X₆₅

ORTHOCENTER OF THE INTOUCH TRIANGLE

X₆₅ is the orthocenter of the intouch triangle.

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

X₆₅ (ORTHOCENTER OF THE INTOUCH TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

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

• Concurrence Lines: T_a–X₆₅, T_b–X₆₅, T_c–X₆₅ concurring at X₆₅

• Related Centers: X₁ (INCENTER)

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

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

Search Value (6, 9, 13): \(0.529726313505\)

▸ Alternate Coordinates3 forms

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

• Trilinears: \(\sin(A/2) \cos(B/2 - C/2) : \sin(B/2) \cos(C/2 - A/2) : \sin(C/2) \cos(A/2 - B/2)\)

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

▸ Incident Central Lines40 lines

{1, 3}, {2, 959}, {4, 158}, {6, 19}, {7, 8}, {10, 12}, {11, 117}, {29, 296}, {31, 1104}, {33, 64}, {37, 71}, {41, 910}, {42, 73}, {44, 374}, {58, 109}, {63, 958}, {68, 91}, {74, 108}, {77, 969}, {79, 80}, {81, 961}, {110, 229}, {169, 218}, {172, 248}, {224, 1004}, {225, 407}, {243, 412}, {257, 894}, {278, 387}, {279, 1002}, {386, 994}, {409, 1098}, {474, 997}, {497, 938}, {516, 950}, {519, 553}, {604, 1100}, {651, 895}, {1039, 1041}, {1061, 1063}

▸ Functional Relations43 relations

• X₆₅ = reflection of X_i in X_j for these (i,j): (1, 942), (72, 10)

• X₆₅ = isogonal conjugate of X₂₁

• X₆₅ = isotomic conjugate of X₃₁₄

• X₆₅ = complement of X₃₈₆₉

• X₆₅ = anticomplement of X₉₆₀

• X₆₅ = circumcircle-inverse of X₅₁₇₂

• X₆₅ = incircle-inverse of X₁₃₁₉

• X₆₅ = X_i-Ceva conjugate of X_j for these (i,j): (1, 73), (4, 225), (7, 226), (10, 227), (109, 513), (226, 37)

• X₆₅ = X₄₂-cross conjugate of X₃₇

• X₆₅ = crosspoint of X_i and X_j for these (i,j): (1, 4), (7, 57)

• X₆₅ = crosssum of X_i and X_j for these (i,j): (1, 3), (9, 55), (56, 1394), (1805, 1806)

• X₆₅ = crossdifference of every pair of points on line X₅₂₁X₆₅₀

• X₆₅ = X₁₂₈₄-Hirst inverse of X₁₄₀₀

• X₆₅ = X_i-beth conjugate of X_j for these (i,j): (1, 65), (8, 72), (10, 10), (65, 1042), (80, 65), (100, 65), (101, 213), (291, 65), (668, 65), (1018, 65)

• X₆₅ = bicentric sum of PU(15)

• X₆₅ = PU(15)-harmonic conjugate of X₆₅₀

• X₆₅ = trilinear product of PU(81)

• X₆₅ = trilinear pole of line X₆₄₇X₆₆₁

• X₆₅ = perspector of intouch triangle and inverse(n(hexyl triangle))

• X₆₅ = orthologic center of inverse(n(hexyl triangle)) to hexyl triangle; the reciprocal orthologic center is X₈₄

• X₆₅ = perspector of ABC and the extangents triangle

• X₆₅ = X₁₉₈₆-of-Fuhrmann-triangle

• X₆₅ = X₄₀ of Mandart-incircle triangle

• X₆₅ = homothetic center of intangents triangle and reflection of extangents triangle in X₄₀

• X₆₅ = homothetic center of extangents triangle and reflection of intangents triangle in X₁

• X₆₅ = reflection of X₃₀₅₇ in X₁

• X₆₅ = {X₁,X₃}-harmonic conjugate of X₂₆₄₆

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

• X₆₅ = {P,Q}-harmonic conjugate of X₁₄₆₃, where P and Q are the intersections of the incircle and line X₇X₈

• X₆₅ = pairwise perspector of: intouch triangle, 4th extouch triangle, 5th extouch triangle

• X₆₅ = perspector of [reflection of incentral triangle in X₁] and tangential triangle, wrt incentral triangle, of circumconic of incentral triangle centered at X₁ (bicevian conic of X₁ and X₅₇)

