X₈₄

ISOGONAL CONJUGATE OF X₄₀

X₈₄ is the isogonal conjugate of X₄₀.

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

X₈₄ (ISOGONAL CONJUGATE OF X₄₀) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 9}, {8, 20}

• Related Centers: X₄₀ (BEVAN POINT), X₃ (CIRCUMCENTER), X₉ (MITTENPUNKT), X₈ (NAGEL POINT), X₂₀ (DE LONGCHAMPS POINT)

Trilinears: \(1/(\cos B + \cos C - \cos A - 1) : :\)

Search Value (6, 9, 13): \(-52.770375222430\)

▸ Alternate Coordinates1 form

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

▸ Linear Combinations1 combination

• X₈₄ = \((r + 2R)\) X₁ - \(6R\) X₂ + \((4R - 2r)\) X₃ (Peter Moses, April 2, 2013)

X₈₄ lies on the Darboux cubic, the circumellipse with center X₉.

▸ Incident Central Lines20 lines

{1, 221}, {3, 9}, {4, 57}, {7, 946}, {8, 20}, {21, 285}, {33, 603}, {36, 90}, {46, 80}, {58, 990}, {64, 3353}, {171, 989}, {256, 988}, {294, 580}, {309, 314}, {581, 941}, {944, 1000}, {2130, 3345}, {3346, 3472}, {3347, 3355}

▸ Functional Relations18 relations

• X₈₄ = reflection of X_i in X_j for these (i,j): (40, 1158), (1490, 3)

• X₈₄ = isogonal conjugate of X₄₀

• X₈₄ = isotomic conjugate of X₃₂₂

• X₈₄ = X_i-Ceva conjugate of X_j for these (i,j): (189, 282), (280, 1)

• X₈₄ = X_i-cross conjugate of X_j for these (i,j): (19, 57), (56, 1)

• X₈₄ = X₂₈₀-aleph conjugate of X₈₄

• X₈₄ = X_i-beth conjugate of X_j for these (i,j): (271, 3), (280, 280), (285, 84)

• X₈₄ = X₆₈-of-the-hexyl-triangle

• X₈₄ = trilinear pole of line X₆₅₀X₁₄₅₉

• X₈₄ = perspector of ABC and the reflection in X₉ of the antipedal triangle of X₉

• X₈₄ = Danneels point of X₁₁₀

• X₈₄ = trilinear product of vertices of hexyl triangle (i.e., the extraversions of X₄₀)

• X₈₄ = hexyl-isotomic conjugate of X₁₂₇₁₇

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

• X₈₄ = perspector of ABC and cross-triangle of extouch and Hutson-extouch triangles

• X₈₄ = Cundy-Parry Phi transform of X₉

• X₈₄ = Cundy-Parry Psi transform of X₅₇

• X₈₄ = intouch-to-excentral similarity image of X₄

Notes

Let J_a,J_b,J_c be the excenters. The perpendiculars from J_b to AB and from J_c to AC meet in a point A″. Points B″ and C″ are determined cyclically. The hexyl triangle, A″B″C″, is perspective to ABC, and X₈₄ is the perspector.

Let N_aN_bN_c be the extouch triangle. Let A″ be the orthocenter of AN_bN_c, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₈₄. (Randy Hutson, September 14, 2016)

Let A₁B₁C₁ be the 1st Conway triangle. Let A' be the crosspoint of B₁ and C₁, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₈₄. (Randy Hutson, December 2, 2017)

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