2,39   8,192   21,385   193,452   350,1107   668,1500

X(1655) = anticomplement of X(274)
X(1655) = X(i)-Ceva conjugate of X(j) for these (i,j): (37,2), (1909,8)

### X(1656) = INTERSECTION OF EULER LINE AND LINEX(17)X(18)

Trilinears    3 cos A + 4 cos B cos C : :
Barycentrics   3 + cot B cot C : :
Barycentrics   3 S^2 + SB SC : :
Barycentrics   a^4 - 3a^2(b^2 + c^2) + 2(b^2 - c^2)^2 : :
X(1656) = 6*X(381) - X(382)

As a point on the Euler line, X(1656) has Shinagawa coefficients (3,1).

Let A' be the reflection of A in X(3), and define B' and C' cyclically. The triangle A'B'C' is homothetic to the medial triangle, and the center of homothety is X(1656).

X(1656) lies on these lines:
2,3   6,17   10,1482   11,498   12,499   49,569   51,1216   125,399   141,1351   302,634   303,633   355,1125   373,568   485,615   486,590   517,1698   567,1147   576,599

X(1656) = midpoint of X(5) and X(632)
X(1656) = reflection of X(i) in X(j) for these (i,j): (3,631), (631,632)
X(1656) = inverse-in-orthocentroidal-circle of X(140)
X(1656) = complement of X(631)
X(1656) = anticomplement of X(632)
X(1656) = {X(3),X(5)}-harmonic conjugate of X(381)
X(1656) = {X(17),X(18)}-harmonic conjugate of X(6)
X(1656) = {X(1506),X(7746)}-harmonic conjugate of X(6)
X(1656) = homothetic center of 2nd Euler triangle and mid-triangle of orthic and circumorthic triangles
X(1656) = homothetic center of submedial triangle and mid-triangle of orthic and circumorthic triangles
X(1656) = X(5)-of-cross-triangle-of-Euler-and-anti-Euler-triangles
X(1656) = X(1385)-of-cross-triangle-of Aquila-and-anti-Aquila-triangles
X(1656) = homothetic center of X(5)-altimedial and X(3)-anti-altimedial triangles
X(1656) = homothetic center of X(140)-altimedial and X(140)-anti-altimedial triangles
X(1656) = endo-homothetic center of Ehrmann mid-triangle and X3-ABC reflections triangle; the homothetic center is X(3843)
X(1656) = radical center of de Longchamps circles of ABC and 1st and 2nd Ehrmann circumscribing triangles

### X(1657) = {X(3),X(4)}-HARMONIC CONJUGATE OFX(1656)

Trilinears    3 cos A - 4 cos B cos C : :
Barycentrics   5a^4 - 2b^4 - 2c^4 - 3a^2b^2 - 3a^2c^2 + 4b^2c^2 : :
X(1657) = 3*X(381) - 2*X(382)

As a point on the Euler line, X(1657) has Shinagawa coefficients (3, -7).

Let La be the polar of X(3) wrt the A-power circle, and define Lb and Lc cyclically. Let A' = Lb∩Lc, B' = Lc∩La, C' = La∩Lb. Triangle A'B'C' is homothetic to ABC, and the orthocenter of A'B'C' is X(1657). (Randy Hutson, December 2, 2017)

X(1657) lies on these lines: 2,3   195,1181   399,1498   516,1482

X(1657) = reflection of X(i) in X(j) for these (i,j): (3,20), (4,550), (382,3)
X(1657) = {X(381),X(382)}-harmonic conjugate of X(5076)
X(1657) = Ehrmann-mid-to-ABC similarity image of X(382)
X(1657) = endo-homothetic center of Ehrmann mid-triangle and ABC-X3 reflections triangle; the homothetic center is X(382)

### X(1658) = CIRCUMCENTER OF KOSNITA TRIANGLE

Trilinears    cos(B - C) + 4 cos 2A cos B cosC : : (Nikolaos Dergiades, Hyacinthos 7752)

As a point on the Euler line, X(1658) has Shinagawa coefficients (E + 8F, -3E - 8F).

The vertices of the Kosnita triangle are the circumcenters of the triangles BOC, COA, AOB, where O is the circumcenter, X(3). (Darij Grinberg, 8/24/03)

Let La be the polar of X(4) wrt the circle centered at A and passing through X(3), and define Lb, Lc cyclically. (Note: X(4) is the perspector of any circle centered at a vertex of ABC.) Let A" = Lb∩Lc, and define B", C" cyclically. Triangle A"B"C" is homothetic to ABC, and its nine-point center is X(1658). (Randy Hutson, July 20, 2016)

X(1658) lies on these lines: 2,3   54,568   143,578   569,973   1092,1511   1147,1154

X(1658) = midpoint of X(3) and X(26)

### X(1659) = YIU-PAASCHE POINT

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1/(1 + sin A + cos A)
Barycentrics    sin A + cos B + cos C : :

For a discussion of this point, see Paul Yiu, Introduction to the Geometry of the Triangle, 2002, Article 3.5.4 Exercise 4d.
(The preceding Exercise 4c presents the Paasche point, X(1123),) (Contributed by Darij Grinberg, 8/24/03)

See Francisco Javier García Capitán, Hyacinthos #21541, 2/14/2013.

X(1659) lies on these lines: 1,4   2,176   57,482   75,491   92,1585   553,1373

X(1659) = isogonal conjugate of X(2066)
X(1659) = X(482)-cross conjugate of X(7)
X(1659) = crosssum of X(48) and X(605)

### X(1660) = 1st GRINBERG MIDPOINTS PERSPECTOR

Trilinears  a3[b8 + c8 - a8 - 2a2(b6 + c6) + 2b2c2(b4 + c4 - 5a4 + 3a2b2 + 3a2c2 - 3b2c2) + 2a6(b2 + c2)]

Let AB be the point in which the line through A perpendicular to CA meets line BC, and define points AC, BC, BA, CA, CB functionally. Let

XA = midpoint{AB, AC},
YA = midpoint{BA, CA},
ZA = midpoint{BC, CB},

and define XB, XC, YB, YC, ZB, ZC functionally.

The lines AXA, BXB, CXC concur in X(20).
The lines AYA, BYB, CYC concur in X(393).
The lines AZA, BZB, CZC concur in X(6).
The lines XAYA, XBYB, XCYC concur in X(1660).
The lines YAZA, YBZB, YCZC concur in X(3).
The lines ZAXA, ZBXB, ZCXC concur in X(1661).

Contributed by Darij Grinberg, August 24, 2003; see Hyacinthos #7225.

X(1660) lies on these lines: 6,25   30,156   110,1370   394,1619   578,1596   1092,1498   1368,1503

X(1660) = midpoint of X(394) and X(1619)
X(1660) = X(20)-Ceva conjugate of X(577)
X(1660) = X(25)-of-A'B'C', as described by Tran Quang Huyng, ADGEOM #2697 (8/26/2015)

### X(1661) = 2nd GRINBERG MIDPOINTS PERSPECTOR

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b/v + c/w - a/u), where
(u,v,w) = (cos A - cos B cos C, cos B - cos C cos A, cos C - cos A cos B).

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1661) is described at X(1660).

X(1661) = X(20)-Ceva conjugate of X(6)

X(1661) lies on these lines: 25,393   154,577   1619,1624

Circle-related Points 1662- 1706

The next cluster of points were contributed by Peter J. C. Moses during August, 2003. Symbols used to represent functions of a,b,c (or A,B,C) include

σ = area of triangle ABC
ω = arccot(cot A + cot B + cot C) = arccot[(a2 + b2 +c2)/(4σ)]; ω is the Brocard angle of ABC
e = sqrt(1 - 4 sin2ω) (as in Gallatly, p. 96, along with other formulas involving ω)
s = (a + b + c)/2 = semiperimeter of ABC
r = σ/s = inradius of ABC

Circles mentioned in this section are the following:

circumcircle X(3) R
incircle X(1) r
nine-point circle X(5) R/2
Brocard circle X(182) eR/(2 cos ω)
1st Lemoine circle X(182) (1/2)(R sec ω)
2nd Lemoine circle (cosine circle) X(6) abc/(a2 + b2 + c2)
Spieker circle X(10) r/2
Apollonius circle X(970) (r2 + s2)/4r
Bevan circle X(40) 2R

For further information on many circles, click MathWorld and scroll down to links to various specific circles.

### X(1662) = 1st INTERSECTION OF BROCARD AXIS AND 1st LEMOINE CIRCLE

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = e cos(A - ω) + cos(A + ω)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(1662) lies on these lines: 3,6   1676,2040   1677,2039

X(1662) = reflection of X(1663) in X(182)
X(1662) = inverse-in-Brocard-circle of X(1664)
X(1662) = {X(1687),X(1688)}-harmonic conjugate of X(1663)

### X(1663) = 2nd INTERSECTION OF BROCARD AXIS AND 1st LEMOINE CIRCLE

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = e cos(A - ω) - cos(A + ω)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(1663) lies on these lines: 3,6   1676,2039   1677,2040

X(1663) = reflection of X(1662) in X(182)
X(1663) = inverse-in-Brocard-circle of X(1665)
X(1663) = {X(1687),X(1688)}-harmonic conjugate of X(1662)

### X(1664) = INVERSE-IN-BROCARD-CIRCLE OF X(1662)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = e cos(A + ω) + cos(A - ω)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(1664) lies on this line: 3,6

X(1664) = reflection of X(1665) in X(182)
X(1664) = inverse-in-Brocard-circle of X(1662)

### X(1665) = INVERSE-IN-BROCARD-CIRCLE OF X(1663)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = e cos(A + ω) - cos(A - ω)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(1665) lies on this line: 3,6

X(1665) = reflection of X(1664) in X(182)
X(1665) = inverse-in-Brocard-circle of X(1663)

### X(1666) = 1st INTERSECTION OF BROCARD AXIS AND 2nd LEMOINE CIRCLE

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(A + ω) + e sin A
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(1666) lies on these lines: 3,6   485,2040   486,2039

X(1666) = reflection of X(1667) in X(6)
X(1666) = inverse-in-Brocard-circle of X(1668)

X(1666) = inverse-in-2nd-Brocard-circle of X(2563)

### X(1667) = 2nd INTERSECTION OF BROCARD AXIS AND 2nd LEMOINE CIRCLE

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(A + ω) - e sin A
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(1667) lies on these lines: 3,6   485,2039   486,2040

X(1667) = reflection of X(1666) in X(6)
X(1667) = inverse-in-Brocard-circle of X(1669)
X(1667) = inverse-in-2nd-Brocard-circle of X(2562)

### X(1668) = EXSIMILICENTER(BROCARD CIRCLE, 2nd LEMOINE CIRCLE)

Trilinears    cos(A - ω) - e sin A : :
X(1668) = |OK|*X(6) + 2R tan ω*X(182)
X(1668) = e*X(6) + 2 sin ω*X(182) X(1668) = sin ω*X(3) + (sin ω + e)*X(6)

X(1668) lies on these lines:
3,6   485,1348   486,1349   1124,1674   1335,1675   1377,1678   1378,1679   1702,1704   1703,1705

X(1668) = inverse-in-Brocard-circle of X(1666)
X(1668) = {X(6),X(182)}-harmonic conjugate of X(1669)

### X(1669) = INSIMILICENTER(BROCARD CIRCLE, 2nd LEMOINE CIRCLE)

Trilinears    cos(A - ω) + e sin A
X(1669) = |OK|*X(6) - 2R tan ω*X(182)
X(1669) = e*X(6) - 2 sin ω*X(182)
X(1669) = sin ω*X(3) + (sin ω - e)*X(6)

X(1669) lies on these lines:
3,6   485,1349   486,1348   1124,1675   1335,1674   1377,1679   1378,1678   1702,1705   1703,1704

X(1669) = inverse-in-Brocard-circle of X(1667)

X(1669) = {X(6),X(182)}-harmonic conjugate of X(1668)

### X(1670) = INVERSE-IN-BROCARD-CIRCLE OF X(1342)

Trilinears    sin A - cos A cot(ω/2) : sin B - cos B cot(ω/2) : sin C - cos C cot(ω/2)
Trilinears    sin A - sin(A + ω) : sin B - sin(B + ω) : sin C - sin(C + ω)
Trilinears    cos A + cos(A + ω) : cos B + cos(B + ω) : cos C + cos(C + ω)
Trilinears    cos(A + ω/2) : cos(B + ω/2) : cos(C + ω/2)
Trilinears    (csc ω + cot ω) cos A - sin A : :
Trilinears    cos A - (csc ω - cot ω) sin A : :

X(1670) is the external center of similitude of the Gallatly circle and the 2nd Lemoine circle. (Peter J. C. Moses, 9/03; cf. X(1342))

X(1670) and X(1671) are the Brocard axis intercepts of the 2nd Brocard circle. (Randy Hutson, August 29, 2018)

X(1670) lies on these lines:
3,6   76,1677   262,1676   485,2009   486,2010   1124,2007   1335,2008   1377,2013   1378,2014   1702,2017   1703,2018

X(1670) = reflection of X(1671) in X(3)
X(1670) = inverse-in-Brocard-circle of X(1342)
X(1670) = X(76)-Ceva conjugate of X(1671)

X(1670) = isogonal conjugate of X(1677)
X(1670) = {X(6),X(39)}-harmonic conjugate of X(1671)
X(1670) = {X(32),X(3094)}-harmonic conjugate of X(1671)
X(1670) = {X(182),X(3095)}-harmonic conjugate of X(1671)

### X(1671) = INVERSE-IN-BROCARD-CIRCLE OF X(1343)

Trilinears    sin A + cos A tan(ω/2) : :
Trilinears    sin A + sin(A + ω) : :
Trilinears    cos A - cos(A + ω) : :
Trilinears    sin(A + ω/2) : :
Trilinears    (sec ω + tan ω) cos A + sin A : :
Trilinears    (csc ω - cot ω) cos A + sin A : :
Trilinears    cos A + (csc ω + cot ω) sin A : :

X(1671) is the internal center of similitude of the Gallatly circle and the 2nd Lemoine circle. (Peter J. C. Moses, 9/03)

X(1671) lies on these lines:
3,6   76,1676   262,1677   485,2010   486,2009   1124,2008   1335,2007   1377,2014   1378,2013   1702,2018   1703,2017

X(1671) = reflection of X(1670) in X(3)
X(1671) = isogonal conjugate of X(1676)
X(1671) = X(76)-Ceva conjugate of X(1670)
X(1671) = X(1)-of-X(6)PU(1)
X(1671) = inverse-in-Brocard-circle of X(1343)
X(1671) = {X(6),X(39)}-harmonic conjugate of X(1670)
X(1671) = {X(32),X(3094)}-harmonic conjugate of X(1670)
X(1671) = {X(182),X(3095)}-harmonic conjugate of X(1670)

### X(1672) = INSIMILICENTER(INCIRCLE, 1st LEMOINE CIRCLE)

Trilinears    1 + cos(A - ω) : :
X(1672) = (R sec ω)*X(1) + 2r*X(182)
X(1672) = (R sec ω)*X(1) + r*X(3) + r*X(6)

X(1672) lies on these lines:
1,182   2,1681   8,1680   11,1676   12,1677   55,1343   56,1342   57,1700   181,1683   371,2008   372,2007   1015,2035   1124,1688   1335,1687   1682,1684   1697,1701

### X(1673) = EXSIMILICENTER(INCIRCLE, 1st LEMOINE CIRCLE)

Trilinears    1 - cos(A - ω) : :

X(1673) = (R sec ω)*X(1) - 2r*X(182)
X(1673) = (R sec ω)*X(1) - r*X(3) - r*X(6)

X(1673) lies on these lines:
1,182   2,1680   8,1681   11,1677   12,1676   55,1342   56,1343   57,1701   181,1684   371,2007   372,2008   1015,2036   1124,1687   1335,1688   1500,2035   1682,1683   1697,1700

### X(1674) = INSIMILICENTER(INCIRCLE, BROCARD CIRCLE)

Trilinears    e + cos(A - ω) : :
X(1674) = |OK|*X(1) + 2r*X(182)
X(1674) = (R e sec ω)*X(1) + 2r*X(182)
X(1674) = (R e sec ω)*X(1) + r*X(3) + r*X(6)

X(1674) lies on these lines:
1,182   2,1679   8,1678   11,1348   12,1349   55,1341   56,1340   57,1704   181,1693   1015,2033   1124,1668   1335,1669   1500,2034   1682,1694   1697,1705   2007,2012   2008,2011

### X(1675) = EXSIMILICENTER(INCIRCLE, BROCARD CIRCLE)

Trilinears    e - cos(A - ω) : :
X(1675) = |OK|*X(1) - 2r*X(182)
X(1675) = (R e sec ω)*X(1) - 2r*X(182)
X(1675) = (R e sec ω)*X(1) - r*X(3) - r*X(6)

X(1675) lies on these lines:
1,182   2,1678   8,1679   11,1349   12,1348   55,1340   56,1341   57,1705   181,1694   1015,2034   1124,1669   1335,1668   1500,2033   1682,1693   1697,1704   2007,2011   2008,2012

### X(1676) = INSIMILICENTER(1st LEMOINE CIRCLE, NINE-POINT CIRCLE)

Trilinears   csc(A + ω/2) : csc(B + ω/2) : csc(C + ω/2)
Trilinears    cos(B - C) + cos(A - ω) : :
X(1676) = (sec ω)*X(5) + X(182)
X(1676) = X(3) + (2 sec ω)*X(5) + X(6)

Let Lab and Lac be the lines obtained by rotating line BC through B and C resp., by an angle of ω/2 away from A. Let A' be Lab∩Lac. Define B' and C'cyclically. The lines AA', BB', CC' concur in X(1676). (Randy Hutson, September 14, 2016)

X(1676) lies on the Kiepert hyperbola and these lines:
2,1343   4,1342   5,182   10,1684   11,1672   12,1673   76,1671   115,2035   262,1670   371,2010   372,2009   485,1688   486,1687   1329,1680   1506,2036   1662,2040   1663,2039   1698,1701   1699,1700

X(1676) = isogonal conjugate of X(1671)
X(1676) = X(32)-Ceva conjugate of X(1677)

### X(1677) = EXSIMILICENTER(1st LEMOINE CIRCLE, NINE-POINT CIRCLE)

Trilinears    sec(A + ω/2) : :
Trilinears    cos(B - C) - cos(A - ω) : :
X(1677) = (sec ω)*X(5) - X(182) = X(3) - (2 sec ω)*X(5) + X(6)

Let A' be the apex of the isosceles triangle BA'C constructed inward on BC such that ∠A'BC = ∠A'CB = ω/2. Define B' and C' cyclically. Let Ha be the orthocenter of BA'C, and define Hb and Hc cyclically. The lines AHa, BHb, CHc concur in X(1677). (Randy Hutson, July 20, 2016)

X(1677) lies on these lines:
2,1342   4,1343   5,182   10,1683   11,1673   12,1672   76,1670   115,2036   262,1671   371,2009   372,2010   485,1687   486,1688   1329,1681   1506,2035   1662,2039   1663,2040   1698,1700   1699,1701

X(1677) = isogonal conjugate of X(1670)
X(1677) = X(32)-Ceva conjugate of X(1676)

### X(1678) = INSIMILICENTER(BROCARD CIRCLE, SPIEKER CIRCLE)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[e(b + c) + a cos(A - ω)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = e(b + c) + a cos(A - ω)

X(1678) lies on these lines:
2,1675   8,1674   9,1705   10,182   958,1341   1329,1348   1340,1376   1377,1668   1378,1669   1573,2034   1574,2033   1704,1706   2011,2014   2012,2013

### X(1679) = EXSIMILICENTER(BROCARD CIRCLE, SPIEKER CIRCLE)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[e(b + c) - a cos(A - ω)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = e(b + c) - a cos(A - ω)

X(1679) lies on these lines:
2,1674   8,1675   9,1704   10,182   958,1340   1329,1349   1341,1376   1377,1669   1378,1668   1573,2033   1574,2034   1705,1706   2011,2013   2012,2014

### X(1680) = INSIMILICENTER(1st LEMOINE CIRCLE, SPIEKER CIRCLE)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[b + c + a cos(A - ω)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = b + c + a cos(A - ω)

X(1680) lies on these lines: 2,1673   8,1672   9,1701   10,182   371,2014   372,2013   958,1343   1329,1676   1342,1376   1377,1688   1378,1687   1573,2036   1574,2035   1700,1706

### X(1681) = EXSIMILICENTER(1st LEMOINE CIRCLE, SPIEKER CIRCLE)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[b + c - a cos(A - ω)]
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = b + c - a cos(A -ω)

X(1681) lies on these lines: 2,1672   8,1673   9,1700   10,182   371,2013   372,2014   958,1342   1329,1677   1343,1376   1377,1687   1378,1688   1573,2035   1574,2036   1701,1706

### X(1682) = INSIMILICENTER(INCIRCLE, APOLLONIUS CIRCLE)

Trilinears    [s cos(A/2) - r sin(A/2)]2 : :
Trilinears    a(b + c - a)(b2 + c2 + ab + ac)2 : :      (M. Iliev, 5/13/07)

The exsimilicenter of the incircle and Apollonius circle is X(181). Also, the triangle A'B'C' formed (as at X(2092) by the intersections of the Apollonius circle and the excircles is perspective to the cevian triangle of X(1), and the perspector is X(1682). (Paul Yiu, Hyacinthos #8076, 10/01/03)

Let JaJbJc be the excentral triangle and PaPbPc be the Apollonius triangle. Let Pa' = {X(970),Ja}-harmonic conjugate of Pa, and define Pb' and Pc' cyclically. The lines APa', BPb', CPc' concur in X(1682); see also X(11). (Randy Hutson, December 10, 2016)

X(1682) lies on these lines: 1,181   3,1397   10,11   43,1697   55,386   56,573   57,1695   73,1362   212,1472   215,501   988,1401   1124,1686   1335,1685   1672,1684   1673,1683   1674,1694   1675,1693   2007,2020   2008,2019

X(1682) = {X(1),X(970)}-harmonic conjugate of X(181)
X(1682) = perspector of ABC and cross-triangle of ABC and Apollonius triangle

### X(1683) = INSIMILICENTER(1st LEMOINE CIRCLE, APOLLONIUS CIRCLE)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = s cos(A -ω/2) - r sin(A - ω/2)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1683) lies on these lines: 3,6   10,1677   43,1700   181,1672   1673,1682   1695,1701

### X(1684) = EXSIMILICENTER(1st LEMOINE CIRCLE, APOLLONIUS CIRCLE)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = r cos(A -ω/2) + s sin(A - ω/2)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1684) lies on these lines: 3,6   10,1676   43,1701   181,1673   1672,1682   1695,1700

### X(1685) = INSIMILICENTER(2nd LEMOINE CIRCLE, APOLLONIUS CIRCLE)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (r - s)sin A - (r + s)cos A
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1685) lies on these lines: 3,6   10,486   43,1702   181,1124   1335,1682   1695,1703

### X(1686) = EXSIMILICENTER(2nd LEMOINE CIRCLE, APOLLONIUS CIRCLE)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (r + s)sin A + (r - s)cos A
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1686) lies on these lines: 3,6   10,485   43,1703   181,1335   1124,1682   1695,1702

### X(1687) = INSIMILICENTER(1st LEMOINE CIRCLE, 2nd LEMOINE CIRCLE)

Trilinears    sin(A - ω/2 + π/4) : : (M. Iliev, 5/13/07)
Trilinears    cos(A - ω/2 - π/4) : :
Trilinears    cos(A - ω) + sin A : :
Trilinears    cos A + (sec ω + tan ω) sin A : :
Trilinears    cos A + sin(A - ω) : :

X(1687) lies on these lines:
3,6   83,2010   98,2009   485,1677   486,1676   1124,1672   1335,1672   1377,1681   1378,1680   1700,1703   1701,1702

X(1687) = reflection of X(1688) in X(1691)
X(1687) = isogonal conjugate of X(2009)
X(1687) = inverse-in-circumcircle of X(1688)
X(1687) = inverse-in-Brocard-circle of X(1690)
X(1687) = inverse-in-1st-Lemoine-circle of X(1688)
X(1687) = X(98)-Ceva conjugate of X(1688)
X(1687) = {X(371),X(372)}-harmonic conjugate of X(1690)
X(1687) = insimilicenter of 2nd Brocard circle and circle{{X(371),X(372),PU(1),PU(39)}}; the exsimilicenter is X(1688)

### X(1688) = EXSIMILICENTER(1st LEMOINE CIRCLE, 2nd LEMOINE CIRCLE)

Trilinears    cos(A - ω) - sin A : :
Trilinears    cos A - (sec ω - tan ω) sin A
Trilinears    cos(A - ω/2 + π/4) : :
Trilinears    sin(A - ω/2 - π/4) : :
Trilinears    cos A - sin(A - ω) : :

X(1688) lies on these lines:
3,6   83,2009   98,2010   485,1676   486,1677   1124,1672   1335,1673   1377,1680   1378,1681   1700,1702   1701,1703

X(1688) = reflection of X(1687) in X(1691)
X(1688) = isogonal conjugate of X(2010)
X(1688) = inverse-in-circumcircle of X(1687)
X(1688) = inverse-in-Brocard-circle of X(1689)
X(1688) = inverse-in-1st-Lemoine-circle of X(1687)
X(1688) = X(98)-Ceva conjugate of X(1687)
X(1688) = {X(371),X(372)}-harmonic conjugate of X(1689)
X(1688) = exsimilicenter of 2nd Brocard circle and circle {{X(371),X(372),PU(1),PU(39)}}; the insimilicenter is X(1687)

### X(1689) = INVERSE-IN-BROCARD-CIRCLE OF X(1687)

Trilinears       cos(A + ω/2 + π/4) : cos(B + ω/2 + π/4) : cos(C + ω/2 + π/4)
Trilinears       sin(A + ω/2 - π/4) : sin(B + ω/2 - π/4) : sin(C + ω/2 - π/4)
Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(A + ω) - sin A
= sin(A + ω) - cos A;

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(1689) is the external center of similitude of the Gallatly circle and the circumcircle. (Peter J. C. Moses, 9/03)

X(1689) lies on these lines: 1,2018   2,2009   3,6   4,2010   55,2007,   56,2008   165,2017   958,2013   1344,2015   1345,2016   1376,2014

X(1689) = reflection of X(1690) in X(6)
X(1689) = inverse-in-Brocard-circle of X(1687)
X(1689) = X(262)-Ceva conjugate of X(1690)
X(1689) = {X(371),X(372)}-harmonic conjugate of X(1688)

### X(1690) = INVERSE-IN-BROCARD-CIRCLE OF X(1688)

Trilinears    cos(A + ω/2 - π/4) : cos(B + ω/2 - π/4) : cos(C + ω/2 - π/4)
Trilinears    sin(A + ω/2 + π/4) : sin(B + ω/2 + π/4) : sin(C + ω/2 + π/4)
Trilinears    cos(A + ω) + sin A : :
Trilinears    sin(A + ω) + cos A : :
Trilinears    cos A - (sec ω + tan ω) sin A : :

X(1690) is the internal center of similitude of the Gallatly circle and the circumcircle. (Peter J. C. Moses, 9/03)

X(1690) lies on these lines: 1,2017   2,2010   3,6   4,2009   55,2008   56,2007   165,2018   958,2014   1344,2016   1345,2015   1376,2013

X(1690) = reflection of X(1689) in X(6)
X(1690) = inverse-in-Brocard-circle of X(1688)
X(1690) = X(262)-Ceva conjugate of X(1689)
X(1690) = {X(371),X(372)}-harmonic conjugate of X(1687)
X(1690) = X(1)-of-X(3)PU(1)

### X(1691) = RADICAL TRACE OF CIRCUMCIRCLE AND 1st LEMOINE CIRCLE

Trilinears    sin(A - 2ω) : :
Trilinears    e^2 cos(A - ω) - cos(A + ω) : :       (c.f. X(2456))
Barycentrics    a2(a4 - b2c2) : :

X(1691) is the perspector of ABC and the reflection of the tangential triangle in the Lemoine axis (i.e., the reflection of the anticevian triangle of X(6) in the trilinear polar of X(6)). (Randy Hutson, September 5, 2015)

Let A'B'C' be the 1st anti-Brocard triangle. X(1691) is the radical center of the circumcircles of A'BC, B'CA, C'AB. (Randy Hutson, July 20, 2016)

X(1691) lies on these lines:
2,1501   3,6   31,893   83,316   98,230   99,698   141,1078   154,1611   184,1613   237,694   249,524   385,732   691,729   695,1176   699,805   1428,1914   1968,1974

X(1691) = midpoint of X(i) and X(j) for these (i,j): (6,2076), (187,1692), (1687,1688)
X(1691) = reflection of X(i) in X(j) for these (i,j): (6,1692), (2076,187)
X(1691) = isogonal conjugate of X(1916)
X(1691) = complement of X(5207)
X(1691) = inverse-in-circumcircle of X(32)
X(1691) = inverse-in-Brocard circle of X(3094)
X(1691) = inverse-in-1st-Lemoine-circle of X(6)
X(1691) = inverse-in-2nd-Lemoine-circle of X(576)
X(1691) = X(i)-Ceva conjugate of X(j) for these (i,j): (237,1971), (699,32), (1976,6)
X(1691) = crosspoint of X(i) and X(j) for these (i,j): (83,98), (385,419)
X(1691) = crosssum of X(39) and X(511)
X(1691) = reflection of X(2076) in the Lemoine axis
X(1691) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (32,182,6), (371, 372, 3095), (1342,1343,39)
X(1691) = crossdifference of every pair of points on the line PU(11)
X(1691) = center of the circle {{X(1687),X(1688),PU(1),PU(2)}} (the circle orthogonal to the circumcircle and passing through the 1st and 2nd Brocard points)
X(1691) = intersection of tangents to 1st Lemoine circle at intersections with 2nd Lemoine circle
X(1691) = centroid of X(6)X(15)X(16)
X(1691) = X(1691) of circumsymmedial triangle
X(1691) = harmonic center of 1st and 2nd Lemoine circles
X(1691) = harmonic center of 2nd Brocard circle and the circle {{X(371),X(372),PU(1),PU(39)}}
X(1691) = perspector of ABC and the reflection of the 2nd Ehrmann triangle in line X(6)X(512) (the perspectrix of ABC and 2nd Ehrmann triangle)

### X(1692) = RADICAL TRACE OF CIRCUMCIRCLE AND 2nd LEMOINE CIRCLE

Trilinears    2 sin(A - 2ω) + sin(A + 2ω) - sin A : :
Barycentrics    a^2(2a^4 + b^4 + c^4 - a^2b^2 - a^2c^2 - 2b^2c^2) : :

X(1692) lies on these lines: 3,6   51,1501   114,230   115,1503   184,1196   698,1569   1015,1428   1627,1994

X(1692) = midpoint of X(i) and X(j) for these (i,j): (6,1691), (187,1570)
X(1692) = reflection of X(i) in X(j) for these (i,j): (39,2024), (187,1570), (1570,6)
X(1692) = inverse-in-circumcircle of X(3053)
X(1692) = inverse-in-1st-Lemoine-circle of X(32)
X(1692) = inverse-in-2nd-Lemoine-circle of X(1351)
X(1692) = crosspoint of X(i) and X(j) for these (i,j): (6,1976), (230,460)
X(1692) = crosssum of X(2) and X(325)
X(1692) = center of inverse-in-Moses-circle-of-Brocard-circle
X(1692) = radical trace of circumcircle and circle {{X(371),X(372),PU(1),PU(39)}}
X(1692) = radical trace of 2nd Lemoine circle and circle {{X(371),X(372),PU(1),PU(39)}}
X(1692) = radical trace of circles {{P(1),U(2),P(39)}} and {{U(1),P(2),U(39)}}
X(1692) = anticenter of cyclic quadrilateral PU(2)PU(39)
X(1692) = crossdifference of every pair of points on line X(69)X(523)
X(1692) = {X(1687),X(1688)}-harmonic conjugate of X(2456)
X(1692) = isogonal conjugate of X(8781)
X(1692) = {X(371),X(372)}-harmonic conjugate of X(9737)
X(1692) = inverse-in-circle-{{X(371),X(372),PU(1),PU(39)}} of X(9737)

### X(1693) = INSIMILICENTER(BROCARD CIRCLE,APOLLONIUS CIRCLE)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = e[2rs sin A + (r2 - s2)cos A] - (r2 + s2)cos(A - ω)

X(1693) lies on these lines:
3,6   10,1349   43,1704   181,1674   1675,1682   1695,1705

### X(1694) = EXSIMILICENTER(BROCARD CIRCLE,APOLLONIUS CIRCLE)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = e[2rs sin A + (r2 - s2)cos A] + (r2 + s2)cos(A - ω)

X(1694) lies on these lines:
3,6   10,1348   43,1705   181,1675   1674,1682   1695,1704

### X(1695) = INSIMILICENTER(BEVAN CIRCLE, APOLLONIUS CIRCLE)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = r(r2 + s2) + 4Rs(r sin A - s cos A)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

The exsimilicenter of the Bevan and Apollonius circles is X(43).

X(1695) lies on these lines: 1,573   10,962   40,43   57,1682   165,386   181,1697   978,1764   1683,1701   1684,1700   1685,1703   1686,1702   1693,1705   1694,1704   2017,2020   2018,2019

X(1695) = X(939)-Ceva conjugate of X(55)

### X(1696) = POINT ELECTRA

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b + c - a cos A + r sin A
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1696) is the trilinear product X(6)*X(1706)

X(1696) lies on these lines: 6,1201   9,56   19,25   220,1400   346,1376   999,1743

### X(1697) = INSIMILICENTER(BEVAN CIRCLE, INCIRCLE)

Trilinears    f(A,B,C) = 3 + cos A - cos B - cos C : :
Trilinears    -1 + sin A/2 cos B/2 cos C/2 : :

The exsimilicenter of the Bevan circle and incircle is X(57).

Randy Hutson (January 29, 2015) gives 3 constructions:
(1) Let A'B'C' be the mixtilinear incentral triangle. Let A" be the cevapoint of B' and C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1697).
(2) Let A'B'C' be the mixtilinear incentral triangle. Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(1697).
Let A'B'C' be the intouch triangle, A"B"C" the extouch triangle, and A*B*C* the excentral triangle. Let OA be the circle through A'A"A*, and define OB and OCc cyclically. X(1697) is the radical center of circles OA, OB, OC.

X(1697) lies on these lines:
1,3   2,1706   8,9   10,497   11,1698   12,1699   33,1831   43,1682   63,145   71,1732   84,944   109,1496   181,1695   200,960   212,595   219,380   221,1419   226,962   388,516   392,936   519,1776   580,1497   1015,1571   1058,1210   1124,1703   1317,1768   1335,1702   1500,1572   1672,1701   1673,1700   1674,1705   1675,1704   2007,2018   2008,2017

X(1697) = reflection of X(3340) in X(1)
X(1697) = isogonal conjugate of X(7091)
X(1697) = X(2339)-Ceva conjugate of X(9)
X(1697) = 2nd-extouch-to-excentral similarity image of X(8)
X(1697) = extangents-to-intangents similarity image of X(1)
X(1697) = {X(1),X(40)}-harmonic conjugate of X(57)
X(1697) = homothetic center of excentral and Hutson-intouch triangles
X(1697) = X(1593)-of-excentral triangle
X(1697) = X(1593)-of-Hutson-intouch triangle
X(1697) = X(11414)-of-intouch triangle

### X(1698) = INSIMILICENTER(BEVAN CIRCLE, NINE-POINTCIRCLE)

Trilinears    2 cos(B - C) + cos A - cos B - cos C + 1 : :
Trilinears    bc(a + 2b + 2c) : ca(b + 2c + 2a) : ab(c + 2a + 2b)     (M. Iliev, 5/13/2007)
Trilinears    bc - rR : ca - rR : ab - rR     (C. Lozada, 9/07/2013)
Trilinears    r - 4 R sin B sin C : :
X(1698) = X(1) - 6 X(2) = X(8) - 6 X(10)

Let A'B'C' be the Aquila triangle. Let A" be the trilinear product B'*C', and define B" and C" cyclically. Then A"B"C" is homothetic to ABC at X(1698) and to A'B'C' at X(10). (Randy Hutson, December 10, 2016)

X(1698) lies on these lines:
1,2   4,165   5,40   9,46   11,1697   12,57   33,451   35,405   36,474   58,750   75,1089   115,1571   116,1282   119,1768   121,1054   140,355   171,1724   210,942   226,1788   281,1838   318,1784   320,1757   404,993   406,1861   443,1478   485,1703   486,1702   515,631   517,1656   595,748   632,952   966,1743   986,1739   1348,1705   1349,1704   1506,1572   1676,1701   1677,1700   2009,2018   2010,2017

X(1698) = isotomic conjugate of X(30598)
X(1698) = {X(2),X(10)}-harmonic conjugate of X(1)
X(1698) = homothetic center of excentral and 4th Euler triangles
X(1698) = crossdifference of every pair of points on line X(649)X(2605)
X(1698) = trilinear product of vertices of Aquila triangle
X(1698) = homothetic center of ABC and cross-triangle of ABC and Aquila triangle
X(1698) = {X(1),X(2)}-harmonic conjugate of X(3624)
X(1698) = perspector of Gemini triangle 26 and cross-triangle of ABC and Gemini triangle 26
X(1698) = isogonal conjugate of isotomic conjugate of X(30596)

### X(1699) = EXSIMILICENTER(BEVAN CIRCLE, NINE-POINT CIRCLE)

Trilinears    2 cos(B - C) - cos A + cos B + cos C - 1 : :
Trilinears    r + 4 R cos B cos C : :
Barycentrics    a^3 + a(b - c)^2 - 2 (b - c)^2 (b + c) : :
X(1699) = X(1) + 2 X(4)

Let A' be the pole of the Gergonne line wrt the circle with BC as diameter, and define B', C' cyclically. X(1699) is the centroid of A'B'C'. (Randy Hutson, June 27, 2018)

Let A' be the orthocenter of BCX(1), and define B' and C' cyclically. A'B'C' is also the anticevian triangle, wrt intouch triangle, of X(1), and X(1699) is the centroid of A'B'C'; see also X(3680). (Randy Hutson, June 27, 2018)

X(1699) is the centroid of triangle formed by reflecting vertices of 1st circumperp triangle in corresponding side of ABC. (If 2nd circumperp triangle is substituted, for 1st, the resulting triangle is the Fuhrmann triangle.) (Randy Hutson, June 27, 2018)

Let A'B'C' be the orthic triangle. Let A" be the A-excenter of AB'C', and define B" and C" cyclically. The centroid of triangle A"B"C" is X(1699). (Randy Hutson, July 31 2018)

X(1699) lies on these lines:
1,4   2,165   5,40   10,962   11,57   12,1697   20,1125   36,1012   55,1538   79,84   80,1537   115,1572   118,1282   200,908   210,381   238,1754   354,971   355,546   382,1385   485,1702   486,1703   499,1770   610,1839   614,990   1329,1706   1348,1704   1506,1571   1676,1700   1677,1701   1730,1985   2009,2017   2010,2018

X(1699) = reflection of X(165) in X(2)
X(1699) = crosspoint of X(92) and X(1088)
X(1699) = crosssum of X(48) and X(1253)
X(1699) = centroid of the six touchpoints of the Johnson circles and the sidelines of the inner Johnson triangle
X(1699) = homothetic center of excentral and 3rd Euler triangles
X(1699) = centroid of triangle formed by reflecting excenters in corresponding vertex of ABC
X(1699) = centroid of anticevian triangle, wrt intouch triangle, of X(1)
X(1699) = homothetic center of circumcevian triangle of X(3) and cross-triangle of Aquila and anti-Aquila triangles
X(1699) = centroid of Garcia reflection triangle (aka Gemini triangle 8)
X(1699) = {X(1),X(4)}-harmonic conjugate of X(5691)

### X(1700) = INSIMILICENTER(BEVAN CIRCLE, 1st LEMOINE CIRCLE)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 2 cos(A - ω) + cos A - cos B - cos C + 1
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(1700) lies on these lines:
1,1342   9,1681   40,182   43,1683   57,1672   165,1343   371,2018   372,2017   1571,2036   1572,2035   1673,1697   1676,1699   1677,1698   1680,1706   1684,1695   1687,1703   1688,1702

### X(1701) = EXSIMILICENTER(BEVAN CIRCLE, 1st LEMOINE CIRCLE)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 2 cos(A - ω) - cos A + cos B + cos C - 1
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(1701) lies on these lines:
1,1343   9,1680   40,182   43,1684   57,1673   165,1342   371,2017   372,2018   1571,2035   1572,2036   1672,1697   1676,1698   1677,1699   1681,1706   1683,1695   1687,1702   1688,1703

### X(1702) = INSIMILICENTER(BEVAN CIRCLE, 2nd LEMOINE CIRCLE)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 2 sin A + cos A - cos B - cos C + 1
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(1702) lies on these lines:
1,371   6,40   9,1378   10,1588   43,1685   57,1124   165,372   485,1699   486,1698   516,1587   580,605   1335,1697   1377,1706   1504,1572   1505,1571   1668,1704   1669,1705   1670,2017   1671,2018   1686,1695   1687,1701   1688,1700

### X(1703) = EXSIMILICENTER(BEVAN CIRCLE, 2nd LEMOINE CIRCLE)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 2 sin A - cos A + cos B + cos C - 1
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(1703) lies on these lines:
1,372   6,40   9,1377   10,1587   43,1686   57,1335   165,371   485,1698   486,1699   516,1588   580,606   1124,1697   1378,1706   1504,1571   1505,1572   1668,1705   1669,1704   1670,2018   1671,2017   1685,1695   1687,1700   1688,1701

### X(1704) = INSIMILICENTER(BEVAN CIRCLE, BROCARD CIRCLE)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 2 cos(A - ω) + e(1 + cos A - cos B - cos C)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(1704) lies on these lines:
1,1340   9,1679   40,182   43,1693   57,1674   165,1341   1348,1699   1349,1698   1571,2034   1572,2033   1668,1702   1669,1703   1675,1697   1678,1706   1694,1695   2011,2018   2012,2017

### X(1705) = EXSIMILICENTER(BEVAN CIRCLE, BROCARD CIRCLE)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 2 cos(A - ω) - e(1 + cos A - cos B - cos C)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(1705) lies on these lines:
1,1341   9,1678   40,182   43,1694   57,1675   165,1340   1348,1698   1349,1699   1571,2033   1572,2034   1668,1703   1669,1702   1674,1697   1679,1706   1693,1695   2011,2017   2012,2018

### X(1706) = EXSIMILICENTER(BEVAN CIRCLE, SPIEKER CIRCLE)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[2(b + c) - a(cos A - cos B - cos C + 1)]
Trilinears       g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a3 + a2(b + c) - a(b + c)2 - (b + c)(b2 - 6bc + c2)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

The insimilicenter of the Bevan and Spieker circles is X(9).

