X₅₆

EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE)

X₅₆ is the perspector of the tangential triangle and the reflection of the intangents triangle in X₁.

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

X₅₆ (EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Tangential Triangle △Tan_aTan_bTan_c — vertices Tan_a, Tan_b, Tan_c (Bounded by tangents to the circumcircle at A, B, C)

• Derived Triangle: Intouch (Contact) Triangle △T_aT_bT_c — vertices T_a, T_b, T_c (Contact triangle of Incircle touchpoints on BC, CA, AB)

• Circle: Circumcircle (circumcircle) — Passes through vertices A, B, C; centered at X₃ CIRCUMCENTER

• Circle: Incircle (incircle) — Inscribed circle tangent to sidelines at T_a, T_b, T_c; centered at X₁ INCENTER

• Central Lines: {1, 3}

• Related Centers: X₃ (CIRCUMCENTER), X₁ (INCENTER)

Trilinears: \(a/(b + c - a) : b/(c + a - b) : c/(a + b - c)\)

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

Search Value (6, 9, 13): \(0.187119023187\)

▸ Alternate Coordinates7 forms

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

• Trilinears: \(\sin^{2}(A/2) : \sin^{2}(B/2) : \sin^{2}(C/2)\)

• Trilinears: \(a(\tan A/2) : :\)

• Trilinears: \(R_{a} - r : R_{b} - r : R_{c} - r\), where \(R_{a}\), \(R_{b}\), \(R_{c}\) are the exradii

• Trilinears: \(a R_{a} : b R_{b} : c R_{c}\), where \(R_{a}\), \(R_{b}\), \(R_{c}\) are the exradii

• Trilinears: \(a \cos A - (c + a) \cos B - (a + b) \cos C : :\)

• Barycentrics: \(\operatorname{area}(A'BC) : :\), where A'B'C' = 2nd circumperp triangle

▸ Linear Combinations1 combination

• X₅₆ = \(R\) X₁ - \(r\) X₃

▸ Incident Central Lines58 lines

{1, 3}, {2, 12}, {4, 11}, {5, 499}, {6, 41}, {7, 21}, {8, 404}, {10, 474}, {19, 207}, {20, 497}, {22, 977}, {25, 34}, {28, 278}, {30, 496}, {31, 154}, {32, 1015}, {33, 963}, {38, 201}, {58, 222}, {61, 202}, {62, 203}, {63, 960}, {72, 997}, {77, 1036}, {78, 480}, {81, 959}, {85, 870}, {87, 238}, {100, 145}, {101, 218}, {105, 279}, {106, 109}, {140, 495}, {181, 386}, {182, 611}, {197, 227}, {212, 939}, {219, 579}, {220, 672}, {223, 937}, {226, 405}, {255, 602}, {266, 289}, {269, 738}, {330, 385}, {376, 1058}, {411, 938}, {511, 613}, {551, 553}, {607, 911}, {631, 1056}, {667, 764}, {946, 1012}, {978, 979}, {1025, 1083}, {1070, 1074}, {1072, 1076}, {1345, 2464}

▸ Functional Relations49 relations

• X₅₆ = midpoint of X₁ and X₄₆

• X₅₆ = reflection of X_i in X_j for these (i,j): (1479, 496), (2098, 1)

• X₅₆ = isogonal conjugate of X₈

• X₅₆ = isotomic conjugate of X₃₅₉₆

• X₅₆ = complement of X₃₄₃₆

• X₅₆ = anticomplement of X₁₃₂₉

• X₅₆ = circumcircle-inverse of X₁₃₁₉

• X₅₆ = antigonal conjugate of X₁₇₁₀₁

• X₅₆ = X_i-Ceva conjugate of X_j for these (i,j): (1, 221), (7, 222), (28, 34), (57, 6), (59, 109), (108, 513)

• X₅₆ = X₃₁-cross conjugate of X₆

• X₅₆ = crosspoint of X_i and X_j for these (i,j): (1, 84), (7, 278), (28, 58), (57, 269), (59, 109)

• X₅₆ = crosssum of X_i and X_j for these (i,j): (1, 40), (2, 144), (6, 197), (9, 200), (10, 72), (11, 522), (55, 219), (175, 176), (519, 1145)

• X₅₆ = crossdifference of every pair of points on line X₅₂₂X₆₅₀

• X₅₆ = X_i-Hirst inverse of X_j for these (i,j): (6, 1458), (34, 1430), (57, 1429), (604, 1428), (1416, 1438)

• X₅₆ = X₂₆₆-aleph conjugate of X₁₀₅₀

• X₅₆ = X_i-beth conjugate of X_j for these (i,j): (1, 1), (21, 3), (56, 1106), (58, 56), and (P,57) for all P on the circumcircle

• X₅₆ = homothetic center of the intouch triangle and the circumcevian triangle of X₁

