X₃₄

X₄-BETH CONJUGATE OF X₄

X₃₄ is the center of perspective of the orthic triangle and the reflection in the incenter of the intangents triangle.

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

X₃₄ (X₄-BETH CONJUGATE OF X₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 4}, {6, 19}

• Related Centers: X₄ (ORTHOCENTER), X₁ (INCENTER), X₆ (SYMMEDIAN POINT), X₁₉ (CLAWSON POINT)

Trilinears: \(1 - \sec A : 1 - \sec B : 1 - \sec C\)

Barycentrics: \(\sin A - \tan A : \sin B - \tan B : \sin C - \tan C\)

Tripolars: \((pending)\)

Search Value (6, 9, 13): \(-0.119171170783\)

▸ Alternate Coordinates4 forms

• Trilinears: \(\tan A \tan(A/2) : \tan B \tan(B/2) : \tan C \tan(C/2)\)

• Trilinears: \(1/[(b + c - a)(b^{2} + c^{2} - a^{2})] : :\)

• Trilinears: \(\sec A \sin^{2}(A/2)\)

• Barycentrics: \(\tan A \sin^{2}(A/2) : :\)

▸ Linear Combinations1 combination

• X₃₄ = \((r + 2R - s)(r + 2R + s)\) X₁ + \(6rR\) X₂ - \(4rR\) X₃ (Peter Moses, April 2, 2013)

▸ Incident Central Lines31 lines

{1, 4}, {2, 1038}, {5, 1060}, {6, 19}, {7, 1039}, {8, 1041}, {9, 201}, {10, 475}, {11, 235}, {12, 427}, {20, 1040}, {24, 36}, {25, 56}, {28, 57}, {29, 77}, {30, 1062}, {35, 378}, {40, 212}, {46, 47}, {55, 227}, {79, 1061}, {80, 1063}, {87, 242}, {106, 108}, {196, 937}, {207, 1042}, {222, 942}, {244, 1106}, {331, 870}, {347, 452}, {860, 997}

▸ Functional Relations18 relations

• X₃₄ = isogonal conjugate of X₇₈

• X₃₄ = isotomic conjugate of X₃₇₁₈

• X₃₄ = anticomplement of X₃₄₈₂₃

• X₃₄ = X_i-Ceva conjugate of X_j for these (i,j): (1, 207), (4, 208), (28, 56), (273, 57), (278, 19)

• X₃₄ = X₂₅-cross conjugate of X₁₉

• X₃₄ = crosssum of X₂₁₉ and X₁₂₆₀

• X₃₄ = crossdifference of every pair of points on line X₅₂₁X₆₅₂

• X₃₄ = circumcircle-inverse of X₃₂₇₅₇

• X₃₄ = X₅₆-Hirst inverse of X₁₄₃₀

• X₃₄ = trilinear pole of polar of X₃₁₂ wrt polar circle (line X₆₄₉X₄₀₁₇)

• X₃₄ = pole wrt polar circle of trilinear polar of X₃₁₂ (line X₅₂₂X₃₇₁₇)

• X₃₄ = polar conjugate of X₃₁₂

• X₃₄ = perspector of ABC and extraversion triangle of X₃₃

• X₃₄ = homothetic center of intangents triangle and reflection of orthic triangle in X₄

• X₃₄ = homothetic center of orthic triangle and anti-tangential midarc triangle

• X₃₄ = X₈₀₇₈-of-orthic-triangle if ABC is acute

• X₃₄ = X_i-beth conjugate of X_j for these (i,j): (1, 221), (4, 4), (28, 34), (29, 1), (107, 158), (108, 34), (110, 47), (162, 34), (811, 34)

• X₃₄ = X_i-isoconjugate of X_j for these {i,j}: {1, 78}, {31, 3718}, {48, 312}

Notes

X₃₄ is the {X₁,X₄}-harmonic conjugate of X₃₃.