X(1706) lies on these lines:
1,474   2,1697   4,9   8,57   46,529   65,200   84,355   165,958   404,1420   517,936   1329,1699   1377,1702   1378,1703   1571,1573   1572,1574   1678,1704   1679,1705   1680,1700   1681,1701

Mimosa Transforms 1707- 1788
The Mimosa transform M(X) of a point X = x : y : z is defined (9/15/03) by

M(X) = - yz cos A + zx cos B + xy cos C : - zx cos B + xy cos C + yz cos A : - xy cos C + yz cos A + xy cos C

and the inverse Mimosa transform M -1(X), by

M -1(X) = (cos A)/(y + z) : (cos B)/(z + x) : (cos C)/(x + y).

As with other names in ETC, the name Mimosa is that of a star. M(X) is the X*X(4)-Ceva conjugate of X(1), where * denotes trilinear product, and M -1(X) is the trilinear quotient X(3)/P(X), where P(X) is the crosssum of X(1) and X.

Let g(P,X) denote the P-gimel conjugate of X. The Mimosa transform M(X) arises in connection with the equation g(P,X) = X. Referring to the definition of gimel conjugate in the Glossary, if

P = p : q : r and X = x : y : z

are triangle centers, then the equation g(P,X) = X is equivalent to the ratio-equation

(2absvS - 8yσ2)/(2abswS - 8zσ2) = y/z,

where v = (cos A)/p - (cos B)/q + (cos C)/r and w = (cos A)/p + (cos B)/q - (cos C)/r.

If S = 0, then the ratio-equation holds. As S = x(bq + cr) + y(cr + ap) + z(ap + bq), it follows that if P is given, then g(P,X) = X if X is on the line S = 0 (regarding x : y : z as variable); and that if X is given, then g(P,X) = P if P is on the line S = 0 (regarding p : q : r as variable). There are too many such cases of gimel conjugates for all to be itemized in ETC. For example, if X = X(1), then g(P,X) = X for every P on the line at infinity; if X = X(513), then g(P,X) = X for every P on the line X(1)X(2); and if X = X(656), then g(P,X) = X for every P on the Euler line, X(2)X(3).

If S ≠ 0, the ratio-equation lends itself to easy simplifications and two Tables conclusions: (1) if P is given then X = M(P) is a solution of g(P,X) = X, and (2) if X is given then P = M*(X) is a solution of g(P,X) = X.

Here is a list of pairs (i,j) for which X(j) = M(X(i)):
1,46   2,19   3,1   4,920   8,1158   21,4   48,43   54,47   59,109   60,580   63,9   69,63   71,846   72,191   73,1046   77,57   78,40   81,579   90,90   95,92   96,91   97,48   99,1577   110,656   219,165   228,1045   248,1580   249,163   250,162   252,564   254,921   271,84   283,3   348,169   394,610   603,978   648,822   651,652   662,1021   895,896   1105,158   1176,31   1259,1490   1297,240   1332,649   1444,2   1459,1054

Of course, reversing the pairs gives a list of (J,I) for which X(i) = M*(X(j)).

### X(1707) = MIMOSA TRANSFORM OF X(6)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(6)
Trilinears       3a2 - b2 - c2 : 3b2 - c2 - a2 : 3c2 - a2 - b2     (M. Iliev, 5/13/2007)
Trilinears       cot B + cot C - cot A : cot C + cot A - cot B : cot A + cot B - cot C      (Randy Hutson, 9/23/2011)
Trilinears       a2 - SA : b2 - SB : c2 - SC      (C. Lozada, 9/07/2013)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1707) lies on these lines:
1,21   9,171   19,1719   33,1776   36,1473   43,165   44,1376   46,1722   57,238   92,1733   109,1395   162,1096   200,1757   204,240   223,1758   326,560   380,1045   484,1774   579,1716   580,1158   610,1740   978,1044   986,1453   1038,1399   1633,1732   1709,1711   1714,1770   1728,1771   1788,1877

X(1707) = {X(6),X(9)}-harmonic conjugate of X(1)

X(1707) = X(i)-Ceva conjugate of X(j) for these (i,j): (19,1), (1778,1724)
X(1707) = X(i)-aleph conjugate of X(j) for these (i,j): (1,610), (4,19), (19,1707), (108,1783), (162,163), (365,1745), (366,1763), (509,223), (1778,1724), (1783,1633)

### X(1708) = MIMOSA TRANSFORM OF X(7)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(7)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1708) lies on these lines:
1,201   2,7   4,46   6,1214   19,1713   33,1736   34,1724   38,1471   40,950   43,1758   44,1427   56,72   58,1038   65,405   109,1395   169,1762   208,860   218,222   223,1743   225,1714   278,1723   354,954   442,1454   518,1260   582,1062   653,1748   1020,1435   115,1864   1396,1778   1412,1812   1711,1738   1712,1715   1750,1768

X(1708) = X(273)-Ceva conjugate of X(1)
X(1708) = cevapoint of X(46) in X(1723)
X(1708) = crosssum of X(652) and X(2170)
X(1708) = X(i)-aleph conjugate of X(j) for these (i,j): (2,1490), (4,1721), (7,223), (27,580), (92,1158), (174,1745), (273,1708), (278,1722), (286,1746), (508,610), (653,109)

### X(1709) = MIMOSA TRANSFORM OF X(9)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(9)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1709) lies on these lines:
1,84   4,46   9,165   11,57   30,40   31,990   33,109   35,1490   55,971   63,516   553,946   774,1448   846,1742   968,991   1707,1711   1719,1744   1730,1889

X(1709) = reflection of X(1) in X(1012)
X(1709) = X(281)-Ceva conjugate of X(1)
X(1709) = X(i)-aleph conjugate of X(j) for these (i,j): (4,1723), (92,1729), (188,610), (281,1709), (366,223)
X(1709) = excentral-isotomic conjugate of X(2947)
X(1709) = antipode of X(1) in circle {{X(1),X(1709),PU(4)}}

### X(1710) = MIMOSA TRANSFORM OF X(10)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(10)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1710) lies on these lines:
1,1437   19,91   28,1725   30,40   35,228   46,407   109,1825   1046,1777   1720,1781   1770,1782

X(1710) = X(i)-aleph conjugate of X(j) for these (i,j): (4,1724), (92,1730)

### X(1711) = MIMOSA TRANSFORM OF X(19)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(19)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1711) lies on these lines:
1,394   9,43   46,407   90,1039   238,1040   920,1714   1707,1709   1708,1738   1720,1722   1756,1763

X(1711) = X(393)-Ceva conjugate of X(1)
X(1711) = X(i)-aleph conjugate of X(j) for these (i,j): (4,9), (19,43), (33,170), (92,1759), (393,1711), (1897,1018)

### X(1712) = MIMOSA TRANSFORM OF (20)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(20)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1712) lies on these lines:
1,204   4,57   9,1249   19,158   63,1895   108,1490   412,1445   774,1096   811,1102   920,1784   1103,1783   1158,1767   1708,1715   1713,1741   1714,1728

X(1712) = X(i)-Ceva conjugate of X(j) for these (i,j): (63,19), (1895,1)
X(1712) = X(i)-aleph conjugate of X(j) for these (i,j): (2,2184), (1895,1712)
X(1712) = eigencenter of cevian triangle of X(63)
X(1712) = eigencenter of anticevian triangle of X(19)

### X(1713) = MIMOSA TRANSFORM OF X(27)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(27)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1713) lies on these lines:
1,6   4,579   19,1708   71,950   284,1006   379,1445   393,1714   580,1172   583,1901   1712,1741

X(1713) = crosspoint of X(765) and X(823)
X(1713) = crosssum of X(244) and X(822)
X(1713) = X(i)-aleph conjugate of X(j) for these (i,j): (4,846), (27,6), (92,1761)

### X(1714) = MIMOSA TRANSFORM OF X(28)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(28)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1714) lies on these lines:
1,2   4,580   6,442   19,46   58,377   100,1612   219,1329   225,1708   238,1479   278,1739   393,1713   405,1834   920,1711   1498,1532   1707,1770   1712,1728   1715,1779

X(1714) = X(i)-aleph conjugate of X(j) for these (i,j): (4,191), (19,1045), (27,2), 29,20)

### X(1715) = MIMOSA TRANSFORM OF X(29)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(29)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1715) lies on these lines:
1,3   4,1730   19,1158   185,851   412,1896  579,1249   1020,1068   1708,1712   1714,1779   1736,1872

X(1715) = X(i)-Ceva conjugate of X(j) for these (i,j): (412,4), (1896,1)
X(1715) = crosssum of X(822) and X(2310)
X(1715) = X(i)-aleph conjugate of X(j) for these (i,j): (4,1046), (29,3), (92,1762), (1896,1715)

### X(1716) = MIMOSA TRANSFORM OF X(31)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(31)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1716) lies on these lines:
1,69   3,238   4,1721   9,43   579,1707   1402,1423   1745,1756

X(1716) = X(25)-Ceva conjugate of X(1)
X(1716) = X(i)-aleph conjugate of X(j) for these (i,j): (1,1763), (4,1726), (6,1745), (19,46), (25,1716), (108,1020), (259,1490), (266,223), (365,610)

### X(1717) = MIMOSA TRANSFORM OF X(35)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(35)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1717) lies on these lines: 1,30   4,1718   33,46   35,37   90,1172   429,1722   1047,1048

X(1717) = X(1770)-Ceva conjugate of X(46)

### X(1718) = MIMOSA TRANSFORM OF X(36)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(36)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1718) lies on these lines:
1,5   4,1717   6,1781   34,46   36,1455   90,1720   106,614   244,1468   1723,1783   1727,1735   1737,1870

X(1718) = X(i)-Ceva conjugate of X(j) for these (i,j): (1737,46), (1870,1)
X(1718) = X(i)-aleph conjugate of X(j) for these (i,j): (4,1727), (1870,1718)

### X(1719) = MIMOSA TRANSFORM OF X(37)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(37)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1719) lies on these lines:
1,1790   10,191   19,1707   27,1733   46,407   165,846   1709,1744

X(1719) = X(1826)-Ceva conjugate of X(1)
X(1719) = X(i)-aleph conjugate of X(j) for these (i,j): (4,579), (1826,1719)

### X(1720) = MIMOSA TRANSFORM OF X(40)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(40)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1720) lies on these lines:
1,84   46,208   90,1718   846,1047   920,1249   1710,1781   1711,1722   1721,1771

X(1720) = X(1158)-Ceva conjugate of X(46)
X(1720) = X(i)-aleph conjugate of X(j) for these (i,j): (4,1728), (366,282)

### X(1721) = MIMOSA TRANSFORM OF X(55)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(55)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1721) lies on these lines:
1,7   4,1716   19,1633   40,984   43,1750   46,1736   165,846   294,1743   1039,1885   1040,1836   1045,1047   1707,1709   1720,1771

X(1721) = reflection of X(1) in X(990)
X(1721) = X(33)-Ceva conjugate of X(1)
X(1721) = X(317)-of-excentral-triangle
X(1721) = excentral isotomic conjugate of X(1490)
X(1721) = {X(8947),X(8949)}-harmonic conjugate of X(1722)
X(1721) = X(i)-aleph conjugate of X(j) for these (i,j): (1,223), (4,1708), (9,1490), (19,1722), (29,1746), (33,1721), (188,1763), (259,1745), (281,1158), (1172,580), (1783,109)

### X(1722) = MIMOSA TRANSFORM OF X(56)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(56)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1722) lies on these lines:
1,2   4,1716   9,986   34,1788   40,238   46,1707   57,1773   87,937   169,1046   171,1453   223,1047   269,979   427,1039   429,1717   920,1772   958,988   1040,1837   1104,1376   1254,1445   1711,1720   1723,1880

X(1722) = X(i)-Ceva conjugate of X(j) for these (i,j): (34,1), (1788,46)
X(1722) = {X(8947),X(8949)}-harmonic conjugate of X(1721)
X(1722) = X(i)-aleph conjugate of X(j) for these (i,j): (1,1490), (4,1158), (19,1721), (27,1746), (28,580), (34,1722), (57,223), (108,109), (174,1763), (266,1745), (278,1708), (509, 610)

### X(1723) = MIMOSA TRANSFORM OF X(57)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(57)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1723) lies on these lines:
1,6   19,46   35,380   36,610   57,1762   90,1172   169,1400   278,1708   281,1737   672,1766   920,1249   928,1047   1707,1709   1718,1783   1722,1880   1729,1744

X(1723) = X(i)-Ceva conjugate of X(j) for these (i,j): (278,1), (1708,46)
X(1723) = crosspoint of X(653) and X(765)
X(1723) = crosssum of X(244) and X(652)
X(1723) = X(i)-aleph conjugate of X(j) for these (i,j): (4,1709), (174,610), (273,1729), (278,1723), (366,1490), (508,1763), (509,1745)

### X(1724) = MIMOSA TRANSFORM OF X(58)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(58)
Trilinears       a2(a + b + c) - bc(b + c) : b2(a + b + c) - ca(c + a) : c2(a + b + c) - ab(a + b)     (M. Iliev, 5/13/07)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1724) lies on these lines:
1,6   2,58   4,580   8,595   10,31   21,286   28,579   30,582   32,1009   34,1708   35,43   36,978   46,1707   47,1737   83,1008   90,1039   109,1788   171,1698   191,986   212,950   226,1451   255,1210   270,469   387,452   515,602   581,1006   748,1125   920,1735   985,1224   993,1193   1020,1398   1445,1448   1726,1829   1738,1770

X(1724) = X(i)-Ceva conjugate of X(j) for these (i,j): (28,1), (579,1754), (1778,1707)
X(1724) = crosspoint of X(162) and X(765)
X(1724) = crosssum of X(244) and X(656)
X(1724) = X(i)-aleph conjugate of X(j) for these (i,j): (4,1710), (27,1730), (28,1724), (266,1047)

### X(1725) = MIMOSA TRANSFORM OF X(74)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(74)
Trilinears       1 + cos(2B) + cos(2C) : 1 + cos(2C) + cos(2A) : 1 + cos(2A) + cos(2B)     (Randy Hutson, 9/23/2011)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Let L denote the line having trilinears of X(1725) as coefficients. Then L is the line passing through X(3) perpendicular to the Euler line.

X(1725) lies on these lines:
1,21   28,1710   33,46   34,90   35,201   40,1775   57,1774   91,158   109,1727   240,522   1172,1744   1776,1870

X(1725) = crosspoint of X(i) and X(j) for these (i,j): (1,2166), (75,2349) X(1725) = crosssum of X(31) and X(2173)

### X(1726) = MIMOSA TRANSFORM OF X(75)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(75)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1726) lies on these lines:
1,184   4,1782   9,440   19,1708   25,1736   46,407   57,1020   63,321   92,1746   1158,1753   1724,1829   1754,1824

X(1726) = X(264)-Ceva conjugate of X(1)
X(1726) = X(i)-aleph conjugate of X(j) for these (i,j): (2,1745), (4,1716), (75,1763), (92,46), (264,1726), (556,1490)

### X(1727) = MIMOSA TRANSFORM OF X(80)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(80)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1727) lies on these lines:
1,1399   4,46   30,80   35,72   36,1768   63,519   109,1725   1718,1735

X(1727) = X(4)-aleph conjugate of X(1718)

### X(1728) = MIMOSA TRANSFORM OF X(84)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(84)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1728) lies on these lines:
1,6   3,1864   4,46   5,57   33,580   36,1490   40,1837   63,1210   84,1750   226,499   1707,1771   1711,1720   1712,1714

X(1728) = X(4)-aleph conjugate of X(1720)

### X(1729) = MIMOSA TRANSFORM OF X(85)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(85)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1729) lies on these lines:
5,9   19,1158   57,1375   63,169   607,1735   920,1752   1723,1744

X(1729) = X(331)-Ceva conjugate of X(1)
X(1729) = X(i)-aleph conjugate of X(j) for these (i,j): (92,1709), (273,1723), (331,1729), (508,1745)

### X(1730) = MIMOSA TRANSFORM OF X(86)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(86)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1730) lies on these lines:
1,228   2,573   4,1715   6,57   19,1708   25,1754   27,1746   28,580   40,405   46,1707   51,851   63,169   165,1011   278,1020   572,1817   1709,1889   1735,1905   1736,1824   1786,1787

X(1730) = X(286)-Ceva conjugate of X(1)
X(1730) = X(i)-aleph conjugate of X(j) for these (i,j): (27,1724), (92,1710), (174,1047), (286,1730)

### X(1731) = MIMOSA TRANSFORM OF X(88)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(88)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1731) lies on these lines:
8,9   19,46   44,517   63,1266   92,1751   243,522   580,1871

X(1731) = crosspoint of X(21) and X(88)
X(1731) = crosssum of X(44) and X(65)

### X(1732) = MIMOSA TRANSFORM OF X(89)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(89)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1732) lies on these lines:
9,1125   19,46   44,56   45,354   48,1743   71,1697   1020,1435   1474,1778   1633,1707

X(1731) = crosspoint of X(75) and X(1821)
X(1731) = crosssum of X(31) and X(1755)
X(1731) = X(98)-aleph conjugate of X(1955)

### X(1733) = MIMOSA TRANSFORM OF X(98)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(98)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1733) lies on these lines:
1,75   19,91   27,1719   92,1707   240,522   896,1109   1580,1821

X(1733) = crosspoint of X(75) and X(1821)
X(1733) = crosssum of X(31) and X(1755)
X(1733) = X(98)-aleph conjugate of X(1955)

### X(1734) = MIMOSA TRANSFORM OF X(101)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(101)
Trilinears       g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = (b - c)(b2 + c2 + bc - ab - ac)     (M. Iliev, 5/13/07)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1734) lies on these lines:
1,905   100,1110   240,522   484,513   512,1491   649,830

X(1734) = reflection of X(1) in X(905)
X(1734) = X(1783)-Ceva conjugate of X(1)
X(1734) = crosspoint of X(75) in X(100)
X(1734) = crosssum of X(i) and X(j) for these (i,j): (31,513), (652,2293), (656,1962)
X(1734) = X(1783)-aleph conjugate of X(1734)

### X(1735) = MIMOSA TRANSFORM OF X(102)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(102)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1735) lies on these lines:
1,3   10,1074   34,1158   109,1870   240,522   607,1729   774,1210   920,1724   946,1393   1711,1720   1718,1727   1730,1905   1765,1880

X(1735) = crosssum of X(31) and X(2182)

### X(1736) = MIMOSA TRANSFORM OF X(103)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(103)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1736) lies on these lines:
1,6   10,774   25,1726   33,1708   46,1721   90,1041   109,1776   201,950   240,522   241,971   307,1210   920,1771   990,1445   1020,1876   1214,1864   1715,1872   1730,1824

X(1736) = crosssum of X(31) and X(910)

### X(1737) = MIMOSA TRANSFORM OF X(104)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(104)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1737) lies on these lines:
1,2   3,1837   4,46   5,65   11,517   12,942   29,1780   30,1155   35,950   36,80   40,1479   47,1724   56,355   57,1478   72,1329   91,225   109,1877   117,1845   119,912   150,1447   240,522   281,1723   354,495   381,1836   427,1905   484,516   579,1826   758,908   952,1319   1718,1870   1747,1890   1782,1842

X(1737) = midpoint of X(36) and X(80)
X(1737) = cevapoint of X(46) and X(1718)
X(1737) = X(2252)-cross conjugate of X(914)
X(1737) = crosssum of X(i) and X(j) for these (i,j): (6,2316), (31,2183)

### X(1738) = MIMOSA TRANSFORM OF X(105)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(105)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)
Barycentrics  g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 2abc - (b + c)[a2 - (b - c)2]

X(1738) lies on these lines:
1,142   2,968   4,1716   10,75   19,46   43,226   225,1788   238,516   240,522   518,1086   527,1757   528,1279   899,908   946,978   1054,1758   1708,1711   1724,1770

X(1738) = crosspoint of X(75) and X(673)
X(1738) = crosssum of X(31) and X(672)

### X(1739) = MIMOSA TRANSFORM OF X(106)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(106)
Trilinears       g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = b3 + c3 + ab2 + ac2 - 2bc2 - 2b2c     (M. Iliev, 5/13/07)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1739) lies on these lines:
1,474   10,38   36,1054   46,1707   238,484   240,522   244,519   278,1714   758,899   986,1698

X(1739) = crosspoint of X(75) and X(88)
X(1739) = crosssum of X(31) and X(44)

### X(1740) = MIMOSA TRANSFORM OF X(184)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2b2 + a2c2 - b2c2     (M. Iliev, 5/13/07)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1740) lies on these lines:
1,75   3,238   6,43   19,1581   31,1582   48,1580   560,662   610,1707   869,894

X(1740) = isogonal conjugate of X(3223)
X(1740) = X(i)-Ceva conjugate of X(j) for these (i,j): (31,1), (1958,610)
X(1740) = X(1613)-cross conjugate of X(1424)
X(1740) = X(i)-aleph conjugate of X(j) for these (i,j): (1,63), (2,1760), (4,1748), (6,1), (19,920), (31,1740), (57,1445), (74,2349), (98,1821), (99,799), (100,190), (101,100), (105,673), (106,88), (107,823), (108,653), (109,651), (110,662), (111,897), (112,162), (259,40), (266,57), (284,411), (365,9), (366,1759), (509,169), (649,1052), (813,660), (825, 1492), (934,658), (1172,412), (2222, 655), (2291,1156)

### X(1741) = MIMOSA TRANSFORM OF X(189)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(189)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1741) lies on these lines:
2,7   19,1158   46,281   920,1249   1712,1713

X(1741) = X(189)-aleph conjugate of X(63)

### X(1742) = MIMOSA TRANSFORM OF X(212)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(212)
Trilinears       g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sec2(B/2) + sec2(C/2) - sec2(A/2)     (Randy Hutson, 9/23/2011)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1742) lies on these lines:
1,7   3,238   35,1745   40,511   43,165   48,1633   87,572   259,503   266,844   376,1064   651,1253   846,1709   971,984

X(1742) = reflection of X(1) in X(991)
X(1742) = X(55)-Ceva conjugate of X(1)
X(1742) = X(i)-aleph conjugate of X(j) for these (i,j): (1,57), (6,978), (9,40), (55,1742), (174,1445), (188,63), (259,1), (365,1743), (366,169)

### X(1743) = MIMOSA TRANSFORM OF X(222)

Trilinears    3a - b - c : :     (M. Iliev, 5/13/2007)
Trilinears    cot(B/2) + cot(C/2) - cot(A/2) : :      (Randy Hutson, 9/23/2011)
Trilinears    aS - rs2 : :    (César Lozada, 9/07/2013)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Let A' be the center of the conic through the contact points of the B- and C-excircles with the sidelines of ABC. Define B' and C' cyclically. The triangle A'B'C' is perspective to the excentral triangle at X(1743). See also X(6), X(25), X(218), X(222), X(940). (Randy Hutson, July 23, 2015)

In the plane of ABC, let a' be the external bisector of A and a" its reflection in BC; define b" and c" cyclically. Then ABC and the triangle bounded by a", b" and c" are perspective with perspector X(15446). (César Lozada, October 10, 2018)

X(1743) lies on these lines:
1,6   10,391   19,1783   31,200   36,198   41,572   43,165   48,1732   57,1122   58,936   71,380   101,604   169,1046   173,266   223,1708   239,1278   241,1419   258,259   269,651   282,1795   284,1778   294,1721   346,519   579,610   580,1490   966,1698   978,1400   999,1696   1249,1785   1750,1754\

X(1743) = isogonal conjugate of X(8056)
X(1743) = X(i)-Ceva conjugate of X(j) for these (i,j): (57,1), (1476,55)
X(1743) = crosspoint of X(i) and X(j) for these (i,j): (57,1420), (651,765)
X(1743) = crosssum of X(244) and X(650)
X(1743) = {X(6),X(9)}-harmonic conjugate of X(1)
X(1743) = trilinear pole wrt excentral triangle of Gergonne line
X(1743) = perspector of ABC and unary cofactor triangle of Triangle T(-1,3)
X(1743) = excentral-isogonal conjugate of X(10860)
X(1743) = X(393)-of-excentral-triangle
X(1743) = X(i)-aleph conjugate of X(j) for these (i,j): (1,165), (2,1766), (7,169), (57,1743), (81,572), (174,9), (259,170), (266,43), (365,1742), (366,40), (507,164), (508,63), (509,1), (513,1053), (651,101)

### X(1744) = MIMOSA TRANSFORM OF X(226)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(226)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1744) lies on these lines: 9,46   19,91   846,1754   1046,1409   1172,1725   1709,1719   1723,1729   1770,1826

X(1744) = X(i)-aleph conjugate of X(j) for these (i,j): (4,1754), (92,1765)

### X(1745) = MIMOSA TRANSFORM OF X(255)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sec B + sec C - sec A    (D. Grinberg, 2/25/04)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1745) lies on the McCay cubic and these lines:
1,4   35,1742   36,978   43,46   78,1330   255,411   579,610   908,1076   920,1758   1464,1836   1716,1756

X(1745) = isogonal conjugate of X(3362)
X(1745) = X(i)-Ceva conjugate of X(j) for these (i,j): (3,1), (1935,1046)
X(1745) = X(1148)-cross conjugate of X(1)
X(1745) = {X(4),X(73)}-harmonic conjugate of X(1)
X(1745) = X(i)-aleph conjugate of X(j) for these (i,j): (1,46), (2,1726), (3,1745), (6,1716), (63,1763), (174,1708), (188,1158), (259,1721), (266,1722), (365,1707), (366,19), (508,1729), (509,1723), (651,1020)

### X(1746) = MIMOSA TRANSFORM OF X(261)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(261)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1746) lies on these lines:
2,572   4,580   27,1730   57,1111   92,1726   333,1764   515,1006   946,1203

X(1746) = X(i)-aleph conjugate of X(j) for these (i,j): (27,1722), (29,1721), (86,223), (286,1708), (333,1490), (648,109)

### X(1747) = MIMOSA TRANSFORM OF X(262)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(262)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1747) lies on these lines:
1,82   19,91   47,240   162,1096   1737,1890

### X(1748) = MIMOSA TRANSFORM OF X(264)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = cot A - tan A
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = (1 + 2 cos A)/(1 + cos A)
Trilinears       cot 2A : cot 2B : cot 2C

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1748) lies on these lines:
19,27   31,240   158,920   162,1096   412,1158   653,1708   1013,1859   1445,1767   1776,1857

X(1748) = isogonal conjugate of X(1820)
X(1748) = cevapoint of X(19) and X(920)
X(1748) = X(i)-cross conjugate of X(j) for these (i,j): (563,1), (2180,47)
X(1748) = X(i)-aleph conjugate of X(j) for these (i,j): (4,1740), (92,1), (264,63), (331,1445), (811,662), (823,162), (1969,1760)

### X(1749) = MIMOSA TRANSFORM OF X(265)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(265)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1749) lies on these lines: 1,21   5,79   30,80

### X(1750) = MIMOSA TRANSFORM OF X(268)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(268)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1750) lies on these lines:
1,4   9,165   20,936   40,210   43,1721   57,971   84,1728   200,329   1708,1768   1743,1754

X(1750) = X(282)-Ceva conjugate of X(1)
X(1750) = X(282)-aleph conjugate of X(1750)

### X(1751) = MIMOSA TRANSFORM OF X(272)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(272)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1751) lies on these lines:
2,272   4,580   6,226   9,321   10,55   19,1708   27,579   57,379   76,333   92,1731   1479,1794

X(1751) = isogonal conjugate of X(579)
X(1751) = cevapoint of X(11) and X(652) X(1751) = X(71)-cross conjugate of X(1)
X(1751) = crosssum of X(1724) and X(1754)

### X(1752) = MIMOSA TRANSFORM OF X(277)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(277)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1752) lies on these lines: 1,41   9,1479   19,46   920,1729

### X(1753) = MIMOSA TRANSFORM OF X(280)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(280)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1753) lies on these lines:
1,947   3,33   4,9   34,517   46,208   55,1887   63,318   204,580   225,1217   475,946   1068,1435   1158,1726   1445,1895   1593,1824   1597,1871   1708,1712

X(1753) = X(92)-aleph conjugate of X(1767)

### X(1754) = MIMOSA TRANSFORM OF X(284)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(284)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1754) lies on these lines:
1,3   4,580   5,582   20,58   25,1730   31,516   33,1708   47,1770   63,990   81,991   109,278   184,851   209,916   212,226   219,1376   238,1699   255,1074   283,377   386,411   394,1004   498,1794   579,1172   595,962   602,946   846,1744   950,1451   1707,1709   1726,1824   1743,1750

X(1754) = X(i)-Ceva conjugate of X(j) for these (i,j): (579,1724), (1172,1)
X(1754) = X(i)-aleph conjugate of X(j) for these (i,j): (4,1744), (29,1765), (365,1047), (1172,1754)

### X(1755) = MIMOSA TRANSFORM OF X(287)

Trilinears  nbsp;     f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(287)
Trilinears    a cos(A + ω) : :
Trilinears    (cot A + cot B + cot C) tan A - (tan A + tan B + tan C) cot A : :
Trilinears    a^2 (a^2 b^2 + a^2 c^2 - b^4 - c^4) : :
Barycentrics    a^2 (a^2 cos B cos C - b c cos^2 A) : :

X(1755) lies on these lines:
6,893   19,27   31,48   44,513   610,1707   1580,1581

X(1755) = isogonal conjugate of X(1821)
X(1755) = X(i)-Ceva conjugate of X(j) for these (i,j): (1581,1964), (1821,1)
X(1755) = cevapoint of X(1580) and X(1955)
X(1755) = crosspoint of X(i) and X(j) for these (i,j): (1,1821), (31,1967), (57,741), (240,1959)
X(1755) = crosssum of X(i) and X(j) for these (i,j): (1,1755), (9,740), (75,1966), (293,1910)
X(1755) = X(i)-aleph conjugate of X(j) for these (i,j): (98,1580), (1821,1755)
X(1755) = X(6)-isoconjugate of X(290)
X(1755) = X(92)-isoconjugate of X(293)
X(1755) = trilinear product X(2)*X(237)
X(1755) = barycentric square root of X(9419)

### X(1756) = MIMOSA TRANSFORM OF X(293)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(293)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1756) lies on these lines: 1,256   9,46   36,238   1711,1763   1716,1745

X(1756) = reflection of X(1) in X(1284)
X(1756) = X(98)-Ceva conjugate of X(1)
X(1756) = crosspoint of X(86) and X(1821)
X(1756) = crosssum of X(42) and X(1755)
X(1756) = X(98)-aleph conjugate of X(1756)

### X(1757) = MIMOSA TRANSFORM OF X(295)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c)2 - (a + b)(a + c)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1757) lies on these lines:
1,6   10,894   42,846   43,63   57,1463   81,756   100,896   171,210   190,740   191,1045   200,1707   209,1762   239,726   240,1783   314,1089   320,1698   333,1215   484,513   527,1738   672,1282   765,1110   899,1054

X(1757) = reflection of X(i) in X(j) for these (i,j): (1,238), (238,44)
X(1757) = isogonal conjugate of X(1929)
X(1757) = X(291)-Ceva conjugate of X(1)
X(1757) = crosspoint of X(660) and X(765)
X(1757) = crosssum of X(i) and X(j) for these (i,j): (244,659), (1931, 1963)
X(1757) = X(i)-aleph conjugate of X(j) for these (i,j): (291,1757), (660,1026)
X(1757) = crossdifference of PU(31)
X(1757) = perspector of conic {A,B,C,X(100),PU(32)}
X(1757) = intersection of trilinear polars of X(100), P(32), and U(32)
X(1757) = inverse-in-circumconic-centered-at-X(9) of X(37)

### X(1758) = MIMOSA TRANSFORM OF X(296)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(296)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1758) lies on these lines:
1,3   21,1254   43,1708   73,1046   108,240   223,1707   225,1247   226,846   238,1465   411,774   651,896   920,1745   1044,1158   1054,1738

X(1758) = isogonal conjugate of X(2648)
X(1758) = X(1937)-Ceva conjugate of X(1)
X(1758) = X(1937)-aleph conjugate of X(1758)
X(1758) = crossdifference of PU(80)
X(1758) = perspector of conic {{A,B,C,PU(81)}}

### X(1759) = MIMOSA TRANSFORM OF X(304)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3 - b3 - c3     (M. Iliev, 5/13/07)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1759) lies on these lines:
1,32   9,46   40,728   41,758   63,169   72,910

X(1759) = X(76)-Ceva conjugate of X(1)
X(1759) = X(i)-aleph conjugate of X(j) for these (i,j): (2,43), (8,170), (75,9), (76,1759), (92,1711), (366,1740), (508,978), (556,165), (668,1018)

### X(1760) = MIMOSA TRANSFORM OF X(305)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4 - b4 - c4     (M. Iliev, 5/13/07)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1760) lies on these lines:
1,82   19,27   190,1766   240,255   326,610   1820,1821

X(1760) = isogonal conjugate of X(2156)
X(1760) = X(561)-Ceva conjugate of X(1)
X(1760) = X(i)-aleph conjugate of X(j) for these (i,j): (2,1740), (75,1), (76,63), (264,920), (314,411), (556,1742), (561,1760), (668,100), (693,1052), (789,1492), (799,662), (811,162), (1969,1748), (1978,190)

### X(1761) = MIMOSA TRANSFORM OF X(306)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(306)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1761) lies on these lines:
1,1333   6,986   9,46   19,27   37,171   40,1503   284,758   896,1778   1158,1766

X(1761) = X(321)-Ceva conjugate of X(1)
X(1761) = X(i)-aleph conjugate of X(j) for these (i,j): (10,846), (92,1713), (321,1761), (556,573)

### X(1762) = MIMOSA TRANSFORM OF X(307)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(307)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1762) lies on these lines:
9,440   10,1782   19,27   30,40   55,846   57,1723   65,1046   71,1654   169,1708   209,1757

X(1762) = X(1441)-Ceva conjugate of X(1)
X(1762) = X(i)-aleph conjugate of X(j) for these (i,j): (2,3), (75,1764), (92,1715), (226,1046), (508,6), (556,20), (1441,1762)

### X(1763) = MIMOSA TRANSFORM OF X(326)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(326)
Trilinears       g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = (sin B)(tan B) + (sin C)(tan C) - (sin A)(tan A)     (Randy Hutson, 9/23/2011)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1763) lies on these lines:
1,25   2,169   6,57   9,440   19,226   40,64   43,46   63,573   73,1452   198,1214   329,1766   1711,1756

