X₆₉

SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE

X₆₉ is the symmedian point of the anticomplementary triangle.

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

X₆₉ (SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Anticomplementary Triangle △K_aK_bK_c — vertices K_a, K_b, K_c (Formed by lines through A, B, C parallel to opposite sides)

• Concurrence Lines: K_a–X₆₉, K_b–X₆₉, K_c–X₆₉ concurring at X₆₉

Trilinears: \((\cos A)/a^{2} : (\cos B)/b^{2} : (\cos C)/c^{2}\)

Barycentrics: \(\cot A : \cot B : \cot C\)

Search Value (6, 9, 13): \(5.902289387335\)

▸ Alternate Coordinates6 forms

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

• Trilinears: \(\sec^{2}(A/2) - \csc^{2}(A/2) : :\)

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

• Barycentrics: \(\cot B + \cot C - \cot \omega : :\)

• Barycentrics: \(\cot B + \cot C - \cot A - \cot \omega : :\)

• Barycentrics: \(S_{A} : S_{B} : S_{C}\)

▸ Linear Combinations2 combinations

• X₆₉ = \(2(r^{2} + 2rR + s^{2})\) X₁ + \(3(r^{2} - s^{2})\) X₂ - \(4r^{2}\) X₃ (Peter Moses, April 2, 2013)

• X₆₉ = \((\cot A)\) A + \((\cot B)\) B + \((\cot C)\) C, where A, B, C denote angles and A, B, C denote vertices

X₆₉ lies on the Lucas cubic.

▸ Incident Central Lines37 lines

{2, 6}, {3, 332}, {4, 76}, {7, 8}, {9, 344}, {10, 969}, {20, 64}, {22, 159}, {54, 95}, {63, 71}, {72, 304}, {73, 77}, {74, 99}, {110, 206}, {125, 895}, {144, 190}, {150, 668}, {189, 309}, {192, 742}, {194, 695}, {200, 269}, {219, 1332}, {248, 287}, {263, 308}, {265, 328}, {274, 443}, {290, 670}, {297, 393}, {347, 664}, {350, 497}, {404, 1014}, {478, 651}, {485, 639}, {486, 640}, {520, 879}, {1225, 2888}, {1369, 3410}

▸ Functional Relations50 relations

• X₆₉ = reflection of X_i in X_j for these (i,j): (2, 599), (4, 1352), (6, 141), (20, 1350), (193, 6), (895, 125), (1351, 5), (1353, 140)

• X₆₉ = isogonal conjugate of X₂₅

• X₆₉ = isotomic conjugate of X₄

• X₆₉ = complement of X₁₉₃

• X₆₉ = anticomplement of X₆

• X₆₉ = anticomplementary conjugate of X₂

• X₆₉ = circumcircle-inverse of X₅₈₆₆

• X₆₉ = cyclocevian conjugate of X₂₅₃

• X₆₉ = X_i-Ceva conjugate of X_j for these (i,j): (76, 2), (304, 345), (314, 75), (332, 326)

• X₆₉ = cevapoint of X_i and X_j for these (i,j): (2, 20), (3, 394), (6, 159), (8, 329), (63, 78), (72, 306), (125, 525)

• X₆₉ = X_i-cross conjugate of X_j for these (i,j): (3, 2), (63, 348), (72, 63), (78, 345), (125, 525), (306, 304), (307, 75), (343, 76)

• X₆₉ = crosspoint of X_i and X_j for these (i,j): (2, 2996), (76, 305), (314, 332)

• X₆₉ = X₂-Hirst inverse of X₃₂₅

• X₆₉ = X_i-beth conjugate of X_j for these (i,j): (69, 77), (99, 347), (314, 7), (332, 69), (645, 69), (668, 69)

• X₆₉ = barycentric product of PU(37)

• X₆₉ = bicentric sum of PU(132)

• X₆₉ = midpoint of PU(132)

• X₆₉ = perspector of the orthic-of-medial triangle and the reference triangle

• X₆₉ = perspector of ABC and the pedal triangle of X₂₀

• X₆₉ = perspector of ABC and (reflection in X₂ of the pedal triangle of X₂)

