X₂₅₅

ISOGONAL CONJUGATE OF X₁₅₈

X₂₅₅ is the isogonal conjugate of X₁₅₈.

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

X₂₅₅ (ISOGONAL CONJUGATE OF X₁₅₈) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 21}, {3, 73}

• Related Centers: X₁₅₈ (-CROSS CONJUGATE OF X₉₂), X₁ (INCENTER), X₂₁ (SCHIFFLER POINT), X₃ (CIRCUMCENTER), X₇₃ (CROSSPOINT OF INCENTER AND CIRCUMCENTER)

Trilinears: \(\cos^{2} A : \cos^{2} B : \cos^{2} C\)

Barycentrics: \(\sin A \cos^{2} A : \sin B \cos^{2} B : \sin C \cos^{2} C\)

Search Value (6, 9, 13): \(2.885490285498\)

▸ Alternate Coordinates2 forms

• Trilinears: \(1 + \cos 2A : 1 + \cos 2B : 1 + \cos 2C\)

• Trilinears: \(\tan B \tan C - 1 : :\)

▸ Incident Central Lines22 lines

{1, 21}, {3, 73}, {35, 991}, {36, 1106}, {40, 109}, {48, 563}, {55, 601}, {56, 602}, {57, 580}, {91, 1109}, {92, 1087}, {158, 775}, {162, 1099}, {165, 1103}, {200, 271}, {201, 1060}, {219, 268}, {293, 304}, {326, 1102}, {411, 651}, {498, 750}, {499, 748}

▸ Functional Relations13 relations

• X₂₅₅ = isogonal conjugate of X₁₅₈

• X₂₅₅ = isotomic conjugate of polar conjugate of X₄₈

• X₂₅₅ = X_i-Ceva conjugate of X_j for these (i,j): (63, 48), (283, 3)

• X₂₅₅ = crosspoint of X₆₃ and X₃₂₆

• X₂₅₅ = crosssum of X_i and X_j for these (i,j): (1, 290), (4, 1068), (19, 1096)

• X₂₅₅ = X_i-aleph conjugate of X_j for these (i,j): (775, 255), (1105, 158)

• X₂₅₅ = trilinear pole of line X₆₈₀X₈₂₂

• X₂₅₅ = trilinear product X₁₁₂₄*X₁₃₃₅

• X₂₅₅ = trilinear square of X₃

• X₂₅₅ = polar conjugate of X₆₅₂₁

• X₂₅₅ = X₁₉-isoconjugate of X₉₂

• X₂₅₅ = {X₁,X₃₁}-harmonic conjugate of X₁₄₉₇

• X₂₅₅ = {X₃₀₇₄,X₃₀₇₅}-harmonic conjugate of X₂

Notes

Let A'B'C' and A″B″C″ be the Lucas and Lucas(-1) central triangles. Let A* be the trilinear product A'*A″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X₂₅₅. (Randy Hutson, October 15, 2018)

Let A'B'C' and A″B″C″ be the Lucas and Lucas(-1) antipodal triangles. Let A* be the trilinear product A'*A″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X₂₅₅. (Randy Hutson, October 15, 2018)

For an artistic design generated by X₂₅₅, see X₂₄₄.