X(1763) = X(69)-Ceva conjugate of X(1)
X(1763) = X(i)-aleph conjugate of X(j) for these (i,j): (1,1716), (2,46), (63,1745), (69,1763), (75,1726), (174,1722), (188,1721), (366,1707), (508,1723), (556,1158), (664,1020)

### X(1764) = MIMOSA TRANSFORM OF X(332)

Trilinears    - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(332)
Trilinears    a^4(b + c) + a^3(b^2 + c^2) - a^2(b^3 + c^3) - a(b^2 + c^2)^2 - bc(b - c)^2(b + c) : :

X(1764) lies on these lines:
1,3   2,573   63,321   81,572   333,1746   345,1018   978,1695

X(1764) = X(314)-Ceva conjugate of X(1)
X(1764) = X(i)-aleph conjugate of X(j) for these (i,j): (2,1046), (75,1762), (188,1045), (314,1764), (333,3), (556,191)
X(1764) = anticomplement of X(2051)
X(1764) = excentral-isogonal conjugate of X(1045)
X(1764) = X(418)-of-excentral-triangle
X(1764) = homothetic center of excentral triangle and 3rd Conway triangle
X(1764) = perspector of 3rd Conway triangle and (cross-triangle of ABC and 4th Conway triangle)
X(1764) = perspector of 3rd Conway triangle and (cross-triangle of ABC and 5th Conway triangle)

### X(1765) = MIMOSA TRANSFORM OF X(333)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(333)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1765) lies on these lines:
1,1409   3,9   6,1012   7,1020   19,1158   20,391   21,572   63,321   71,515   580,1778   608,1777   1707,1709   1735,1880   1768,1781

X(1765) = X(i)-aleph conjugate of X(j) for these (i,j): (29,1754), (92,1744), (366,1047)

### X(1766) = MIMOSA TRANSFORM OF X(345)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(345)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1766) lies on these lines:
1,572   3,37   4,9   6,517   20,346   46,1400   63,321   101,610   165,846   190,1760   329,1763   355,594   672,1723   971,1350   1100,1482   1158,1761

X(1766) = reflection of X(990) in X(3)
X(1766) = X(312)-Ceva conjugate of X(1)
X(1766) = X(i)-aleph conjugate of X(j) for these (i,j): (2,1743), (8,165), (75,169), (188,43), (190,101), (312,1766), (333,572), (366,978), (522,1053), (556,9)

X(1766) = excentral-isogonal conjugate of X(43)

### X(1767) = MIMOSA TRANSFORM OF X(347)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(347)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1767) lies on these lines:
3,207   19,57   46,208   63,653   65,1498   108,165   109,204   1158,1712   1445,1748

X(1767) = X(522)-Ceva conjugate of X(1)
X(1767) = X(i)-aleph conjugate of X(j) for these (i,j): (92,1753), (342,1767), (508,282), (653,108), (522,1768)

### X(1768) = MIMOSA TRANSFORM OF X(521)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(521)

X(1768) lies on the Bevan circle, the excentral-hexyl ellipse, and these lines:
1,104   3,191   10,153   11,57   36,1727   40,550   46,80   63,100   119,1698   149,516   484,515   971,1155   1156,1445   1317,1697   1708,1750   1765,1781

X(1768) = reflection of X(i) in X(j) for these (i,j): (1,104), (153,10)
X(1768) = X(i)-aleph conjugate of X(j) for these (i,j): (2,514), (174,905), (366,650), (508,657), (522,1768)

X(1768) = trilinear pole wrt excentral triangle of line X(4)X(9)
X(1768) = excentral-isogonal conjugate of X(513)
X(1768) = X(110)-of-excentral-triangle

### X(1769) = MIMOSA TRANSFORM OF X(901)

Trilinears    - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(901)

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

X(1769) lies on these lines: 11,244   104,106   108,109   240,522   513,663

X(1769) = X(162)-Ceva conjugate of X(1845)
X(1769) = crosspoint of X(i) and X(j) for these (i,j): (88,934), (162,759)
X(1769) = crosssum of X(i) and X(j) for these (i,j): (31,1635), (522,1737), (656,758)
X(1769) = trilinear product of circumcircle intercepts of Sherman line

### X(1770) = MIMOSA TRANSFORM OF X(943)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(943)

Let IaIbIc be the reflection triangle of X(1). Let A' be the cevapoint of Ib and Ic, and define B', C' cyclically. The lines AA', BB', CC' concur in X(1770). (Randy Hutson, July 20, 2016)

X(1770) lies on these lines:
1,7   3,1836   4,46   5,1155   10,191   27,1780   28,1633   30,65   35,79   36,946   40,1478   47,1754   57,1479   109,225   165,498   382,1837   1707,1714   1710,1782   1724,1738   1744,1826   1771,1785   1885,1905

X(1770) = cevapoint of X(46) and X(1717)

### X(1771) = MIMOSA TRANSFORM OF X(947)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(947)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1771) lies on these lines:
1,3   4,109   10,255   31,1210   515,603   516,1076   580,1788   601,950   920,1736   1399,1837   1707,1728   1720,1721   1770,1785

### X(1772) = MIMOSA TRANSFORM OF X(953)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(953)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1772) lies on these lines:
1,88   3,1411   34,46   40,1421   240,522   498,986   920,1722   1068,1788

### X(1773) = MIMOSA TRANSFORM OF X(1038)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(1038)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1773) lies on these lines:
1,25   10,46   40,984   57,1722   169,1400   244,1468   388,1452

X(1773) = X(i)-Ceva conjugate of X(j) for these (i,j): (388,1), (1452,46)

### X(1774) = MIMOSA TRANSFORM OF X(1061)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(1061)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1774) lies on these lines: 4,46   40,47   57,1725   378,1061   484,1707

### X(1775) = MIMOSA TRANSFORM OF X(1063)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(1063)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1775) lies on these lines: 4,46   24,1063   40,1725   47,57

### X(1776) = MIMOSA TRANSFORM OF X(1156)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(1156)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1776) lies on these lines: 4,46   21,60   33,1707   63,497   109,1736   191,950   243,522   411,1898   519,1697   1005,1864   1155,1156   1725,1870   1748,1857

X(1776) = crosspoint of X(21) and X(1156)
X(1776) = crosssum of X(65) and X(1155)

### X(1777) = MIMOSA TRANSFORM OF X(1167)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(1167)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1777) lies on these lines:
1,84   4,109   34,1158   35,1742   36,1044   46,1707   47,1754   90,1041   226,601   255,516   603,946   608,1765   1046,1710   1399,1836

### X(1778) = MIMOSA TRANSFORM OF X(1171)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(1171)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1778) lies on these lines:
6,21   9,58   19,1707   28,579   37,81   44,1333   90,1172   284,1743   580,1765   896,1761   966,1010   1396,1708   1474,1732

X(1778) = cevapoint of X(1707) and (1724)

### X(1779) = MIMOSA TRANSFORM OF X(1172)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(1172)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1779) lies on these lines:
3,1780   4,46   35,47   43,165   378,580   579,1172   1714,1715

X(1779) = X(i)-aleph conjugate of X(j) for these (i,j): (4,1781), (29,573)

### X(1780) = MIMOSA TRANSFORM OF X(1175)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(1175)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1780) lies on these lines:
1,21   3,1779   4,580   27,1770   28,46   29,1737   35,71   90,1172   219,1333   579,1474   1010,1098   1214,1399   1408,1617

X(1780) = X(27)-Ceva conjugate of X(284)

### X(1781) = MIMOSA TRANSFORM OF X(1214)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(1214)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1781) lies on these lines:
1,19   6,1718   9,46   35,37   57,1723   71,484   165,846   169,1046   281,1478   1710,1720   1765,1768

X(1781) = X(226)-Ceva conjugate of X(1)
X(1781) = X(i)-aleph conjugate of X(j) for these (i,j): (2,573), (4,1779), (174,6), (226,1781), (366,3), (508,2)

### X(1782) = MIMOSA TRANSFORM OF X(1257)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(1257)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1782) lies on these lines:
4,1726   8,20   10,1762   19,46   58,65   71,191   580,1829   1710,1770   1737,1842

### X(1783) = MIMOSA TRANSFORM OF X(1262)

Trilinears        (tan A)/(b - c) : (tan B)/(c - a) : (tan C)/(a - b)
Barycentrics   a(tan A)/(b - c) : b(tan B)/(c - a) : c(tan C)/(a - b)

X(1783) lies on these lines:
4,218   6,281   19,1743   28,291   80,1172   100,112   101,108   150,1814   200,204   219,1249   240,1757   644,648   650,1415   651,653   899,1430   905,934   1103,1712   1718,1723   1785,1886

X(1783) = isogonal conjugate of X(905)
X(1783) = X(i)-Ceva conjugate of X(j) for these (i,j): (648,1897), (653,108)
X(1783) = cevapoint of X(i) and X(j) for these (i,j): (1,1734), (6,650), (513,614)
X(1783) = crosspoint of X(i) and X(j) for these (i,j): (162,648), (653,1897)
X(1783) = crosssum of X(i) and X(j) for these (i,j): (647,656), (652,1459), (513,614)
X(1783) = X(i)-aleph conjugate of X(j) for these (i,j): (108,1707), (651,610), (653,19)

### X(1784) = MIMOSA TRANSFORM OF X(1294)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(1294)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1784) lies on these lines: 1,29   4,79   36,243   240,522   318,1698   451,498   484,653   920,1712   1118,1479   1478,1857

### X(1785) = MIMOSA TRANSFORM OF X(1295)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(1295)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1785) lies on these lines:
1,4   2,1074   9,393   10,158   20,1076   25,1324   36,108   37,53   40,1118   46,208   65,1872   106,1309   240,522   406,498   407,1844   475,499   517,1361   519,1897   942,1887   1210,1895   1249,1743   1465,1532   1712,1714   1770,1771   1783,1886   1824,1894   1830,1835   1867,1904

X(1785) = reflection of X(1845) in X(1875)
X(1785) = isogonal conjugate of X(1795)
X(1785) = inverse-in-incircle of X(946)
X(1785) = X(4)-Ceva conjugate of X(1845)

### X(1786) = MIMOSA TRANSFORM OF X(1442)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(1442)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1786) lies on these lines: 33,46   57,77   1730,1787

### X(1787) = MIMOSA TRANSFORM OF X(1443)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(1443)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1787) lies on these lines: 34,46   57,88   1020,1435   1730,1786

### X(1788) = MIMOSA TRANSFORM OF X(1476)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = - (cos A)/x + (cos B)/y + (cos C)/z,  x : y : z = X(1476)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1788) lies on these lines:
1,631   2,65   4,46   7,12   8,56   10,57   11,962   20,1155   34,1722   36,944   40,497   43,73   55,938   109,1724   145,1319   165,950   171,1451   200,1467   201,986   208,1861   225,1738   226,1698   227,241   278,1714   281,579   329,1329   344,1284   345,1403   377,1454   387,1214   412,1857   484,1479   519,1420   580,1771   651,1406   653,1118   899,1042   958,1466   961,1150   978,1457   1068,1772   1707,1877

X(1788) = cevapoint of X(46) and X(1722)

### X(1789) = INVERSE MIMOSA TRANSFORM OF X(5)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos A)/(y + z),  x : y : z = X(5)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1789) lies on these lines: 3,125   21,36

X(1789) = isogonal conjugate of X(1825)

### X(1790) = INVERSE MIMOSA TRANSFORM OF X(6)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 + c2 - a2)/(b + c)     (M. Iliev, 5/13/07)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1790) lies on these lines:
1,1719   2,572   3,49   6,967   21,84   22,991   27,86   36,58   48,63   57,77   71,1796   73,1798   103,110   199,511   222,1804   228,295   306,332   333,662   1214,1813   1408,1470

X(1790) = isogonal conjugate of X(1826)
X(1790) = X(i)-Ceva conjugate of X(j) for these (i,j): (86,58), (1444,283)
X(1790) = cevapoint of X(3) and X(48)
X(1790) = X(222)-cross conjugate of X(81)
X(1790) = crosssum of X(1) and X(1719)

### X(1791) = INVERSE MIMOSA TRANSFORM OF X(10)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos A)/(y + z);x : y : z = X(10)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1791) lies on these lines:
1,1472   2,12   3,345   8,197   21,37   48,78   63,201   72,1437   228,1792   280,1436   975,993

X(1791) = isogonal conjugate of X(1829)
X(1791) = cevapoint of X(i) and X(j) for these (i,j): (3,72), (37,197), (219,228)

### X(1792) = INVERSE MIMOSA TRANSFORM OF X(20)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos A)/(y + z),  x : y : z = X(20)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1792) lies on these lines:
3,69   8,21   78,212   81,1257   86,939   99,972   228,1791   271,1819   314,943   1260,1265

X(1792) = isogonal conjugate of X(1426)
X(1792) = X(332)-Ceva conjugate of X(1812)
X(1792) = cevapoint of X(i) and X(j) for these (i,j): (78,1259), (283,1819)

### X(1793) = INVERSE MIMOSA TRANSFORM OF X(30)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos A)/(y + z),  x : y : z = X(30)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1793) lies on these lines: 3,125   10,21   72,283   307,1444

X(1793) = isogonal conjugate of X(1835)

### X(1794) = INVERSE MIMOSA TRANSFORM OF X(35)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos A)/(y + z),  x : y : z = X(35)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1794) lies on these lines:
1,201   10,29   35,71   36,951   77,255   498,1754   1059,1617   1479,1751

X(1794) = isogonal conjugate of X(1838)
X(1794) = cevapoint of X(71) and X(212)
X(1794) = crosssum of X(1841) and X(1859)

### X(1795) = INVERSE MIMOSA TRANSFORM OF X(36)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos A)/(y + z),  x : y : z = X(36)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1795) lies on these lines:
1,104   3,1364   29,58   35,947   36,102   56,945   57,1845   78,255   117,1478   124,499   163,284   171,1065   219,577   282,1743   912,1807   999,1361

X(1795) = isogonal conjugate of X(1785)

### X(1796) = INVERSE MIMOSA TRANSFORM OF X(37)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos A)/(y + z),  x : y : z = X(37)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1796) lies on these lines: 27,1268   35,42   57,1255   71,1790

X(1796) = isogonal conjugate of X(1839)
X(1796) = X(1268)-Ceva conjugate of X(1126)
X(1796) = cevapoint of X(3) and X(71)

### X(1797) = INVERSE MIMOSA TRANSFORM OF X(44)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos A)/(y + z),  x : y : z = X(44)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1797) lies on these lines:
3,1331   27,648   57,88   58,106   63,1332   84,1320   103,677   222,1813   320,908

X(1797) = X(903)-Ceva conjugate of X(106)

### X(1798) = INVERSE MIMOSA TRANSFORM OF X(58)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos A)/(y + z),  x : y : z = X(58)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1798) lies on these lines:
6,60   54,970   58,1245   65,81   71,283   72,1437   73,1790   110,960   285,1903

X(1798) = isogonal conjugate of X(429)
X(1798) = cevapoint of X(3) and X(1437)

### X(1799) = INVERSE MIMOSA TRANSFORM OF X(75)

Trilinears    (cos A)/(y + z),  x : y : z = X(75)
Trilinears    cot A csc(A + ω) : :
Barycentrics    (b^2 + c^2 - a^2)/(b^2 + c^2) : :

Let A'B'C' be the circummedial triangle (the circumcevian triangle of X(2)). Let A" be the cevapoint of B' and C', and define B" and C" cyclically. AA", BB", and CC" concur in X(1799). (Randy Hutson, December 10, 2016)

X(1799) lies on these lines:
2,32   3,305   22,76   25,183   69,184   95,325   98,689   287,343   385,1194   1402,1441

X(1799) = isogonal conjugate of X(1843)
X(1799) = isotomic conjugate of X(427)
X(1799) = X(308)-Ceva conjugate of X(83)
X(1799) = cevapoint of X(i) and X(j) for these (i,j): (2,22), (3,69)
X(1799) = complement of X(8878)
X(1799) = crosspoint of X(2) and X(22) wrt both the anticomplementary and tangential triangles
X(1799) = X(92)-isoconjugate of X(3051)
X(1799) = perspector of circummedial triangle and cross-triangle of ABC and circummedial triangle

### X(1800) = INVERSE MIMOSA TRANSFORM OF X(155)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos A)/(y + z),  x : y : z = X(155)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1800) lies on these lines:
1,921   3,49   21,90   29,662   46,453   65,1813

X(1800) = X(21)-Ceva conjugate of X(283)

### X(1801) = INVERSE MIMOSA TRANSFORM OF X(159)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos A)/(y + z),  x : y : z = X(159)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1801) lies on these lines: 3,49   58,78   171,306

### X(1802) = INVERSE MIMOSA TRANSFORM OF X(170)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos A)/(y + z),  x : y : z = X(170)
Trilinears        (1 + cos A)2 cot A : (1 + cos B)2 cot B : (1 + cos C)2 cot C     (M. Iliev, 4/12/07)
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = cos A cot(A/2) cos2(A/2) (P. Moses, 12/6/11)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1802) lies on these lines:
3,48   6,939   9,943   40,101   41,55   63,1803   255,906

X(1802) = isogonal conjugate of X(1847)
X(1802) = X(i)-Ceva conjugate of X(j) for these (i,j): (200,1253), (219,212)
X(1802) = crosspoint of X(i) and X(j) for these (i,j): (219,1260), (906,1110)
X(1802) = crosssum of X(i) and X(j) for these (i,j): (269,1435), (278,1119)

### X(1803) = INVERSE MIMOSA TRANSFORM OF X(218)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos A)/(y + z),  x : y : z = X(218)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1803) lies on these lines:
35,103   41,57   58,1458   63,1802

X(1803) = isogonal conjugate of X(1855)
X(1803) cevapoint of X(48) and X(222)

### X(1804) = INVERSE MIMOSA TRANSFORM OF X(223)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos A)/(y + z),  x : y : z = X(223)
Trilinears        (1 + cos 2A)/(1 + cos A) : (1 + cos 2B)/(1 + cos B) : (1 + cos 2C)/(1 + cos C)     (M. Iliev, 4/12/07)
Trilinears        [cos A sec(A/2)]2 : [cos B sec(B/2)]2 : [cos C sec(C/2)]2     (M. Iliev, 4/12/07)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1804) lies on these lines:
3,77   7,21   20,1440   36,269   55,1442   63,268   69,1809   198,651   219,1813   222,1790   326,1259   347,934   573,1461

X(1804) = isogonal conjugate of X(1857)
X(1804) = X(i)-Ceva conjugate of X(j) for these (i,j): (348,222), (1444,77)

### X(1805) = INVERSE MIMOSA TRANSFORM OF X(371)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[1 - cot(A/2)]/(b + c)     (M. Iliev, 5/13/07)
Trilinears       g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sec[(B - C)/2)][sin(A/2) - cos(A/2)]     (M. Iliev, 5/13/07)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1805) lies on this line: 3,6

X(1805) = X(21)-Ceva conjugate of X(1806)

### X(1806) = INVERSE MIMOSA TRANSFORM OF X(372)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[1 + cot(A/2)]/(b + c)     (M. Iliev, 5/13/07)
Trilinears       g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sec[(B - C)/2)][sin(A/2) + cos(A/2)]     (M. Iliev, 5/13/07)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1806) lies on this line: 3,6

X(1806) = X(21)-Ceva conjugate of X(1805)

### X(1807) = INVERSE MIMOSA TRANSFORM OF X(484)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos A)/(y + z),  x : y : z = X(484)

Trilinears       1/(2 - sec A) : 1/(2 - sec B) : 1/(2 - sec C)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1807) lies on these lines:
1,5   3,201   29,1897   37,101   72,283   73,265   77,1060   78,1062   102,517   296,916   912,1795   942,951   945,1482   947,1385   976,1036   999,1037

X(1807) = isogonal conjugate of X(1870)
X(1807) = crosssum of X(i) and X(j) for these (i,j): (1,1718), (1464,1835)

### X(1808) = INVERSE MIMOSA TRANSFORM OF X(511)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos A)/(y + z),  x : y : z = X(511)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1808) lies on these lines: 41,60   42,81   228,295

X(1808) = isogonal conjugate of X(1874)

### X(1809) = INVERSE MIMOSA TRANSFORM OF X(515)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos A)/(y + z),  x : y : z = X(515)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1809) lies on these lines: 3,8   69,1804   78,255   1259,1265   1295,1309

X(1809) = isogonal conjugate of X(1875)

### X(1810) = INVERSE MIMOSA TRANSFORM OF X(518)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos A)/(y + z),  x : y : z = X(518)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1810) lies on these lines: 57,100   222,1331

X(1810) = cevapoint of X(3) and X(1818)

### X(1811) = INVERSE MIMOSA TRANSFORM OF X(519)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos A)/(y + z),  x : y : z = X(519)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1811) lies on these lines: 56,100   603,1331

X(1811) = isogonal conjugate of X(1878)

### X(1812) = INVERSE MIMOSA TRANSFORM OF X(573)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos A)/(y + z),  x : y : z = X(573)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1812) lies on these lines:
2,6   21,60   48,63   58,997   72,1437   78,212   219,332   222,348   274,1231   280,285   306,1332   314,1172   662,1817   860,1330   1006,1092   1412,1708

X(1812) = isogonal conjugate of X(1880)
X(1812) = X(i)-Ceva conjugate of X(j) for these (i,j): (314,21), (332,1792)
X(1812) = cevapoint of X(i) and X(j) for these (i,j): (63,394), (78,219)

### X(1813) = INVERSE MIMOSA TRANSFORM OF X(650)

Trilinears        (cos A)/(cos B - cos C) : (cos B)/(cos C - cos A) : (cos C)/(cos A - cos B)
Barycentrics   (sin 2A)/(cos B - cos C) : (sin 2B)/(cos C - cos A) : (sin 2C)/(cos A - cos B)

X(1813) lies on these lines:
48,77   59,677   65,1800   73,895   101,651   109,110   219,1804   222,1797   224,1420   283,296   284,1442   287,307   347,1630   604,1445   648,653   1214,1790

X(1813) = isogonal conjugate of X(3064)
X(1813) = X(i)-Ceva conjugate of X(j) for these (i,j): (662,651), (664,109)
X(1813) = cevapoint of X(i) and X(j) for these (i,j): (3,652), (48,1459), (905,1214)
X(1813) = X(219)-cross conjugate of X(59)

### X(1814) = INVERSE MIMOSA TRANSFORM OF X(672)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos A)/(y + z),  x : y : z = X(672)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1814) lies on these lines:
6,7   48,77   63,212   69,219   81,105   150,1783   286,648   518,677   1438,1449

X(1814) = reflection of X(651) in X(6)

### X(1815) = INVERSE MIMOSA TRANSFORM OF X(910)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos A)/(y + z),  x : y : z = X(910)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1815) lies on these lines:
9,77   86,648   103,110   219,1804   326,1332   394,1260

X(1815) = isogonal conjugate of X(1886)

X(1815) = cevapoint of X(219) and X(1818)

### X(1816) = INVERSE MIMOSA TRANSFORM OFX(1075)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos A)/(y + z),  x : y : z = X(1075)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

As a point on the Euler line, X(1816) has Shinagawa coefficients (4EF3abc - \$aSA\$[S2 - (2E + 3F)F][S2 - (2E + F)F] + 2\$aSBSC\$[S2 - (2E + F)F]F + 4\$a(SA)3\$F2, [S4 - (2E - F)FS2 + F4]abc + \$aSA\$[S2 - (2E + 3F)F][S2 - (2E + F)F] - 2\$aSBSC\$[S2 - (2E + F)F]F - 4\$a(SA)3\$F2).

X(1816) lies on this line: 2,3
X(1816) = X(283)-Ceva conjugate of X(21)

### X(1817) = INVERSE MIMOSA TRANSFORM OFX(1249)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos A)/(y + z),  x : y : z = X(1249)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

As a point on the Euler line, X(1817) has Shinagawa coefficients (4EF2S2 - \$aSA\$(S2 - 4F2)S2 + 3\$bcSBSC\$S2 + 4\$abSC\$FS2 - 2\$bc(SB)2(SC)2\$, [S2 - 4(E + F)F]ES2 + \$bc\$(S2 - 4F2)S2 - 3\$bcSBSC\$S2 - 4\$abSC\$FS2 + 2\$bc(SB)2(SC)2\$).

X(1817) lies on these lines:
2,3   40,1819   57,77   58,937   63,610   100,306   110,972   189,333   196,347   198,329   572,1730   662,1812   1172,1214   1396,1465

X(1817) = isogonal conjugate of X(1903)
X(1817) = X(i)-Ceva conjugate of X(j) for these (i,j): (333,81), (1444,21)
X(1817) = cevapoint of X(i) and X(j) for these (i,j): (3,610), (40,198)

### X(1818) = INVERSE MIMOSA TRANSFORM OFX(1282)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos A)/(y + z),  x : y : z = X(1282)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1818) lies on these lines:
1,142   3,48   9,991   42,750   69,73   198,1350   212,394   222,1260   241,518   386,1449   521,656   581,936   603,1259   997,1064   1193,1386

X(1818) = X(i)-Ceva conjugate of X(j) for these (i,j): (1810,3), (1815,219)
X(1818) = crosspoint of X(3) and X(295)
X(1818) = crosssum of X(i) and X(j) for these (i,j): (1,1738), (4,242)

### X(1819) = INVERSE MIMOSA TRANSFORM OF X(1498)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos A)/(y + z),  x : y : z = X(1498)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1819) lies on these lines:
3,49   9,21   40,1817   58,1167   271,1792

X(1819) = X(1792)-Ceva conjugate of X(283)

### X(1820) = ISOGONAL CONJUGATE OF MIMOSA TRANSFORM OF X(264)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[- (cos A)/x + (cos B)/y + (cos C)/z],  x : y : z = X(264)

Trilinears       tan 2A : tan 2B : tan 2C

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1820) lies on these lines:
1,563   19,91   68,71   1400,1454   1760,1821

X(1820) = isogonal conjugate of X(1748)
X(1820) = crosspoint of X(63) and X(921)
X(1820) = crosssum of X(19) and X(920)

### X(1821) = ISOGONAL CONJUGATE OF MIMOSA TRANSFORM OF X(287)

Trilinears    1/[a^2(a^2b^2 + a^2c^2 - b^4 - c^4) : :
Barycentrics    sec(A + ω) : :
Barycentrics    1/(a^2 cos B cos C - bc cos^2 A) : :

X(1821) lies on these lines:
19,823   31,92   48,75   63,561   71,190   98,100   287,651   653,1400   1580,1733   1760,1820

X(1821) = isogonal conjugate of X(1755)
X(1821) = isotomic conjugate of X(1959)
X(1821) = cevapoint of X(i) and X(j) for these (i,j): (1,1755), (9,740), (75,1966), (293,1910)
X(1821) = crosspoint of X(1581) and X(1956)
X(1821) = crosssum of X(1580) and X(1955)
X(1821) = X(i)-aleph conjugate of X(j) for these (i,j): (98,1740), (290,73), (1821,1)
X(1821) = trilinear pole of line X(1)X(810)
X(1821) = pole wrt polar circle of trilinear polar of X(240)
X(1821) = X(48)-isoconjugate (polar conjugate) of X(240)
X(1821) = X(6)-isoconjugate of X(511)

### X(1822) =  ISOGONAL CONJUGATE OF X(2588)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[- (cos A)/x + (cos B)/y + (cos C)/z],  x : y : z = X(1113)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1822) lies on these lines: 1,21   109,1113

X(1822) = isogonal conjugate of X(2588)
X(1822) = X(1101)-Ceva conjugate of X(1823)
X(1822) = X(i)-aleph conjugate of X(j) for these (i,j): (162,1823), (1113,1)
X(1822) = trilinear quotient X(110)/X(1114)

### X(1823) = ISOGONAL CONJUGATE OF X(2589)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[- (cos A)/x + (cos B)/y + (cos C)/z],  x : y : z = X(1114)

X(1823) lies on these lines: 1,21   109,1114

X(1823) = isogonal conjugate of X(2589)
X(1823) = X(1101)-Ceva conjugate of X(1822)
X(1823) = X(i)-aleph conjugate of X(j) for these (i,j): (162,1822), (1114,1)
X(1823) = trilinear quotient X(110)/X(1113)

Zosma Transforms 1824- 1907

The Zosma transform of a point X = x : y : z is the isogonal conjugate of the inverse Mimosa transform of X, given by trilinears
(y + z) sec A : (z + x) sec B : (x + y)sec C.

(Zosma is another star name.)

### X(1824) = ZOSMA TRANSFORM OF X(2)

Trilinears    (b + c)tan A : (c + a) tan B : (a + b) tan C
Barycentrics    a(b + c) tan A : b(c + a) tan B : c(a + b) tan C

X(1824) lies on these lines:
4,8   10,429  12,431   19,25   27,295   28,1255   34,1887   42,1880   51,1864   65,225   209,1865   210,430   213,607   240,444   278,1002   427,1848   428,528   518,1889   674,1839   756,862   851,1214   942,1068   989,1039   990,1473   1593,1753   1726,1754   1730,1736   1785,1894\

X(1824) = isogonal conjugate of X(1444)
X(1824) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,1826), (33,42), (225,1826)
X(1824) = X(i)-cross conjugate of X(j) for these (i,j): (181,42), (213,37)
X(1824) = crosspoint of X(i) and X(j) for these (i,j): (4,19), (33,1857), (65, 1903), (225,1826)
X(1824) = crosssum of X(i) and X(j) for these (i,j): (3,63), (21,1817), (77,1804), (283,1790)

X(1824) = intersection of tangents to hyperbola {{A,B,C,X(4),X(19)}} at X(4) and X(19)
X(1824) = pole wrt polar circle of trilinear polar of X(274) (line X(320)X(350))
X(1824) = polar conjugate of X(274)
X(1824) = barycentric product of vertices of 2nd extouch triangle
X(1824) = X(174)-of-orthic-triangle if ABC is acute

### X(1825) = ZOSMA TRANSFORM OF X(5)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(5)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(5)

X(1825) lies on these lines:
4,80   19,41   33,40   34,1126   35,186   65,225   109,1710   250,270   319,340   1829,1877   1859,1902   1872,1905   1875,1900

X(1825) = isogonal conjugate of X(1789)

### X(1826) = ZOSMA TRANSFORM OF X(6)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(6)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(6)

X(1826) lies on these lines:
4,9   6,1837   11,1108   12,37   25,1631   27,1268   28,1224   29,1220   33,42   48,515   53,1904   65,1868   80,1172   92,264   101,1300   209,1859   210,430   219,355   286,334   407,1213   427,1841   579,1737   608,1877   857,1441   1744,1770   1836,1853

X(1826) = isogonal conjugate of X(1790)
X(1826) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,1824), (281,37)
X(1826) = cevapoint of X(1) and X(1719)
X(1826) = X(42)-cross conjugate of X(10)
X(1826) = crosspoint of X(4) and X(92)
X(1826) = crosssum of X(3) and X(48)

### X(1827) = ZOSMA TRANSFORM OF X(7)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(7)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(7)

X(1827) lies on these lines: 4,7   19,25   46,1721   208,1887   294,1172   430,1856   1425,1547   1828,1843   1845,1905

X(1827) = crosspoint of X(4) and X(33)
X(1827) = crosssum of X(3) and X(77)

### X(1828) = ZOSMA TRANSFORM OF X(8)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(8)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(8)

X(1828) lies on these lines:
4,8   10,1883   19,44   25,34   28,88   46,1707   51,65   225,1846   427,1329   428,529   1827,1843   1838,1894   1844,1884   1848,1904

X(1828) = crosspoint of X(4) and X(34)
X(1828) = crosssum of X(3) and X(78)
X(1828) = inverse-in-Fuhrmann-circle of X(5101)

### X(1829) = ZOSMA TRANSFORM OF X(10)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(10)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(10)

X(1829) lies on these lines:
1,25   4,8   6,19   10,427   24,1385   27,239   28,60   29,242   40,1593   52,912   56,1452   57,1398   209,1869   225,1866   235,946   278,959   388,1892   392,406   407,1838   428,519   429,960   444,1193   468,1125   516,1885   518,1843   580,1782   1100,1474   1395,1468   1482,1598   1724,1726   1825,1877   1831,1842   1852,1858   1861,1883

X(1829) = reflection of X(1902) in X(4)
X(1829) = isogonal conjugate of X(1791)
X(1829) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,429), (19,444)
X(1829) = crosspoint of X(i) and X(j) for these (i,j): (4,28), (278,286))
X(1829) = crosssum of X(i) and X(j) for these (i,j): (3,72), (37,197), (219,228)

X(1829) = inverse-in-Fuhrmann-circle of X(5090)
X(1829) = X(177)-of-orthic triangle if ABC is acute
X(1829) = polar conjugate of X(30710)

### X(1830) = ZOSMA TRANSFORM OF X(11)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(11)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(11)

X(1830) lies on these lines: 4,80   33,57   65,1846   225,1872   517,1877   900,1862   908,1861   1785,1835   1887,1902   1888,1900

X(1830) = polar conjugate of isogonal conjugate of X(21742)

### X(1831) = ZOSMA TRANSFORM OF X(12)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(12)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(12)

X(1831) lies on these lines: 4,80   28,501   33,1697   225,1871   1829,1842   1835,1838   1839,1858   1844,1870

X(1831) = crosspoint of X(4) and X(270)
X(1831) = crosssum of X(3) and X(201)

### X(1832) = ZOSMA TRANSFORM OF X(15)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(15)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(15)

X(1832) lies on these lines: 4,1251   12,37

### X(1833) = ZOSMA TRANSFORM OF X(16)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(16)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(16)

X(1833) lies on this line: 12,37

### X(1834) = ZOSMA TRANSFORM OF X(28)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(28)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(28)

X(1834) lies on these lines:
1,442   2,1043   4,6   5,386   8,1211   10,37   11,1193   12,42   30,58   33,429   43,1329   56,851   65,225   115,118   377,940   405,1714   440,950   496,995   497,1191   524,1330   942,1086   1058,1616

X(1834) = complement of X(1043)
X(1834) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,1842), (1897,523)
X(1834) = crosspoint of X(4) and X(10)
X(1834) = crosssum of X(i) and X(j) for these (i,j): (3,58), (21,404) X(1834) = crossdifference of every pair of points on line X(520)X(3733)

### X(1835) = ZOSMA TRANSFORM OF X(30)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(30)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(30)

X(1835) lies on these lines:
4,79   28,34   36,186   65,225   320,340   513,1874   758,860   1398,1470   1785,1830   1831,1838

X(1835) = isogonal conjugate of X(1793)
X(1835) = X(1870)-Ceva conjugate of X(1464)

### X(1836) = ZOSMA TRANSFORM OF X(33)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(33)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(33)

X(1836) lies on these lines:
1,30   2,1155   3,1770   4,65   5,46   6,1839   7,354   11,57   12,40   19,1901   33,1892   34,1852   55,226   56,946   210,329   221,225   235,1452   278,1456   377,960   381,1737   388,962   517,1478   614,1086   908,1376   942,1479   1040,1721   1158,1454   1399,1777   1470,1519   1826,1853

X(1836) = reflection of X(i) in X(j) for these (i,j): (55,226), (1012,946)
X(1836) = crosspoint of X(4) and X(7)
X(1836) = crosssum of X(3) and X(55)

### X(1837) = ZOSMA TRANSFORM OF X(34)

Trilinears    (y + z) sec A : (z + x) sec B : (x + y) sec C, where x : y : z = X(34)
Trilinears    cos B + cos C - 2 cos B cos C : :
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C, where x : y : z = X(34)

X(1837) lies on these lines:
1,5   3,1737   4,65   6,1826   8,210   10,55   19,1852   20,1155   30,46   33,429   40,1728   56,515   78,1329   221,1877   354,388   382,1770   499,1385   517,1479   942,1478   944,1319   1040,1722   1399,1771   1464,1745   1853,1854

X(1837) = midpoint of X(65) and X(1898)
X(1837) = reflection of X(i) in X(j) for these (i,j): (1,496), (56,1210), (78,1329)
X(1837) = inverse-in-Fuhrmann-circle of X(11)
X(1837) = crosspoint of X(4) and X(8)
X(1837) = crosssum of X(3) and X(56)
X(1837) = X(24)-of-Fuhrmann-triangle
X(1837) = inverse-in-Feuerbach-hyperbola of X(355)
X(1837) = {X(1),X(80)}-harmonic conjugate of X(355)
X(1837) = {X(4),X(65)}-harmonic conjugate of X(1836)
X(1837) = Ursa-major-to-Ursa-minor similarity image of X(1)

### X(1838) = ZOSMA TRANSFORM OF X(35)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(35)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(35)

X(1838) lies on these lines:
1,4   2,1076   5,1214   10,92   11,133   19,46   20,1074   27,58   28,36   29,1125   30,1852   47,1754   53,1108   57,1118   65,1243   79,1172   158,273   235,1893   281,1698   403,1873   407,1829   412,516   427,1867   442,1841   517,1888   942,1844   1426,1905   1598,1617   1828,1894   1831,1835

X(1838) = isogonal conjugate of X(1794)
X(1838) = X(4)-Ceva conjugate of X(1844)
X(1838) = cevapoint of X(1841) and X(1859)
X(1838) = crosspoint of X(27) and X(273)
X(1838) = crosssum of X(71) and X(212)

### X(1839) = ZOSMA TRANSFORM OF X(37)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(37)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(37)

X(1839) lies on these lines:
4,9   6,1836   27,86   48,946   52,916   79,1172   193,3187   225,608   278,1419   430,1213   534,1441   579,1770   610,1699   674,1824   1831,1858   1840,1900   1841,1852   1877,1880

X(1839) = isogonal conjugate of X(1796)
X(1839) = X(4)-Ceva conjugate of X(430)
X(1839) = crosspoint of X(4) and X(27)
X(1839) = crosssum of X(3) and X(71)
X(1839) = polar conjugate of X(1268)

### X(1840) = ZOSMA TRANSFORM OF X(39)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(39)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(39)

X(1840) lies on these lines: 12,37   19,318   419,1215   1839,1900

### X(1841) = ZOSMA TRANSFORM OF X(71)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(71)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(71)

X(1841) lies on these lines:
4,37   6,19   28,1104   53,225   71,1888   216,1465   241,273   278,393   281,475   427,1826   442,1838   581,1871   594,1861   1100,1172   1119,1418   1400,1875   1839,1852

X(1841) = X(1838)-Ceva conjugate of X(1859)
X(1841) = crosspoint of X(28) and X(278)
X(1841) = crosssum of X(72) and X(219)

### X(1842) = ZOSMA TRANSFORM OF X(72)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(72)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(72)

