X₁₅₇₆

ISOGONAL CONJUGATE OF X₈₅₀

X₁₅₇₆ is the isogonal conjugate of X₈₅₀.

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

X₁₅₇₆ (ISOGONAL CONJUGATE OF X₈₅₀) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {648, 1632}

• Related Centers: X₆₄₈ (TRILINEAR POLE OF EULER LINE), X₁₆₃₂ (TCC-PERSPECTOR OF X₆₄₈)

Trilinears: \(a^{3}/(b^{2} - c^{2}) : b^{3}/(c^{2} - a^{2}) : c^{3}/(a^{2} - b^{2})\)

Barycentrics: \(a^{4}/(b^{2} - c^{2}) : b^{4}/(c^{2} - a^{2}) : c^{4}/(a^{2} - b^{2})\)

Search Value (6, 9, 13): \(0.193589623399\)

▸ Incident Central Lines12 lines

{3, 1177}, {6, 157}, {32, 1084}, {50, 237}, {99, 827}, {107, 933}, {110, 351}, {160, 206}, {163, 692}, {250, 523}, {338, 1316}, {662, 1492}

▸ Functional Relations19 relations

• X₁₅₇₆ = midpoint of X₆₄₈ and X₁₆₃₂

• X₁₅₇₆ = isogonal conjugate of X₈₅₀

• X₁₅₇₆ = X_i-Ceva conjugate of X_j for these (i,j): (249, 1501), (250, 6), (827, 110), (933, 112)

• X₁₅₇₆ = cevapoint of X_i and X_j for these (i,j): (32, 669), (39, 647), (51, 512)

• X₁₅₇₆ = X_i-cross conjugate of X_j for these (i,j): (669, 32), (1501, 249)

• X₁₅₇₆ = crosspoint of X₁₁₀ and X₁₁₂

• X₁₅₇₆ = crosssum of X₅₂₃ and X₅₂₅

• X₁₅₇₆ = crossdifference of every pair of points on line X₁₁₅X₁₂₇

• X₁₅₇₆ = barycentric product of PU(2)

• X₁₅₇₆ = barycentric product of vertices of circumtangential triangle

• X₁₅₇₆ = trilinear pole of line X₃₂X₁₈₄

• X₁₅₇₆ = X₉₂-isoconjugate of X₅₂₅

• X₁₅₇₆ = X₁₅₇₇-isoconjugate of X₂

• X₁₅₇₆ = barycentric product X₁₃₇₉*X₁₃₈₀

• X₁₅₇₆ = barycentric product X₃*X₁₁₂

• X₁₅₇₆ = barycentric product X₆*X₁₁₀

• X₁₅₇₆ = polar conjugate of isotomic conjugate of X₃₂₆₆₁

• X₁₅₇₆ = vertex conjugate of X₃₆₈₃₉ and X₃₆₈₄₀

• X₁₅₇₆ = X₆₃-isoconjugate of X₁₄₆₁₈

Notes

X₁₅₇₆ is the center of the conic transform of the Stammler quartic (Q066 in Bernard Gibert’ catalogue) by X₃₁-isoconjugation. This conic is given by the barycentric equation b^4c^4(b^2-c^2)x^2+c^4a^4(c^2-a^2)y^2+a^4b^4(a^2-b^2)z^2 = 0, and it passes through the following triangle centers: X₆, X₃₁, X₄₈, X₁₅₄, X₁₆₁₃, X₂₅₇₈, X₂₅₇₉, X₅₆₃₈, X₅₆₃₉. (Angel Montesdeoca, May 7, 2016)

Let A'B'C' be the circumcevian triangle of X₅₁₂. Let A″ be the barycentric product B'*C', and define B″ and C″ cyclically. A″, B″, C″ are collinear on line X₆₆₉X₆₈₈. The lines AA″, BB″, CC″ concur in X₁₅₇₆. (Randy Hutson, August 19, 2019)