• X₆₅ = inverse-in-{incircle, circumcircle}-inverter of X₂₀₇₈

• X₆₅ = pedal-isogonal conjugate of X₁

• X₆₅ = X₅ of reflection triangle of X₁

• X₆₅ = radical trace of circumcircle and circumcircle of reflection triangle of X₁

• X₆₅ = X₁₈₈-of-orthic-triangle if ABC is acute

• X₆₅ = perspector of ABC and cross-triangle of ABC and 4th extouch triangle

• X₆₅ = perspector of ABC and cross-triangle of ABC and 5th extouch triangle

• X₆₅ = polar conjugate of X₃₁₆₂₃

• X₆₅ = pole wrt polar circle of trilinear polar of X₃₁₆₂₃ (line X₅₂₁X₁₉₄₈)

• X₆₅ = perspector of ABC and anti-tangential midarc triangle

• X₆₅ = homothetic center of extangents triangle and anti-tangential midarc triangle

• X₆₅ = excentral-to-intouch similarity image of X₁

Notes

Let A' be the intersections of the tangents to the Yiu conic at the points where they meet the A-excircle. Define B' and C' similarly. The lines AA', BB', CC' concur in X₆₅. (Randy Hutson, July 20, 2016)

Let A_b, A_c be the points where the A-excircle touches lines CA and AB resp., and define B_c, B_a, C_a, C_b cyclically. Let T_a be the intersection of the tangents to the Yiu conic (defined at X₄₇₈) at B_c and C_a, and define T_b, T_c cyclically. Let T_a' be the intersection of the tangents to the Yiu conic at B_a and C_b, and define T_b', T_c' cyclically. Let S_a = T_bT_c∩T_b'T_c', S_b = T_cT_a∩T_c'T_a', S_c = T_aT_b∩T_a'T_b'. The lines AS_a, BS_b, CS_c concur in X₆₅. (See also X₁₉₀₃.) (Randy Hutson, July 20, 2016)

Let A' be the homothetic center of the orthic triangles of the intouch and A-extouch triangles, and define B' and C' cyclically. The triangle A'B'C' is perspective to the extouch triangle at X₆₅. (Randy Hutson, July 20, 2016)

Let H_aH_bH_c be the orthic triangle. Let H_bH_cA″ be the triangle similar to ABC such that segment H_aA″ crosses the line H_bH_c. Define B″ and C″ cyclically. Equivalently, A″ is the reflection of A in H_bH_c, and cyclically for B″ and C″. Let I_a be the incenter of H_bH_cA″, and define I_b and I_c cyclically. The circumcenter of triangle I_aI_bI_c is X₆₅. Let A* be the intersection of lines A″I_a and H_bH_c, and define B* and C* cyclically. The lines H_aA*, H_bB*, H_cC* concur in X₆₅. (Randy Hutson, July 20, 2016)

Let A_b, A_c be the points where the A-excircle touches lines CA and AB resp., and define B_c, B_a, C_a, C_b cyclically. Let I_aI_bI_c be the intouch triangle. Let O_a be the circle through A_b, A_c, I_b, I_c, and define O_b, O_c cyclically. X₆₅ is the radical center of O_a, O_b, O_c. (Randy Hutson, July 20, 2016)

Let A'B'C' be the intouch triangle of the extangents triangle, if ABC is acute. Then A'B'C' is perspective to the intouch triangle and 4th and 5th extouch triangles at X₆₅. (Randy Hutson, December 2 2017)

Let O_A be the circle centered at the A-vertex of the Wasat triangle and passing through A; define O_B and O_C cyclically. X₆₅ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)

Let A' be the isogonal conjugate of A wrt the A-extouch triangle. Define B' and C' cyclically. The lines AA', BB', CC' concur in X₆₅. (Randy Hutson, August 30, 2020)

In the plane of a triangle ABC, let

• B_a = reflection of A in the external angular bisector of angle B, and define C_b and A_c cyclically;

• C_a = reflection of A in the external angular bisector of angle C, and define A_b and B_c cyclically;

• P_a = A_cB_c∩A_bC_b, and define P_b and P_c cyclically;

• K_a = A_bB_a∩A_cC_a, and define K_b and K_c cyclically.

Then P_aP_bP_c and ABC, and also K_aK_bK_c and ABC, are perspective, and the perspector is X₆₅. (Dasari Naga Vijay Krishna, June 19, 2021)

X₆₅ is the {X₁,X₄₀}-harmonic conjugate of X₅₅.