X(1842) lies on these lines:
4,9   25,225   28,36   29,1848   33,976   34,207   51,65   92,1891   278,1420   1737,1782   1829,1831   1878,1888

X(1842) = X(4)-Ceva conjugate of X(1834)
X(1842) = crosspoint of X(27) and X(1119)
X(1842) = crosssum of X(71) and X(1260)

### X(1843) = ZOSMA TRANSFORM OF X(75)

Trilinears    (y + z) sec A : (z + x) sec B : (x + y) sec C,    where x : y : z = X(75)
Trilinears    tan A sin(A + ω) : :
Barycentrics    a^2(b^2 + c^2)/(b^2 + c^2 - a^2) : :

Let E = Euler line of ABC. Let g = isogonal conjugate, and t = isotomic conjugate. Then X(1843) = g(t(E))∩t(g(E)). (Randy Hutson, December 2, 2017)

X(1843) lies on these lines:
4,69   6,25   24,182   34,1469   112,755   113,1596   125,1205   141,427   143,1353   155,1351   157,571   160,570   181,1395   185,1503   216,237   263,393   373,468   428,524   518,1829   674,1824   1350,1593   1827,1828

X(1843) = reflection of X(i) in X(j) for these (i,j): (1205,125), (1289,647)
X(1843) = isogonal conjugate of X(1799)
X(1843) = X(i)-Ceva conjugate for these (i,j): (4,427), (427,39), (1289,647)
X(1843) = crosspoint of X(i) and X(j) for these (i,j): (4,25), (6,66)
X(1843) = crosssum of X(i) and X(j) for these (i,j): (2,22), (3,69)

X(1843) = orthic isotomic conjugate of X(185)
X(1843) = X(7)-of-orthic-triangle if ABC is acute
X(1843) = orthic-isogonal conjugate of X(427)
X(1843) = anticomplement of X(6) wrt orthic triangle
X(1843) = pole wrt polar circle of trilinear polar of X(308) (line X(316)X(512))
X(1843) = X(48)-isoconjugate (polar conjugate) of X(308)
X(1843) = inverse-in-polar-circle of X(316)
X(1843) = Ehrmann-vertex-to-orthic similarity image of X(3818)
X(1843) = perspector of [reflection of symmedial triangle in X(6)] and tangential triangle, wrt symmedial triangle, of circumconic of symmedial triangle centered at X(6) (bicevian conic of X(6) and X(25))

### X(1844) = ZOSMA TRANSFORM OF X(79)

Trilinears    (y + z) sec A : (z + x) sec B : (x + y) sec C, where x : y : z = X(79)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C, where x : y : z = X(79)

Let A'B'C' be the orthic triangle. Let A" be the incenter of AB'C', and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(1844). (Randy Hutson, January 29, 2018)

X(1844) lies on these lines:
1,19   4,79   29,758   33,46   35,186   354,1871   407,1785   942,1838   1828,1884   1831,1870   1846,1887

X(1844) = X(4)-Ceva conjugate of X(1838)

### X(1845) = ZOSMA TRANSFORM OF X(80)

Trilinears    (y + z) sec A : (z + x) sec B : (x + y) sec C, where x : y : z = X(80)
Barycentrics   (y + z) tan A : (z + x) tan B : (x + y) tan C, where x : y : z = X(80)

The following ten points lie on a circle: X(i) for i = 11, 36, 65, 80, 108, 759, 1354, 1845, 2588, 2589. (Chris Van Tienhoven, Hyacinthos, January 4, 2011)

Let A'B'C' be the orthic triangle. Let A" be the A-excenter of AB'C', and define B" and C" cyclically. The lines A'A", B'B", C'C" concur in X(1845). (Randy Hutson, January 29, 2018)

X(1845) lies on these lines:
1,102   4,80   19,1743   34,46   36,186   57,1795   65,389   92,994   117,1737   162,759   407,1829   517,1361   942,1354   1146,1901   1827,1905

X(1845) = reflection of X(i) in X(j) for these (i,j): (1364,942), (1785,1875)
X(1845) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,1785), (162,1769)
X(1845) = crosspoint of X(4) and X(1870)
X(1845) = crosssum of X(3) and X(1807)

### X(1846) = ZOSMA TRANSFORM OF X(104)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(104)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(104)

X(1846) lies on these lines: 4,11   19,53   65,1830   225,1828   517,1361   1319,1877   1844,1887

X(1846) = X(4)-Ceva conjugate of X(1877)
X(1846) = crosspoint of X(4) and X(1785)
X(1846) = crosssum of X(3) and X(1795)

### X(1847) = ZOSMA TRANSFORM OF X(170)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(170)
Trilinears        (tan A)/(1 + cos A)2 : (tan B)/(1 + cos B)2 : (tan C)/(1 + cos C)2     (M. Iliev, 4/12/07)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(170)

X(1847) lies on these lines:
4,7   27,1088   85,92   158,1111   224,664   278,279   917,934

X(1847) = isogonal conjugate of X(1802)
X(1847) = cevapoint of X(i) and X(j) for these (i,j): (269,1435), (278,1119)
X(1847) = isotomic conjugate of X(3692)
X(1847) = polar conjugate of X(200)

### X(1848) = ZOSMA TRANSFORM OF X(171)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(171)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(171)

X(1848) lies on these lines:
1,4   2,19   5,1871   7,1435   11,132   25,1001   27,86   28,1125   29,1842   92,264   286,350   427,1824   429,960   608,940   1595,1872   1828,1904   1883,1900

X(1848) = isogonal conjugate of X(2359)
X(1848) = crosspoint of X(27) and X(92)
X(1848) = crosssum of X(i) and X(j) for these (i,j): (42,205), (48,71)
X(1848) = pole wrt polar circle of trilinear polar of X(1220) (line X(522)X(649))
X(1848) = polar conjugate of X(1220)
X(1848) = perspector of Gemini triangle 38 and cross-triangle of Gemini triangles 37 and 38

### X(1849) = ZOSMA TRANSFORM OF X(175)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(175)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(175)

X(1849) lies on this line: 4,7

### X(1850) = ZOSMA TRANSFORM OF X(176)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(176)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(176)

X(1850) lies on this line: 4,7

### X(1851) = ZOSMA TRANSFORM OF X(200)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(200)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(200)

X(1851) lies on these lines:
2,242   4,8   19,672   25,105   34,207   196,1876   281,427   286,310   479,1119   497,1863   1146,1853   1395,1430

X(1851) = X(4)-Ceva conjugate of X(1863)
X(1851) = crosspoint of X(4) and X(1119)
X(1851) = crosssum of X(3) and X(1260)
X(1851) = polar conjugate of X(30701)

### X(1852) = ZOSMA TRANSFORM OF X(201)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(201)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(201)

X(1852) lies on these lines:
4,12   11,28   19,1837   30,1838   34,1836   516,1888   950,1859   1829,1858   1839,1841

### X(1853) = ZOSMA TRANSFORM OF X(204)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(204)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(204)

Let A' be the orthocenter of BCX(3), and define B' and C' cyclically; then X(1853) is the centroid of A'B'C'.

X(1853) lies on these lines:
2,154   3,161   4,64   5,1498   6,66   12,221   25,125   122,1073   157,426   343,1350   394,858   1146,1851   1181,1594   1352,1368   1826,1836   1837,1854

X(1853) = reflection of X(154) in X(2)
X(1853) = crosspoint of X(4) and X(253)
X(1853) = crosssum of X(i) and X(j) for these (i,j): (3,154), (206,577)
X(1853) = centroid of pedal triangle of X(64)

### X(1854) = ZOSMA TRANSFORM OF X(208)

Trilinears    (y + z) sec A : (z + x) sec B : (x + y) sec C, where x : y : z = X(208)
Trilinears    (b + c - a)[a^5 + 2a^2 (b - c)^2 (b + c) - a(b^2 - c^2)^2 - 2(b - c)^2 (b + c) (b^2 + c^2)] : :
Barycentrics    (y + z) tan A : (z + x) tan B : (x + y) tan C, where x : y : z = X(208)

X(1854) lies on these lines:
1,84   6,1858   33,64   55,976   56,774   154,968   227,1490   960,1040   1192,1452   1837,1853

X(1854) = reflection of X(221) in X(1)
X(1854) = crosspoint of X(4) and X(280)
X(1854) = crosssum of X(i) and X(j) for these (i,j): (1,1394), (3,221), (55,478)

### X(1855) = ZOSMA TRANSFORM OF X(218)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(218)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(218)

X(1855) lies on these lines:
1,1886   4,9   55,1856   65,1146   85,92

X(1855) = isogonal conjugate of X(1803)
X(1855) = crosspoint of X(92) and X(281)
X(1855) = crosssum of X(48) and X(222)

### X(1856) = ZOSMA TRANSFORM OF X(222)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(222)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(222)

X(1856) lies on these lines:
4,57   11,1427   33,42   55,1855   225,235   430,1827

### X(1857) = ZOSMA TRANSFORM OF X(223)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(223)
Trilinears        [cos(A/2) sec A]2 : [cos(B/2) sec B]2 : [cos(C/2) sec C]2 ( M. Iliev, 4/12/07)
Trilinears        (1 + cos A)/(1 + cos 2A) : (1 + cos B)/(1 + cos 2B) : (1 + cos C)/(1 + cos 2C)     (M. Iliev, 4/12/07)

Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(223)

X(1857) lies on these lines:
2,243   4,65   8,1896   11,278   33,42   55,281   92,497   189,1364   388,1895   403,1068   412,1788   1478,1784   1748,1776

X(1857) = isogonal conjugate of X(1804)
X(1857) = X(158)-Ceva conjugate of X(393)
X(1857) = pole wrt polar circle of trilinear polar of X(348) (line X(4025)X(4131))
X(1857) = polar conjugate of X(348)

### X(1858) = ZOSMA TRANSFORM OF X(225)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(225)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(225)

X(1858) lies on these lines:
1,90   3,920   4,65   6,1854   11,113   21,60   52,517   55,72   73,774   144,145   411,1155   758,950   1829,1852   1831,1839

X(1858) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,431), (648,650)
X(1858) = crosspoint of X(4) and X(21)
X(1858) = crosssum of X(i) and X(j) for these (i,j): (1,1935), (3,65), (478,1402)

### X(1859) = ZOSMA TRANSFORM OF X(226)

Trilinears    (y + z) sec A : (z + x) sec B : (x + y) sec C, where x : y : z = X(226)
Trilinears    tan A [(b + c) cos A + b cos B + c cos C] : :
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C, where x : y : z = X(226)

X(1859) lies on these lines:
1,1871   4,65   6,1096   11,132   19,25   27,243   29,960   40,1872   92,518   209,1826   210,281   278,354   942,1838   950,1852   1013,1748   1825,1902   1829,1831   1869,1894

X(1859) = X(i)-Ceva conjugate of X(j) for these (i,j): (4,1865), (107,650), 1838,1841)
X(1859) = crosspoint of X(i) and X(j) for these (i,j): (4,1172), (281,1896)
X(1859) = crosssum of X(3) and X(1214)

### X(1860) = ZOSMA TRANSFORM OF X(228)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(228)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(228)

X(1860) lies on these lines:
4,42   6,1836   25,225   27,58   92,984   278,1458

### X(1861) = ZOSMA TRANSFORM OF X(238)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(238)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(238)

X(1861) lies on these lines:
1,475   2,33   4,9   5,1872   8,34   12,1887   25,1376   75,225   208,1788   232,1575   235,1329   240,522   378,993   406,1698   427,1824   429,1900   468,1862   518,1876   519,1870   594,1841   765,1877   908,1830   958,1593   960,1902   1528,1532   1595,1871   1829,1883

X(1861) = complement of X(3100)
X(1861) = inverse-in-circumconic-centered-at-X(9) of X(19)
X(1861) = pole wrt polar circle of trilinear polar of X(673) (line X(1)X(514))
X(1861) = X(48)-isoconjugate (polar conjugate) of X(673)
X(1861) = perspector of circumconic through the polar conjugates of PU(47) and PU(51)
X(1861) = X(8076)-of-orthic-triangle if ABC is acute

### X(1862) = ZOSMA TRANSFORM OF X(244)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(244)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(244)

X(1862) lies on these lines:
4,145   11,33   25,100   34,1317   80,1039   104,1593   119,235   428,528   468,1861   519,1878   900,1830   1387,1883   1484,1595

### X(1863) = ZOSMA TRANSFORM OF X(269)

Trilinears    (y + z) sec A : : , where x : y : z = X(269)
Barycentrics    (a^2 + b^2 + c^2 - 2 a b - 2 a c)/(b^2 + c^2 - a^2) : :

X(1863) lies on these lines: 4,7   25,1604   33,42   242,390   497,1851

X(1863) = X(4)-Ceva conjugate of X(1851)
X(1863) = polar conjugate of X(30705)

### X(1864) = ZOSMA TRANSFORM OF X(278)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(278)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(278)

X(1864) lies on these lines:
3,1728   4,65   6,33   9,55   11,118   44,212   51,1824   56,1490   57,971   72,519   84,1466   227,774   329,497   381,942   389,1872   405,997   430,1827   452,960   1005,1776   1155,1708   1210,1532   1214,1736

X(1864) = X(i)-Ceva conjugate of X(j) for these (i,j): (1210,1108), (1897,650)
X(1864) = crosspoint of X(i) and X(j) for these (i,j): (4,9), (8,282)
X(1864) = crosssum of X(i) and X(j) for these (i,j): (3,57), (56,223)

### X(1865) = ZOSMA TRANSFORM OF X(284)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(284)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(284)

X(1865) lies on these lines:
4,6   12,37   19,407   33,430   92,1211   209,1824   281,860   286,297   442,1838   1474,1884

X(1865) = X(4)-Ceva conjugate of X(1859)
X(1865) = crosssum of X(577) and X(1437)

### X(1866) = ZOSMA TRANSFORM OF X(355)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(355)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(355)

X(1866) lies on these lines:
4,80   19,1405   28,34   51,65   52,1905   225,1829   1878,1887

### X(1867) = ZOSMA TRANSFORM OF X(386)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(386)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(386)

X(1867) lies on these lines: 4,8   10,407   12,37   427,1838   1118,1892   1785,1904   1884,1891

### X(1868) = ZOSMA TRANSFORM OF X(387)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(387)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(387)

X(1868) lies on these lines:
4,8   29,894   34,37   65,1826   210,1869   226,429   228,1593

### X(1869) = ZOSMA TRANSFORM OF X(405)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(405)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(405)

X(1869) lies on these lines:
4,9   27,306   28,35   34,42   65,225   209,1829   210,1868   1710,1770   1859,1894

### X(1870) = ZOSMA TRANSFORM OF X(484)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(484)

Trilinears       2 - sec A : 2 - sec B : 2 - sec C

Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(484)

X(1870) lies on these lines:
1,4   2,1060   3,1398   7,1061   8,475   11,403   12,1594   19,1449   20,1062   24,56   25,999   28,60   36,186   54,65   55,378   59,517   77,1119   104,1455   108,953   109,1735   208,1420   221,1181   232,1015   235,496   242,514   273,1442   354,1905   376,1040   389,1425   427,495   451,1125   459,614   519,1861   631,1038   651,912   982,1395   1000,1041   1006,1214   1100,1172   1318,1878   1385,1426   1718,1737   1725,1776   1831,1844

X(1870) = isogonal conjugate of X(1807)
X(1870) = cevapoint of X(i) and X(j) for these (i,j): (1,1718), (1464,1835)
X(1870) = homothetic center of circumorthic triangle and anti-tangential midarc triangle

### X(1871) = ZOSMA TRANSFORM OF X(498)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(498)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(498)

X(1871) lies on these lines:
1,1859   3,19   4,8   5,1848   28,1385   29,392   33,1598   52,916   65,1243   225,1831   273,1148   278,942   354,1844   580,1731   581,1841   952,1891   1595,1861   1597,1753

### X(1872) = ZOSMA TRANSFORM OF (499)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(499)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(499)

X(1872) lies on these lines:
1,1887   3,33   4,8   5,1861   19,1598   34,1482   40,1859   65,1785   225,1830   389,1864   1068,1876   1595,1848   1715,1736   1825,1905

### X(1873) = ZOSMA TRANSFORM OF X(500)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(500)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(500)

X(1873) lies on these lines: 4,79   12,37   403,1838

### X(1874) = ZOSMA TRANSFORM OF X(511)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(511)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(511)

X(1874) lies on these lines: 4,240   12,37   29,34   238,242   513,1835   862,1284

X(1874) = isogonal conjugate of X(1808)
X(1874) = X(242)-Ceva conjugate of X(1284)

### X(1875) = ZOSMA TRANSFORM OF X(515)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(515)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(515)

X(1875) lies on these lines:
1,945   4,65   25,34   108,953   225,1829   278,957   513,1835   517,1361   859,1465   1119,1122   1400,1841   1452,1454   1825,1900

X(1875) = isogonal conjugate of X(1809)
X(1875) = midpoint of X(1785) and X(1845)

### X(1876) = ZOSMA TRANSFORM OF X(516)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(516)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(516)

X(1876) lies on these lines:
1,1037   4,7   6,19   25,57   28,1170   33,354   59,517   72,475   108,840   196,1851   225,1887   226,427   235,1210   242,653   278,1002   428,553   513,1835   518,1861   614,3195   851,1465   950,1885   1011,1214   1020,1736   1068,1872   1458,2356   1471,2212

X(1876) = midpoint of X(65) and X(1456)

### X(1877) = ZOSMA TRANSFORM OF X(517)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(517)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(517)

X(1877) lies on these lines:
1,4   11,1455   12,1883   25,1470   30,1465   51,65   109,1737   221,1837   513,1835   517,1830   603,1210   608,1826   751,1890   765,1861   1319,1846   1707,1788   1825,1829   1839,1880

X(1877) = X(4)-Ceva conjugate of X(1846)
X(1877) = pole wrt polar circle of trilinear polar of X(4997) (line X(8)X(522))
X(1877) = polar conjugate of X(4997)

### X(1878) = ZOSMA TRANSFORM OF X(519)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(519)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(519)

X(1878) lies on these lines: 4,8   25,36   34,1319   428,535   513,1835   519,1862   855,1465   1318,1870   1842,1888   1866,1887

X(1878) = isogonal conjugate of X(1811)
X(1878) = inverse-in-polar-circle of X(8)

### X(1879) = ZOSMA TRANSFORM OF X(563)

Trilinears    (y + z) sec A : (z + x) sec B : (x + y) sec C, where x : y : z = X(563)
Trilinears    sin A cos(2B - 2C) : :
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C, where x : y : z = X(563)

X(1879) lies on these lines:
4,96   5,570   6,13   53,235   230,428   233,566   1598,1609

X(1879) = crosssum of X(6) and X(156)

X(1879) = X(48)-of-orthic-triangle if ABC is acute

### X(1880) = ZOSMA TRANSFORM OF X(573)

Trilinears    (y + z) sec A : : , where x : y : z = X(573)

Trilinears    tan A tan(A/2)(b + c) : :

X(1880) lies on these lines:
2,92   4,941   6,19   12,37   25,1096   28,961   42,1824   57,967   108,111   331,1218   1171,1396   1254,1400   1411,1474   1722,1723   1735,1765   1839,1877

X(1880) = isogonal conjugate of X(1812)
X(1880) = X(i)-Ceva conjugate of X(j) for these (i,j): (19,1400), (225,1824), (278,225)
X(1880) = crosspoint of X(i) and X(j) for these (i,j): (19,393), (34,278)
X(1880) = crosssum of X(i) and X(j) for these (i,j): (63,394), (78,219)
X(1880) = polar conjugate of X(314)

### X(1881) = ZOSMA TRANSFORM OF X(577)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(577)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(577)

X(1881) lies on these lines: 4,48   12,37   71,860

### X(1882) = ZOSMA TRANSFORM OF X(581)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(581)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(581)

X(1882) lies on these lines:
4,65   5,1214   12,37   92,960

### X(1883) = ZOSMA TRANSFORM OF X(595)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(595)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(595)

As a point on the Euler line, X(1883) has Shinagawa coefficients (\$a\$F, \$a\$(E + F) - 2abc).

X(1883) lies on these lines: 2,3   10,1828   12,1877   1387,1862   1829,1861   1848,1900

### X(1884) = ZOSMA TRANSFORM OF X(758)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z =X(758)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(758)

As a point on the Euler line, X(1884) has Shinagawa coefficients (\$a\$F, -(E + F)3 + \$aSA\$ + 3ES2).

X(1884) lies on these lines: 2,3   34,1464   513,1835   1474,1865   1828,1844   1829,1831   1867,1891

### X(1885) = ZOSMA TRANSFORM OF X(774)

Trilinears    (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(774)
Barycentrics    (2 a^6 - 3 a^4 (b^2 + c^2) + 8 a^2 b^2 c^2 + (b^2 - c^2)^2 (b^2 + c^2)) / (b^2 + c^2 - a^2) : :

As a point on the Euler line, X(1885) has Shinagawa coefficients (F, E - 3F).

Let A'B'C' be the orthic triangle. X(1885) is the radical center of the 2nd Droz-Farny circles of triangles AB'C', BC'A', CA'B'. (Randy Hutson, July 31 2018)

X(1885) lies on these lines: 2,3   64,1899   389,974   497,1398   515,1902   516,1829   950,1876   1039,1721   1770,1905

X(1885) = crosspoint of X(4) and X(1105)
X(1885) = crosssum of X(3) and X(185)
X(1885) = intersection of tangents to Hatzipolakis-Lozada hyperbola at X(4) and X(185)
X(1885) = crosspoint, wrt orthic triangle, of X(4) and X(185)
X(1885) = X(3057)-of-orthic-triangle if ABC is acute

### X(1886) = ZOSMA TRANSFORM OF X(910)

Trilinears    (tan A)(a^2 - b^2 cos C - c^2 cos B) : :
Trilinears    (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(910)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(910)

X(1886) lies on these lines:
1,1855   6,1836   19,57   33,42   225,607   230,231   910,1360   1783,1785

X(1886) = isogonal conjugate of X(1815)
X(1886) = X(917)-Ceva conjugate of X(25)
X(1886) = crosssum of X(219) and X(1818)

### X(1887) = ZOSMA TRANSFORM OF X(946)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(946)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(946)

X(1887) lies on these lines:
1,1872   4,65   12,1861   33,56   34,1824   55,1753   208,1827   225,1876   318,518   942,1785   1825,1829   1830,1902   1844,1846   1866,1878

### X(1888) = ZOSMA TRANSFORM OF X(950)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(950)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(950)

X(1888) lies on these lines:
4,65   19,44   28,1155   33,1426   34,55   71,1841   209,1829   225,1902   516,1852   517,1838   1830,1900   1842,1878

### X(1889) = ZOSMA TRANSFORM OF X(968)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(968)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(968)

As a point on the Euler line, X(1889) has Shinagawa coefficients (F, -E - F - 2\$bc\$).

X(1889) lies on these lines: 2,3   6,1836   57,1893   518,1824   1709,1730

X(1889) = inverse-in-orthocentroidal-circle of X(430)

### X(1890) = ZOSMA TRANSFORM OF X(984)

Trilinears    (y + z) sec A : (z + x) sec B : (x + y) sec C, where x : y : z = X(984)

X(1890) lies on these lines:
4,9   7,34   25,1001   27,162   28,142   33,390   82,225   428,528   518,1829   751,1877   1445,1452   1724,1738   1737,1747

X(1890) = X(29)-beth conjugate of X(1826)

### X(1891) = ZOSMA TRANSFORM OF X(986)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(986)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(986)

X(1891) lies on these lines:
1,4   8,19   10,28   25,958   27,306   29,1220   65,1503   92,1842   428,529   518,1829   952,1871   1867,1884

### X(1892) = ZOSMA TRANSFORM OF X(990)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(990)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(990)

X(1892) lies on these lines:
4,7   12,1452   25,226   33,1836   57,427   65,66   79,1041   208,429   225,608   388,1829   1118,1867   1478,1905

X(1892) = X(4)-beth conjugate of X(608)

### X(1893) = ZOSMA TRANSFORM OF X(991)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(991)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(991)

X(1893) lies on these lines:
4,7   11,1427   12,37   57,1889   226,430   235,1838

X(1893) = X(4)-beth conjugate of X(1880)

### X(1894) = ZOSMA TRANSFORM OF X(993)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(993)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(993)

As a point on the Euler line, X(1894) has Shinagawa coefficients (\$a\$F, -2\$aSA\$ - \$a\$(E + F)).

X(1894) lies on these lines: 2,3   19,53   225,1829   1785,1824   1828,1838   1859,1869

### X(1895) = ZOSMA TRANSFORM OF X(1044)

Trilinears    (y + z) sec A : : , where x : y : z = X(1044)
Trilinears    (cos A - cos B cos C) sec A : :
Trilinears    (sec A - sec B sec C) sec A : :

X(1895) lies on these lines:
1,29   2,280   4,7   8,1054   40,653   48,821   56,243   57,412   63,1712   78,1897   108,411   162,255   196,962   204,1097   240,774   304,811   388,1857   497,1118   517,1148   1210,1785   1445,1753

X(1895) = X(75)-Ceva conjugate of X(92)
X(1895) = cevapoint of X(i) and X(j) for these (i,j): (1,1712), (204,610)
X(1895) = polar conjugate of X(2184)

### X(1896) = ZOSMA TRANSFORM OF X(1047)

Trilinears        (sec A)/(sec B + sec C) : (sec B)/(sec C + sec A) : (sec C)/(sec A + sec B)
= (sec2A)(b + c - a)/(b + c) : (sec2B)(c + a - b)/(c + a) : (sec2C)(a + b - c)/(a + b)
Barycentrics  (tan A)/(sec B + sec C) : (tan B)/(sec C + sec A) : (tan C)/(sec A + sec B)

X(1896) lies on these lines:
1,29   4,51   7,286   8,1857   9,318   21,243   27,84   28,104   393,941   412,1715   823,1156

X(1896) = cevapoint of X(i) and X(j) for these (i,j): (1,1715), (4,158)
X(1896) = X(4)-cross conjugate of X(29)
X(1896) = isogonal conjugate of X(22341)
X(1896) = pole wrt polar circle of trilinear polar of X(1214) (line X(520)X(656))
X(1896) = polar conjugate of X(1214)

### X(1897) = ZOSMA TRANSFORM OF X(1054)

Trilinears        (sec A)/(b - c) : (sec B)/(c - a) : (sec C)/(a - b)
Barycentrics   (tan A)/(b - c) : (tan B)/(c - a) : (tan C)/(a - b)

X(1897) lies on these lines:
1,318   4,145   27,295   29,1807   33,92   34,1120   78,1895   100,108   101,107   109,522   112,835   162,190   192,1013   243,518   278,1280   346,1249   519,1785   644,1783   726,1430

X(1897) = isogonal conjugate of X(1459)
X(1897) = anticomplement of X(2968)
X(1897) = X(648)-Ceva conjugate of X(1783)
X(1897) = cevapoint of X(i) and X(j) for these (i,j): (1,522), (523,1834), (650,1864)
X(1897) = X(101)-cross conjugate of X(190)
X(1897) = crosspoint of X(648) and X(811)
X(1897) = crosssum of X(647) and X(810)
X(1897) = isotomic conjugate of X(4025)
X(1897) = trilinear pole of line X(4)X(9) (complement of Soddy line, and Brocard axis of excentral triangle)
X(1897) = pole wrt polar circle of trilinear polar of X(514) (line X(11)X(244), the Feuerbach tangent line)
X(1897) = polar conjugate of X(514)

### X(1898) = ZOSMA TRANSFORM OF X(1068)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(1068)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(1068)

X(1898) lies on these lines:
3,90   4,65   21,662   56,971   84,1470   354,496   411,1776   912,1479   920,1155

X(1898) = reflection of X(65) in X(1837)
X(1898) = crosspoint of X(4) and X(90)
X(1898) = crosssum of X(3) and X(46)

### X(1899) = ZOSMA TRANSFORM OF X(1096)

Trilinears    (y + z) sec A : : , where x : y : z = X(1096)
Trilinears    2 b c - a b sec B - a c sec C : :
Barycentrics    (-a^2+b^2+c^2) (a^4+b^4-2 b^2 c^2+c^4) : :

X(1899) lies on these lines:
{2,98}, {3,68}, {4,51}, {5,1181}, {6,66}, {20,1204}, {22,3580}, {25,1503}, {54,70}, {64,1885}, {65,5130}, {67,5486}, {69,305}, {154,468}, {217,2548}, {235,1498}, {265,974}, {315,3978}, {388,1425}, {394,1368}, {407,5786}, {429,5706}, {442,5810}, {462,5869}, {463,5868}, {497,3270}, {511,1370}, {578,3541}, {686,804}, {858,1993}, {860,5767}, {940,5820}, {1092,3546}, {1147,3548}, {1321,3070}, {1322,3071}, {1495,6353}, {1587,3127}, {1588,3128}, {1591,6289}, {1592,6290}, {1593,6247}, {1824,5928}, {1864,5101}, {1974,5596}, {2072,5654}, {2450,3767}, {2549,3269}, {2550,3611}, {2888,3523}, {2892,5095}, {2992,2993}, {3134,5877}, {3136,5816}, {3142,5713}, {3332,4207}, {3549,5449}, {3818,5943}, {3851,5644}, {3926,4121}, {5064,5480}, {5133,5422}, {5200,5870}

X(1899) = reflection of X(394) in X(1368)
X(1899) = crosspoint of X(4) and X(69)
X(1899) = crosssum of X(i) and X(j) for these (i,j): (3,25), (52,418), (206,571)
X(1899) = anticomplement of X(9306)
X(1899) = crossdifference of every pair of points on line X(3569)X(6753)
X(1899) = X(1370)-of-1st-Brocard-triangle
X(1899) = X(200)-of-orthic-triangle if ABC is acute

### X(1900) = ZOSMA TRANSFORM OF X(1125)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(1125)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(1125)

X(1900) lies on these lines:
4,8   10,1904   19,45   25,35   225,1876   407,1785   429,1861   1825,1875   1830,1888   1839,1840   1848,1883   1859,1869

### X(1901) = ZOSMA TRANSFORM OF X(1172)

Trilinears    (y + z) sec A : : , where x : y : z =X(1172)
Barycentrics    (b + c) (2 a^4 + a^3 (b + c) - a^2 (b - c)^2 - a (b - c)^2 (b + c) - (b^2 - c^2)^2) : :

X(1901) lies on these lines:
4,6   5,579   7,857   9,46   12,71   19,1836   30,284   37,226   65,1826   72,594   115,117   198,851   208,429   219,1478   329,1211   377,965   430,1827   583,1713   946,1108   950,1100   1146,1845

X(1901) = X(653)-Ceva conjugate of X(523)
X(1901) = crosspoint of X(4) and X(226)
X(1901) = crosssum of X(3) and X(284)
X(1901) = complement of X(8822)

### X(1902) = ZOSMA TRANSFORM OF X(1210)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(1210)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(1210)

X(1902) lies on these lines:
1,1037   4,8   10,235   19,220   25,40   33,64   125,429   225,1888   378,1385   392,475   427,946   515,1885   960,1861   1482,1597   1825,1859   1830,1887

X(1902) = reflection of X(1829) in X(4)

### X(1903) = ZOSMA TRANSFORM OF X(1249)

Trilinears    (y + z) sec A : : , where x : y : z = X(1249)
Barycentrics    a (b + c)/(a^3 + a^2 (b + c) - a (b + c)^2 - (b - c)^2 (b + c)) : :

Let Ab, Ac be the points where the A-excircle touches lines CA and AB resp., and define Bc, Ba, Ca, Cb cyclically. Let Ta be the intersection of the tangents to the Yiu conic (defined at X(478)) at Bc and Ca, and define Tb, Tc cyclically. Let Ta' be the intersection of the tangents to the Yiu conic at Ba and Cb, and define Tb', Tc' cyclically. Let Va = TbTb'∩TcTc', Vb = TcTc'∩TaTa', Vc = TaTa'∩TbTb'. The lines AVa, BVb, CVc concur in X(1903). (See also X(65).) (Randy Hutson, July 20, 2016)

X(1903) lies on these lines:
3,9   6,33   19,64   37,73   65,1826   69,189   71,210   226,1439   285,1798   478,1413   1419,1422

X(1903) = isogonal conjugate of X(1817)
X(1903) = crosspoint of X(84) and X(189)
X(1903) = crosssum of X(i) and X(j) for these (i,j): (3,610), (40,198)

### X(1904) = ZOSMA TRANSFORM OF X(1468)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(1468)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(1468)

As a point on the Euler line, X(1904) has Shinagawa coefficients (FS2, -(E - F)S2 - 2\$bc\$S2 + 2\$bcSBSC\$).

X(1904) lies on these lines:
2,3   10,1900   12,968   53,1826   80,1039   1785,1867   1828,1848

### X(1905) = ZOSMA TRANSFORM OF X(1478)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(1478)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(1478)

X(1905) lies on these lines:
1,25   3,1452   4,65   33,517   34,222   46,1593   52,1866   169,607   225,1831   354,1870   378,1155   406,960   427,1737   1426,1838   1478,1892   1730,1735   1770,1885   1785,1824   1825,1872   1827,1845   1828,1844

X(1905) = reflection of X(222) in X(942)
X(1905) = crosspoint of X(4) and X(1061)
X(1905) = crosssum of X(3) and X(1060)

### X(1906) = ZOSMA TRANSFORM OF X(1496)

Trilinears       (y + z) sec A : (z + x) sec B : (x + y) sec C,  x : y : z = X(1496)
Barycentrics  (y + z) tan A : (z + x) tan B : (x + y) tan C,  x : y : z = X(1496)

As a point on the Euler line, X(1906) has Shinagawa coefficients (F, -2E + F).

X(1906) lies on this line: 2,3

### X(1907) = ZOSMA TRANSFORM OF X(1497)

Trilinears    (y + z) sec A : :, where x : y : z = X(1497)
Trilinears    (sec A) (2 + sin^2 B + sin^2 C) : :
Barycentrics    (tan A) (2 + sin^2 B + sin^2 C) : :

As a point on the Euler line, X(1907) has Shinagawa coefficients (F, 2E + F).

X(1907) lies on this line: 2,3

Centers from Bicentric Pairs, 1908- 1982
For a definition of a bicentric pair (e.g., the 1st and 2nd Brocard points) click Tables at the top of this page. Suppose P and U are a bicentric pair. Many operations on P and U result in triangle centers. Among these are trilinear and barycentric product, bicentric sum, bicentric difference, crosssum, and crossdifference. For definitions of these, click Tables. At the time this section is added to ETC (September 15, 2003), bicentric pairs

P(1),U(1); P(2),U(2); ...; P(42),U(42)

are defined in Tables. In this present section, the abbreviation PU(n) means the bicentric pair P(n),U(n).

### X(1908) = MIDPOINT OF PU(8)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b3 + c3) + 2abc(a2 + bc)     (Wimalasiri Perera, August 29, 2011)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1908) lies on these lines: 39,1155   42,649   43,2235   171,292   551,2666   2243,2276

### X(1909) = CROSSSUM OF PU(9)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1 + bc/a2
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1909) lies on these lines:
1,76   2,330   7,8   10,274   12,325   34,264   35,99   36,1078   37,1655   42,310   56,183   73,290   86,313   172,385   190,1334   226,1432   256,1221   257,335   286,1891   305,612   315,1478   538,1500   732,894   1215,1237   1235,1870

X(1909) = isogonal conjugate of X(904)
X(1909) = isotomic conjugate of X(256)
X(1909) = anticomplement of X(1107)
X(1909) = X(i)-Ceva conjugate of X(j) for these (i,j): (335,350), (1221,2)
X(1909) = cevapoint of X(8) and X(1655)
X(1909) = crosspoint of PU(10)
X(1909) = intersection of tangents at PU(10) to hyperbola {{A,B,C,PU(10)}}

### X(1910) = TRILINEAR POLE OF PU(23)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(a2cos B cos C - bc cos2A)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1910) lies on these lines:
1,163   10,98   19,560   37,692   48,75   65,172   1580,1581

X(1910) = isogonal conjugate of X(1959)
X(1910) = X(1821)-Ceva conjugate of X(293)
X(1910) = cevapoint of X(i) and X(j) for these (i,j): (1,1580), (240,1957)
X(1910) = barycentric product of PU(88)

### X(1911) = TRILINEAR POLE OF PU(9)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2/(a2 - bc)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1911) lies on these lines:
1,335   6,292   42,81   86,334   172,694   238,660   692,1333   739,813   875,890   1403,1407   1429,1458

X(1911) = isogonal conjugate of X(350)
X(1911) = isotomic conjugate of X(18891)
X(1911) = X(741)-Ceva conjugate of X(292)
X(1911) = cevapoint of X(172) and X(1914)
X(1911) = crosspoint of X(727) and X(1438)
X(1911) = trilinear pole of line X(213)X(667)

### X(1912) = IDEAL POINT OF PU(9)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b - c - a(b/c + c/b)(b/c - c/b)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

As the isogonal conjugate of a point on the circumcircle, X(1912) lies on the line at infinity.