• X₆₉ = intersection of extended sides P(11)U(45) and U(11)P(45) of the trapezoid PU(11)PU(45)

• X₆₉ = perspector of ABC and 4th extouch triangle

• X₆₉ = antipode of X₂₈₇ in hyperbola {A, B, C,X₂,X₆₉}}

• X₆₉ = trilinear pole of line X₄₄₁X₅₂₅

• X₆₉ = pole wrt polar circle of trilinear polar of X₃₉₃ (line X₄₆₀X₅₁₂)

• X₆₉ = X₄₈-isoconjugate (polar conjugate) of X₃₉₃

• X₆₉ = X₆-isoconjugate of X₁₉

• X₆₉ = X₉₂-isoconjugate of X₃₂

• X₆₉ = antigonal image of X₈₉₅

• X₆₉ = crosssum of X_i and X_j for these (i,j): (6, 3053), (32, 1974)

• X₆₉ = perspector of ABC and the 2nd pedal triangle of X₃

• X₆₉ = crosspoint of X₆ and X₁₅₉ wrt both the excentral and tangential triangles

• X₆₉ = crosspoint of X₂ and X₂₀ wrt both the excentral and anticomplementary triangles

• X₆₉ = homothetic center of anticomplementary triangle and 2nd antipedal triangle of X₄ (i.e., of 1st and 2nd antipedal triangles of X₄)

• X₆₉ = perspector of the complement of the polar circle

• X₆₉ = perspector of the inconic with center X₃

• X₆₉ = pole, wrt de Longchamps circle, of trilinear polar of X₉₅

• X₆₉ = perspector of the extraversion triangles of X₇ and X₈

• X₆₉ = {X₂,X₆}-harmonic conjugate of X₃₆₁₈

• X₆₉ = perspector of ABC and anticomplement of submedial triangle

• X₆₉ = perspector of ABC and mid-triangle of orthic and dual of orthic triangles

• X₆₉ = perspector of ABC and cross-triangle of ABC and 2nd Brocard triangle

• X₆₉ = perspector of 2nd Conway triangle and cross-triangle of ABC and 2nd Conway triangle

• X₆₉ = Lucas-isogonal conjugate of X₃₇₆

• X₆₉ = anticevian-isogonal conjugate of X₂

• X₆₉ = inverse-in-MacBeath-circumconic of X₂₂₁₅₁

• X₆₉ = {X₇,X₈}-harmonic conjugate of X₇₅

• X₆₉ = intersection of van Aubel lines of outer and inner Vecten triangles

• X₆₉ = orthic-isogonal conjugate of X₁₉₅₈₃

• X₆₉ = X₄-Ceva conjugate of X₁₉₅₈₃

Notes

Let K_aK_bK_c be the anticomplementary triangle. Let A″ be the inverse-in-anticomplementary-circle of A, and define B″ and C″ cyclically. The lines K_aA″, K_bB″, K_cC″ concur in X₆₉. (Randy Hutson, February 10, 2016)

Let K_aK_bK_c be the anticomplementary triangle. Let A″ be the orthogonal projection of K_a on line BC, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₆₉. (Randy Hutson, February 10, 2016)

Let A'B'C' be the half-altitude triangle. Let A″ be the trilinear pole of line B'C', and define B″ and C″ cyclically. Let A* be the trilinear pole of line B″C″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X₆₉. (Randy Hutson, February 10, 2016)

Let A₂B₂C₂ be the 2nd Conway triangle. Let A' be the cevapoint of B₂ and C₂, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₆₉. (Randy Hutson, December 10, 2016)

X₆₉ is the barycentric multiplier for the MacBeath circumconic. (The barycentric product of X₆₉ and the circumcircle is the MacBeath circumconic.) (Randy Hutson, August 19, 2019)

Let O_A be the circle centered at the A-vertex of the 1st Ehrmann triangle and passing through A; define O_B and O_C cyclically. X₆₉ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)

X₆₉ is the {X₇,X₈}-harmonic conjugate of X₇₅.