X₁₈₅₇

ZOSMA TRANSFORM OF X₂₂₃

X₁₈₅₇ = X₁₅₈-Ceva conjugate of X₃₉₃

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

X₁₈₅₇ (ZOSMA TRANSFORM OF X₂₂₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 65}, {33, 42}

• Related Centers: X₁₈₀₄ (INVERSE MIMOSA TRANSFORM OF X₂₂₃), X₄ (ORTHOCENTER), X₆₅ (ORTHOCENTER OF THE INTOUCH TRIANGLE), X₃₃ (PERSPECTOR OF THE ORTHIC AND INTANGENTS TRIANGLES), X₄₂ (CROSSPOINT OF INCENTER AND SYMMEDIAN POINT)

Trilinears: (y + z) sec A : (z + x) sec B : (x + y) sec C,\(x : y : z\) = X₂₂₃

Barycentrics: (y + z) tan A : (z + x) tan B : (x + y) tan C,\(x : y : z\) = X₂₂₃

Search Value (6, 9, 13): \(1.584417234156\)

▸ Alternate Coordinates2 forms

• Trilinears: \([\cos(A/2) \sec A]^{2} : [\cos(B/2) \sec B]^{2} : [\cos(C/2) \sec C]^{2} ( M. Iliev, 4/12/07)\)

• Trilinears: \((1 + \cos A)/(1 + \cos 2A) : (1 + \cos B)/(1 + \cos 2B) : (1 + \cos C)/(1 + \cos 2C)\) (M. Iliev, 4/12/07)

▸ Incident Central Lines13 lines

{2, 243}, {4, 65}, {8, 1896}, {11, 278}, {33, 42}, {55, 281}, {92, 497}, {189, 1364}, {388, 1895}, {403, 1068}, {412, 1788}, {1478, 1784}, {1748, 1776}

▸ Functional Relations5 relations

• X₁₈₅₇ = isogonal conjugate of X₁₈₀₄

• X₁₈₅₇ = isotomic conjugate of X₇₀₅₅

• X₁₈₅₇ = X₁₅₈-Ceva conjugate of X₃₉₃

• X₁₈₅₇ = pole wrt polar circle of trilinear polar of X₃₄₈ (line X₄₀₂₅X_4131)

• X₁₈₅₇ = polar conjugate of X₃₄₈