X₂₁₉

X₈-CEVA CONJUGATE OF X₅₅

X₂₁₉ = X₉₂-isoconjugate of X₅₆

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

X₂₁₉ (X₈-CEVA CONJUGATE OF X₅₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 6}, {8, 29}

• Related Centers: X₈ (NAGEL POINT), X₅₅ (INSIMILICENTER), X₁ (INCENTER), X₆ (SYMMEDIAN POINT), X₂₉ (CEVAPOINT OF INCENTER AND ORTHOCENTER)

Trilinears: \(\cos A \cot A/2 : :\)

Barycentrics: \(\sin 2A \cot A/2 : :\)

Search Value (6, 9, 13): \(4.717066795933\)

▸ Alternate Coordinates4 forms

• Trilinears: \((\sin A)/(1 - \sec A) : :\)

• Trilinears: \(1/(\csc A - 2 \csc 2A) : :\)

• Trilinears: \(a(b + c - a)(b^{2} + c^{2} - a^{2}) : :\)

• Trilinears: \((b + c - a) \cos A : :\)

▸ Incident Central Lines23 lines

{1, 6}, {3, 48}, {8, 29}, {10, 965}, {19, 517}, {40, 610}, {41, 1036}, {55, 284}, {56, 579}, {63, 77}, {69, 1332}, {101, 102}, {144, 347}, {200, 282}, {206, 692}, {255, 268}, {278, 329}, {332, 345}, {346, 644}, {572, 947}, {577, 906}, {604, 672}, {1993, 3219}

▸ Functional Relations15 relations

• X₂₁₉ = isogonal conjugate of X₂₇₈

• X₂₁₉ = isotomic conjugate of X₃₃₁

• X₂₁₉ = X_i-Ceva conjugate of X_j for these (i,j): (8, 55), (63, 3), (283, 212)

• X₂₁₉ = X_i-cross conjugate of X_j for these (i,j): (48, 268), (71, 9), (212, 3)

• X₂₁₉ = crosspoint of X_i and X_j for these (i,j): (8, 345), (64, 78)

• X₂₁₉ = crosssum of X_i and X_j for these (i,j): (19, 34), (56, 608)

• X₂₁₉ = X_i-beth conjugate of X_j for these (i,j): (101, 478), (219, 48), (644, 219)

• X₂₁₉ = trilinear pole of line X₆₅₂X₁₉₄₆

• X₂₁₉ = crossdifference of every pair of points on line X₅₁₃X₁₈₃₅

• X₂₁₉ = X₉₂-isoconjugate of X₅₆

• X₂₁₉ = perspector of extouch triangle and unary cofactor triangle of intouch triangle

• X₂₁₉ = perspector of ABC and unary cofactor triangle of Gemini triangle 37

• X₂₁₉ = polar conjugate of isogonal conjugate of X₆₀₅₆

• X₂₁₉ = polar conjugate of isotomic conjugate of X₁₂₅₉

• X₂₁₉ = X₆₃-isoconjugate of X₁₁₁₈

Notes

Let N_aN_bN_c be the extouch triangle. Let A″ be the barycentric product of the circumcircle intercepts of line N_bN_c, and define B″ and C″ cyclically. The lines N_aA″, N_bB″, N_cC″ concur in X₂₁₉. (Randy Hutson, July 31 2018)