X(1912) lies on these (parallel) lines:
30,511   213,667   1166,1203

X(1912) = crossdifference of every pair of points on line X(6)X(350)

### X(1913) = MIDPOINT OF PU(9)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2[2a3bc + ab4 + ac4 + b2c2(b + c)]
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1913) lies on this line: 213,667

### X(1914) = CROSSDIFFERENCE OF PU(10)

Trilinears    1 - a2/(bc) : :
Trilinears    a^3 - abc : :

X(1914) is the perspector of triangle ABC and the tangential triangle of the conic that passes through A, B, C, and the pair P(9) and U(9) of bicentric points (see the notes just before X(1908). (Randy Hutson, 9/23/2011)

X(1914) lies on these lines:
1,32   6,31   9,983   11,230   21,1107   35,39   36,187   37,82   44,765   48,1613   81,593   100,1575   105,910   112,1870   213,595   284,893   292,1438   350,385   577,1040   584,1185   604,1403   649,834   727,813   739,901   741,1326   999,1384   1055,1149   1319,1415   1428,1691

X(1914) = isogonal conjugate of X(335)
X(1914) = isotomic conjugate of X(18895)
X(1914) = X(i)-Ceva conjugate of X(j) for these (i,j): (727,31), (1429,1428), (1438,6), (1911,172)
X(1914) = crosspoint of X(i) and X(j) for these (i,j): (81,105), (238,1429), (239,242), (904,1911), (919,1252)
X(1914) = crosssum of X(i) and X(j) for these (i,j): (37,518), (292,295), (350,1909), (918,1086)
X(1914) = {X(1),X(32)}-harmonic conjugate of X(172)
X(1914) = intersection of trilinear polars of P(9) and U(9)
X(1914) = X(92)-isoconjugate of X(295)
X(1914) = barycentric product of PU(134)
X(1914) = homothetic center of intangents triangle and mid-triangle of 1st and 2nd Kenmotu diagonals triangles

### X(1915) = CROSSSUM OF PU(11)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(a4 + b2c2)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1915) lies on these lines:
2,1501   6,25   31,292   32,1613   110,251   141,1799

### X(1916) = TRILINEAR POLE OF PU(11)

Trilinears    bc/(a4 - b2c2) : :
Trilinears    csc(A - 2ω) : :

Let A'B'C' be the 1st Brocard triangle. Let A" be the reflection of A' in BC, and define B" and C" cyclically; then X(1916) is the radical center of the circumcircles of A"BC, B"CA, C"AB. Let A* be the reflection of A in B'C', and define B* and C* cyclically; then X(1916) is the radical center of the circumcircles of A*BC, B*CA, C*AB. The first set of circles equals the second set. (Randy Hutson, February 10, 2016)

X(1916) lies on these lines:
2,694   4,147   10,257   39,83   76,115   98,385   114,262   226,335   256,291   316,736   325,698   538,671   543,598   690,882   804,881

X(1916) = midpoint of X(148) and X(194)
X(1916) = reflection of X(i) in X(j) for these (i,j): (76,115), (99,39)
X(1916) = isogonal conjugate of X(1691)
X(1916) = isotomic conjugate of X(385)
X(1916) = cevapoint of X(39) and X(511)

X(1916) = complement of X(8782)
X(1916) = trilinear pole of line X(141)X(523)
X(1916) = pole wrt polar circle of trilinear polar of X(419)
X(1916) = X(48)-isoconjugate (polar conjugate) of X(419)
X(1916) = antigonal image of X(76)
X(1916) = X(76) of 1st anti-Brocard triangle
X(1916) = intersection, other than A, B, C, of the 1st and 2nd isobarycs of the circumcircle
X(1916) = perspector of ABC and 1st anti-Brocard triangle

### X(1917) = TRILINEAR 6th POWER POINT

Trilinears    a6 : :

Let A'B'C' and A"B"C" be the 5th Brocard and 5th anti-Brocard triangles, resp. Let A* be the trilinear product A'*A", and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X(1917). (Randy Hutson, November 30, 2018)

X(1917) lies on these lines: 1,3409 31,2085 560,9247 922,4020 1501,9448

X(1917) = barycentric product of PU(12) X(1917) = isogonal conjugate of X(1928)
X(1917) = trilinear product of PU(13)
X(1917) = trilinear square of X(32)
X(1917) = trilinear cube of X(31)

### X(1918) = BICENTRIC SUM OF PU(12)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(b + c)

X(1918) lies on these lines:
6,31   9,981   10,82   32,560   86,171   100,715   101,729   213,872   313,983   393,465   692,1333

X(1918) = isogonal conjugate of X(310)
X(1918) = X(i)-Ceva conjugate of X(j) for these (i,j): (31,213), (692,1919), (983,37)
X(1918) = crosspoint of X(i) and X(j) for these (i,j): (31,32), (213,1402)
X(1918) = crosssum of X(i) and X(j) for these (i,j): (75,76), (274,314)
X(1918) = PU(12)-harmonic conjugate of X(1919)

### X(1919) = BICENTRIC DIFFERENCE OF PU(12)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(b - c)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1919) lies on these lines: 101,765   649,834   667,788   669,688

X(1919) = isogonal conjugate of X(1978)
X(1919) = X(i)-Ceva conjugate of X(j) for these (i,j): (32,1977), (692,1333)
X(1919) = cevapoint of X(669) and X(1924)
X(1919) = crosspoint of X(i) and X(j) for these (i,j): (31,101), (81,932), (692,1333)
X(1919) = crosssum of X(i) and X(j) for these (i,j): (75,514), (321,693), (646,668), (850,1230)
X(1919) = PU(12)-harmonic conjugate of X(1918)
X(1919) = barycentric product of PU(25)
X(1919) = trilinear product of PU(42)

### X(1920) = CROSSSUM OF PU(12)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b3c3(a2 + bc)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1920) lies on these lines:
2,561   37,1221   75,982   76,85   210,668   310,321   334,1581   1002,1611   1215,1237   1240,1441

X(1920) = isotomic conjugate of X(893)
X(1920) = perspector of Gemini triangle 31 and cross-triangle of ABC and Gemini triangle 31
X(1920) = trilinear pole of perspectrix of ABC and Gemini triangle 32
X(1920) = X(i)-Ceva conjugaute of X(j) for these (i,j): (334,1921), (1240,76)

### X(1921) = CROSSDIFFERENCE OF PU(12)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b3c3(a2 - bc)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1921) lies on these lines:
2,561   10,75   37,308   274,1107   350,740   518,668

X(1921) = isogonal conjugate of X(1922)
X(1921) = isotomic conjugate of X(292)
X(1921) = X(334)-Ceva conjugate of X(1920)
X(1921) = perspector of Gemini triangle 32 and cross-triangle of ABC and Gemini triangle 32
X(1921) = trilinear pole of perspectrix of ABC and Gemini triangle 31

### X(1922) = TRILINEAR POLE OF PU(12)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3/(a2 - bc)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1922) lies on these lines: 6,291   58,101   81,335   727,813   1416,1428

X(1922) = isogonal conjugate of X(1921)
X(1922) = perspector of ABC and unary cofactor triangle of Gemini triangle 31

### X(1923) = BICENTRIC SUM OF PU(13)

Trilinears    a4(b2 + c2) : :
Trilinears    1 - sec^2 ω cos^2 A : :

X(1923) lies on these lines: 1,21   110,719

X(1923) = X(i)-Ceva conjugate of X(j) for these (i,j): (31,1964), (163,1924)
X(1923) = crosspoint of X(31) and X(560)
X(1923) = crosssum of X(75) and X(561)
X(1923) = trilinear product of X(32) and X(39)
X(1923) = PU(13)-harmonic conjugate of X(1924)

### X(1924) = BICENTRIC DIFFERENCE OF PU(13)

Trilinears       a4(b2 - c2) : b4(c2 - a2) : c4(a2 - b2)
Barycentrics  a5(b2 - c2) : b5(c2 - a2) : c5(a2 - b2)

X(1924) lies on these lines: 661,830   667,788   681,1612

X(1924) = X(i)-Ceva conjugate of X(j) for these (i,j): (163,1923), (662,31), (1919,669)
X(1924) = crosspoint of X(i) and X(j) for these (i,j): (31,662), (1919, 1980)
X(1924) = crosssum of X(i) and X(j) for these (i,j): (75,661), (1577,1930)
X(1924) = isogonal conjugate of X(4602)
X(1924) = PU(13)-harmonic conjugate of X(1923)
X(1924) = trilinear product of PU(91)

### X(1925) = CROSSSUM OF PU(13)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1 + b2c2/a4
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1925) lies on these lines: 76,335   92,304   469,1601

X(1925) = X(1934)-Ceva conjugate of X(1926)
X(1925) = crosspoint of PU(14)
X(1925) = intersection of tangents at PU(14) to conic {{A,B,C,PU(14)}}

### X(1926) = CROSSDIFFERENCE OF PU(13)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1 - b2c2/a4
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1926) lies on these lines: 38,75   76,257   661,786   799,1755   1590,1636

X(1926) = isogonal conjugate of X(1927)
X(1926) = isotomic conjugate of X(1967)
X(1926) = X(1934)-Ceva conjugate of X(1925)
X(1926) = crosssum of X(1932) and X(1933)
X(1926) = perspector of conic {A,B,C,PU(14)}
X(1926) = intersection of trilinear polars of P(14) and U(14)

### X(1927) = TRILINEAR POLE OF PU(13)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4/(a4 - b2c2)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1927) lies on these lines: 82,662   172,694   715,805   733,787

X(1927) = isogonal conjugate of X(1926)
X(1927) = cevapoint of X(1932) and X(1933)

### X(1928) = ISOGONAL CONJUGATE OF X(1917)

Trilinears       1/a6 : 1/b6 : 1/c6
Barycentrics  1/a5 : 1/b5 : 1/c5

X(1928) is the Brianchon point (perspector) of the inellipse that is the trilinear square of the de Longchamps line. This inellipse has center X(21235). (Randy Hutson, October 15, 2018)

X(1928) = isogonal conjugate of X(1917)
X(1928) = isotomic conjugate of X(560)
X(1928) = trilinear product of PU(14)
X(1928) = trilinear product of vertices of Gemini triangle 31
X(1928) = trilinear product of vertices of Gemini triangle 32

### X(1929) = ISOGONAL CONJUGATE OF X(1757)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/[-(cos A)/x + (cos B)/y + (cos C)/z], where x : y : z = X(295)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1929) lies on ths line: 1,3125   2,846   105,2702   1758,2006   2640,3122

X(1929) = isogonal conjugate of X(1757)
X(1929) = cevapoint of X(i) and X(j) for these (i,j): (244,659), (1966,1909)
X(1929) = X(238)-cross conjugate of X(1)
X(1929) = trilinear pole of PU(31) (line X(513)X(1100))

### X(1930) = BICENTRIC SUM OF PU(14)

Trilinears    b2c2(b2 + c2) : :
Trilinears    |AP(1)|^2 + |AU(1)|^2 : :

X(1930) lies on these lines: 1,75   8,150   76,334   213,742

X(1930) = isotomic conjugate of X(82)
X(1930) = X(75)-Ceva conjugate of X(38)
X(1930) = X(1194)-cross conjugate of X(251)
X(1930) = crosspoint of X(75) and X(561)
X(1930) = crosssum of X(31) and X(560)

### X(1931) = CROSSDIFFERENCE OF PU(32)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [(b + c)2 - (a + b)(a + c)]/(b + c)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1931) lies on these lines:
1,21   37,757   44,662   99,239   172,593   241,1414   261,894   661,1019   1014,1423   1326,1757   1444,1778   2641,2642

X(1931) = X(1929)-Ceva conjugate of X(1963)
X(1931) = crosssum of X(1757) and X(1961)
X(1931) = perspector of conic {{A,B,C,PU(31)}}
X(1931) = intersection of trilinear polars of P(31) and U(31)

### X(1932) = CROSSSUM OF PU(14)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a4 + b2c2)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1932) lies on this line: 293,774

X(1932) = X(1927)-Ceva conjugate of X(1933)
X(1932) = intersection of tangents at PU(13) to conic {A,B,C,PU(13)}
X(1932) = crosspoint of PU(13)

### X(1933) = CROSSDIFFERENCE OF PU(14)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(a4 - b2c2)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1933) lies on these lines: 31,48   32,904   887,1737   896,1101

X(1933) = isogonal conjugate of X(1934)
X(1933) = X(1927)-Ceva conjugate of X(1932)
X(1933) = crosspoint of X(82) and X(1910)
X(1933) = crosssum of X(i) and X(j) for these (i,j): (38,1959), (1925,1926)
X(1933) = perspector of conic {A,B,C,PU(13)}
X(1933) = intersection of trilinear polars of P(13) and U(13)

### X(1934) = TRILINEAR POLE OF PU(14)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b2c2/(a4 - b2c2)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1934) lies on these lines:
38,799   75,1581   257,335   334,1441   561,1109   764,1244   1821,1959

X(1934) = isogonal conjugate of X(1933)
X(1934) = isotomic conjugate of X(1580)
X(1934) = cevapoint of X(i) and X(j) for these (i,j): (38,1959), (1925,1926)

### X(1935) = CROSSSUM OF PU(15)

Trilinears    cos2A + cos B cos C : :
Trilinears    (a^4 - a^2 (b^2 + b c + c^2) + b c (b + c)^2)/(a - b - c) : :

X(1935) lies on these lines:
1,90   2,603   3,1745   4,255   7,1451   9,478   10,109   12,171   20,212   21,73   31,388   34,63   40,1777   47,1478   56,87   57,1724   58,226   65,1046   84,1040   221,958   222,405   225,283   415,1098   497,1496   748,1106   774,1776   896,1254   940,1806   960,1455   978,1470   1056,1497   1400,1778   1448,1708   1761,1880

X(1935) = X(296)-Ceva conjugate of X(1936)
X(1935) = cevapoint of X(1046) and X(1745)
X(1935) = crosspoint of PU(16)
X(1935) = intersection of tangents at PU(16) to conic {A,B,C,PU(16)}
X(1935) = perspector of ABC and the side-triangle of the 1st and 2nd bicentrics of the orthic triangle

### X(1936) = CROSSDIFFERENCE OF PU(15)

Trilinears    cos2A - cos B cosC : :
Trilinears    (a - b - c) (a^4 - a^2 (b^2 - b c + c^2) - b c (b - c)^2) : :
X(1936) is the perspector of triangle ABC and the tangential triangle of the conic that passes through A, B, C, and the pair P(16) and U(16) of bicentric points; This conic also passes through X(21) and X(651); see the notes just before X(1908). (Randy Hutson, 9/23/2011)

Let L be the line X(1)X(3) = trilinear polar of X(651). Let V be the trilinear polar of the cevapoint of X(1) and X(3), so that V = X(521)X(650); let M = X(3157) = X(1)-Ceva conjugate of X(3), and let N = X(1745) = X(3)-Ceva conjugate of X(1). The lines L, V, MN concur in X(1936). (Randy Hutson, December 26, 2015)

Let A1B1C1 and A2B2C2 be the 1st and 2nd bicentrics of the orthic triangle. The six vertices A1, B1, C1, A2, B2, C2 lie on a conic, denoted here by H. Let A' be the intersection of the tangents to H at A1 and A2. Define B' and C' cyclically. The lines AA', BB', CC' concur in X(1936). Let A" be the intersection of the tangents to H at B2 and C1. Define B' and C' cyclically. The lines AA", BB", CC" concur in X(1936). (Randy Hutson, March 25, 2016)

X(1936) lies on these lines:
1,3   2,212   4,255   11,238   20,603   29,270   31,497   33,63   47,1479   58,950   73,411   100,1818   109,516   225,412   243,1430   388,1496   495,738   511,1364   521,650   580,1210   750,1253   896,1776   908,1331   938,1451   1044,1406   1046,1858   1058,1497   1762,1859

X(1936) = isogonal conjugate of X(1937)
X(1936) = X(296)-Ceva conjugate of X(1935)
X(1936) = crosssum of X(i) and X(j) for these (i,j): (1,1758), (243,1940)
X(1936) = intersection of trilinear polars of P(16) and U(16) (the 1st and 2nd bicentrics of the orthic axis)
X(1936) = crossdifference of every pair of points on line X(65)X(650)
X(1936) = inverse-in-circumconic-centered-at-X(1) of X(55)
X(1936) = X(92)-isoconjugate of X(1949)
X(1936) = perspector of orthic triangle and the side-triangle of the 1st and 2nd bicentrics of the orthic triangle

### X(1937) = TRILINEAR POLE OF PU(15)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/(cos2A - cos B cos C)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1937) lies on these lines:
1,185   4,774   8,201   21,73   80,1736   90,1745   104,1458   108,1172   225,1896   307,314   515,1694   851,1758   885,1769

X(1937) = isogonal conjugate of X(1936)
X(1937) = cevapoint of X(i) and X(j) for these (i,j): (1,1758), (243,1940)
X(1937) = trilinear pole of line X(65)X(650)
X(1937) = pole wrt polar circle of trilinear polar of X(1948)
X(1937) = X(48)-isoconjugate (polar conjugate) of X(1948)
X(1937) = point of intersection, other than A, B, C, of 1st and 2nd bicentrics of the MacBeath circumconic

### X(1938) = IDEAL POINT OF PU(15)

Trilinears    (cos A cos B - cos2C) sin B - (cos A cos C - cos2B) sin C

As the isogonal conjugate of a point on the circumcircle, X(1938) lies on the line at infinity.

X(1938) lies on these lines: 30,511   65,650

X(1938) = crossdifference of every pair of points on line X(6)X(1936)
X(1938) = ideal point of PU(i) for these i: 15, 110

### X(1939) = MIDPOINT OF PU(15)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = (a cos B + b cos C + c cos A) cos C + (a cos C + b cos A + c cos B) cos B
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1939) lies on these lines: 65,650   169,1575   1385,2649

### X(1940) = CROSSSUM OF PU(16)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = sec2A + sec B sec C
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1940) lies on these lines: 1,1075   2,1118   3,158   4,46   20,1857   27,1882   29,65   34,87   35,1784   55,1895   56,92   73,1047   162,1399   201,240   225,1247   281,388   318,1376   331,1447   412,1155   425,1098   471,580   1038,1096   1816,1896

X(1940) = X(1937)-Ceva conjugate of X(243)
X(1940) = cevapoint of X(46) and X(1047)
X(1940) = crosspoint of PU(15)
X(1940) = intersection of tangents at PU(15) to conic {A,B,C,PU(15)}
X(1940) = pole wrt polar circle of trilinear polar of X(7108)
X(1940) = X(48)-isoconjugate (polar conjugate) of X(7108)

### X(1941) = CROSSSUM OF PU(17)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = cos B cos C (cos4A + cos2B cos2C)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1941) lies on these lines: 3,1075   4,155   185,648   194,1593   450,1092

X(1941) = X(1942)-Ceva conjugate of X(450)
X(1941) = crosspoint of PU(17)
X(1941) = intersection of tangents at PU(17) to conic {A,B,C,PU(17)}

### X(1942) = TRILINEAR POLE OF U(17)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos A)/(cos4A - cos2B cos2C)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1942) lies on these lines: 6,1624   852,895

X(1942) = isogonal conjugate of X(450)
X(1942) = cevapoint of X(450) and X(1941)

### X(1943) = CROSSSUM OF PU(18)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(cos2A + cos B cos C)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1943) lies on these lines:
2,914   57,239   69,278   75,222   81,1441   92,394   225,1330   321,651   333,664   637,1659   1231,1396

### X(1944) = CROSSDIFFERENCE OF PU(18)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(cos2A - cos B cos C)
Barycentrics   cos2A - cos B cos C : cos2B - cos C cos A : cos2C - cos A cos B

X(1944) lies on these lines:
{2, 7}, {69, 281}, {75, 219}, {92, 394}, {220, 4363}, {239, 2323}, {242, 511}, {314, 1172}, {448, 662}, {522, 663}, {524, 1146}, {534, 5195}, {666, 1814}, {960, 1010}, {990, 997}, {1212, 4670}, {1332, 3262}, {1737, 1757}, {1958, 2289}, {2324, 3729}, {2607, 3792}

X(1944) = isogonal conjugate of X(1945)
X(1944) = isotomic conjugate of X(1952)
X(1944) = X(i)-complementary conjugate of X(j) for these (i,j): (2648,141), (2701,4885)
X(1944) = X(i)-cross conjugate of X(j) for these (i,j): (1936,5088), (1951,243)
X(1944) = X(i)-isoconjugate of X(j) for these (i,j): (1,1945), (4,1949), (6,1937), (19,296), (31,1952), (65,2249)
X(1944) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (92,394,1943), (2323,4858,239)

### X(1945) = TRILINEAR POLE OF PU(18)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a/(cos2A - cos B cos C)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1945) lies on these lines:
19,800   37,1569   109,284   333,664   673,1465

X(1945) = isogonal conjugate of X(1944)
X(1945) = X(1952)-Ceva conjugate of X(296)

### X(1946) = BICENTRIC DIFFERENCE OF PU(19)

Trilinears    (sin 2A)(cos B - cos C) : :

X(1946) lies on these lines: 3,905   35,1734   105,2724   110,2714   187,237   650,2202   810,822

X(1946) = bicentric difference of PU(19)
X(1946) = PU(19)-harmonic conjugate of X(1409)
X(1946) = trilinear pole of PU(101)
X(1946) = crossdifference of every pair of points on line X(2)X(92)
X(1946) = X(92)-isoconjugate of X(651)
X(1946) = barycentric product of PU(77)

### X(1947) = CROSSSUM OF PU(19)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos2A + cos B cos C) csc 2A
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1947) lies on these lines: 57,264   278,330   318,377

X(1947) = X(1952)-Ceva conjugate of X(1948)
X(1947) = crosspoint of PU(20)
X(1947) = intersection of tangents at PU(20) to conic {A,B,C,PU(20)}
X(1947) = pole wrt polar circle of trilinear polar of X(7105)
X(1947) = X(48)-isoconjugate (polar conjugate) of X(7105)

### X(1948) = CROSSDIFFERENCE OF PU(19)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (cos2A - cos B cos C) csc 2A
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1948) lies on these lines: 2,92   9,264   24,547

X(1948) = isogonal conjugate of X(1949)
X(1948) = X(1952)-Ceva conjugate of X(1947)
X(1948) = crosssum of X(1950) and X(1951)
X(1948) = perspector of conic {A,B,C,PU(20)}
X(1948) = intersection of trilinear polars of P(20) and U(20)
X(1948) = pole wrt polar circle of line X(65)X(650), PU(15))
X(1948) = X(48)-isoconjugate (polar conjugate) of X(1937)

### X(1949) = TRILINEAR POLE OF PU(19)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (sin 2A)/(cos2A - cos B cos C)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1949) lies on these lines:
108,1172   219,296

X(1949) = isogonal conjugate of X(1948)
X(1949) = cevapoint of X(1950) and X(1951)

### X(1950) = CROSSSUM OF PU(20)

Trilinears    a(cos2A + cos B cos C) : :

X(1950) lies on these lines:
3,608   6,1195   19,577   37,1415   65,1333   109,284   604,1403   1011,1395

X(1950) = X(1949)-Ceva conjugate of X(1951)
X(1950) = crosspoint of X(1940) and X(1943)
X(1950) = crosspoint of PU(19)
X(1950) = intersection of tangents at PU(19) to conic {A,B,C,X(109),PU(19)}

### X(1951) = CROSSDIFFERENCE OF PU(20)

Trilinears    a(cos2A - cos B cos C)

X(1951) lies on these lines:
1,1729   3,607   6,41   19,577   21,270   104, 294   517,906   652,663   851,1430   910,1415   1262,1465   1409,1630

X(1951) = isogonal conjugate of X(1952)
X(1951) = X(1949)-Ceva conjugate of X(1950)
X(1951) = crosspoint of X(243) and X(1944)
X(1951) = crosssum of X(i) and X(j) for these (i,j): (296,1945), (1947,1948)
X(1951) = perspector of conic {A,B,C,X(109),PU(19)}
X(1951) = intersection of trilinear polars of X(109), P(19), and U(19)
X(1951) = X(92)-isoconjugate of X(296)

### X(1952) = TRILINEAR POLE OF PU(20)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc/(cos2A - cos B cos C)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1952) lies on these lines: 8,201   29,65   92,1146   232,1148   333,664

X(1952) = reflection of X(i) in X(j) for these (i,j): (92,1146), (664,1214)
X(1952) = isogonal conjugate of X(1951)
X(1952) = isotomic conjugate of X(1944)
X(1952) = cevapoint of X(i) and X(j) for these (i,j): (296,1945), (1947,1948)

### X(1953) = BICENTRIC SUM OF PU(21)

Trilinears    sin 2B + sin 2C : :
Trilinears    sin A cos(B - C) : :
Trilinears    a^2(b^2 + c^2) - (b^2 - c^2)^2 : :
Trilinears    SBSC + S2 : SCSA + S2 : :

X(1953) lies on these lines:
1,19   6,1411   9,1389   31,1820   38,1755   65,1108   71,517   73,1841   216,1393   219,1482   515,1839   946,1826   991,1414   1457,1880

X(1953) = {X(1),X(19)}-harmonic conjugate of X(48)
X(1953) = isogonal conjugate of X(2167)
X(1953) = X(i)-Ceva conjugate of X(j) for these (i,j): (163,661), (823,656)
X(1953) = crosspoint of X(1) and X(92)
X(1953) = crosssum of X(1) and X(48)
X(1953) = crossdifference of every pair of points on line X(656)X(1955)
X(1953) = bicentric sum of PU(21)
X(1953) = PU(21)-harmonic conjugate of X(656)
X(1953) = barycentric product of PU(69)

### X(1954) = CROSSSUM OF PU(21)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = sin 2B sin 2C + sin22A
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1954) lies on this line: 1,21

### X(1955) = CROSSDIFFERENCE OF PU(21)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = sin 2B sin 2C - sin22A
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1955) lies on these lines:
1,19   47,1740   58,1047   293,1755   1580,1733

X(1955) = isogonal conjugate of X(1956)
X(1955) = X(i)-Ceva conjugate of X(j) for these (i,j): (293,1), (1755,1580)
X(1955) = X(i)-aleph conjugate of X(j) for these (i,j): (98,1733), (293,1955)

### X(1956) = TRILINEAR POLE OF PU(21)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/(sin 2B sin 2C - sin22A)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1956) lies on these lines: 293,1755

X(1956) = isogonal conjugate of X(1955)
X(1956) = X(240)-cross conjugate of X(1)

### X(1957) = CROSSSUM OF PU(22)

Trilinears    tan2A + tan B tan C : :

X(1957) lies on these lines:
1,204   2,1430   19,1707   29,1468   31,92   42,1013   43,1783   63,240   158,255   171,281   212,243   238,278   242,1395   896,1748   1496,1895   1724,1838

X(1957) = X(1910)-Ceva conjugate of X(240)
X(1957) = crosspoint of PU(23)
X(1957) = intersection of tangents at PU(23) to conic {{A,B,C,PU(23)}}

### X(1958) = CROSSSUM OF PU(23)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cot2A + cot B cot C
Trilinears       g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a4 - a2b2 - a2c2 + 2b2c2     (M. Iliev, 5/13/2007)
Trilinears       SASA + SBSC : SBSB + SCSA : SCSC + SASB      (C. Lozada, 9/07/2013)
Trilinears       a2SA - b2c2 : b2SB - c2a2 : c2SC - a2b2      (C. Lozada, 9/07/2013)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(1958) lies on these lines:
19,326   31,1582   41,894   48,75   63,610   100,1253   239,604

X(1958) = X(i)-Ceva conjugate of X(j) for these (i,j): (293,1959), (775,63)
X(1958) = cevapoint of X(610) and X(1740)

### X(1959) = CROSSDIFFERENCE OF PU(23)

Trilinears    cot2A - cot B cot C : :
Trilinears    b4 + c4 - a2b2 - a2c2 : :      (M. Iliev, 5/13/2007)
Trilinears    SASA - SBSC : :      (C. Lozada, 9/07/2013)

X(1959) lies on these lines:
1,21   2,257   19,326   48,1760   92,304   329,1655   514,661   1444,1761   1762,1812

X(1959) = isogonal conjugate of X(1910)
X(1959) = isotomic conjugate of X(1821)
X(1959) = X(i)-Ceva conjugate of X(j) for these (i,j): (293,1958), (1934,38)
X(1959) = X(i)-cross conjugate of X(j) for these (i,j): (6,2065), (114,2), (230,98), (1692,1976), (1733,1821)
X(1959) = crosspoint of X(1) and X(1581)
X(1959) = crosssum of X(i) and X(j) for these (i,j): (1,1580), 240,1957)

### X(1960) = BICENTRIC SUM OF PU(25)

Trilinears    a(b - c)(2a - b - c) : :

X(1960) is the center of the circle V(X(101)) = {{15,16,101,106}}; see the preamble to X(6137). (Randy Hutson, December 26, 2015)

X(1960) lies on these lines:
1,659   101,692   187,237   214,900   292,875   660,898   678,1635   884,1438

X(1960) = midpoint of X(i) and X(j) for these (i,j): (1,659), (663,667), (1635,3251)
X(1960) = X(i)-Ceva conjugate of X(j) for these (i,j): (101,1017), (106,1015), (901,6), (1319,2087)
X(1960) = crosspoint of X(i) and X(j) for these (i,j): (6,901), (101,106)
X(1960) = crosssum of X(i) and X(j) for these (i,j): (2,900), 514,519)
X(1960) = bicentric sum of PU(25)
X(1960) = PU(25)-harmonic conjugate of X(1015)
X(1960) = bicentric difference of PU(99)
X(1960) = PU(99)-harmonic conjugate of X(1017)
X(1960) = isogonal conjugate of X(4555)
X(1960) = crossdifference of every pair of points on line X(2)X(45)

### X(1961) = CROSSSUM OF PU(31)

Trilinears    (a + b)(a + c) + (b + c)2

X(1961) lies on these lines:
1,2   35,199   37,171   81,756   86,1215   100,1255   111,831   940,984   1051,1100

X(1961) = cevapoint of X(846) and X(1051)
X(1961) = crosspoint of PU(32)
X(1961) = intersection of tangents at PU(32) to conic {{A,B,C,X(100),PU(32)}}

### X(1962) = BICENTRIC SUM OF PU(32)

Trilinears    (b + c)(2a + b + c) : :

X(1962) lies on these lines:
1,21   2,740   37,42   55,199   100,1255   351,1635

X(1962) = reflection of X(1635) in X(351)
X(1962) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,1100), (100,661), (1125,1213)
X(1962) = crosspoint of X(i) and X(j) for these (i,j): (1,37), (1100,1125)
X(1962) = crosssum of X(i) and X(j) for these (i,j): (1,81), (1126,1255)
X(1962) = X(2)-of incentral triangle
X(1962) = bicentric sum of PU(32)
X(1962) = PU(32)-harmonic conjugate of X(661)
X(1962) = complement of X(17163)
X(1962) = homothetic center of Gemini triangle 17 and cross-triangle of Gemini triangles 15 and 17

### X(1963) = CROSSSUM OF PU(32)

Trilinears    1/[(a + b)(a + c)] + 1/(b + c)2 : :

X(1963) lies on these lines:
1,1326   2,6   37,757   662,1100   894,1509

X(1963) = X(1929)-Ceva conjugate of X(1931)
X(1963) = crosspoint of PU(31)
X(1963) = intersection of tangents at PU(31) to conic {{A,B,C,PU(31)}}

### X(1964) = BICENTRIC SUM OF PU(36)

Trilinears    a2(b2 + c2) : :
Trilinears    SASA - SωSω : SBSB - SωSω : SCSC - SωSω      (C. Lozada, 9/07/2013)
Trilinears    a2(SA + Sω : :      (C. Lozada, 9/07/2013)
Trilinears    Area(BCP(1)) + Area(BCU(1)) : :

X(1964) lies on these lines:
1,75   6,292   31,48   42,1100   82,662   99,719   110,745   214,995   313,730   501,595   741,757   1042,1360   1193,1386   1201,1279

X(1964) = isogonal conjugate of X(3112)
X(1964) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,38), (31,1923), (662,798), (1178,6), (1581,1755)
X(1964) = crosspoint of X(i) and X(j) for these (i,j): (1,31), (39,1401)
X(1964) = crosssum of X(1) and X(75)
X(1964) = PU(36)-harmonic conjugate of X(798)

### X(1965) = CROSSSUM OF PU(36)

Trilinears    b2c2(a4 + b2c2) : :

X(1965) lies on these lines: 2,292   19,27   31,561   38,799   332,375

X(1965) = X(1581)-Ceva conjugate of X(1966)
X(1965) = crosspoint of PU(35)
X(1965) = intersection of tangents at PU(35) to conic {A,B,C,PU(35)}

### X(1966) = CROSSDIFFERENCE OF PU(36)

Trilinears    b2c2(a4 - b2c2) : :
Trilinears    directed distance of A to line PU(1) : :

Let A'B'C' be the 1st anti-Brocard triangle. Let A" be the trilinear product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1966). (Randy Hutson, December 26, 2015)

X(1966) lies on these lines:
1,75   2,893   31,561   240,811   350,1281   668,1757   732,894   798,812   799,896   1821,1934

X(1966) = isogonal conjugate of X(1967)
X(1966) = isotomic conjugate of X(1581)
X(1966) = X(i)-Ceva conjugate of X(j) for these (i,j): (874,804), (1581,1965), (1821,75)
X(1966) = X(i)-cross conjugate of X(j) for these (i,j): (698,1916), (1691,699)
X(1966) = crosssum of X(1580) and X(1582)
X(1966) = perspector of conic {{A,B,C,PU(35)}}
X(1966) = intersection of trilinear polars of P(35) and U(35)
X(1966) = trilinear product of PU(133)

### X(1967) = TRILINEAR POLE OF PU(36)

Trilinears    a2/(a4 - b2c2)

X(1967) lies on these lines:
1,1581   38,799   42,694   213,904   256,291   733,813   741,805   875,881

X(1967) = isogonal conjugate of X(1966)
X(1967) = isotomic conjugate of X(1926)
X(1967) = cevapoint of X(1580) and X(1582)
X(1967) = X(i)-cross conjugate of X(j) for these (i,j): (32,699), (698,76)
X(1967) = trilinear product of circumcircle intercepts of line PU(1)
X(1967) = trilinear product X(292)*X(893)

### X(1968) = CROSSSUM OF PU(37)

Trilinears    a(tan2A + tan B tan C) : :
Trilinears    2 cos A + sin A (tan A - cot ω) : :

X(1968) lies on these lines:
3,232   4,32   6,64   20,393   24,187   25,1611   33,172   39,378   53,571   194,648   217,578   230,235   264,384   1147,1625   1384,1598   1691,1974

X(1968) = X(1976)-Ceva conjugate of X(232)
X(1968) = crosspoint of PU(39)
X(1968) = intersection of tangents at PU(39) to hyperbola {A,B,C,X(4),X(112),PU(39)}
X(1968) = crossdifference of every pair of points on line X(684)X(8057)

### X(1969) = TRILINEAR PRODUCT OF PU(38)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b2c2 sin 2B sin 2C
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = sec A csc3A
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1969) lies on these lines: 1,336   75,158   76,331   92,304   273,1240

X(1969) = isotomic conjugate of X(48)
X(1969) = cevapoint of X(75) and X(92)

X(1969) = trilinear pole of polar of X(31) wrt polar circle (line X(14208)X(20948))
X(1969) = pole wrt polar circle of trilinear polar of X(31) (line X(667)X(788))
X(1969) = polar conjugate of X(31)
X(1969) = trilinear product of vertices of Gemini triangle 37
X(1969) = trilinear product of vertices of Gemini triangle 38

### X(1970) = CROSSSUM OF PU(38)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(sin22A + sin 2B sin 2C)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1970) lies on these lines: 3,6   49,1625   54,112

X(1970) = perspector of ABC and 1st Brocard triangle of orthic triangle

### X(1971) = CROSSDIFFERENCE OF PU(38)

Trilinears    a(sin22A - sin 2B sin 2C): :

X(1971) lies on these lines:
6,25   50,647   53,1629   98,230   217,1614   237,248   571,1613   1609,1619

X(1971) = isogonal conjugate of X(1972)
X(1971) = X(i)-Ceva conjugate of X(j) for these (i,j): (237,1691), (248,6)
X(1971) = crosspoint of X(98) and X(275)
X(1971) = crosssum of X(216) and X(511)
X(1971) = homothetic center of X(3)-Ehrmann triangle and mid-triangle of 1st and 2nd Kenmotu diagonals triangles

### X(1972) = TRILINEAR POLE OF PU(38)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc/(sin22A - sin 2B sin 2C)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1972) lies on these lines:
95,216   287,401   925,1298

X(1972) = reflection of X(648) in X(216)
X(1972) = isogonal conjugate of X(1971)
X(1972) = isotomic conjugate of X(401)
X(1972) = cevapoint of X(216) and X(511)

X(1972) = antipode of X(264) in hyperbola {{A,B,C,X(2),X(69)}}

### X(1973) = TRILINEAR PRODUCT OF PU(39)

Trilinears    a3cos B cos C : :
Trilinears    tan A sin2A : :

X(1973) lies on these lines:
1,19   6,1245   25,41   32,1395   34,1438   47,163   112,741   255,1755   278,1429   604,608   1148,1283   1842,1886

X(1973) = isogonal conjugate of X(304)
X(1973) = X(i)-Ceva conjugate of X(j) for these (i,j): (19,31), (608,1395), (1474,25)
X(1973) = X(i)-cross conjuguate of X(j) for these (i,j): (4,683), (682,3), (1196,2), (1368,305)
X(1973) = crosspoint of X(i) and X(j) for these (i,j): (19,1096), (25,608)
X(1973) = crosssum of X(i) and X(j) for these (i,j): (63,326), (69,345), (312,322), (525,1565)

X(1973) = barycentric product of PU(18)
X(1973) = trilinear product of intersections of circumcircle and 2nd Lemoine circle
X(1973) = pole wrt polar circle of trilinear polar of X(561)
X(1973) = X(48)-isoconjugate (polar conjugate) of X(561)
X(1973) = X(75)-isoconjugate of X(63)
X(1973) = X(92)-isoconjugate of X(326)

### X(1974) = BARYCENTRIC PRODUCT OF PU(39)

Trilinears    a4cos B cos C : :
Trilinears    tan A sin3A : :
Trilinears    tan A sin(A - ω) : :

X(1974) is the X(i)-isoconjugate of X(j) for these (i,j): (48,1502), (92,3926); also X(1974) is the pole with respect to the polar cirle of the trilinear polar of X(1502).    Randy Hutson, August 15, 2013

X(1974) lies on these lines:
4,83   6,25   24,511   32,682   34,1428   53,460   66,125   69,459   110,193   112,729   141,468   156,1353   235,1503   237,577   264,419   428,597   571,1576   981,1172   1147,1351   1386,1829   1395,1397

X(1974) = isogonal conjugate of X(305)
X(1974) = X(25)-Ceva conjugate of X(32)
X(1974) = crosspoint of X(1395) and X(1973)
X(1974) = crosssum of X(i) and X(j) for these (i,j): (2,1370), (339,3267)
X(1974) = crossdifference of every pair of points on the line X(525)X(3267)
X(1974) = X(92)-isoconjugate of X(3926)
X(1974) = trilinear product of vertices of Ara triangle
X(1974) = barycentric product of intersections of circumcircle and 2nd Lemoine circle

### X(1975) = CROSSSUM OF PU(39)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = cos B cos C + (bc cos2A)/a2
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1975) lies on these lines:
3,76   4,325   6,194   20,64   25,305   30,315   32,538   56,350   75,958   190,220   221,664   264,1105   274,405   310,1011   316,382   378,1235   394,401   543,626   801,1073

X(1975) = midpoint of X(489) and X(490)
X(1975) = X(i)-Ceva conjugate of X(j) for these (i,j): (287,325), (801,69)
X(1975) = cevapoint of X(20) and X(194)
X(1975) = anticomplement of X(5254)
X(1975) = crosspoint of PU(37)
X(1975) = intersection of tangents at PU(37) to hyperbola {A,B,C,X(99),PU(37)}
X(1975) = crosspoint of X(20) and X(194) wrt excentral triangle
X(1975) = crosspoint of X(20) and X(194) wrt anticomplementary triangle
X(1975) = X(32) of 6th Brocard triangle
X(1975) = 5th-Brocard-to-6th-Brocard similarity image of X(32)

### X(1976) = TRILINEAR POLE OF PU(39)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2/(a2cos B cos C - bc cos2A)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1976) lies on these lines:
2,98   6,157   25,1501   32,263   37,692   51,251   111,1495   237,694   290,308   351,878   419,685   879,1177   1492,1821

X(1976) = isogonal conjugate of X(325)
X(1976) = X(i)-Ceva conjugate of X(j) for these (i,j): (98,248), (2065,6)
X(1976) = cevapoint of X(i) and X(j) for these (i,j): (6,1691), (232,1968)
X(1976) = crosssum of X(2) and X(147)
X(1976) = trilinear pole of line X(32)X(512)
X(1976) = crossdifference of every pair of points on line X(2799)X(3569)

### X(1977) = BICENTRIC DIFFERENCE OF PU(42)

Trilinears    a3(b - c)2 : :

X(1977) lies on the Brocard inellipse and on these lines: 6,100   213,1017   291,1017   1397,1501

X(1977) = X(i)-Ceva conjugate of X(j) for these (i,j): (6,667), (31,669), (32,1919), (739,890), (1397,1980)
X(1977) = crosspoint of X(i) and X(j) for these (i,j): (6,667), (32,1919), (87,1019)
X(1977) = crosssum of X(i) and X(j) for these (i,j): (2,668), (43,1018), (76,1978)
X(1977) = trilinear pole wrt symmedial triangle of line X(1)X(6)
X(1977) = crossdifference of every pair of points on line X(668)X(891) (the tangent to the Steiner circumellipse at X(668))
X(1977) = barycentric square of X(649)

### X(1978) = TRILINEAR PRODUCT OF PU(41)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b3c3/(b - c)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1978) lies on these lines:
75,244   99,835   100,789   101,689   190,670   310,321   312,561   668,891   811,1897

X(1978) = isogonal conjugate of X(1919)
X(1978) = isotomic conjugate of X(649)
X(1978) = X(670)-Ceva conjugate of X(668)
X(1978) = cevapoint of X(i) and X(j) for these (i,j): (75,514), (321,693), (646,668), (850,1230)
X(1978) = crosssum of X(669) and X(1924)
X(1978) = anticomplement of X(6377)
X(1978) = trilinear pole of line X(10)X(75) (the isotomic conjugate of the isogonal conjugate of the Nagel line)
X(1978) = trilinear product of intercepts of Steiner circumellipse and line X(2)X(37)

### X(1979) = CROSSSUM OF PU(41)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a3(b - c)2 - abc(a - b)(a - c)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1979) lies on this line: 6,100

X(1979) = X(667)-Ceva conjugate of X(6)
X(1979) = crosspoint of PU(42)
X(1979) = polar conjugate of isotomic conjugate of X(22158)

### X(1980) = BARYCENTRIC PRODUCT OF PU(42)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a4(b - c)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1980) lies on these lines: 667,838   669,688   692,1252   813,929

X(1980) = X(i)-Ceva conjugate of X(j) for these (i,j): (692,32), (1397,1977)
X(1980) = crosspoint of X(32) and X(692)
X(1980) = crosssum of X(i) and X(j) for these (i,j): (76,693), (850,1228)

X(1980) = isogonal conjugate of X(6386)
X(1980) = crossdifference of every pair of points on line X(76)X(321)

### X(1981) = VEGA TRANSFORM OF X(647)

Trilinears    [sin 2B sin(C - A) - sin 2C sin(A - B)]/[sin 2A sin(B - C)] : :

As a point on the Euler line, X(1981) has Shinagawa coefficients (\$aSBSC\$(E+F)F+\$aSA\$FS2 -\$a\$(E-2F)FS2, -\$aSBSC\$S2 -3\$aSA\$FS2-\$a\$[(E+F)F-S2]S2).

