X₃₁₂

ISOTOMIC CONJUGATE OF X₅₇

X₃₁₂ is the isotomic conjugate of X₅₇.

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

X₃₁₂ (ISOTOMIC CONJUGATE OF X₅₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 37}, {29, 33}

• Related Centers: X₅₇ (ISOGONAL CONJUGATE OF X₉), X₂ (CENTROID), X₃₇ (CROSSPOINT OF INCENTER AND CENTROID), X₂₉ (CEVAPOINT OF INCENTER AND ORTHOCENTER), X₃₃ (PERSPECTOR OF THE ORTHIC AND INTANGENTS TRIANGLES)

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

Barycentrics: \(bc(b + c - a) : ca(c + a - b) : ab(a + b - c)\)

Search Value (6, 9, 13): \(5.350193890803\)

▸ Alternate Coordinates3 forms

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

• Trilinears: \((\csc A)/(1 - \cos A) : (\csc B)/(1 - \cos B) : (\csc C)/(1 - \cos C)\) (M. Iliev, 4/12/07)

• Trilinears: \(bc \tan A' : ca \tan B' : ab \tan C'\), where J_aJ_bJ_c is the excentral triangle

▸ Incident Central Lines14 lines

{1, 1089}, {2, 37}, {8, 210}, {9, 314}, {29, 33}, {63, 190}, {69, 189}, {76, 85}, {92, 264}, {212, 643}, {223, 664}, {726, 982}, {894, 940}, {975, 1010}

▸ Functional Relations14 relations

• X₃₁₂ = isogonal conjugate of X₆₀₄

• X₃₁₂ = isotomic conjugate of X₅₇

• X₃₁₂ = complement of X₃₂₁₀

• X₃₁₂ = anticomplement of X₃₇₅₂

• X₃₁₂ = X_i-Ceva conjugate of X_j for these (i,j): (76, 75), (304, 322), (314, 8)

• X₃₁₂ = cevapoint of X_i and X_j for these (i,j): (2, 329), (8, 346), (9, 78), (306, 321)

• X₃₁₂ = X_i-cross conjugate of X_j for these (i,j): (8, 75), (9, 318), (306, 345), (346, 341)

• X₃₁₂ = crosssum of X_i and X_j for these (i,j): (32, 1397), (56, 1403), (57, 1424)

• X₃₁₂ = polar conjugate of X₃₄

• X₃₁₂ = perspector of ABC and extraversion triangle of X₈₅

• X₃₁₂ = BSS(A→J_a) of X₄, where J_aJ_bJ_c is the excentral triangle

• X₃₁₂ = barycentric product of vertices of Gemini triangle 27

• X₃₁₂ = perspector of ABC and cross-triangle of ABC and Gemini triangle 27

• X₃₁₂ = trilinear pole of line X₅₂₂X₃₇₁₇ (the polar of X₃₄ wrt polar circle, and the radical axis of the circumcircles of the outer and inner Garcia triangles)

Notes

Let A₃₇B₃₇C₃₇ be Gemini triangle 37. Let A' be the perspector of conic {A, B, C,B₃₇,C₃₇}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₃₁₂. (Randy Hutson, January 15, 2019)

Let A₃₉B₃₉C₃₉ be Gemini triangle 39. Let A' be the perspector of conic {A, B, C,B₃₉,C₃₉}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₃₁₂. (Randy Hutson, January 15, 2019)