X(63) = ISOGONAL CONJUGATE OF X(19)

Trilinears

\(cot A : cot B : cot C\)

\(b2 + c2 - a2 : c2 + a2 - b2 : a2 + b2 - c2\)

\(SA : SB : SC\)

\(csc A - tan(A/2) : :\)

\(csc A - cot(A/2) : :\)

\(tan(A/2) - cot(A/2) : :\)

\(d(a,b,c) : : , where d(a,b,c) = directed distance from A to the orthic axis\)

\(2 csc 2A - tan A : :\)

Barycentrics

\(cos A : cos B : cos C\)

Notes

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. Triangle A’B’C’ is perspective to the excentral triangle at X(63). (Randy Hutson, February 10, 2016)

Let A’B’C’ be the 2nd Brocard triangle. Let A″ be the trilinear product B’*C’, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(63). (Randy Hutson, February 10, 2016)

Let A’B’C’ be the hexyl 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(63). (Randy Hutson, February 10, 2016)

Let A’B’C’ be the side-triangle of ABC and hexyl triangle. Let A″ be the {B,C}-harmonic conjugate of A’, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X(63). (Randy Hutson, February 10, 2016)

Let A’B’C’ be the excentral triangle. Let A″ be the isotomic conjugate, wrt triangle A’BC, of X(1). Define B″ and C″ cyclically. The lines A’A″, B’B″, C’C″ concur in X(63). (Randy Hutson, July 31 2018)

X(63) lies on these lines: 1,21 2,7 3,72 6,2221 8,20 10,46 19,27 33,1013 36,997 37,940 48,326 55,518 56,960 65,958 69,71 77,219 91,921 100,103 162,204 169,379 171,612 190,312 194,239 201,603 210,1004 212,1040 213,980 220,241 223,651 238,614 240,1096 244,748 304,1102 318,412 354,1001 392,999 404,936 405,942 452,938 484,535 517,956 544,1018 561,799 654,918 750,756

X(63) is the {X(9),:ref:X(57) <X(57)>}-harmonic conjugate of X(2). For a list of other harmonic conjugates of X(63), click Tables at the top of this page.

X(63) = reflection of X(i) in X(j) for these (i,j): (1,993), (1478,10)

X(63) = isogonal conjugate of X(19)

X(63) = isotomic conjugate of X(92)

X(63) = complement of X(5905)

X(63) = anticomplement of X(226)

X(63) = anticomplementary conjugate of X(2893)

X(63) = X(i)-Ceva conjugate of X(j) for these (i,j): (7,224), (69,78), (75,1), (304,326), (333,2), (348,77)

X(63) = cevapoint of X(i) and X(j) for these (i,j): (3,219), (9,40), (48,255), (71,72)

X(63) = X(i)-cross conjugate of X(j) for these (i,j): (3,77), (9,271), (48,1), (71,3), (72,69), (219,78), (255,326)

X(63) = crosspoint of X(i) and X(j) for these (i,j): (69,348), (75,304)

X(63) = crosssum of X(25) and X(607)

X(63) = crossdifference of every pair of points on line X(661)X(663)

X(63) = trilinear product X(2)

X(63) = trilinear product of PU(22)

X(63) = bicentric sum of PU(i) for these i: 128, 129

X(63) = PU(128)-harmonic conjugate of X(661)

X(63) = midpoint of PU(129)

X(63) = {X(1),:ref:X(1707) <X(1707)>}-harmonic conjugate of X(31)

X(63) = {X(2),:ref:X(9) <X(9)>}-harmonic conjugate of X(3305)

X(63) = {X(2),:ref:X(57) <X(57)>}-harmonic conjugate of X(3306)

X(63) = {X(92),:ref:X(1748) <X(1748)>}-harmonic conjugate of X(19)

X(63) = trilinear pole of line X(521)X(656)

X(63) = pole wrt polar circle of trilinear polar of X(158)

X(63) = X(48)-isoconjugate (polar conjugate) of X(158)

X(63) = X(i)-isoconjugate of X(j) for these {i,j}: {4,6}, {31,92}, {75,1973}

X(63) = excentral isogonal conjugate of X(1742)

X(63) = homothetic center of excentral triangle and anticomplement of the intouch triangle

X(63) = X(161)-of-intouch-triangle

X(63) = X(184)-of-excentral-triangle

X(63) = inverse-in-circumconic-centered-at-X(9) of X(908)

X(63) = trilinear square of X(5374)

X(63) = perspector of excentral triangle and Gemini triangle 2

X(63) = homothetic center of excentral triangle and Gemini triangle 30

X(63) = perspector of ABC and cross-triangle of Gemini triangles 35 and 36

X(63) = perspector of ABC and cross-triangle of ABC and Gemini triangle 35

X(63) = perspector of ABC and cross-triangle of ABC and Gemini triangle 36

X(63) = barycentric product of vertices of Gemini triangle 35

X(63) = barycentric product of vertices of Gemini triangle 36

X(63) = X(i)-aleph conjugate of X(j) for these (i,j): (2,1), (75,63), (92,920), (99,662), (174,978), (190,100), (333,411), (366,43), (514,1052), (556,40), (648,162), (664,651), (668,190), (670,799), (671,897), (903,88)

X(63) = X(i)-beth conjugate of X(j) for these (i,j): (63,222), (190,63), (333,57), (345,345), (643,63), (645,312), (662,223)

X(63) = perspector of ABC and extraversion triangle of X(63), which is also the anticevian triangle of X(63)