If X = x : y : z is a triangle center other than X(1), then the Vega transform of X, defined by trilinears

(y - z)/x : (z - x)/y : (x - y)/z

lies on the line [x:y:z] that has x,y,z as coefficients. (The line [x:y:z] is the trilinear polar of the isogonal conjugate of X.) Thus, the Vega transform of X(647) lies on the Euler line.

In general, the bicentrics of X also lie on [x:y:z]; for details, click Tables at the top of this page.

X(1981) lies on these lines: 2,3   651,653   662,811

X(1981) = bicentric difference of PU(30)
X(1981) = PU(30)-harmonic conjugate of X(1982)
X(1981) = intersection of lines P(15)U(16) and U(15)P(16)

### X(1982) = PU(22)-HARMONIC CONJUGATE OF X(1981)

Trilinears    (2x - y - z)/x : : , where x = x(a,b,c) = sin 2A sin(B - C)

As a point on the Euler line, X(1982) has Shinagawa coefficients (2\$aSA2SB\$F -2\$aSCSA2\$F +\$aSC2SA\$F -\$aSASB2\$F, \$aSCSA\$S2 -\$aSASB\$S2+3\$aSB\$FS2 -3\$aSC\$FS2).

X(1982) lies on these lines: 1,648   2,3   255,1098

X(1982) = bicentric sum of PU(30)
X(1982) = PU(30)-harmonic conjugate of X(1981)

### X(1983) = VEGA TRANSFORM OF X(523)

Trilinears    [sin(C - A) - sin(A - B)]/sin(B - C) : :

X(1983) lies on these lines: 3,6   101,109   919,1027   1023,1252   1258,1497

X(1983) = X(i)-Ceva conjugate of X(j) for these (i,j): (59,215), (901,692)
X(1983) = X(215)-cross conjugate of X(110)
X(1983) = crosspoint of X(651) and X(1290)
X(1983) = crosssum of X(661) and X1769)
X(1983) = bicentric difference of PU(29)
X(1983) = PU(29)-harmonic conjugate of X(9275)

### X(1984) = VEGA TRANSFORM OF X(1020)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (cos2A - cos B cos C)(cos B - cos C)2/(cos B + cos C)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(1984) has Shinagawa coefficients (\$a(SA)3\$ - \$a(SA)2\$E + \$aSA\$(E - F)F - 2abc(E - 2F)F, -\$a(SA)3\$ + \$a(SA)2\$E + \$aSBSC\$(E - 2F) - \$aSA\$(E + F)F - 2abc[(E + F)F - S2]).

X(1984) lies on this line: 2,3

X(1984) = crosssum of X(851) and X(1020)

### X(1985) = EULER LINE INTERCEPT OF LINE X(6)X(11)

Trilinears    bc[bca4 + (b3 + c3)a3 - ua - v] : : , where u = u(a,b,c) = (b + c)(b - c)2(b2 + bc + c2) and v = v(a,b,c) = bc(b - c)2(b + c)2
Trilinears    (sec A + sec B + sec C) csc A + (csc A + csc B + csc C) sec A : :

As a point on the Euler line, X(1985) has Shinagawa coefficients (\$bcSBSC\$, \$bc\$S2).

X(1985) lies on these lines:
2,3   6,11   42,1837   184,1746   1465,1893   1699,1730

X(1985) = inverse-in-orthocentroidal-circle of X(851)

### X(1986) = HATZIPOLAKIS REFLECTION POINT

Trilinears    (1 + cos 2B + cos 2C) sin 3A csc 2A : :

Let A'B'C' be the orthic triangle of triangle ABC. Let AB be the reflection of A in C', and define AC, BC, BA, CA, CB functionally. Then the nine-point circles of the triangles

AABAC,    BBCBA,    CCACB,

concur in X(1986). (Antreas Hatzipolakis, Hyacinthos 7868, 9/12/03; coordinates by Barry Wolk, Hyacinthos 7876, 9/13/03)

Let A'B'C' = cevian triangle of X(186). Let A", B", C" be the inverse-in-circumcircle of A', B', C'. The lines AA", BB", CC" concur in X(1986). (Randy Hutson, December 2, 2017)

Let A'B'C' = orthic triangle. Let B'C'A" be the triangle similar to ABC such that segment A'A" crosses the line B'C'. Define B" and C" cyclically. Equivalently, A" is the reflection of A in B'C', and cyclically for B" and C". Equivalently, A" is the isogonal conjugate of A' wrt AB'C', and cyclically for B"and C". The lines A'A", B'B", C'C" concur in X(1986). (Randy Hutson, December 2, 2017)

Let Ha be the foot of the A-altitude. Let Ba, Ca be the feet of perpendiculars from Ha to CA, AB, resp. Let Na be the nine-point center of HaBaCa. Define Nb and Nc cyclically. The lines HaNa, HbNb, HcNc concur in X(1986). (Randy Hutson, December 2, 2017)

Let A'B'C' be the orthic triangle. Let Oa be the A-Johnson circle of triangle AB'C', and define Ob and Oc cyclically. The circles Oa, Ob, Oc concur in X(1986). (Randy Hutson, July 31 2018)

X(1986) lies on these lines:
4,94   6,74   24,110   25,399   113,403   125,389   186,323   542,1843   648,1300   1844,1845

X(1986) = reflection of X(i) in X(j) for these (i,j): (4,1112), (74,974), (125,389)
X(1986) = X(4)-Ceva conjugate of X(403)
X(1986) = crosspoint of X(4) and X(186)
X(1986) = crosssum of X(3) and X(265)
X(1986) = X(80)-of-orthic-triangle if ABC is acute
X(1986) = antigonal conjugate of X(4) wrt orthic triangle
X(1986) = antipode of X(4) in Hatzipolakis-Lozada hyperbola
X(1986) = perspector of orthic triangle and Hatzipolakis-Moses triangle
X(1986) = X(11)-of-circumorthic-triangle if ABC is acute

### X(1987) = 1st LEMOINE ANTIPARALLELS POINT

Trilinears    (sin A)/(sin22A - sin 2B sin 2C) : :

X(1987) is discussed in Lemoine's paper cited at X(19). Contributed by Darij Grinberg.

X(1987) lies on these lines:
3,1625   54,112   69,1972   72,1956   237,248   290,297

X(1987) = isogonal conjugate of X(401)
X(1987) = cevapoint of X(217) and X(237)
X(1987) = X(232)-cross conjugate of X(6)
X(1987) = trilinear pole of line X(51)X(647)
X(1987) = trilinear pole of PU(157)

### X(1988) = 2nd LEMOINE ANTIPARALLELS POINT

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin A)/(csc 2B + csc 2C - csc 2A)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(1988) is discussed in Lemoine's paper cited at X(19). Contributed by Darij Grinberg.

X(1988) lies on these lines: 6,436   184,1968   394,401   577,1971

X(1988) = isogonal conjugate of X(3164)
X(1988) = X(4)-cross conjugate of X(6)

### X(1989) = ISOGONAL CONJUGATE OF X(323)

Trilinears    sin2A csc 3A : :
Trilinears    a/(1 - 4 cos^2 A) : :
Trilinears    csc(A + π/3) + csc(A - π/3) : :
Barycentrics  sin3A csc 3A : sin3B csc 3B : sin3C csc 3C
Barycentrics   1/((a^2 - b^2 - c^2)^2 - b^2 c^2) : :

X(1989) plays a major role in the theory of special isocubics, as presented in Chapter 6 of Jean-Pierre Ehrmann and Bernard Gibert,, "Special Isocubics in the Triangle Plane," downloadable from Bernard Gibert, Cubics in the Triangle Plane.

X(1989) is the barycentric product X(13)*X(14) of the Fermat points. The line through X(50) parallel to the Eular line X(2)X(3) passes through X(1989).

Let A1B1C1 and A2B2C2 be the 1st and 2nd Ehrmann circumscribing triangles. X(1989) is the barycentric product A1*A2 = B1*B2 = C1*C2. (Randy Hutson, June 27, 2018)

Let VaVbVc be the Ehrmann vertex-triangle. Let A' be the barycentric product Vb*Vc, and define B', C' cyclically. The lines AA', BB', CC' concur in X(1989). (Randy Hutson, June 27, 2018)

Let VaVbVc and SaSbSc be the Ehrmann vertex-triangle and Ehrmann side-triangle, resp. Let A' be the barycentric product Va*Sa, and define B', C' cyclically. The lines AA', BB', CC' concur in X(1989). (Randy Hutson, June 27, 2018)

X(1989) lies on these lines:
2,94   6,13   30,50   53,112   67,868   111,230   403,1990   1427,2006

X(1989) = isogonal conjugate of X(323)
X(1989) = complement of X(1272)
X(1989) = X(94)-Ceva conjugate of X(265)
X(1989) = cevapoint of X(i) and X(j) for these (i,j): (53,1990), (115,1637), (395,396)
X(1989) = crosspoint of X(2) and X(1138)
X(1989) = crosssum of X(6) and X(399)
X(1989) = barycentric product of X(13) and X(14)
X(1989) = isotomic conjugate of X(7799)
X(1989) = inverse-in-Kiepert-hyperbola of X(265)
X(1989) = {X(13),X(14)}-harmonic conjugate of X(265)
X(1989) = trilinear pole of line X(51)X(512)
X(1989) = pole wrt polar circle of trilinear polar of X(340)
X(1989) = X(48)-isoconjugate (polar conjugate) of X(340)
X(1989) = X(50)-of-orthocentroidal-triangle
X(1989) = perspector of ABC and unary cofactor triangle of Trinh triangle
X(1989) = barycentric product X(79)*X(80)
X(1989) = barycentric product of circumcircle intercepts of Johnson circle (or line PU(5), X(5)X(523))

### X(1990) = ORTHIC-AXIS INTERCEPT OF LINE X(4)X(6)

Trilinears    bc[a2(b2 + c2 - 2a2) + (b2 - c2)2]/(b2 + c2 - a2) : :
Trilinears    (tan A)(cos A - 2 cos B cos C) : :
Trilinears    sin A - 2 tan A cos B cos C : :
Barycentrics    [a2(b2 + c2 - 2a2) + (b2 - c2)2]/(b2 + c2 - a2) : :

X(1990) is described in section 6.4.2 of the downloadable article cited at X(1989).

X(1990) lies on these lines:
4,6   44,1785   50,112   140,216   186,1138   230,231   297,340   395,471   396,470   403,1989   458,597   550,577   1033,1609

X(1990 = midpoint of X(297) and X(648)
X(1990) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,133), (1300,25), (1989,53)
X(1990) = crosspoint of X(2) and X(1294)
X(1990) = perspector of circumconic centered at X(133)
X(1990) = center of circumconic that is locus of trilinear poles of lines passing through X(133)
X(1990) = pole wrt polar circle of trilinear polar of X(1494) (line X(2)X(525))
X(1990) = polar conjugate of X(1494)
X(1990) = X(44)-of-orthic-triangle if ABC is acute
X(1990) = inverse of X(4) in circumconic centered at X(1249)
X(1990) = PU(4)-harmonic conjugate of X(9209)

### X(1991) = 2nd VAN LAMOEN PERPENDICULAR BISECTORS POINT

Trilinears    bc[b2 + c2 - 2a2 - 4*area(ABC)]

Erect squares inwardly on the sides of triangle ABC. Two edges emanate from A; let P and Q be their endpoints. Let a' be the perpendicular bisector of PQ, and define b' and c' cyclically. Then a', b', c' concur in X(1991). See X(591) for the 1st Van Lamoen perpendicular bisectors point, constructed from outwardly drawn squares.

If you have The Geometer's Sketchpad, you can view 2nd Van Lamoen Perpendicular Bisectors Point.

X(1991) lies on these lines: 2,6   371,754   487,3070   638,1151

X(1991) = reflection of X(591) in X(2)
X(1991) = centroid of AbAcBcBaCaCb used in construction of 3rd Lozada circle
X(1991) = perspector of outer Vecten triangle and outer Vecten of inner Vecten triangle

Orthocorrespondents, 1992- 2006
Suppose P is a point in the plane of triangle ABC. The perpendiculars through P to the lines AP, BP, CP meet the lines BC, CA, AB, respectively, in collinear points. Let L denote their line. The trilinear pole of L is the orthocorrespondent of P. This definition is introduced in

Bernard Gibert, Orthocorrespondence and Orthopivotal Cubics, Forum Geometricorum 3 (2003) pages 1-27.

If P is given in barycentrics by P = p : q : r, then the orthocorrespondent of P has barycentrics
f(a,b,c) : f(b,c,a) : f(c,a,b), where

f(a,b,c) = a2qr + (-pSA + qSB + rSC)p,

where SA = (b2 + c2 - a2)/2, and SB and SC are defined cyclically.

If follows that if X = x : y : z in trilinears, then the orthocorrespondent of X has trilinears
g(a,b,c) : g(b,c,a) : g(c,a,b), where

g(a,b,c) = yz + (-x cos A + y cos B + z cos C)x.

Pairs (i,j) for which the orthocorrespondent of X(i) is X(j) include the following:
(1,57), (4,2), (11,651), (13,13), (14,14), (15,62), (16,61), (30,2), (98,287), (100,1332), (101,1331), (103,1815), (105,1814), (106,1797), (107,648), (108,651), (109,1813), (111,895), (112,110), (115,110), (125,648), (132,287), (1560,895), (1566,677), (1785,57)

The orthocorrespondent of every point on the line at infinity is the centroid. Two orthoassociate points (i.e., an inverse pair in the polar circle, such as X(112) and X(115)) share the same orthocorrespondent.

### X(1992) = ORTHOCORRESPONDENT OF X(2)

Trilinears    bc(b2 + c2 - 5a2) : :
Barycentrics    b2 + c2 - 5a2 : :

Let A'B'C' be the 1st Parry 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(1992). (Randy Hutson, February 10, 2016)

Let A'B'C' be the anti-Artzt triangle. Let A" be the barycentric product B'*C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1992). (Randy Hutson, December 10, 2016)

Let A'B'C' be the orthic triangle. X(1992) is the radical center of the 2nd Lemoine circles of triangles AB'C', BC'A', CA'B'. (Randy Hutson, July 31 2018)

X(1992) lies on these lines:
2,6   4,542   30,1351   145,190   218,1332   317,1249   344,1743   376,511   575,631

X(1992) = midpoint of X(2) and X(193)
X(1992) = reflection of X(i) in X(j) for these (i,j): (2,6), (69,2), (599, 597)
X(1992) = isotomic conjugate of X(5485)
X(1992) = anticomplement of X(599)
X(1992) = X(598)-Ceva conjugate of X(2)
X(1992) = perspector of ABC and unary cofactor triangle of 4th anti-Brocard triangle
X(1992) = X(4)-of-anti-Artzt-triangle
X(1992) = {X(597),X(599)}-harmonic conjugate of X(2)
X(1992) = trilinear pole of line X(1499)X(8644) (the perspectrix of ABC and 1st Parry triangle, and the orthic axis of the Thomson triangle)

### X(1993) = ORTHOCORRESPONDENT OF X(3)

Trilinears       csc A cos 2A : csc B cos 2B : csc C cos 2C
Barycentrics  cos 2A : cos 2B : cos 2C

X(1993) lies on these lines:
2,6   3,54   4,155   20,1181   22,184   23,154   24,52   25,110   26,49   51,576   63,2003   68,1594   194,401   264,275   278,651   283,581   317,467   371,1599   372,1600   389,1092   399,1539   458,1235   493,588   494,589   569,1216   573,1790   631,1199   858,1899   1196,1570   1353,1368

X(1993) = reflection of X(22) in X(184)
X(1993) = isogonal conjugate of X(2165)
X(1993) = isotomic conjugate of X(5392)
X(1993) = anticomplement of X(343)
X(1993) = X(i)-Ceva conjugate of X(j) for these (i,j): (264,3), (275,2), (317,24), (1585,1599), (1586,1600)
X(1993) = cevapoint of X(i) and X(j) for these (i,j): (6,155), (571,1147)
X(1993) = crosspoint of X(i) and X(j) for these (i,j): (249,648), (1585,1586)
X(1993) = crosssum of X(115) and X(647)

### X(1994) = ORTHOCORRESPONDENT OF X(5)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a[b2c2 - 16(area(ABC))2]
Trilinears        cos 3A csc 2A : cos 3B csc 2B : cos 3C csc 2C ( M. Iliev, 4/12/07)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1994) lies on these lines:
2,6   3,1199   5,195   22,1351   23,184   49,143   51,110   52,54   94,275   97,216   186,568   427,1353   567,1154   1194,1570   1627,1692

X(1994) = isogonal conjugate of X(2963)
X(1994) = cevapoint of X(6) and X(195)
X(1994) = X(2965)-cross conjugate of X(3518)
X(1994) = crosspoint of X(588) and X(589)
X(1994) = crosssum of X(590) and X(615)

### X(1995) = ORTHOCORRESPONDENT OF X(6)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b4 + c4 - a4 - 4b2c2)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

As a point on the Euler line, X(1995) has Shinagawa coefficients (E - 2F, 2E + 2F).

X(1995) lies on these lines:
2,3   6,110   51,576   98,1302   100,344   107,264   157,1624   169,2000   182,373   184,575   197,1621   251,1184   323,1351   1383,1384   1611,1627   1915,2001

X(1995) = isogonal conjugate of X(5486)
X(1995) = inverse-in-orthocentroidal-circle of X(858)
X(1995) = X(598)-Ceva conjugate of X(6)
X(1995) = crossdifference of every pair of points on line X(647)X(690)
X(1995) = harmonic center of circumcircle and {circumcircle, nine-point circle}-inverter
X(1995) = Euler line intercept, other than X(378), of circle {X(378),PU(4)}

### X(1996) = ORTHOCORRESPONDENT OF X(7)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = yz + (-x cos A + y cos B + z cos C)x, where x : y : z = X(7)

X(1996) lies on these lines: 2,85   7,11

### X(1997) = ORTHOCORRESPONDENT OF X(8)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = yz + (-x cos A + y cos B + z cos C)x, where x : y : z = X(8)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1997) lies on these lines: 2,37   1210,1265

### X(1998) = ORTHOCORRESPONDENT OF X(9)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = yz + (-x cos A + y cos B + z cos C)x, where x : y : z = X(9)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1998) lies on these lines:
1,2   57,1004   63,1005   100,1445   224,1467   273,1897   1331,1743

### X(1999) = ORTHOCORRESPONDENT OF X(10)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = yz + (-x cos A + y cos B + z cos C)x, where x : y : z = X(10)
Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc(a3 + a2b + a2c + abc - b2c - bc2)      (M. Iliev, 5/13/07)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(1999) lies on these lines:
1,2   6,312   27,295   37,333   63,192   75,940   81,314   100,1402   171,740   193,329   226,1943   319,1211   350,1965   553,1266   664,1427

X(1999) = X(65)-Ceva conjugate of X(894)

### X(2000) = ORTHOCORRESPONDENT OF X(19)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = yz + (-x cos A + y cos B + z cos C)x, where x : y : z = X(19)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2000) lies on these lines:
1,2   9,1331   33,63   75,1897   169,1995   241,1004

### X(2001) = ORTHOCORRESPONDENT OF X(32)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = yz + (-x cos A + y cos B + z cos C)x, where x : y : z = X(32)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2001) lies on these lines: 2,98   1915,1995

### X(2002) = ORTHOCORRESPONDENT OF X(33)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = yz + (-x cos A + y cos B + z cos C)x, where x : y : z = X(33)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2002) lies on these lines:
1,1813   2,7   19,77   36,990   85,653   169,651   269,1781   1172,1790

### X(2003) = ORTHOCORRESPONDENT OF X(36)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = yz + (-x cos A + y cos B + z cos C)x, where x : y : z = X(36)
Trilinears        sin(3A/2) sec(A/2) : sin(3B/2) sec(B/2) : sin(3C/2) sec(C/2)    (M. Iliev, 4/12/07)
Trilinears        (sin A + sin 2A)/(1 + cos A) : (sin B + sin 2B)/(1 + cos B) : (sin C + sin 2C)/(1 + cos C)    (M. Iliev, 4/12/07)
Trilinears        (1 + 2 cos A) tan(A/2) : (1 + 2 cos B) tan(B/2) : (1 + 2 cos C) tan(C/2)     (M. Iliev, 4/12/07)
Trilinears       g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a(b2 + c2 - a2 + bc)/(b + c - a)      (M. Iliev, 5/13/07)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2003) lies on these lines:
1,90   6,57   9,394   35,500   42,109   56,1203   58,73   63,1993   65,267   77,1708   81,226   84,1181   212,991   255,581   323,1442   354,1421   386,603   648,1947   894,1943   1171,1400   1397,1469   1401,1428

X(2003) = X(1442)-Ceva conjugate of X(35)

### X(2004) = ORTHOCORRESPONDENT OF X(61)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = yz + (-x cos A + y cos B + z cos C)x, where x : y : z = X(61)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2004) lies on these lines: 2,13   32,2005   51,61

### X(2005) = ORTHOCORRESPONDENT OF X(62)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = yz + (-x cos A + y cos B + z cos C)x, where x : y : z = X(62)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2005) lies on these lines: 2,14   32,2004   51,62

### X(2006) = ORTHOCORRESPONDENT OF X(80)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = yz + (-x cos A + y cos B + z cos C)x, where x : y : z = X(80)
Trilinears       sin(A/2) sec(3A/2) : sin(B/2) sec(3B/2) : sin(C/2) sec(3C/2)    (M. Iliev, 4/12/07)
Trilinears       (tan A/2)/(1 - 2 cos A) : (tan B/2)/(1 - 2 cos B): (tan C/2)/(1 - 2 cos C)    (M. Iliev, 4/12/07)
Trilinears       g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = bc/[(b + c - a)(b2 + c2 - a2 - bc)]     (M. Iliev, 5/13/07)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2006) lies on these lines:
1,5   7,89   28,108   57,1020   79,1399   81,226   88,655   274,349   1427,1989   1758,1929

X(2006) = isogonal conjugate of X(2323)
X(2006) = cevapoint of X(1400) and X(1457)
X(2006) = X(2)-beth conjugate of X(651)

Gallatly Circle, etc., 2007- 2040
In his book The Modern Geometry of the Triangle, 2nd edition (Francis Hodgson, London, 1913), William Gallatly, on page 117, introduces the pedal cicle of the 1st and 2nd Brocard points. The circle is here named the Gallatly circle. Its center is the Brocard midpoint, X(39), and the radius, R sin ω, where ω denotes the Brocard angle, given by

tan ω = 4[area(ABC)]/(a2 + b2 + c2).

Other special symbols used in this section are identified just before X(1662).

Centers X(2007)-X(2040) were contributed with coordinates by Peter J. C. Moses during September, 2003. For centers of similitude of the Gallatly circle and the circumcircle, see X(1689) and X(1690); for those of the Gallatly circle and the 2nd Lemoine circle, see X(1671) and X(1670).

### X(2007) = INSIMILICENTER(GALLATLY CIRCLE, INCIRCLE)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1 + sin(A + ω)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(2007) lies on these lines:
1,39   11,2009   12,2010   55,1689   56,1690   57,2017   181,2019   371,1673   372,1672   1124,1670   1335,1671   1682,2020   1697,2018

### X(2008) = EXSIMILICENTER(GALLATLY CIRCLE, INCIRCLE)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1 - sin(A + ω)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(2008) lies on these lines:
1,39   11,2010   12,2009   55,1689   56,1690   57,2018   181,2020   371,1672   372,1673   1124,1671   1335,1670   1682,2019   1697,2017

### X(2009) = INSIMILICENTER(GALLATLY CIRCLE, NINE-POINT CIRCLE)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(B - C) + sin(A + ω)
Trilinears       h(A,B,C) : h(B,C,A): h(C,A,B), where h(A,B,C) = csc(A - ω/2 + π/4)     (M. Iliev, 5/13/07)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(2009) lies on the Kiepert hyperbola and these lines:
2,1689   4,1690   5,39   10,2020   11,2007   12,2008   83,1688   98,1687   485,1670   486,1671   1346,2016   1347,2015   1348,2012   1349,2011   1698,2018   1699,2017   2026,2040   2027,2039

X(2009) = reflection of X(2010) in X(115)
X(2009) = isogonal conjugate of X(1687)
X(2009) = antigonal conjugate of X(2010)

### X(2010) = EXSIMILICENTER(GALLATLY CIRCLE, NINE-POINT CIRCLE)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(B - C) - sin(A + ω)
Trilinears       h(A,B,C) : h(B,C,A): h(C,A,B), where h(A,B,C) = csc(A - ω/2 - π/4)     (M. Iliev, 5/13/07)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(2010) lies on the Kiepert hyperbola and these lines:
2,1690   4,1689   5,39   10,2019   11,2008   12,2007   83,1687   98,1688   485,1671   486,1670   1346,2015   1347,2016   1348,2011   1349,2012   1698,2017   1699,2018   2026,2039   2027,2040

X(2010) = reflection of X(2009) in X(115)
X(2010) = isogonal conjugate of X(1688)
X(2010) = antigonal conjugate of X(2009)

### X(2011) = INSIMILICENTER(GALLATLY CIRCLE, BROCARD CIRCLE)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(A - ω) + e sin(A + ω)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(2011) lies on these lines: 3,6   1348,2010   1349,2009

### X(2012) = EXSIMILICENTER(GALLATLY CIRCLE, BROCARD CIRCLE)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos(A - ω) - e sin(A + ω)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(2012) lies on these lines: 3,6   1348,2009   1349,2010

### X(2013) = INSIMILICENTER(GALLATLY CIRCLE, SPIEKER CIRCLE)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = [sin B + sin C + sin A sin(A + ω)](csc A)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(2013) lies on this line:
2,2008   8,2007   9,2018   10,39   371,1681   372,1680   958,1689   1329,2009   1376,1690   1377,1670   1378,1671   1678,2012   1679,2011   1706,2017

### X(2014) = EXSIMILICENTER(GALLATLY CIRCLE, SPIEKER CIRCLE)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = [sin B + sin C - sin A sin(A + ω)](csc A)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(2014) lies on these lines:
2,2007   8,2008   9,2017   10,39   371,1680   372,1681   958,1690   1329,2010   1376,1689   1377,1671   1378,1670   1678,2011   1679,2012   1706,2018

### X(2015) = INSIMILICENTER(GALLATLY CIRCLE, ORTHOCENTROIDAL CIRCLE)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = cos A + 4 cos B cos C + J sin(A + ω), J = |OH|/R; see X(1113)
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2015) lies on these lines: 39,381   1344,1689   1345,1690   1346,2010   1347,2009

### X(2016) = EXSIMILICENTER(GALLATLY CIRCLE, ORTHOCENTROIDAL CIRCLE)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = cos A + 4 cos B cos C - J sin(A + ω), J = |OH|/R; see X(1113)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2016) lies on these lines:
39,381   1344,1690   1345,1689   1346,2009   1347,2010

### X(2017) = INSIMILICENTER(GALLATLY CIRCLE, BEVAN CIRCLE)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1 + cos A - cos B - cos C + 2 sin(A + ω)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2017) lies on these lines:
1,1690   9,2014   10,2545   39,40   43,2019   57,2007   165,1689   371,1701   372,1700   516,2544   1670,1702   1671,1703   1695,2020   1697,2008   1698,2010   1699,2009   1704,2012   1705,2011 nbsp;  1706,2013   2562,2573   2563,2572

### X(2018) = EXSIMILICENTER(GALLATLY CIRCLE, BEVAN CIRCLE)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1 + cos A - cos B - cos C - 2 sin(A + ω)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(2018) lies on these lines:
1,1689   9,2013   10,2544   39,40   43,2020   57,2008   165,1690   371,1700   372,1701   516,2545   1670,1703   1671,1702   1695,2019   1697,2007   1698,2009   1699,2010   1704,2011   1705,2012   1706,2014   2562,2572   2563,2573

### X(2019) = INSIMILICENTER(GALLATLY CIRCLE, APOLLONIUS CIRCLE)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (s2 - r2) cos A - 2rs sin A + (s2 + r2) sin(A + ω),
where s = (a + b + c)/2 and r = inradius = [area(ABC)]/s

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2019) lies on these lines:
3,6   10,2010   43,2017   181,2007   1682,2008   1695,2018

X(2019) = reflection of X(2020) in X(2092)

### X(2020) = EXSIMILICENTER(GALLATLY CIRCLE, APOLLONIUS CIRCLE)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (s2 - r2) cos A - 2rs sin A - (s2 + r2) sin(A + ω),
where s = (a + b + c)/2 and r = inradius = [area(ABC)]/s

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2020) lies on these lines:
3,6   10,2009   43,2018   181,2008   1682,2007   1695,2017

X(2020) = reflection of X(2019) in X(2092)

### X(2021) = RADICAL TRACE OF GALLATLY CIRCLE AND CIRCUMCIRCLE

Trilinears    3 sin(A - ω) - 2 sin(A + ω) + sin(A + 3ω) : :
Trilinears    a(b6 + c6 + 3a4b2 + 3a4c2 - 2a2b4 - 2a2c4 - 2a2b2c2 - b2c4 - b4c2) : :     (M. Iliev, 5/13/07)

X(2021) lies on these lines:
3,6   30,2025   115,1513   538,1569   620,736   625,1506

X(2021) = midpoint of X(39) and X(187)
X(2021) = reflection of X(2025) in X(2024)
X(2021) = centroid of PU(1)PU(2)
X(2021) = X(230)-of-X(3)PU(1)

### X(2022) = RADICAL TRACE OF GALLATLY AND BROCARD CIRCLES

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) =
cos 5ω sin A + 3 sin(A - 3ω) + (cos A)(8 sin ω + sin 5ω)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(2022) lies on this line: 3,6

### X(2023) = RADICAL TRACE OF GALLATLY AND NINE-POINT CIRCLES

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin ω cos(B - C) - cos 2ω sin(A + ω)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(2023) lies on these lines:
2,694   5,39   6,98   30,2021   194,1007   230,511   325,732   625,736   1503,2024   1575,1738

X(2023) = midpoint of X(39) and X(115)
X(2023) = crosssum of X(1662) and X(1663)

### X(2024) = RADICAL TRACE OF GALLATLY AND 1st LEMOINE CIRCLES

Trilinears    sin A - 3 sin(A - 2ω) - 2 cos ω sin(A + 3ω) : :

X(2024) lies on these lines: 3,6   230,732   1503,2023

X(2024) = midpoint of X(i) and X(j) for these (i,j): (39,1692), (2021,2025)
X(2024) = radical trace of 1st Lemoine circle and circle {{X(371),X(372),PU(1),PU(39)}}
X(2024) = radical trace of Gallatly circle and circle {{X(371),X(372),PU(1),PU(39)}}

### X(2025) = RADICAL TRACE OF GALLATLY AND 2ndLEMOINE CIRCLES

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos 2ω sin(A + ω) - 2 sin ω tan ω sin A
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(2025) lies on this line: 3,6

X(2025) = midpoint of X(39) and X(1570)
X(2025) = reflection of X(2021) in X(2024)

### X(2026) = 1st BROCARD-AXIS-GALLATLY-CIRCLE INTERSECTION

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = e sin(A + ω) + cos(A + ω)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Of the two points of intersection of the Brocard axis, X(3)X(6) with the Gallatly circle, X(2026) is the one nearer to X(3).

X(2026) lies on these lines: 3,6   2009,2040   2010,2039

X(2026) = reflection of X(2027) in X(39)

### X(2027) = 2nd BROCARD-AXIS-GALLATLY-CIRCLE INTERSECTION

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = e sin(A + ω) - cos(A + ω)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(2027) lies on these lines: 3,6   2009,2039   2010,2040

X(2027) = reflection of X(2026) in X(39)

### X(2028) = 1st BROCARD-AXIS-MOSES-CIRCLE INTERSECTION

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A - e sin(A + ω) - sin(A + 2ω)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Of the two points of intersection of the Brocard axis, X(3)X(6) with the Moses circle, X(2028) is the one nearer to X(3). Points X(2028) and X(2029) are the points of intersection of the Brocard axis and the asymptotes of the Kiepert hyperbola; see X(2039) and X(2040). For a description of the Moses circle and others, see the notes just above X(1662).

X(2028) lies on the Brocard inellipse and these lines: 3,6   115,2039   1506,2040

X(2028) = crosssum of the minor vertices of the Steiner circumellipse

### X(2029) = 2nd BROCARD-AXIS-MOSES-CIRCLE INTERSECTION

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + e sin(A + ω) - sin(A + 2ω)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(2029) lies on the Brocard inellipse and these lines: 3,6   115,2040   1506,2039

X(2029) = crosssum of the vertices of the Steiner circumellipse

### X(2030) = RADICAL TRACE OF MOSES CIRCLE AND CIRCUMCIRCLE

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A - 4 sin(A - 2ω) - sin(A + 2ω)
Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(4a4 + b4 + c4 - a2b2 - a2c2 - 4b2c2)     (M. Iliev, 5/13/07)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(2030) lies on these lines: 3,6   111,1495   112,843   230,542   524,620

### X(2031) = RADICAL TRACE OF MOSES AND BROCARD CIRCLES

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 2 sin(A - 3ω) + (sin ω)[5 cos A + cos(A + 2ω)]
Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(4a6 + b6 + c6 - 5a4b2 - 5a4c2 + 4a2b4 + 4a2c4 - 6a2b2c2 + b2c4 + b4c2)     (M. Iliev, 5/13/07)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(2031) lies on these lines: 3,6   230,625

### X(2032) = RADICAL TRACE OF MOSES AND 1st LEMOINE CIRCLES

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (3 + 5 cos 4ω)sin A + (2 sin 2ω - 3 sin 4ω)cos A
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(2032) lies on this line: 3,6

### X(2033) = INSIMILICENTER(MOSES CIRCLE, BROCARD CIRCLE)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + e sin(A + ω) - sin(A - 2ω)
= 2 sin ω cos(A - ω) + e sin(A + ω)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(2033) lies on these lines: 3,6   115,1348   1349,1506

### X(2034) = EXSIMILICENTER(MOSES CIRCLE, BROCARD CIRCLE)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A - e sin(A + ω) - sin(A - 2ω)
= 2 sin ω cos(A - ω) - e sin(A + ω)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(2034) lies on these lines: 3,6   115,1349   1348,1506

### X(2035) = INSIMILICENTER(MOSES CIRCLE, 1st LEMOINE CIRCLE)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A + sin(A + ω) - sin(A - 2ω)
= 2 sin ω cos(A - ω) + sin(A + ω)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(2035) lies on these lines:
3,6   1015,1672   1500,1673   1571,1701   1572,1700   1573,1681   1574,1680

### X(2036) = EXSIMILICENTER(MOSES CIRCLE, 1st LEMOINE CIRCLE)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A - sin(A + ω) - sin(A - 2ω)
= 2 sin ω cos(A - ω) - sin(A + ω)

Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(2036) lies on these lines:
3,6   1015,1673   1500,1672   1571,1700   1572,1701   1573,1680   1574,1681

### X(2037) = 1st BROCARD-AXIS-APOLLONIUS-CIRCLE INTERSECTION

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = ea cot A - (2r' - es csc ω) cos(A + ω),
where r' = (r2 + s2)/(4r) = radius of Apollonius circle (where r = inradius)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Of the two points of intersection of the Brocard axis and Apollonius circle, X(2037) is the one nearer to X(3) and also nearer to X(6).

