X₃₆

INVERSE-IN-CIRCUMCIRCLE OF INCENTER

X₃₆ = circumcircle-inverse of X₁

5, 6, 7 3, 4, 5 Kepler Kimberling

X₃₆ (INVERSE-IN-CIRCUMCIRCLE OF INCENTER) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 484}

• Related Centers: X₁ (INCENTER), X₄₈₄ (1st EVANS PERSPECTOR)

Trilinears: \(1 - 2 \cos A : 1 - 2 \cos B : 1 - 2 \cos C\)

Barycentrics: \(\sin A - \sin 2A : :\)

Tripolars: \(\sqrt{b c (b + c - a)} : :\)

Search Value (6, 9, 13): \(-2.575286786660\)

▸ Alternate Coordinates4 forms

• Trilinears: \(a(b^{2} + c^{2} - a^{2} - bc)\)

• Trilinears: \(\sec(A/2) \cos(3A/2) : :\)

• Barycentrics: \(a^{2}(b^{2} + c^{2} - a^{2} - bc) : :\)

• Tripolars: \(\sec A' : :\), where J_aJ_bJ_c is the excentral triangle

▸ Incident Central Lines41 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}

▸ Functional Relations34 relations

• X₃₆ = midpoint of X₁ and X₄₈₄

• X₃₆ = reflection of X_i in X_j for these (i,j): (1, 1319), (484, 1155) (2077, 3)

• X₃₆ = isogonal conjugate of X₈₀

• X₃₆ = isotomic conjugate of X₂₀₅₆₆

• X₃₆ = complement of X₅₀₈₀

• X₃₆ = anticomplement of X₃₈₁₄

• X₃₆ = circumcircle-inverse of X₁

• X₃₆ = inccircle-inverse of X₉₄₂

• X₃₆ = Bevan-circle-inverse of X₄₆

• X₃₆ = polar conjugate of isotomic conjugate of X₂₂₁₂₈

• X₃₆ = X_i-Ceva conjugate of X_j for these (i,j): (88, 6), (104, 1)

• X₃₆ = crosspoint of X₅₈ and X₁₀₆

• X₃₆ = crosssum of X_i and X_j for these (i,j): (1, 484), (10, 519), (11, 900)

• X₃₆ = crossdifference of every pair of points on line X₃₇X₆₅₀

• X₃₆ = X₁₀₄-aleph conjugate of X₃₆

• X₃₆ = X_i-beth conjugate of X_j for these (i,j): (21, 36), (100, 36), (643, 519)

• X₃₆ = X₂₀₇₀-of-intouch-triangle

• X₃₆ = X₁₈₆-of-2nd circumperp-triangle

• X₃₆ = {X₅₅,X₅₆}-harmonic conjugate of X₉₉₉

• X₃₆ = reflection of X₄₈₄ in the antiorthic axis

• X₃₆ = inverse-in-{circumcircle, nine-point circle}-inverter of X₃₅₄

• X₃₆ = perspector of ABC and extraversion triangle of X₃₅

• X₃₆ = homothetic center of intangents and Trinh triangles

• X₃₆ = perspector of ABC and the reflection of the 2nd circumperp triangle in line X₁X₃

• X₃₆ = X₁₈₆-of-reflection-triangle-of-X₁

• X₃₆ = Cundy-Parry Psi transform of X₁₅₄₄₆

• X₃₆ = exsimilicenter of circumcircle and circumcircle of reflection triangle of X₁; insimilicenter is X₃₅

• X₃₆ = homothetic center of medial triangle and cross-triangle of ABC and 2nd isogonal triangle of X₁

• X₃₆ = perspector of ABC and the reflection of the excentral triangle in the antiorthic axis (the reflection of the anticevian triangle of X₁ in the trilinear polar of X₁)

• X₃₆ = Cundy-Parry Phi transform of X₅₉₀₃

• X₃₆ = homothetic center of Kosnita triangle and anti-tangential midarc triangle

• X₃₆ = orthocenter of cross-triangle of ABC and outer Yff triangle

• X₃₆ = exsimilicenter of circumcircles of ABC and outer Yff triangle; the insimilicenter is X₁

• X₃₆ = outer-Yff-isogonal conjugate of X₃₄₇₈₉

Notes

Let A' be the isogonal conjugate of A with respect to BCX₁, and define B' and C' cyclically. Let A″ be the circumcenter of BCX₁, and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X₃₆. Also, X₃₆ is the QA-P₄ center (Isogonal Center) of quadrangle ABCX₁ (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 = X₁, the 2nd isogonal triangle of X₁ is homothetic to ABC at X₃₆; see also X₃₅, X₁₄₇₈, X₁₄₇₉, X₃₅₈₃, X₃₅₈₅, X₅₉₀₃, X₇₇₄₁, X₇₉₅₁. (Randy Hutson, November 18, 2015)

Let I_aI_bI_c be the incentral triangle. Let A″ be the reflection of A in line I_bI_c, and define B″, C″ cyclically. The lines I_aA″, I_bB″, I_cC″ concur in X₃₆. (Randy Hutson, June 27, 2018)

X₃₆ is the {X₃,X₅₆}-harmonic conjugate of X₁.