X(83) = X(83) CEVAPOINT OF CENTROID AND SYMMEDIAN POINT¶

Trilinears

\(bc/(b2 + c2) : ca/(c2 + a2) : ab/(a2 + b2)\)

Barycentrics

\(1/(b2 + c2) : 1/(c2 + a2) : 1/(a2 + b2) Let K denote the symmedian point, :ref:`X(6) <X(6)>\). Let A’B’C’ be the cevian triangle of K. Let KA be K of the triangle AB’C’; let KB be K of A’BC’ and let KC be K of A’B’C. The lines AKA, BKB, CKC concur in X(83). (Randy Hutson, 9/23/2011) Let A’B’C’ be the 1st Brocard triangle. Let A″ be the reflection of A’ in BC, and define B″ and C″ cyclically. AA″, BB″, CC″ concur in X(83). (Randy Hutson, December 26, 2015) Let Ba, Ca be the intersections of lines CA, AB, resp., and the antiparallel to BC through X(2). Define Cb, Ab, Ac, Bc cyclically. Triangles ABaCa, AbBCb, AcBcC are similar to each other and inversely similar to ABC. Let Sa be the similitude center of triangles AbBCb and AcBcC. Define Sb and Sc cyclically. The lines ASa, BSb, CSc concur in X(83). (Randy Hutson, December 26, 2015) Let (Oa) be the circle whose diameter is the orthogonal projections of PU(1) on line BC. Define (Ob) and (Oc) cyclically. X(83) is the radical center of circles (Oa), (Ob), (Oc). (Randy Hutson, December 26, 2015) Let A’B’C’ be the circummedial 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(83). (Randy Hutson, December 26, 2015) X(83) lies on these lines: 2,32 3,262 4,182 5,98 6,76 10,82 17,624 18,623 39,99 213,239 217,287 275,297 597,671 689,729 X(83) = isogonal conjugate of X(39) <X(39)>`

Notes

Barycentrics 1/(b2 + c2) : 1/(c2 + a2) : 1/(a2 + b2)

Let K denote the symmedian point, X(6). Let A’B’C’ be the cevian triangle of K. Let KA be K of the triangle AB’C’; let KB be K of A’BC’ and let KC be K of A’B’C. The lines AKA, BKB, CKC concur in X(83). (Randy Hutson, 9/23/2011)

Let A’B’C’ be the 1st Brocard triangle. Let A″ be the reflection of A’ in BC, and define B″ and C″ cyclically. AA″, BB″, CC″ concur in X(83). (Randy Hutson, December 26, 2015)

Let Ba, Ca be the intersections of lines CA, AB, resp., and the antiparallel to BC through X(2). Define Cb, Ab, Ac, Bc cyclically. Triangles ABaCa, AbBCb, AcBcC are similar to each other and inversely similar to ABC. Let Sa be the similitude center of triangles AbBCb and AcBcC. Define Sb and Sc cyclically. The lines ASa, BSb, CSc concur in X(83). (Randy Hutson, December 26, 2015)

Let (Oa) be the circle whose diameter is the orthogonal projections of PU(1) on line BC. Define (Ob) and (Oc) cyclically. X(83) is the radical center of circles (Oa), (Ob), (Oc). (Randy Hutson, December 26, 2015)

Let A’B’C’ be the circummedial 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(83). (Randy Hutson, December 26, 2015)

X(83) lies on these lines: 2,32 3,262 4,182 5,98 6,76 10,82 17,624 18,623 39,99 213,239 217,287 275,297 597,671 689,729

X(83) = isogonal conjugate of X(39)

X(83) = isotomic conjugate of X(141)

X(83) = complement of X(2896)

X(83) = cevapoint of X(2) and X(6)

X(83) = X(i)-cross conjugate of X(j) for these (i,j): (2,308), (6,251), (512,99)

X(83) = trilinear pole of line X(23)X(385) (line is the polar of X(2) wrt the circumcircle, and also the anticomplement of the de Longchamps line, and also the polar of X(5) wrt {circumcircle, nine-point circle}-inverter)

X(83) = crossdifference of every pair of points on line X(688)X(3005)

X(83) = pole wrt polar circle of trilinear polar of X(427)

X(83) = X(48)-isoconjugate (polar conjugate) of X(427)

X(83) = perspector of ABC and medial triangle of 1st Brocard triangle

X(83) = crosspoint of X(2) and X(6) wrt both the anticomplementary and tangential triangles

X(83) = trilinear product of vertices of circummedial triangle

X(83) = midpoint of PU(137)

X(83) = bicentric sum of PU(i) for these i: 137, 141

X(83) = homothetic center of 5th anti-Brocard triangle and medial triangle

X(83) = X(8290)-of-1st-Brocard-triangle

X(83) = perspector of ABC and 1st Brocard triangle of medial triangle

X(83) = perspector of ABC and 1st Brocard triangle of 5th anti-Brocard triangle

X(83) = homothetic center of ABC and cross-triangle of ABC and 5th anti-Brocard triangle

X(83) = Cundy-Parry Phi transform of X(262)

X(83) = Cundy-Parry Psi transform of X(182)

X(83) = barycentric product of circumcircle intercepts of line X(316)X(512)