X(2037) lies on these lines: 3,6   10,2039   2040,2051

X(2037) = reflection of X(2038) in X(970)

### X(2038) = 2nd BROCARD-AXIS-APOLLONIUS-CIRCLE INTERSECTION

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = e cos(A + 2 arctan(r/s)) + cos(A + ω),
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2038) lies on these lines: 3,6   10,2040   2039,2051

X(2038) = reflection of X(2037) in X(970)

### X(2039) = 1st NINE-POINT-CIRCLE-KIEPERT-ASYMPTOTES INTERSECTION

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = e cos A + 2e cos B cos C + cos(A + ω)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Of the points other than X(115) in which the nine-point circle meets the asymptotes of the Kiepert hyperbola, X(2039) is the one nearer to X(3).

The points X(2039) and X(2040) are endpoints of the diameter of the nine-point circle that is parallel to the line X(25)X(394). (M. Iliev, 5/13/07)

X(2039) lies on these lines:
2,1379   3,1349   4,1380   5,141   6,1348   10,2037   115,2028   485,1667   486,1666   1506,2029   1662,1677   1663,1676   1670,2567   1671,2566   2009,2027   2010,2026   2038,2051

X(2039) = midpoint of X(4) and X(1380)
X(2039) = reflection of X(2040) in X(5)
X(2039) = complement of X(1379)
X(2039) = complementary conjugate of X(3413)

### X(2040) = 2nd NINE-POINT-CIRCLE-KIEPERT-ASYMPTOTES INTERSECTION

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = e cos A + 2e cos B cos C - cos(A + ω)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(2040) lies on these lines:
2,1380   3,1348   4,1379   5,141   6,1349   10,2038   115,2029   485,1666   486,1667   1506,2028   1662,1676   1663,1677   1670,2566   1671,2567   2009,2026   2010,2027   2037,2051

X(2040) = midpoint of X(4) and X(1379)
X(2040) = reflection of X(2039) in X(5)
X(2040) = complement of X(1380)
X(2040) = complementary conjugate of X(3414)

Euler-Vecten-Gibert Points, 2041 - 2046

On August 13, 2003, Bernard Gibert contributed six centers that lie on the Euler line and are related to the Vecten points, X(485) and X(486).

>

### X(2041) = 1st EULER-VECTEN-GIBERT POINT

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[(b2 - c2)2 - a2(b2 + c2) + 31/2a2(b2 + c2 - a2)]
Trilinears       g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin(B + π/4)sin(C + π/4) - cos(A - π/6)      (M. Iliev, 5/13/07)
Trilinears       h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = sin A + cos(B - C) - 2 cos(A - π/6)      (M. Iliev, 5/13/07)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2041) = 3X(2) - (1 + sqrt(3))*X(3)

As a point on the Euler line, X(2041) has Shinagawa coefficients (31/2 - 1, -31/2 - 1).

X(2041) lies on these lines: 2,3   15,485   16,486   489,622   490,621   491,628   492,627

X(2041) = reflection of X(2042) in X(382)

### X(2042) = 2nd EULER-VECTEN-GIBERT POINT

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[(b2 - c2)2 - a2(b2 + c2) - 31/2a2(b2 + c2 - a2)]
Trilinears       g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = sin(B + π/4)sin(C + π/4) + cos(A + π/6)      (M. Iliev, 5/13/07)
Trilinears       h(A,B,C) : h(B,C,A) : h(C,A,B), where h(A,B,C) = sin A + cos(B - C) + 2 cos(A + π/6)      (M. Iliev, 5/13/07)

Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2042) = 3X(2) + (-1 + sqrt(3))*X(3)

As a point on the Euler line, X(2042) has Shinagawa coefficients (31/2 + 1, -31/2 + 1).

X(2042) lies on these lines: 2,3   15,486   16,485   489,621   490,622   491,627   492,628

X(2042) = reflection of X(2041) in X(382)

### X(2043) = 3rd EULER-VECTEN-GIBERT POINT

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[- 2(b2 - c2)2 + a2(a2 + b2 + c2) + 31/2[(b2 - c2)2 - a2(b2 + c2)]]
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2043) = 3X(2) - (3 + sqrt(3))*X(3)

As a point on the Euler line, X(2043) has Shinagawa coefficients (31/2 - 1, -3 + 31/2).

X(2043) lies on these lines:
2,3   13,485   14,486   17,1327   18,1328   298,637   299,638   489,616   490,617   491,622   492,621

X(2043) = reflection of X(2044) in X(3)
X(2043) = inverse-in-orthocentroidal-circle of X(2044)
X(2043) = {X(2),X(4)}-harmonic conjugate of X(2044)
X(2043) = {X(3),X(4)}-harmonic conjugate of X(2042)
X(2043) = {X(5),X(381)}-harmonic conjugate of X(2044)
X(2043) = {X(20),X(376)}-harmonic conjugate of X(2044)
X(2043) = {X(382),X(549)}-harmonic conjugate of X(2044)

### X(2044) = 4th EULER-VECTEN-GIBERT POINT

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[- 2(b2 - c2)2 + a2(a2 + b2 + c2) - 31/2[(b2 - c2)2 - a2(b2 + c2)]]
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2044) = 3X(2) + (-3 + sqrt(3))*X(3)

As a point on the Euler line, X(2044) has Shinagawa coefficients (31/2 + 1,3 + 31/2).

X(2044) lies on these lines:
2,3   13,486   14,485   17,1328   18,1327   298,638   299,637   489,617   490,616   491,621   492,622

X(2044) = reflection of X(2043) in X(3)
X(2044) = inverse-in-orthocentroidal-circle of X(2043)

### X(2045) = 5th EULER-VECTEN-GIBERT POINT

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[- 2(b2 - c2)2 + a2(3b2 + 3c2 - a2) + 31/2[(b2 - c2)2 - a2(b2 + c2)]]
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2045) = 3X(2) + (1 + sqrt(3))*X(3)

As a point on the Euler line, X(2045) has Shinagawa coefficients (3 - 31/2,1 - 31/2).

X(2045) lies on these lines:
2,3   17,485   18,486   302,637   303,638   397,590   398,615   491,634   492,633

X(2045) = inverse-in-orthocentroidal-circle of X(2046)
X(2044) = X(1)-of-X(381)PU(5)
X(2044) = {X(2),X(4)}-harmonic conjugate of X(2043)
X(2044) = {X(3),X(4)}-harmonic conjugate of X(2041)
X(2044) = {X(5),X(381)}-harmonic conjugate of X(2043)
X(2044) = {X(20),X(376)}-harmonic conjugate of X(2043)
X(2044) = {X(382),X(549)}-harmonic conjugate of X(2043)

### X(2046) = 6th EULER-VECTEN-GIBERT POINT

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = bc[- 2(b2 - c2)2 + a2(3b2 + 3c2 - a2) - 31/2[(b2 - c2)2 - a2(b2 + c2)]]
Barycentrics  af(a,b,c) : bf(b,c,a) : cf(c,a,b)
X(2046) = 3X(2) + (1 - sqrt(3))*X(3)

As a point on the Euler line, X(2046) has Shinagawa coefficients (3 + 31/2,1 + 31/2).

X(2046) lies on these lines:
2,3   17,486   18,485   302,638   303,637   397,615   398,590   491,633   492,634

X(2046) = inverse-in-orthocentroidal-circle of X(2045)

### X(2047) = EULER LINE INTERCEPT OF LINE X(10)X(485)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (s - r)cos A + (s + r)cos(B -C),
where r = inradius, s = semiperimeter
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(2047) has Shinagawa coefficients (s,r).

X(2047) lies on these lines:
2,3   10,485   86,637   492,1330   590,1834   966,1587

### X(2048) = {X(3),X(5)}-HARMONIC CONJUGATE OF X(2047)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (s - r)cos A - (s + r)cos(B - C),
where r = inradius, s = semiperimeter
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(2048) has Shinagawa coefficients (r, s).

X(2048) lies on these lines: 2,3   486,1686

### X(2049) = EULER LINE INTERCEPT OF LINE X(6)X(10)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (s2 - r2)cos A + (s2 + r2)cos(B - C),
where r = inradius, s = semiperimeter
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(2049) has Shinagawa coefficients (s2,r2).

X(2049) lies on these lines: 2,3   6,10   12,1460   171,1698

### X(2050) = {X(3),X(5)}-HARMONIC CONJUGATE OFX(2049)

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (s2 - r2)cos A - (s2 + r2)cos(B - C),
where r = inradius, s = semiperimeter
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

As a point on the Euler line, X(2050) has Shinagawa coefficients (r2, s2).

X(2050) lies on these lines: 2,3   11,1460   171,1699   1482,1999

### X(2051) = EXSIMILICENTER(NINE-POINT CIRCLE, APOLLONIUS CIRCLE)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = (s2 - r2)cos A - (s2 + r2)cos(B - C) - 2rs sin A (Peter J. C. Moses)
Barycentrics   g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 1/[a3 - a(b2 - bc + c2) - bc(b + c)] (Paul Yiu)

X(2051) is the external center of similitude of the nine-point and Apollonius circles. Trilinears for the internal center, X(10), result from g(a,b,c) by changing the "-" just after "cos A" to "+". (The two circles are described just before X(1662).)

X(2051) lies on these lines:
2,573   4,386   5,10   6,2050   11,181   12,1682   27,275   43,1699   226,1465   321,908   469,2052   485,1685   486,1686   572,2185   1348,1693   1349,1694   1676,1683   1677,1684   1695,1698   1766,2339   2009,2019   2101,2020   2037,2040

### X(2052) = ISOGONAL CONJUGATE OF X(577)

Trilinears    csc A sec2A : :
Trilinears    sec A csc 2A : :
Trilinears    sec(A - T) : : , T as at X(389)
Barycentrics    sec2A : :
Barycentrics    b^2c^2/(b^2 + c^2 - a^2)^2 : :

Let A'B'C' be the orthic triangle of a triangle ABC, and let
O(A) = circle with center A and radius AA', and define O(B) and O(C) cyclically
p(A) = polar of X(4) wrt O(A), and define p(B) and p(C) cyclically
A'' = p(B)∩p(C), and define B'' and C'' cyclically.
Then A''B''C'' is homothetic to ABC, and the center of homothety is X(2052). (Angel Montesdeoca, September 30, 2016)

X(2052) lies on these lines: 2,216   4,51   6,275   10,158   13,473   14,472   17,470   18,471   24,96   25,98   63,1947   76,297   83,458   92,226   184,436   262,427   331,1446   394,801   401,1968   485,1585   486,1586   648,1993

X(2052) = isogonal conjugate of X(577)
X(2052) = isotomic conjugate of X(394)
X(2052) = cevapoint of X(i) and X(j) for these (i,j): (4,393), (6,24), (324,467), (1585,1586)
X(2052) = X(i)-cross conjugate of X(j) for these (i,j): (4,264), (6,847), (53,4), (393,1093), (686,1304)
X(2052) = trilinear pole of line X(403)X(523) (polar of X(3) wrt polar circle)
X(2052) = pole wrt polar circle of trilinear polar of X(3) (line X(520)X(647))
X(2052) = polar conjugate of X(3)
X(2052) = X(43)-of-orthic-triangle if ABC is acute

### X(2053) = X(2)-ISOCONJUGATE OF X(1423)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c - a)/(ab + ac - bc)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2053) lies on these lines:
3,2108   6,904   31,172   41,2330   55,2319   56,87   105,330   1403,1580   2115,2144

X(2053) = isogonal conjugate of X(3212)
X(2053) = X(9)-cross conjugate of X(55)
X(2053) = crosssum of X(43) and X(1423)

### X(2054) = X(2)-ISOCONJUGATE OF X(1931)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b + c)/[(b + c)2 - (a + b)(a + c)]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2054) lies on these lines: 2,846   110,1171   111,902   238,1931

Orion Transforms, 2055 - 2065
On September 24, 2003, Jean-Pierre Ehrmann suggested in Hyacinthos 7999 a transformation: suppose P = x : y : z (trilinears); the Orion transform of P is given by OT(P) = f(a,b,c) : f(b,c,a) : f(c,a,b), where

f(a,b,c) = x[- y2z2 + x2(y2 + z2 + 2yz cos A)].

Among pairs (i,j) for which OT(X(j)) = X(i) are these:

(1,35), (2,69), (4,24), (7,57), (99,249), (100,59), (110,250)

Ehrmann's construction of OT(P) is as the perspector of triangles ABC and A"B"C", where A" is the reflection of P in the sideline B'C' of the cevian triangle A'B'C' of P, and B" and C' are defined cyclically. Alternative and related constructions are given in other Hyacinthos messages prompted by #7999.

### X(2055) = ORION TRANSFORM OF X(3)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is given just above X(2055), using X = X(3)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2055) is the perspector of triangle ABC and the orthic triangle of the cevian triangle of the circumcenter. (Randy Hutson, 9/23/2011)

X(2055) lies on these lines: 3,6   5,275   54,418

### X(2056) = ORION TRANSFORM OF X(6)

Trilinears    a(a4 - 2a2b2 - 2a2c2 + b2c2) : :      (M. Iliev, 5/13/07)

X(2056) lies on these lines: 6,1196   110,251   184,1613

X(2056) = perspector of ABC and orthic triangle of symmedial triangle

### X(2057) = ORION TRANSFORM OF X(8)

Trilinears    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is given just above X(2055), using X = X(8)

X(2057) lies on these lines: 1,2   84,1259   100,1490

X(2057) = perspector of ABC and orthic triangle of extouch triangle

### X(2058) = ORION TRANSFORM OF X(15)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is given just above X(2055), using X = X(15)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

### X(2059) = ORION TRANSFORM OF X(16)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is given just above X(2055), using X = X(16)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

### X(2060) = ORION TRANSFORM OF X(20)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is given just above X(2055), using X = X(20)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

As a point on the Euler line, X(2060) has Shinagawa coefficients (4(E - 2F)F - S2,8F2).

X(2060) lies on this line: 2,3

### X(2061) = ORION TRANSFORM OF X(40)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is given just above X(2055), using X = X(40)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2061) = R(4R2 - s2 - S)(4R2 - s2 + S)*X(1) + 2(4rR + 4R2 - s2)(4R3 + rs2 - Rs2)*X(3)    (Peter Moses, April 2, 2012)

X(2061) lies on this line: 1,3

### X(2062) = ORION TRANSFORM OF X(63)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is given just above X(2055), using X = X(63)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2062) lies on these lines: 3,77   21,307   284,2003   1167,1672

### X(2063) = ORION TRANSFORM OF X(69)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is given just above X(2055), using X = X(69)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2063) lies on these lines: 2,6   64,207   110,1619   264,801   378,1092

### X(2064) = ORION TRANSFORM OF X(75)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b2c2(a4 - b4 - c4 + a2bc - b3c - bc3)     (M. Iliev, 5/13/07)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2064) lies on these lines: 76,85   81,314   278,345   329,1264

### X(2065) = ORION TRANSFORM OF X(98)

Trilinears    f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is given just above X(2055), using X = X(98)

X(2065) lies on these lines: 98,325   232,1692   248,1570   460,685   511,1976

X(2065) = isogonal conjugate of X(114)
X(2065) = X(3)-cross conjugate of X(98)
X(2065) = cevapoint of X(6) and X(1976)
X(2065) = perspector of ABC and cross-triangle of ABC and circumcevian triangle of X(511)

### X(2066) = POINT CAROLI I

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 1 + sin A + cos A
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(2066) lies on these lines: 1,371   3,1124   6,31   11,590   35,372   56,1151   283,1806   405,1378   485,1479   486,498

X(2066) = isogonal conjugate of X(1659)
X(2066) = cevapoint of X(48) and X(605)

### X(2067) = POINT CAROLI II

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = - 1 + sin A + cos A
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(2067) lies on these lines: 1,371   3,1335   6,41   12,590   27,1659   36,372   55,1151   474,1377   485,1478   486,499   603,606   956,1378   999,1124   1790,1805

### X(2068) = POINT CURSA I

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a + b1/2c1/2
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2068) lies on these lines: 6,366   43,365   364,1743

### X(2069) = POINT CURSA II

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a - b1/2c1/2
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2069) lies on these lines: 1,364   6,366

Inverses in Circumcircle, 2070 - 2080
Suppose X is a point in the plane of triangle ABC, other than the circumcenter O and not on the line at infinity. The inverse-in-circumcircle of X is the point X' on ray OX whose distance from O is R2/|OX|2. In normalized trilinears, if

X = (x, y, z)     O = (x3, y3, z3),    R = circumradius,

then X' is given as (f(a,b,c), f(b,c,a), f(c,a,b)) by f(a,b,c) = x3 + (x - x3)(R/|OX|)2.

In this section, representations using the circumradius R, inradius r, Brocard angle ω and J as defined at X(1113), were contributed by Peter J. C . Moses.

### X(2070) = INVERSE-IN-CIRCUMCIRCLE OF X(5)

Trilinears    (J2 - 2) cos A + 4 cos B cos C, where J is as at X(1113)

As a point on the Euler line, X(2070) has Shinagawa coefficients (J2 - 2, -J2 + 6).

X(2070) lies on these lines:
2,3   49,52   51,567   54,143   98,1287   110,1154   184,568   187,2079   231,1989   399,1495   476,1141   500,501   827,842

X(2070) = midpoint of X(23) and X(186)
X(2070) = reflection of X(i) in X(j) for these (i,j): (3,186), (2072,468)
X(2070) = inverse-in-circumcircle of X(5)
X(2070) = X(94)-Ceva conjugate of X(6)
X(2070) = crosspoint of X(i) and X(j) for these (i,j): (250,476), (1141,1166)
X(2070) = crosssum of X(i) and X(j) for these (i,j): (125,526), (1154,1209)
X(2070) = center of circle {{X(23),X(186),PU(4)}}, which is the inverse-in-circumcircle-of-orthocentroidal-circle
X(2070) = inverse-in-tangential-circle of X(3)
X(2070) = pole wrt circumcircle of line PU(5) (line X(5)X(523))
X(2070) = tangential isogonal conjugate of X(399)
X(2070) = X(36)-of-tangential-triangle if ABC is acute
X(2070) = {X(3),X(26)}-harmonic conjugate of X(2937)
X(2070) = crossdifference of every pair of points on line X(570)X(647)

### X(2071) = INVERSE-IN-CIRCUMCIRCLE OF X(20)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (J2 + 1) cos A - 2 cos B cos C, where J is as at X(1113)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

As a point on the Euler line, X(2071) has Shinagawa coefficients (J2 + 1, -J2 - 3).

Let A'B'C' be the antipedal triangle of X(64). The circumcircles of AA'X(64), BB'X(64), CC'X(64) concur in two points: X(64) and X(2071). Also, let A'B'C' be the Trinh triangle. Let A" be the cevapoint of B' and C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(2071). (Randy Hutson, October 13, 2015)

X(2071) lies on these lines: 2,3   64,2063   74,323   99,1236   476,1294   477,925   691,1297   1290,1295

X(2071) = reflection of X(i) in X(j) for these (i,j): (4,2072), (23,186), (186,3)
X(2071) = inverse-in-circumcircle of X(20)
X(2071) = anticomplement of X(403)
X(2071) = crosspoint of X(3) and X(2935) wrt both the excentral and tangential triangles
X(2071) = crossdifference of every pair of points on line X647)X(800)
X(2071) = inverse-in-de-Longchamps-circle of X(4)
X(2071) = inverse-in-first-Droz-Farny-circle of X(5)
X(2071) = X(36) of Trinh triangle if ABC is acute
X(2071) = Trinh-isogonal conjugate of X(74)
X(2071) = insimilicenter of circumcircle and Trinh circle (the exsimilicenter is X(3520))

### X(2072) = INVERSE-IN-CIRCUMCIRCLE OF X(26)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (J2 + 1) cos A + 2(J2 - 1) cos B cos C, where J is as at X(1113)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

As a point on the Euler line, X(2072) has Shinagawa coefficients (J2 + 1, J2 - 3).

X(2072) lies on these lines: 2,3   125,1568   127,625   325,339   842,1286

X(2072) = midpoint of X(i) and X(j) for these (i,j): (4,2071), (125,1568), (403,858)
X(2072) = reflection of X(i) in X(j) for these (i,j): (403,5), (2070,468)
X(2072) = inverse-in-circumcircle of X(26)
X(2072) = inverse-in-nine-point-circle of X(3)
X(2072) = complement of X(186)
X(2072) = complementary conjugate of X(1511)
X(2072) = crosspoint of X(2) and X(328)

X(2072) = center of inverse-in-nine-point-circle-of-orthocentroidal-circle
X(2072) = crossdifference of every pair of points on line X(571)X(647)
X(2072) = inverse-in-polar-circle of X(24)
X(2072) = inverse-in-complement-of-polar-circle of X(140)
X(2072) = inverse-in-{circumcircle, nine-point circle}-inverter of X(22)
X(2072) = inverse-in-first-Droz-Farny-circle of X(3)

### X(2073) = INVERSE-IN-CIRCUMCIRCLE OF X(27)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is as noted just above X(2070) using X = X(27)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

As a point on the Euler line, X(2073) has Shinagawa coefficients (EF(R/|OX|)2 - (\$a\$\$bc\$ - abc)F{1 - (R/|OX|)2}, -[(E + F) + \$bc\$]E(R/|OX|)2 + (\$a\$\$bc\$ - abc)F{1 - (R/|OX|)2}).

X(2073) lies on these lines: 2,3   35,270   103,1304   110,916   476,917   675,935

X(2073) = inverse-in-circumcircle of X(27)

### X(2074) = INVERSE-IN-CIRCUMCIRCLE OF X(28)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is as noted just above X(2070) using X = X(28)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

As a point on the Euler line, X(2074) has Shinagawa coefficients (abc\$a\$F(R/|OX|)2 - [2\$a\$(E + F) - \$a3\$ + 2abc]FS{1 - (R/|OX|)2}, -[abc\$a\$(E+F) + ES2](R/|OX|)2 + [2\$a\$(E+F) - \$a3\$ + 2abc]FS{1 - (R/|OX|)2}).

X(2074) lies on these lines: 2,3   104,1304   105,935   110,912   162,1870   476,915

X(2074) = inverse-in-circumcircle of X(28)

### X(2075) = INVERSE-IN-CIRCUMCIRCLE OF X(29)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is as noted just above X(2070) using X = X(29)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

As a point on the Euler line, X(2075) has Shinagawa coefficients ([\$a\$S2 - \$aSBSC\$ - 2\$aSA\$F]FS2{1 - (R/|OX|)2} + abc[2F2S2 + (E - 2F)FS2](R/|OX|)2, -[\$a\$S2 + \$aSBSC\$ + 2\$aSA\$F]FS2{1 - (R/|OX|)2} + abc[2F2S2 + {(E - 2F)FS2 - E\$bcSBSC\$}](R/|OX|)2).

X(2075) lies on these lines: 2,3   36,162   102,1304   250,1101   935,1311

X(2075) = inverse-in-circumcircle of X(29)

### X(2076) = INVERSE-IN-CIRCUMCIRCLE OF X(39)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin A - 2 cos A sin 2ω
Trilinears       g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a(a4 - b4 - c4 + a2b2 + a2c2 - b2c2)     (M. Iliev,5/13/07)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

Let T1 be the circumcevian triangle of the 1st Brocard point, and let T2 be the circumcevian triangle of the 2nd Brocard point. Let VAVBVC be the vertex triangle of T1 and T2, and let SASBSC be the side triangle of T1 and T2. The lines VASA, VBSB, VCSC concur in X(2076).

Let T be the reflection of the symmedial triangle in the Lemoine axis; i.e., the reflection of the cevian triangle of X(6) in the trilinear polar of X(6). The triangle T is perspective to ABC, and the perspector is X(2076. (Randy Hutson, February 16, 2015)

X(2076) lies on these lines:
3,6   22,1613   99,732   141,384   385,698   599,1003   691,755   733,805   904,1964

X(2076) = reflection of X(i) in X(j) for these (i,j): (6,1691), (1691,187)
X(2076) = inverse-in-circumcircle of X(39)
X(2076) = X(694)-Ceva conjugate of X(6)
X(2076) = crosspoint of X(249) and X(805)
X(2076) = crosssum of X(115) and X(804)
X(2076) = reflection of X(1691) in the Lemoine axis
X(2076) = radical center of Lucas(-csc 2ω) circles
X(2076) = intersection of tangents to circumcircle at intersections with line X(39)X(512)
X(2076) = crossdifference of every pair of points on line X(523)X(3589)
X(2076) = {X(3),X(6)}-harmonic conjugate of X(5116)
X(2076) = perspector of tangential triangle and the vertex triangle of the anticevian triangles of PU(1)
X(2076) = Brocard axis intercept, other than X(3), of circle {X(3),PU(2)}

### X(2077) = INVERSE-IN-CIRCUMCIRCLE OF X(40)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = r + 2(r - R) cos A
= g(A,B,C) : g(B,C,A) : g(C,A,B),
where g(A,B,C) = cos 2A + cos B + cos C + (2 cos B + 2 cos C - 3) cos A

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2077) lies on these lines:
1,3   24,1753   30,119   78,1158   84,1259   100,515   102,901   104,519   122,856   376,535   386,601   404,946   516,1519   912,1768   953,1293   972,1308   1012,1376   1593,1878   1618,1818

X(2077) = reflection of X(36) in X(3)
X(2077) = inverse-in-circumcircle of X(40)
X(2077) = X(186) of 1st circumperp triangle
X(2077) = crossdifference of every pair of points on line X(650)X(1108)

### X(2078) = INVERSE-IN-CIRCUMCIRCLE OF X(57)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = r - 2R + 2(r + R) cos A
= g(A,B,C) : g(B,C,A) : g(C,A,B),
where g(A,B,C) = cos B + cos C - 3 + (1 + 2 cos A + 2 cos B + 2 cos C) cos A
Trilinears       g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = a(a2 + b2 + c2 - 2ab - 2ac + bc)/(b+c-a)     (M. Iliev, 5/13/07)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2078) lies on these lines:
1,3   31,2003   59,672   73,595   105,2006   109,840   226,1005   388,535   581,1497   901,1477   1174,1202   1279,1421   1283,1284   1308,1323

X(2078) = isogonal conjugate of X(3254)
X(2078) = inverse-in-circumcircle of X(57)
X(2078) = crosssum of X(142) and X(527)
X(2078) = inverse-in-{circumcircle, incircle}-inverter of X(65)
X(2078) = {X(3513),X(3514)}-harmonic conjugate of X(36)

### X(2079) = INVERSE-IN-CIRCUMCIRCLE OF X(115)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is as noted just above X(2070) using X = X(115)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2079) lies on the tangential circle and these lines:
3,115   6,1511   22,111   24,112   25,1560   186,230   187,2070

X(2079) = inverse-in-circumcircle of X(115)
X(2079) = center of bicevian conic of X(1113) and X(1114)

### X(2080) = INVERSE-IN-CIRCUMCIRCLE OF X(182)

Trilinears    2 cos A - cos(A - 2ω)

The locus of X(23) in a Brocard porism (triangles sharing circumcircle and Brocard inellipse with ABC) is a circle with center X(2080) and passing through X(23), X(385) and X(11676). This circle is the reflection of the circumcircle in X(187). (Randy Hutson, August 29, 2018)

X(2080) lies on these lines:
3,6   5,316   30,98   83,140   110,237   114,754   625,1656

X(2080) = reflection of X(i) in X(j) for these (i,j): (3,187), (316,5)
X(2080) = inverse-in-circumcircle of X(182)
X(2080) = reflection of X(3) in the Lemoine axis
X(2080) = pole of line X(23)X(352) wrt Parry circle
X(2080) = crossdifference of every pair of points on line X(523)X(3815)
X(2080) = {X(1687),X(1688)}-harmonic conjugate of X(187)
X(2080) = harmonic center of circumcircle and Ehrmann circle
X(2080) = pole of Lemoine axis wrt circle {{X(1687),X(1688),PU(1),PU(2)}}
X(2080) = intersection of Brocard axes of ABC and Artzt triangle
X(2080) = X(23)-of-X(3)PU(1)

PK and NK Transforms, 2081 -  2088
On October 1, 2003, Bernard Gibert described two transforms, given for X = u : v : w (barycentrics) by

PK(X) = a2(c2v2 - b2w2)u : b2(a2w2 - c2u2)v : c2(b2u2 - a2v2)w;

NK(X) = a2(c2v2 + b2w2)u : b2(a2w2 + c2u2)v : c2(b2u2 + a2v2)w.

Let X -1 denote the isogonal conjugate of X. Then PK(X) is the point of intersection of the trilinear polar of X and the trilinear polar of X -1, and NK(P) is the pole of the line XX -1 with respect to the conic that passes through points A, B, C, X, and X -1.

Also, PK(X) is the crossdifference, and NK(X) the crosssum, of X and X -1.

The transform PK carries any point on the cubic pK(X6,X) to the trilinear polar of X -1, and NK carries any point on the cubic nK0(X6,X) to the trilinear polar of X -1. (The notations for these cubics are given at Bernard Gibert's Cubics in the Triangle Plane.)

PK(X(i)) = X(j) for these (i,j):
(2,512), (3,647), (4,647), (6,512), (9,663), (19,810), (57,663), (63,810)

NK(X(i)) = X(j) for these (i,j):
(1,1), (2,30), (3,185), (4,185), (6,39), (98,446), (511,446)

In trilinears: suppose X = x : y : z; then

PK(X) = (y2 - z2)x : (z2 - x2)y : (x2 - y2)z;

NK(X) = (y2 + z2)x : (z2 + x2)y : (x2 + y2)z.

The self-isogonal cubic pK(X6,X) is denoted as Z(X) in TCCT (p. 240), and nK0(K6,X), as Z+(L). Extending examples on p. 240-241,

Z(X2) passes through X(i) for I = 1,2,3,4,6,9,57, 223, 282, 1073, 1249.
Z(X4) passes through X(i) for I = 1,3,4,46,90,155,254,371,372,485,486,487,488.
Z(X5) passes through X(i) for I = 1,3,4,5,17,18,54,61,62,195,627,628.
Z(X20) passes through X(i) for I = 1,3,4,20,40,64,84,1490,1498.
Z(X30) passes through X(i) for I = 1,3,4,13,14,15,16, 30, 74, 399, 484, 616,617,1138,1157,1263,1276,1277,1337,1338.
Z(X98) passes through X(i) for I = 1,98,511,1687,1688,1756,2009,2010.
Z(X99) passes through X(i) for I = 1,39,83,99,512,1018,1019,1379,1380.
Z(X100) passes through X(i) for I = 1,100,513,1381,1382.
Z(X110) passes through X(i) for I = 1,5,54,110,523,1113,1114.

Z+(X1X6) passes through X(i) for I = 1,44,88,239,241,292,294,1931.
Z+(X3X6) passes through X(i) for I = 2,6,13,14,15,16,111,524
Z+(X1X2) passes through X(i) for I = 1,238,291,899
Z+(X1X3) passes through X(i) for I = 1,105,243,296,518,1155,1156

Additional pass-through points for these and other cubics are given at Bernard Gibert's Cubics in the Triangle Plane.

### X(2081) = PK-TRANSFORM OF X(5)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = [cos2(A - B) - cos2(A - C)]cos(B - C)
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(2081) lies on these lines: 50,647   115,125

X(2081) = X(50)-Ceva conjugate of X(2088)
X(2081) = crosspoint of X(94) and X(110)
X(2081) = crosssum of X(50) and X(523)

### X(2082) = NK-TRANSFORM OF X(9)

Trilinears    (b + c - a)[a2 + (b - c)2] : :
Trilinears    SBSC - a2bc : :
Trilinears    [a^2 + (b - c)^2]/[a^2 - (b - c)^2] : :

X(2082) lies on these lines:
1,41   6,19   8,9   32,1951   40,672   55,1212   57,279   63,194   85,673   198,1108   213,1572   218,517   284,1800   354,1190   573,1723   579,1195   604,610   614,1184   1039,1172   1055,1420   1146,1837   1743,1766

X(2082) = X(664)-Ceva conjugate of X(663)
X(2082) = crosspoint of X(i) and X(j) for these (i,j): (4,277), (9,57), (333,1172)
X(2082) = crosssum of X(i) and X(j) for these (i,j): (3,218), (9,57), (664,663)

X(2082) = bicentric sum of PU(115)
X(2082) = PU(115)-harmonic conjugate of X(663)
X(2082) = eigencenter of cevian triangle of X(664)
X(2082) = eigencenter of anticevian triangle of X(663)

### X(2083) = NK-TRANSFORM OF X(19)

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (tan2B + tan2C) tan A
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(2083) lies on these lines:
1,163   19,158   48,1820   63,304   774,1973   920,1755   1497,1953   1760,1966   1821,1969

X(2083) = X(811)-Ceva conjugate of X(810)
X(2083) = crosspoint of X(19) and X(63)
X(2083) = crosssum of X(19) and X(63)

### X(2084) = PK-TRANSFORM OF X(31)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b4 - c4)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2084) lies on these lines: 512,1500   514,661   667,788

X(2084) = X(i)-cross conjugate of X(j) for these (i,j): (782,880), (881,783)
X(2084) = crosspoint of X(661) and X(798)
X(2084) = crosssum of X(i) and X(j) for these (i,j): (662,799), (2084,2085)

### X(2085) = NK-TRANSFORM OF X(31)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b4 + c4)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2085) lies on these lines: 1,1581   31,1917   75,1928   1755,1923

X(2085) = crosspoint of X(31) and X(75)
X(2085) = crosssum of X(31) and X(75)

### X(2086) = PK-TRANSFORM OF X(99)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 - c2)2(a4 - b2c2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2086) lies on these lines: 2,6   351,865   671,729

X(2086) = X(i)-Ceva conjugate of X(j) for these (i,j): (98,669), (694,512)
X(2086) = crosspoint of X(512) and X(694)
X(2086) = crosssum of X(99) and X(385)

### X(2087) = PK-TRANSFORM OF X(100)

Trilinears    (b + c - 2a)(b - c)2 : :

X(2087) lies on these lines: 1,6   244,665   1914,1983

X(2087) = X(i)-Ceva conjugate of X(j) for these (i,j): (44,1635), (88,513), (104,667), (1022,764), (1319,1960)
X(2087) = crosspoint of X(i) and X(j) for these (i,j): (1,1022), (44,1635), (88,513)
X(2087) = crosssum of X(i) and X(j) for these (i,j): (1,1023), (44,100)
X(2087) = bicentric difference of PU(114)
X(2087) = PU(114)-harmonic conjugate of X(6161)

### X(2088) = PK-TRANSFORM OF X(110)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a(b2 - c2)2[(b2 + c2 - a2)2 - b2c2]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2088) lies on these lines: 3,6   115,125

X(2088) = X(i)-Ceva conjugate of X(j) for these (i,j): (50,2081), (74,512), (94,523), (323,526)
X(2088) = crosspoint of X(i) and X(j) for these (i,j): (94,523), (323,526)
X(2088) = crosssum of X(i) and X(j) for these (i,j): (50,110), (112,403), (476,1989)

### X(2089) = 3rd MID-ARC POINT

Trilinears    (cos B/2 + cos C/2 - cos A/2) sec A/2 : :
Trilinears    1/(1 - sin A/2) : :
Trilinears    csc^2(A'/2) : :, where A'B'C' = excentral triangle

Let A', B', C' be the first points of intersection of the angle bisectors of triangle ABC with its incircle. Let A" B" C" be the triangle formed by the lines tangent to the incircle at A', B', C'. Then A"B"C" is perspective to the intouch triangle of ABC, and the perspector is X(2089). (Darij Grinberg, Hyacinthos #8072, 10/01/03)

X(2089) lies on these lines: 1,167   2,178   7,1488

X(2089) = X(7)-Ceva conjugate of X(174)
X(2089) = X(173)-cross conjugate of X(174)
X(2089) = X(56)-of-intouch triangle
X(2089) = SS(A->A')-of-X(8), where A'B'C' is the excentral triangle
X(2089) = trilinear pole of perspectrix of ABC and Yff central triangle

### X(2090) = 1st STEVANOVIC PERSPECTOR

Trilinears        f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = (sin B/2 + sin C/2) csc A
Barycentrics   (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

The bisector AX(1) of angle A meets the A-excircle in two points; let A' be the point closer to I, and define B' and C' cyclically. Then triangle A'B'C' is perspective to the medial triangle, and the perspector is X(2090). (Milorad Stevanovic, Hyacinthos #8088, 10/02/03)

X(2090) lies on these lines: 2,174   8,188   9,362   85,555   177,178   312,556

X(2090) = complement of X(174)
X(2090) = crosspoint of X(2) and X(556)

### X(2091) = 2nd STEVANOVIC PERSPECTOR

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = [1/(b + c - a)](sin B/2 + sin C/2) sec A/2
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

The bisector AX(1) of angle A meets the A-excircle in two points; let A' be the point closer to I, and define B' and C' cyclically. Then triangle A'B'C' is perspective to the intouch triangle, and the perspector is X(2091). (Milorad Stevanovic, Hyacinthos #8088, 10/02/03)

X(2091) lies on this line: {7,4146}, {57,173}

### X(2092) = DANNEELS-APOLLONIUS PERSPECTOR

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = a2(b + c)2 + a(b + c)(b2 + c2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

Let A' be the point common to Apollonius circle and the A-excircle, and define B' and C' cyclically. Then triangle A'B'C' is perspective to the cevian triangle of X(6), and the perspector is X(2092). (Eric Danneels, Hyacinthos, #8070, 10/01/03). (Triangle A'B'C' is also perspective to the cevian triangle of X(1); see X(1682).)