• X₅₆ = trilinear pole of line X₆₄₉X₈₅₄ (the isogonal conjugate of the isotomic conjugate of the Gergonne line)

• X₅₆ = homothetic center of ABC and the reflection of the Mandart-incircle triangle in X₁

• X₅₆ = {X₁,X₄₀}-harmonic conjugate of X₃₀₅₇

• X₅₆ = {X₁,X₅₇}-harmonic conjugate of X₆₅

• X₅₆ = trilinear square of X₂₆₆

• X₅₆ = trilinear square root of X₁₁₀₆

• X₅₆ = X₉₂-isoconjugate of X₂₁₉

• X₅₆ = vertex conjugate of PU(93)

• X₅₆ = inverse-in-{circumcircle, incircle}-inverter of X₃₆₆₀

• X₅₆ = perspector of ABC and extraversion triangle of X₅₅

• X₅₆ = perspector of ABC and unary cofactor triangle of Gemini triangle 15

• X₅₆ = homothetic center of intangents triangle and reflection of tangential triangle in X₃

• X₅₆ = homothetic center of tangential triangle and reflection of intangents triangle in X₁

• X₅₆ = Brianchon point (perspector) of inellipse that is isogonal conjugate of isotomic conjugate of incircle

• X₅₆ = pole wrt polar circle of trilinear polar of X₇₀₁₇ (line X₂₈₀₄X₄₃₉₇)

• X₅₆ = X₄₈-isoconjugate (polar conjugate) of X₇₀₁₇

• X₅₆ = barycentric product of PU(46)

• X₅₆ = bicentric sum of PU(60)

• X₅₆ = PU(60)-harmonic conjugate of X₆₅₀

• X₅₆ = trilinear product of PU(92)

• X₅₆ = perspector of ABC and cross-triangle of ABC and Apus triangle

• X₅₆ = perspector of ABC and cross-triangle of ABC and Hutson intouch triangle

• X₅₆ = homothetic center of ABC and cross-triangle of ABC and 1st Johnson-Yff triangle

• X₅₆ = homothetic center of midarc triangle and 1st-circumperp-of-1st-circumperp triangle (which is also 2nd-circumperp-of-2nd-circumperp triangle)

• X₅₆ = homothetic center of 2nd midarc triangle and 2nd-circumperp-of-1st-circumperp triangle (which is also 1st-circumperp-of-2nd-circumperp triangle)

• X₅₆ = Cundy-Parry Phi transform of X₅₁₇

• X₅₆ = Cundy-Parry Psi transform of X₁₀₄

• X₅₆ = {X₃₅₁₃,X₃₅₁₄}-harmonic conjugate of X₅₅

• X₅₆ = X₄-of-2nd-Johnson-Yff-triangle

• X₅₆ = homothetic center of tangential triangle and anti-tangential midarc triangle

• X₅₆ = Ursa-major-to-Ursa-minor similarity image of X₄

• X₅₆ = barycentric product of (nonreal) circumcircle intercepts of the Gergonne line

Notes

Let A'B'C' be the Fuhrmann triangle. Let L_a be the line through A' parallel to BC, and define L_b and L_c cyclically. Let A″ = L_b∩L_c, B″ = L_c∩L_a, C″ = L_a∩L_b. The lines AA″, BB″, CC″ concur in X₅₆. Also, let A_aB_aC_a be the poristic triangle (i.e., a triangle with common circumcircle and incircle as ABC) such that B_aC_a is parallel to BC. Define A_bB_bC_b and A_cB_cC_c cyclically. The lines AA_a, BB_b, CC_c concur in X₅₆. (Randy Hutson, November 18, 2015)

Let T_aT_bT_c be the intouch triangle. Let A″ be the barycentric product of the circumcircle intercepts of line T_bT_c, and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X₅₆. (Randy Hutson, June 27, 2018)

See Dasari Naga Vijay Krishna, On a Conic Through Twelve Notable Points, Int. J. Adv. Math. and Mech. 7(2) (2019) 1-15.

In the plane of a triangle ABC, let

• A'B'C' = circumcevian triangle of X₁;

• T_a = line tangent to circumcircle at A', and define T_b and T_c cyclically;

• V_a = T_b∩T_c, and define V_b and V_c cyclically.

The triangle V_aV_bV_c is perspective to ABC, and the perspector is X₅₆. (Dasari Naga Vijay Krishna, June 19, 2021)

In the plane of a triangle ABC, let (O_a) be the circle tangent internally to the incircle and tangent internally to the circumcircle at A. Define (O_b) and (O_c) cyclically. The radical center of (O_a), (O_b), (O_c) is X₅₆. (Ivan Pavlov, February 24, 2022)

X₅₆ is the {X₁,X₃}-harmonic conjugate of X₅₅.