X₂₅₇₅

ISOGONAL CONJUGATE OF X₁₁₁₄

As the isogonal conjugate of a point on the circumcircle, X₂₅₇₅ lies on the line at infinity.

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

X₂₅₇₅ (ISOGONAL CONJUGATE OF X₁₁₁₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {30, 511}, {74, 1113}

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

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

X₂₅₇₅ lies on the Simson quartic (Q101).

▸ Incident Central Lines12 lines

{4, 2593}, {6, 1345}, {30, 511}, {65, 2589}, {71, 2579}, {72, 2583}, {73, 2585}, {74, 1113}, {110, 1114}, {113, 1313}, {125, 1312}, {895, 2105}

▸ Functional Relations13 relations

• X₂₅₇₅ = isogonal conjugate of X₁₁₁₄

• X₂₅₇₅ = isotomic conjugate of X₁₅₁₆₅

• X₂₅₇₅ = complementary conjugate of X₁₃₁₂

• X₂₅₇₅ = X_i-Ceva conjugate of X_j for these (i,j): (2, 15167), (4, 1312), (110, 2574), (1114, 3)

• X₂₅₇₅ = X₁₂₅-cross conjugate of X₂₅₇₄

• X₂₅₇₅ = crosspoint of X₄ and X₁₁₁₄

• X₂₅₇₅ = crosssum of X_i and X_j for these (i,j): (3, 2575), (523, 1313)

• X₂₅₇₅ = infinite point of major axis of orthic inconic

• X₂₅₇₅ = 2nd point of intersection of Jerabek hyperbola and line at infinity

• X₂₅₇₅ = Thomson-isogonal conjugate of X₁₁₁₃

• X₂₅₇₅ = Lucas-isogonal conjugate of X₁₁₁₃

• X₂₅₇₅ = infinite point of minor axis of ellipse that is locus of centroids of (degenerate) pedal triangles of points on the circumcircle

• X₂₅₇₅ = infinite point of major axis of ellipse that is locus of radical center of circles centered at vertices of ABC and tangent to lines through X₃