X₂₅₇₄

ISOGONAL CONJUGATE OF X₁₁₁₃

As the isogonal conjugate of a point on the circumcircle, X₂₅₇₄ lies on the line at infinity.

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

X₂₅₇₄ (ISOGONAL CONJUGATE OF X₁₁₁₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {30, 511}, {74, 1114}

• Related Centers: X₁₁₁₃ (1st EULER-LINE-CIRCUMCIRCLE INTERSECTION), X₃₀ (EULER INFINITY POINT), X₅₁₁ (ISOGONAL CONJUGATE OF X₉₈), X₇₄ (ISOGONAL CONJUGATE OF EULER INFINITY POINT), X₁₁₁₄ (2nd EULER-LINE-CIRCUMCIRCLE INTERSECTION)

Trilinears: \(1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b)\), where f(a,b,c) = X₁₁₁₃

Barycentrics: \(a/f(a,b,c) : b/f(b,c,a) : c/f(c,a,b)\)

X₂₅₇₄ lies on the Simson quartic (Q101).

▸ Incident Central Lines12 lines

{4, 2592}, {6, 1344}, {30, 511}, {65, 2588}, {71, 2578}, {72, 2582}, {73, 2584}, {74, 1114}, {110, 1113}, {113, 1312}, {125, 1313}, {895, 2104}

▸ Functional Relations13 relations

• X₂₅₇₄ = isogonal conjugate of X₁₁₁₃

• X₂₅₇₄ = isotomic conjugate of X₁₅₁₆₄

• X₂₅₇₄ = complementary conjugate of X₁₃₁₃

• X₂₅₇₄ = X_i-Ceva conjugate of X_j for these (i,j): (2, 15166), (4, 1313), (110, 2575), (1113, 3)

• X₂₅₇₄ = X₁₂₅-cross conjugate of X₂₅₇₅

• X₂₅₇₄ = crosspoint of X₄ and X₁₁₁₃

• X₂₅₇₄ = crosssum of X_i and X_j for these (i,j): (3, 2574), (523, 1312)

• X₂₅₇₄ = infinite point of minor axis of orthic inconic

• X₂₅₇₄ = 1st point of intersection of Jerabek hyperbola and line at infinity

• X₂₅₇₄ = Thomson-isogonal conjugate of X₁₁₁₄

• X₂₅₇₄ = Lucas-isogonal conjugate of X₁₁₁₄

• X₂₅₇₄ = infinite point of major axis of ellipse that is locus of centroids of (degenerate) pedal triangles of points on the circumcircle

• X₂₅₇₄ = infinite point of minor axis of ellipse that is locus of radical center of circles centered at vertices of ABC and tangent to lines through X₃

Notes

The asymptotes of the Jerabek hyperbola meet the infinity line in X₂₅₇₄ and X₂₅₇₅. In general, the infinite points on the right circumhyperbola {{A, B, C, X₄,P}}, where P =\(p : q : r\) (barycentrics) are given first barycentrcis u - v and u + v, where

u = p ((a^2-b^2+c^2) q+(-a^2-b^2+c^2) r) (c^2 (a^2-b^2-c^2) p q+b^2 (a^2-b^2-c^2) p r+(a^2 b^2-b^4+a^2 c^2+2 b^2 c^2-c^4) q r

v = (a^2-b^2-c^2) \(\sqrt{(a^{4}+b^{4}-2 a^{2} c^{2}-2 b^{2} c^{2}+c^{4}) p q r^{2}+q^{2} ((a^{4}-2 a^{2} b^{2}+b^{4}-2 b^{2} c^{2}+c^{4}) p r+a^{4} r^{2})+p^{2} (c^{4} q^{2}+(a^{4}-2 a^{2} b^{2}+b^{4}-2 a^{2} c^{2}+c^{4}) q r+b^{4} r^{2})}\)).

The infinite points of the asymptotes of the Feuerbach hyperbola at X₃₃₀₇ and X₃₃₀₈, and for the Kiepert hyperbola, by X₃₄₁₃ and X₃₄₁₄. (Peter Moses, Ocober 8, 2018)