X(2092) lies on these lines:
{1,2277}, {2,314}, {3,6}, {4,3597}, {9,43}, {10,37}, {41,2273}, {42,181}, {44,3647}, {69,980}, {71,213}, {100,2298}, {115,119}, {142,3752}, {192,3596}, {214,1015}, {232,1172}, {291,2663}, {407,1880}, {442,1738}, {538,3770}, {656,3126}, {665,2642}, {851,2285}, {869,3688}, {900,2511}, {966,5283}, {968,4204}, {992,3216}, {1017,3031}, {1107,3686}, {1193,1682}, {1194,5276}, {1196,5275}, {1211,3666}, {1449,2275}, {2171,4642}, {2178,2242}, {2183,2653}, {2240,5279}, {2294,3125}, {2303,5277}, {2309,3271}, {2610,3310}, {2667,3122}, {3030,3124}, {3136,3914}, {3247,4050}, {3454,3821}, {3662,3936}, {3728,4111}, {3949,3954}, {4016,4053}

X(2092) = midpoint of X(i) and X(j) for these (i,j): (256, 1045), (2019,2020)
X(2092) = complement of X(314)
X(2092) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,960), (100,512), (1415,647)
X(2092) = crosspoint of X(i) and X(j) for these (i,j): (2,65), (6,37), (429,1211)
X(2092) = crosssum of X(i) and X(j) for these (i,j): (2,81), (6,21), (58,572), (1169,1798) X(2092) = perspector of circumconic centered at X(960)
X(2092) = center of circumconic that is locus of trilinear poles of lines passing through X(960)
X(2092) = harmonic center of Apollonius and Gallatly circles

Reflections, 2093 -  2105
Suppose X = x : y : z and U = u : v : w (trilinears). Then the reflection of X in U is the point

(au - bv - cw)x + 2u(by + cz) : (bv - cw - au)y + 2v(cz + ax) : (cw - au - bv)z + 2w(ax + by).

Thus, if x = x(a,b,c), y = x(b,c,a), z = x(c,a,b) and u = u(a,b,c), v = u(b,c,a), w = u(c,a,b), then the reflection of X in U is given by f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (au - bv - cw)x + 2u(by + cz).

In this section, several coordinate representations were contributed by Peter J. C . Moses.

### X(2093) = REFLECTION OF X(1) IN X(57)

Trilinears    a3 + 3(b + c)a2 - (b + c)2a - 3(b + c)(b - c)2
Trilinears    cos A - 3 cos B - 3 cos C + 1 : :

Let A'B'C' be the intouch triangle of the extangents triangle, if ABC is acute. Then A'B'C' is homothetic to the 2nd isogonal triangle of X(1) at X(2093). (Randy Hutson, December 2, 2017)

X(2093) lies on these lines:
1,3   10,329   19,1743   72,1706   196,1785   200,758   515,2096   518,2097   519,2094   527,1478   553,1056   946,1788   962,1210   1046,1719   1103,1254   1699,1737

X(2093) = reflection of X(i) in X(j) for these (i,j) : (1,57), (329,10)
X(2093) = X(25)- of-reflection-triangle-of-X(1)

### X(2094) = REFLECTION OF X(2) IN X(57)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is as noted just above X(2093)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2094) lies on these lines:
2,7   30,2095   189,1121   376,517   519,2093   524,2087   999,1621

X(2094) = reflection of X(i) in X(j) for these (i,j): (2,57), (329,2)

### X(2095) = REFLECTION OF X(3) IN X(57)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is as noted just above X(2093)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2095) lies on these lines: 1,3   5,329   30,2094   381,527   511,2097

X(2095) = reflection of X(i) in X(j) for these (i,j) : (3,57), (329,5)

### X(2096) = REFLECTION OF X(4) IN X(57)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is as noted just above X(2093)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2096) lies on these lines:
3,329   4,57   7,104   20,145   30,2094   376,527   388,1158   515,2093   1478,1768   1503,2097

X(2096) = reflection of X(i) in X(j) for these (i,j): (4,57), (329,3)

### X(2097) = REFLECTION OF X(6) IN X(57)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is as noted just above X(2093)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2097) lies on these lines:
6,57   19,1122   141,329   511,2095   517,990   518,2093   524,2094   527,599   1503,2096

X(2097) = reflection of X(i) in X(j) for these (i,j): (6,57), (329,141)

### X(2098) = REFLECTION OF X(56) IN X(1)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - a)(a2 - 2b2 - 2c2 + 4bc - ab - ac)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2098) lies on these lines:
1,3   8,11   21,1392   33,1828   145,497   219,1731   499,1387   519,1837   944,1317   952,1479   1000,1389   1108,1732

X(2098) = reflection of X(i) in X(j) for these (i,j): (8,1329), (56,1)
X(2098) = crosssum of X(1) and X(1420)
X(2098) = {X(1),X(40)}-harmonic conjugate of X(1319)
X(2098) = homothetic center of intangents triangle and reflection of tangential triangle in X(1)
X(2098) = homothetic center of Mandart-incircle triangle and 5th mixtilinear triangle
X(2098) = X(56)-of-Mandart-incircle-triangle
X(2098) = X(56)-of-5th-mixtilinear-triangle
X(2098) = Ursa-major-to-Ursa-minor similarity image of X(8)

### X(2099) = REFLECTION OF X(55) IN X(1)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (a + b - c)(a - b + c)(a - 2b - 2c)
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 1 - cos A - 2 cos B - 2 cos C     (Randy Hutson, 9/23/2011)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2099) lies on these lines:
1,3   4,1389   6,1411   7,528   8,12   33,1875   34,1824   42,1457   45,1405   80,381   145,388   226,519   515,1836   674,1469   758,956   946,1837   952,1478   959,1255   1149,1450   1392,1476   1399,1468

X(2099) = reflection of X(55) in X(1)
X(2099) = isogonal conjugate of X(2320)

### X(2100) = REFLECTION OF X(1) IN X(1113)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is as noted just above X(2093)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2100) lies on the Bevan circle and these lines: 1,1113   30,40   165,1114   1312,1699   1313,1698

X(2100) = reflection of X(i) in X(j) for these (i,j): (1,1113), (2101,40)
X(2100) = X(188)-aleph conjugate of X(2575)

### X(2101) = REFLECTION OF X(1) IN X(1114)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is as noted just above X(2093)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2101) lies on the Bevan circle and these lines: 1,1114   30,40   165,1113   1312,1698   1313,1699

X(2101) = reflection of X(i) in X(j) for these (i,j): (1,1114), (2100,40)
X(2101) = X(188)-aleph conjugate of X(2574)

### X(2102) = REFLECTION OF X(1113) IN X(1)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is as noted just above X(2093)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2102) lies on these lines: 1,1113   8,1313   30,944   517,1114   518,2104

X(2102) = reflection of X(i) in X(j) for these (i,j): (8,1313), (1113,1), (2103,1482)

### X(2103) = REFLECTION OF X(1114) IN X(1)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is as noted just above X(2093)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2103) lies on these lines: 1,1114   8,1312   30,944   517,1113   518,2105

X(2103) = reflection of X(i) in X(j) for these (i,j): (8,1312), (1114,1), (2102,1482)

### X(2104) = REFLECTION OF X(1113) IN X(6)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is as noted just above X(2093)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2104) lies on these lines: 6,1113   30,1351   69,1313   511,1114   518,2102

X(2104) = reflection of X(i) in X(j) for these (i,j) : (69,1313), (1113,6), (2105,1351)

### X(2105) = REFLECTION OF X(1114) IN X(6)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is as noted just above X(2093)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2105) lies on these lines: 6,1114   30,1351   69,1312   511,1113   518,2103

X(2105) = reflection of X(i) in X(j) for these (i,j): (69,1312), (1114,6), (2104,1351)

Centers 2106 - 2119
are on the 2nd equal-areas cubic, EAC2. (For the definition, click Tables and then Bicentric Pairs.)

For any point P on EAC2, the X(2)-isoconjugate of P is also on EAC2. In the following table, for each (i,j), the centers X(i) and X(j) are on EAC2 and are an X(2)-isoconjugate pair. Each pair are collinear with the pivot X(238).

IJ
16
231
105672
238292
365365
14232053
19312054
21062107
21082109
21102111
21122113
21142115
21162117
21182119
21442145
21462147
300920332

Triangle centers on a cubic yield non-central points on the cubic; e.g., if P and P' are on EAC2, then the line of PP' meets EAC2 in third point, L(P,P'), on EAC2. (Possibly L(P,P') is P or P'.) If A', B', C' are central triangle vertices and X is a triangle center, and all four points A', B', C', and X lie on EAC2, then the points

L(A',X),    L(B',X),    L(C',X)

are central triangle vertices on EAC2. (A definition of central triangle is given in the Glossary.) In the following table, column 1 indicates the A-vertex of such a triangle; column 2 two points collinear with that vertex; and column 3, the A-vertex of the X(2)-isoconjugate of the triangle. Taking row 1 as an example, - a : c : represents the point -a : c : b, which is the A-vertex of the central triangle with B-vertex c : - b : a and C-vertex b : a : - c. The A-vertex is collinear with A and X(2) (so that the B-vertex is collinear with B and X(2), etc.). Then, in column 3, - bc : a2b : represents the A-vertex of the X(2)-isoconjugate of - a : c : . The vertices - a : c : and - bc : a2b : are collinear with the pivot, X(238).

A-vertexLineA-vertex (iso-)
- a : c :A, X(2)- bc : b2 :
- a1/2 : b1/2 :A, X(365)- a1/2 : b1/2 :
- a(a + b + c) : bc + ca + ab :A, X(1)- 1/(a + b + c) : b/(bc + ca + ab) :
bc + ab + ac : - (a + b + c)b :A, X(6)a/(bc + ab + ac) : - 1/(a + b + c) :
- bc : b2 :A, X(31)- a : c :
a(a + b + c) : a2 + b2 - (a + b)c:X(6), - a : c : 1/(a + b + c) : b/[a2 + b2 - (a + b)c] :
2a(b2 + ca)(c2 + ab) :
(b2 + ca)(a3 + b3 - c3 - abc) :
X(31), - a : c : 1/[2a(b2 + ca)(c2 + ab)] :
b/[(b2 + ca)(a3 + b3 - c3 - abc)] :
(a+b)(a+c)(a2+b2+c2+bc+ca+ab) :
b(a+c)(a2+b2+ab-c(a+b+c)) :
X(1), - a : c : a/[(a+b)(a+c)(a2+b2+c2+bc+ca+ab)] :
1/[(a+c)(a2+b2+ab-c(a+b+c))] :
a(bc + ab + ac) : (c - b)a2 - b2a + b2c : X(1), - bc : b2 : 1/[(bc + ab + ac) : b/[(c - b)a2 - b2a + b2c] :
2a2bc(b2 + ca)(c2 + ba) :
(b2 + ca)(b3c3 + a3c3 - a3b3 - a2b2c2) :
X(2), - bc : b2 : 1/[2abc(b2 + ca)(c2 + ba)] :
b/[(b2 + ca)(b3c3 + a3c3 - a3b3 - a2b2c2)] :
(a+b)(a+c)[(b2+bc+c2)a2 + abc(b+c) + b2c2] :
b(a+c)[(c2-bc-b2)a2 + abc(c-b) + b2c2] :
X(6), - bc : b2 : 1/[(a+b)(a+c)[(b2+bc+c2)a2 + abc(b+c) + b2c2]] :
1/[(a+c)[(c2-bc-b2)a2 + abc(c-b) + b2c2]] :

### X(2106) = POINT KEPLER I

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = (a + b)(a + c)(b2c2 - a2b2 - a2c2 + abc(b + c - a))
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2106) lies on these lines:
2,6   105,805   213,1509   238,741   292,1931   662,1914   873,894   1621,1964   2111,2112   2114,2117

X(2106) = X(i)-Ceva conjugate of X(j) for these (i,j): (238,1931), (741,81)
X(2106) = X(2664)-cross conjugate of X(2669)

### X(2107) = X(2)-ISOCONJUGATE OF X(2106)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = a/[(a + b)(a + c)(b2c2 - a2b2 - a2c2 + abc(b + c - a))]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2107) lies on these lines: 1,1655   31,1979   238,741   672,1967   1423,2116

X(2107) = isogonal conjugate of X(2669)
X(2107) = cevapoint of X(2663) and X(2664)
X(2107) = X(i)-cross conjugate of X(j) for these (i,j): (292,2054), (2238,42)

### X(2108) = POINT KEPLER I

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = (b + c)a3 - (b2 - bc + c2)a2 - (b + c)(b2 + c2)a + bc(b2 + bc + c2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2108) lies on these lines:
1,39   2,846   3,2053   31,43   42,1051   170,411   238,1575   672,1282   1011,1283

X(2108) = X(i)-Ceva conjugate of X(j) for these (i,j): (238,1), (1575,43)
X(2108) = X(i)-aleph conjugate of X(j) for these (i,j): (1,1757), (100,1026), (238,2108), (365,2664)

X(2108) = perspector of 2nd Sharygin triangle and unary cofactor triangle of 1st Sharygin triangle

### X(2109) = X(2)-ISOCONJUGATE OF X(2108)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = a/[(b + c)a3 - (b2 - bc + c2)a2 - (b + c)(b2 + c2)a + bc(b2 + bc + c2)]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2109) lies on these lines: 238,1575   1914,2112

X(2109) = X(i)-cross conjugate of X(j) for these (i,j): (292,6), (727,2162), (1929,2248)

### X(2110) = POINT KEPLER III

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = a[(b2 + bc + c2)a3 - (b + c)(b2 + c2)a2 - bc(b2 - bc + c2)a + b2c2(b + c)]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2110) lies on these lines:
2,11   6,292   31,1979   238,2111   365,2119   672,2117   1185,1977   1423,1740   1931,2113

X(2110) = X(i)-Ceva conjugate of X(j) for these (i,j): (238,6), (2223,55)

### X(2111) = X(2)-ISOCONJUGATE OF X(2110)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = 1/[(b2 + bc + c2)a3 - (b + c)(b2 + c2)a2 - bc(b2 - bc + c2)a + b2c2(b + c)]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2111) lies on these lines: 238,2110   239,672   2106,2112

X(2111) = X(i)-cross conjugate of X(j) for these (i,j): (292,1), (673,57)

### X(2112) = POINT KEPLER IV

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = a[a4 - bca2 - (b3 + c3)a + 2b2c2]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2112) lies on these lines:
1,41   6,2054   31,292   172,1201   238,2113   604,651   672,2115   1015,1468   1914,2109   2106,2111

X(2112) = X(238)-Ceva conjugate of X(31)

X(2112) = eigencenter of extouch triangle
X(2112) = eigencenter of anticevian triangle of X(56)

### X(2113) = X(2)-ISOCONJUGATE OF X(2112)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = 1/[a4 - bca2 - (b3 + c3)a + 2b2c2]
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2113) lies on these lines: 238,2112   672,1282   1458,2114   1931,2110

X(2113) = X(292)-cross conjugate of X(2)

### X(2114) = POINT KEPLER V

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = (c + a - b)(a + b - c)[a4 + (b + c)a3 - u(a,b,c)a2 + v(a,b,c)a - w(a,b,c)],
where u(a,b,c) = 2b2 + 3bc + 2c2,    v(a,b,c) = (b + c)(b2 + c2),    w(a,b,c) = (b - c)2(b2 + bc + c2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2114) lies on these lines:
9,77   31,57   238,241   269,292   1279,1429   1458,2113   2106,2117

### X(2115) = X(2)-ISOCONJUGATE OF X(2114)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = (b + c - a)a/[a4 + (b + c)a3 - u(a,b,c)a2 + v(a,b,c)a - w(a,b,c)],
where u(a,b,c) = 2b2 + 3bc + 2c2,    v(a,b,c) = (b + c)(b2 + c2),    w(a,b,c) = (b - c)2(b2 + bc + c2)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2115) lies on these lines: 238,241   672,2112   1931,2116

### X(2116) = POINT KEPLER VI

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where
f(a,b,c) = u(b,c,a)u(c,a,b)(d1a5 - d2a4 - d3a3 - d4a2 - d5a - d6),
where u(a,b,c) = a2 - 2bc - ac - bc,
d1(a,b,c) = 2(b + c),
d2(a,b,c) = b2 - bc + c2,
d3(a,b,c) = 3bc(b + c),
d4(a,b,c) = b4 + b2c2 + c4 - b3c - c3b,
d5(a,b,c) = bc(b + c)(b2 - 4bc + c2),
d6(a,b,c) = b2c2(b - c)2

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2116) lies on these lines: 6,105   238,2117   1423,2107   1931,2115

### X(2117) = X(2)-ISOCONJUGATE OF X(2116)

Trilinears        a/f(a,b,c) : b/f(b,c,a) : c/f(c,a,b), where f(a,b,c) is as at X(2116)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2117) lies on these lines: 238,2116   672,2110   2106,2114

### X(2118) = POINT KEPLER VII

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = a(a + b + c)(bc)1/2 + (bc + ca + ab)[a - (ab)1/2 - (ac)1/2]     (M. Iliev, 5/13/07)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2118) lies on these lines: 6,365   238,2119   2108,2147   2111,2146

### X(2119) = X(2)-ISOCONJUGATE OF X(2118)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b),
where f(a,b,c) = a1/2/[(a + b + c)(abc)1/2 + (bc + ca + ab)[a1/2 - b1/2 - c1/2)]     (M. Iliev, 5/13/07)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2119) lies on these lines: 1,2146   238,2118   365,2110

Eigencenters I, 2120 - 2143

The eigencenter of a triangle is defined in the Glossary and in TCCT, p. 192. In case the triangle is the cevian triangle of a point U = u : v : w, the eigencenter is given by

f(u,v,w) : f(v,w,u) : f(w,u,v), where f(u,v,w) = vw(u2v2 + u2w2 - v2w2).

This point is now named the eigentransform of U, denoted by ET(U). Pairs (i,j) such that X(j) = ET(X(i)) include

(1,1), (2,3), (3,1075), (4,155), (6,194), (7,218), (13,62), (14,61), (63,1712), (88,1), (92,47), (94,49), (99,39), (100,1), (110,5), (162,1), (174,266), (190,1), (648,185), (651,1), (653,1), (655,1), (658,1), (660,1), (662,1), (664,2082), (673,1), (694,384), (771,1), (799,1), (811,2083), (823,1), (897,1), (1113,3), (1114,3), (1156,1), (1492,1), (1821,1), (1942,1941)

For any point P not on a sideline of triangle ABC, let P -1 denote the isogonal conjugate of P. Easily verified properties of eigentransforms include the following.

1.    ET(U) = U-Ceva conjugate of U -1.

2.    ET(U) = X(1) if and only if U = X(1) or U lies on the Steiner circumellipse: yz + zx + xy = 0.

3.    ET(U) = eigencenter of the anticevian triangle of U -1.

4.    ET(U) is on the self-isogonal cubic Z(U). That is, if x : y : z denotes a trilinear variable point, then ET(U) is on the cubic

ux(y2 - z2) + vy(z2 - x2) + wz(x2 - y2) = 0.

5.    The U-Ceva conjugate of every X on Z(U) is on Z(U); thus, the U-Ceva conjugate, V, of X(1), lies on Z(U); trilinears for V -1 are

1/(1/v + 1/w - 1/u) : 1/(1/w + 1/u - 1/v) : 1/(1/u + 1/v - 1/w);

ET(U) is the third point of intersection of the line X(1)-to-V -1 and the cubic Z(U).

6.    As U is the pivot of Z(U), the following points on Z(U) are collinear: U, ET(U), [ET(U)] -1 .

For discussions and generalizations using barycentric coordinates, see Section 1.4 of "Special Isocubics in the Triangle Plane," downloadable from Bernard Gibert's Cubics in the Triangle Plane. The cubic Z(U) is there denoted by pK(X6,U), and ET(U) is = (cevian quotient of U and U*) = (U-Ceva conjugate of U -1); ET(U) is the tangential of U* in pK(X6,U).

This section was added to ETC on 10/(14-17)/03.

### X(2120) = EIGENTRANSFORM OF X(5)

Trilinears        f(u,v,w) : f(v,w,u) : f(w,u,v), where f is as given just before X(2120) and u : v : w = X(5)
Barycentrics   af(u,v,w) : bf(v,w,u) : cf(w,u,v)

X(2120) lies on the Napoleon cubic and these lines: 1,3461   3,3463   4,1157   5,2121   3467,3469

X(2120) = isogonal conjugate of X(2121)
X(2120) = X(5)-Ceva conjugate of X(54)

### X(2121) = ISOGONAL CONJUGATE OF X(2120)

Trilinears        1/f(u,v,w) : 1/f(v,w,u) : 1/f(w,u,v), where f is as given just before X(2120) and u : v : w = X(5)
Barycentrics   a/f(u,v,w) : b/f(v,w,u) : c/f(w,u,v)

X(2121) lies on the Napoleon cubic and these lines: 5,2120   195,3462

X(2121) = isogonal conjugate of X(2120)
X(2121) = X(54)-cross conjugate of X(5)

### X(2122) = EIGENTRANSFORM OF X(8)

Trilinears        f(u,v,w) : f(v,w,u) : f(w,u,v), where f is as given just before X(2120) and u : v : w = X(8)
Barycentrics   af(u,v,w) : bf(v,w,u) : cf(w,u,v)

X(2122) lies on the cubic K692 and these lines:
1,84   8,2123   34,1767   109,1035   603,1450   608,1249   1406,1456

X(2122) = isogonal conjugate of X(2123)
X(2122) = X(8)-Ceva conjugate of X(56)

### X(2123) = ISOGONAL CONJUGATE OF X(2122)

Trilinears        1/f(u,v,w) : 1/f(v,w,u) : 1/f(w,u,v), where f is as given just before X(2120) and u : v : w = X(8)
Barycentrics   a/f(u,v,w) : b/f(v,w,u) : c/f(w,u,v)

X(2123) lies on the cubic K692 and this line: 8,2122

X(2123) = isogonal conjugate of X(2122)
X(2123) = X(56)-cross conjugate of X(89)

### X(2124) = EIGENTRANSFORM OF X(9)

Trilinears        f(u,v,w) : f(v,w,u) : f(w,u,v), where f is as given just before X(2120) and u : v : w = X(9)
Barycentrics   af(u,v,w) : bf(v,w,u) : cf(w,u,v)

X(2124) lies on the cubic K351 and these lines: 1,971   8,2125   57,279   223,1212   664,728

X(2124) = isogonal conjugate of X(2125)
X(2124) = X(9)-Ceva conjugate of X(57)
X(2124) = perspector of 7th mixtilinear triangle and unary cofactor triangle of 4th mixtilinear triangle

### X(2125) = ISOGONAL CONJUGATE OF X(2124)

Trilinears        1/f(u,v,w) : 1/f(v,w,u) : 1/f(w,u,v), where f is as given just before X(2120) and u : v : w = X(9)
Barycentrics   a/f(u,v,w) : b/f(v,w,u) : c/f(w,u,v)

X(2125) lies on the cubic K351 and these lines: 9,2124   165,220

X(2125) = isogonal conjugate of X(2124)
X(2125) = X(57)-cross conjugate of X(9)

### X(2126) = EIGENTRANSFORM OF X(10)

Trilinears        f(u,v,w) : f(v,w,u) : f(w,u,v), where f is as given just before X(2120) and u : v : w = X(10)
Barycentrics   af(u,v,w) : bf(v,w,u) : cf(w,u,v)

X(2126) lies on these lines: 1,229   5,572   10,2127   49,970

X(2126) = isogonal conjugate of X(2127)
X(2126) = X(10)-Ceva conjugate of X(58)

### X(2127) = ISOGONAL CONJUGATE OF X(2126)

Trilinears        f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) is as given just before X(2120)
Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2127) lies on this line: 10,2126

X(2127) = isogonal conjugate of X(2126)
X(2127) = X(58)-cross conjugate of X(10)

### X(2128) = EIGENTRANSFORM OF X(19)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b2 + c2 - a2)[(a2 + b2 + c2)2 - 8b2c2]     (M. Iliev, 5/13/07)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2128) lies on the cubic Z(X(19)) and these lines: 1,1958   2,2082   19,2129   63,304

X(2128) = isogonal conjugate of X(2129)
X(2128) = X(19)-Ceva conjugate of X(63)

### X(2129) = ISOGONAL CONJUGATE OF X(2128)

Trilinears       f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = 1/{(b2 + c2 - a2)[(a2 + b2 + c2)2 - 8b2c2]}     (M. Iliev, 5/13/07)

Barycentrics   af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(2129) lies on the cubic Z(X(19)) and these lines: 19,2128   1707,1973

X(2129) = isogonal conjugate of X(2128)
X(2129) = X(63)-cross conjugate of X(19)

### X(2130) = EIGENTRANSFORM OF X(20)

Trilinears        f(u,v,w) : f(v,w,u) : f(w,u,v), where f is as given just before X(2120) and u : v : w = X(20)
Barycentrics   af(u,v,w) : bf(v,w,u) : cf(w,u,v)

X(2130) lies on the Darboux cubic (K004) and these lines: 1,3347   3,3348   4,1073   20,2131   40,3354   84,3345   1490,3473

X(2130) = reflection of X(3348) in X(3)
X(2130) = isogonal conjugate of X(2131)
X(2130) = X(20)-Ceva conjugate of X(64)

### X(2131) = ISOGONAL CONJUGATE OF X(2130)

Trilinears        1/f(u,v,w) : 1/f(v,w,u) : 1/f(w,u,v), where f is as given just before X(2120) and u : v : w = X(20)
Barycentrics   a/f(u,v,w) : b/f(v,w,u) : c/f(w,u,v)

X(2131) lies on the Darboux cubic (K004) and these lines: 3,3355   20,2130   40,3472   1490,3353   1498,3183

X(2131) = reflection of X(3355) in X(3)
X(2131) = isogonal conjugate of X(2130)
X(2131) = X(64)-cross conjugate of X(20)

### X(2132) = EIGENTRANSFORM OF X(30)

Trilinears        f(u,v,w) : f(v,w,u) : f(w,u,v), where f is as given just before X(2120) and u : v : w = X(30)
Barycentrics   af(u,v,w) : bf(v,w,u) : cf(w,u,v)

X(2132) is the tangential of X(74) on the Neuberg cubic (K001).

X(2132) lies on the Neuberg cubic and these lines: 4,1138   30,2133   3065,3466   3440,3441

X(2132) = isogonal conjugate of X(2133)
X(2132) = X(30)-Ceva conjugate of X(74)
X(2132) = Miquel associate of X(30)

### X(2133) = ISOGONAL CONJUGATE OF X(2132)

Trilinears        1/f(u,v,w) : 1/f(v,w,u) : 1/f(w,u,v), where f is as given just before X(2120) and u : v : w = X(30)
Barycentrics   a/f(u,v,w) : b/f(v,w,u) : c/f(w,u,v)

X(2133) lies on the Neuberg cubic (K001) and this line: 30,2132

X(2133) = isogonal conjugate of X(2132)
X(2133) = X(74)-cross conjugate of X(30)

### X(2134) = EIGENTRANSFORM OF X(37)

Trilinears        f(u,v,w) : f(v,w,u) : f(w,u,v), where f is as given just before X(2120) and u : v : w = X(37)
Barycentrics   af(u,v,w) : bf(v,w,u) : cf(w,u,v)

X(2134) lies on the these lines: 1,1326   37,2135   39,1931   81,763

X(2134) = isogonal conjugate of X(2135)
X(2134) = X(37)-Ceva conjugate of X(81)

### X(2135) = ISOGONAL CONJUGATE OF X(2134)

Trilinears        1/f(u,v,w) : 1/f(v,w,u) : 1/f(w,u,v), where f is as given just before X(2120) and u : v : w = X(37)
Barycentrics   a/f(u,v,w) : b/f(v,w,u) : c/f(w,u,v)

X(2135) lies on the cubic Z(X(37)) and this line: 37,2134

X(2135) = isogonal conjugate of X(2134)
X(2135) = X(81)-cross conjugate of X(37)

### X(2136) = EIGENTRANSFORM OF X(57)

Trilinears    f(u,v,w) : f(v,w,u) : f(w,u,v), where f is as given just before X(2120) and u : v : w = X(57)
Trilinears    (s - a)[(s - b)^2 + (s - c)^2 - (s - a)^2] : :

X(2136) lies on the cubics K201 and K372, and on these lines:
1,474   8,9   10,1058   40,376   43,1050   57,145   84,952   100,1420   188,258   517,1490   664,738

X(2136) = isogonal conjugate of X(2137)
X(2136) = X(145)-Ceva conjugate of X(1)
X(2136) = X(145)-aleph conjugate of X(2136)
X(2136) = SS(a->s-a) of X(3)
X(2136) = X(64)-of-excentral-triangle
X(2136) = orthologic center of these triangles: excentral to excenters-midpoints
X(2136) = orthologic center of these triangles: excentral to 2nd Schiffler

### X(2137) = ISOGONAL CONJUGATE OF X(2136)

Trilinears        1/f(u,v,w) : 1/f(v,w,u) : 1/f(w,u,v), where f is as given just before X(2120) and u : v : w = X(57)
Barycentrics   a/f(u,v,w) : b/f(v,w,u) : c/f(w,u,v)

X(2137) lies on the cubic K830 and these lines:
44,380   56,1743   57,145   1018,1400   1149,1450   1407,1420

X(2137) = isogonal conjugate of X(2136)
X(2137) = X(9)-cross conjugate of X(57)

### X(2138) = EIGENTRANSFORM OF X(69)

Trilinears        f(u,v,w) : f(v,w,u) : f(w,u,v), where f is as given just before X(2120) and u : v : w = X(69)
Barycentrics   af(u,v,w) : bf(v,w,u) : cf(w,u,v)

X(2138) lies on the cubic K169 and these lines: 6,64   69,2139   1249,1941

X(2138) = isogonal conjugate of X(2139)
X(2138) = X(69)-Ceva conjugate of X(25)

### X(2139) = ISOGONAL CONJUGATE OF X(2138)

Trilinears        1/f(u,v,w) : 1/f(v,w,u) : 1/f(w,u,v), where f is as given just before X(2120) and u : v : w = X(69)
Barycentrics   a/f(u,v,w) : b/f(v,w,u) : c/f(w,u,v)

X(2139) lies on the cubic, K169 and these lines: 20,159   69,2138

X(2139) = isogonal conjugate of X(2138)
X(2139) = X(25)-cross conjugate of X(69)

### X(2140) = EIGENTRANSFORM OF X(101)

Trilinears        f(u,v,w) : f(v,w,u) : f(w,u,v), where f is as given just before X(2120) and u : v : w = X(101)
Barycentrics   af(u,v,w) : bf(v,w,u) : cf(w,u,v)

X(2140) lies on the cubic K258 and these lines:
3,142   5,116   39,1086   85,514   101,2141   170,1699   218,226   244,2085

X(2140) = isogonal conjugate of X(2141)
X(2140) = X(101)-Ceva conjugate of X(514)

### X(2141) = ISOGONAL CONJUGATE OF X(2140)

Trilinears        1/f(u,v,w) : 1/f(v,w,u) : 1/f(w,u,v), where f is as given just before X(2120) and u : v : w = X(101)
Barycentrics   a/f(u,v,w) : b/f(v,w,u) : c/f(w,u,v)

X(2141) lies on this line: 101, 2140

X(2141) = isogonal conjugate of X(2140)
X(2141) = X(514)-cross conjugate of X(101)

### X(2142) = EIGENTRANSFORM OF X(512)

Trilinears        f(u,v,w) : f(v,w,u) : f(w,u,v), where f is as given just before X(2120) and u : v : w = X(512)
Barycentrics   af(u,v,w) : bf(v,w,u) : cf(w,u,v)

X(2142) lies on the 1st equal-areas cubic (K021) and this line: 512,2143

X(2142) = isogonal conjugate of X(2143)
X(2142) = X(512)-Ceva conjugate of X(99)

### X(2143) = ISOGONAL CONJUGATE OF X(2142)

Trilinears        1/f(u,v,w) : 1/f(v,w,u) : 1/f(w,u,v), where f is as given just before X(2120) and u : v : w = X(512)
Barycentrics   a/f(u,v,w) : b/f(v,w,u) : c/f(w,u,v)

X(2143) lies on the 1st equal-areas cubic (K021) and this line: 512,2142

X(2143) = isogonal conjugate of X(2142)
X(2143) = X(99)-cross conjugate of X(512)

### X(2144) = X(2)-EIGENTRANSFORM OF X(238)

Trilinears        f(u,v,w) : f(v,w,u) : f(w,u,v), where f is as given just before X(2120) and u : v : w = X(2382)
Barycentrics   af(u,v,w) : bf(v,w,u) : cf(w,u,v)

Suppose U = u : v : w and P = p : q : r are triangle centers. The P-eigentransform of U, denoted by ET(U,P), is the point given by first trilinear

qrvw(pqu2v2 + pru2w2 - qrv2w2).

Thus, ET(U) = ET(U,X(1)), and, extending the on-cubic property, ET(U,P) lies on the cubic Z(U,P) given by

upx(qy2 - rz2) + vqy(rz2 - pu2) + wrz(px2 - qv2) = 0.

X(2144) lies on the 2nd equal-areas cubic (K155) and these lines: 1,2111   2,2113   6,2109   238,2145   2053,2115   2054,2107

X(2144) = X(238)-Ceva conjugate of X(292)

### X(2145) = X(2)-ISOCONJUGATE OF X(2144)

Trilinears        1/f(u,v,w) : 1/f(v,w,u) : 1/f(w,u,v), where f is as given just before X(2120) and u : v : w = X(238)
Barycentrics   a/f(u,v,w) : b/f(v,w,u) : c/f(w,u,v)

X(2145) lies on the 2nd equal-areas cubic (K155): 238,2144   2108,2110

X(2145) = X(292)-cross conjugate of X(238)

### X(2146) = X(238)-CEVA CONJUGATE OF X(365)

Trilinears        a1/2(b1/2/v + c1/2/w - a1/2/u) : b1/2(c1/2/w + a1/2/u - b1/2/v) : c1/2(a1/2/u + b1/2/v - c1/2/w),
where u : v : w = a2 - bc : b2 - ca : c2 - ab

Barycentrics   a3/2(b1/2/v + c1/2/w - a1/2/u) : b3/2(c1/2/w + a1/2/u - b1/2/v) : c3/2(a1/2/u + b1/2/v - c1/2/w)

X(2146) lies on the 2nd equal-areas cubic (K155) and these lines: 1,2119   238,2147   292,365   2111,2118

X(2146) = X(238)-Ceva conjugate of X(365)

### X(2147) = X(2)-ISOCONJUGATE OF X(2146)

Trilinears        a/f(a,b,c) : b/f(b,c,a) : c/f(c,a,b),
where f(a,b,c) : f(b,c,a) : f(c,a,b) = X(2146)

Barycentrics   a2/f(a,b,c) : b2/f(b,c,a) : c2/f(c,a,b)

X(2147) lies on the 2nd equal-areas cubic (K155) and these lines: 238,2146   365,2144   2108,2118

X(2147) = X(292)-cross conjugate of X(365)

### X(2148) = X(2)-ISOCONJUGATE OF X(5)

Trilinears        a sec(B - C) : b sec(C - A): c sec(A - B)
Barycentrics   a2sec(B - C) : b2sec(C - A): c2sec(A - B)

X(2148) lies on these lines:
1,563   19,2190   47,48   54,71   63,2167   163,1953   228,2361   933,2249   2083,2158

X(2148) = X(2167)-Ceva conjugate of X(2169)
X(2148) = cevapoint of X(48) and X(563)
X(2148) = X(i)-cross conjugate of X(j) for these (i,j): (31,2190), (48,2168), (661,163)
X(2148) = crosspoint of X(2167) and X(2190)

### X(2149) = X(2)-ISOCONJUGATE OF X(11)

Trilinears    (sin A)/[1 - cos(B - C)] : :

X(2149) lies on these lines:
59,672   101,652   109,649   163,1625   661,2222   1055,1262   1110,2223

X(2149) = isogonal conjugate of X(4858)
X(2149) = X(59)-Ceva conjugate of X(1110)
X(2149) = cevapoint of X(i) and X(j) for these (i,j): (41,692), (101,572), (604,1415)
X(2149) = X(i)-cross conjugate of X(j) for these (i,j): (6,163), (31,109), (41,692), (48,101), (604,1415), (1755,813), (2352,108)
X(2149) = crosspoint of X(59) and X(1262)
X(2149) = crosssum of X(11) and X(1146)

### X(2150) = X(2)-ISOCONJUGATE OF X(12)

Trilinears        a/f(a,b,c) : b/f(b,c,a) : c/f(c,a,b), where f(a,b,c) is as in X(12)
Barycentrics   a2/f(a,b,c) : b2/f(b,c,a): c2/f(c,a,b)

X(2150) lies on these lines:
6,163   9,1098   58,1474   60,283   593,1412   759,1953   849,1333   2193,2194

X(2150) = isogonal conjugate of X(6358)
X(2150) = crossdifference of every pair of points on line X(4036)X(4064)
X(2150) = X(593)-Ceva conjugate of X(849)
X(2150) = X(i)-cross conjugate of X(j) for these (i,j): (1333,2189), (2194,60)
X(2150) = crosspoint of X(i) and X(j) for these (i,j): (60,593), (270,2185)
X(2150) = crosssum of X(i) and X(j) for these (i,j): (12,594), (201,2171)

### X(2151) = X(2)-ISOCONJUGATE OF X(13)

Trilinears        a/f(a,b,c) : b/f(b,c,a) : c/f(c,a,b), where f(a,b,c) is as in X(13)
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = sin A sin(A + π/3)
Barycentrics   a2/f(a,b,c) : b2/f(b,c,a): c2/f(c,a,b)

X(2151) lies on these lines: 1,1095   31,48   2153,2173

X(2151) = X(2159)-cross conjugate of X(2152)

### X(2152) = X(2)-ISOCONJUGATE OF X(14)

Trilinears    sin A sin(A - π/3) : :

X(2152) lies on these lines: 1,1094   31,48   2154,2173

X(2152) = X(2159)-cross conjugate of X(2151)

### X(2153) = X(2)-ISOCONJUGATE OF X(15)

Trilinears        a/f(a,b,c) : b/f(b,c,a) : c/f(c,a,b), where f(a,b,c) is as in X(15)
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = sin A csc(A + π/3)
Barycentrics   a2/f(a,b,c) : b2/f(b,c,a): c2/f(c,a,b)

X(2153) lies on these lines: 1,1094   10,13   31,2154   37,1250   65,2306   2151,2173

X(2153) = X(2173)-cross conjugate of X(2154)
X(2153) = trilinear product of circumcircle intercepts of inner Napoleon circle

### X(2154) = X(2)-ISOCONJUGATE OF X(16)

Trilinears        a/f(a,b,c) : b/f(b,c,a) : c/f(c,a,b), where f(a,b,c) is as in X(16)
Trilinears        g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A, B, C) = sin A csc(A - π/3)
Barycentrics   a2/f(a,b,c) : b2/f(b,c,a): c2/f(c,a,b)

X(2154) lies on these lines: 1,1095   10,14   31,2153   2152,2173

X(2154) = X(2173)-cross conjugate of X(2153)
X(2154) = trilinear product of circumcircle intercepts of outer Napoleon circle

Tri