X(36) = INVERSE-IN-CIRCUMCIRCLE OF INCENTER¶
Trilinears
\(1 - 2 cos A : 1 - 2 cos B : 1 - 2 cos C\)
\(a(b2 + c2 - a2 - bc)\)
\(sec(A/2) cos(3A/2) : :\)
Barycentrics
\(sin A - sin 2A : :\)
\(a2(b2 + c2 - a2 - bc) : :\)
Notes
If you have The Geometer’s Sketchpad, you can view X(36). If you have GeoGebra, you can view X(36).
Let A’ be the isogonal conjugate of A with respect to BC:ref:X(1) <X(1)>, and define B’ and C’ cyclically. Let A″ be the circumcenter of BC:ref:X(1) <X(1)>, and define B″ and C″ cyclically. The lines A’A″, B’B″, C’C″ concur in X(36). Also, X(36) is the QA-P4 center (Isogonal Center) of quadrangle ABC:ref:X(1) <X(1)> (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/25-qa-p4.html)
Let P be a point in the plane of triangle ABC, not on a sideline of ABC. Let A1 be the isogonal conjugate of A with respect to triangle BCP, and define B1, C1 cyclically. Call triangle A1B1C1 the 1st isogonal triangle of P. A1B1C1 is also the reflection triangle of the isogonal conjugate of P. A1B1C1 is perspective to ABC iff P lies on the Neuberg cubic. The perspector lies on cubic K060 (pK(X1989, X265), O(X5) orthopivotal cubic). Let A2 be the isogonal conjugate of A1 with respect to triangle B1C1P, and define B2, C2 cyclically. Call triangle A2B2C2 the 2nd isogonal triangle of P. Continuing, let An be the isogonal conjugate of A(n-1) with respect to triangle B(n-1)C(n-1)P, and define B(n), C(n) cyclically. Call triangle AnBnCn the nth isogonal triangle of P. For n >= 2, all triangles AnBnCn are perspective to A(n-1)B(n-1)C(n-1). Call the perspector, Pn, the nth isogonal perspector of P. Pn is the orthocenter of A(n-1)B(n-1)C(n-1) and either the incenter or an excenter of AnBnCn. The triangles AnBnCn are all concyclic, with P as center. Call the circle the isogonal circle of P. For P = X(1), the 2nd isogonal triangle of X(1) is homothetic to ABC at X(36); see also X(35), X(1478), X(1479), X(3583), X(3585), X(5903), X(7741), X(7951). (Randy Hutson, November 18, 2015)
Let A’B’C’ be the incentral triangle. Let A″ be the reflection of A in line B’C’, and define B″, C″ cyclically. The lines A’A″, B’B″, C’C″ concur in X(36). (Randy Hutson, June 27, 2018)
X(36) is the {X(3),:ref:X(56) <X(56)>}-harmonic conjugate of X(1). For a list of other harmonic conjugates of X(36), click Tables at the top of this page.
X(36) lies on these lines: 1,3 2,535 4,499 6,609 10,404 11,30 12,140 15,202 16,203 21,79 22,614 24,34 31,995 33,378 39,172 47,602 48,579 54,73 58,60 59,1110 63,997 80,104 84,90 99,350 100,519 101,672 106,901 109,953 187,1015 191,960 214,758 226,1006 238,513 255,1106 376,497 388,498 474,958 495,549 496,550 573,604 1030,1100
X(36) = midpoint of X(1) and X(484)
X(36) = reflection of X(i) in X(j) for these (i,j): (1,1319), (484,1155) (2077,3)
X(36) = isogonal conjugate of X(80)
X(36) = isotomic conjugate of X(20566)
X(36) = complement of X(5080)
X(36) = anticomplement of X(3814)
X(36) = circumcircle-inverse of X(1)
X(36) = inccircle-inverse of X(942)
X(36) = Bevan-circle-inverse of X(46)
X(36) = polar conjugate of isotomic conjugate of X(22128)
X(36) = X(i)-Ceva conjugate of X(j) for these (i,j): (88,6), (104,1)
X(36) = crosspoint of X(58) and X(106)
X(36) = crosssum of X(i) and X(j) for these (i,j): (1,484), (10,519), (11,900)
X(36) = crossdifference of every pair of points on line X(37)X(650)
X(36) = X(104)-aleph conjugate of X(36)
X(36) = X(i)-beth conjugate of X(j) for these (i,j): (21,36), (100,36), (643,519)
X(36) = X(2070)-of-intouch-triangle
X(36) = X(186)-of-2nd circumperp-triangle
X(36) = {X(55),:ref:X(56) <X(56)>}-harmonic conjugate of X(999)
X(36) = reflection of X(484) in the antiorthic axis
X(36) = inverse-in-{circumcircle, nine-point circle}-inverter of X(354)
X(36) = perspector of ABC and extraversion triangle of X(35)
X(36) = homothetic center of intangents and Trinh triangles
X(36) = perspector of ABC and the reflection of the 2nd circumperp triangle in line X(1)X(3)
X(36) = X(186)-of-reflection-triangle-of-X(1)
X(36) = Cundy-Parry Psi transform of X(15446)
X(36) = exsimilicenter of circumcircle and circumcircle of reflection triangle of X(1); insimilicenter is X(35)
X(36) = homothetic center of medial triangle and cross-triangle of ABC and 2nd isogonal triangle of X(1)
X(36) = perspector of ABC and the reflection of the excentral triangle in the antiorthic axis (the reflection of the anticevian triangle of X(1) in the trilinear polar of X(1))
X(36) = Cundy-Parry Phi transform of X(5903)
X(36) = homothetic center of Kosnita triangle and anti-tangential midarc triangle
X(36) = orthocenter of cross-triangle of ABC and outer Yff triangle
X(36) = exsimilicenter of circumcircles of ABC and outer Yff triangle; the insimilicenter is X(1)
X(36) = outer-Yff-isogonal conjugate of X(34789)