X₃₃₃

CEVAPOINT OF X₈ AND X₉

X₃₃₃ is the cevapoint of X₈ and X₉.

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

X₃₃₃ (CEVAPOINT OF X₈ AND X₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 6}, {8, 21}

• Related Centers: X₈ (NAGEL POINT), X₉ (MITTENPUNKT), X₂ (CENTROID), X₆ (SYMMEDIAN POINT), X₂₁ (SCHIFFLER POINT)

Trilinears: \(bc(b + c - a)/(b + c) : ca(c + a - b)/(c + a) : ab(a + b - c)/(a + b)\)

Barycentrics: \((b + c - a)/(b + c) : (c + a - b)/(c + a) : (a + b - c)/(a + b)\)

Search Value (6, 9, 13): \(4.136357530042\)

▸ Incident Central Lines16 lines

{2, 6}, {8, 21}, {9, 312}, {10, 58}, {19, 27}, {29, 270}, {57, 85}, {190, 321}, {239, 257}, {261, 284}, {306, 319}, {310, 673}, {662, 909}, {740, 846}, {859, 956}, {1021, 1024}

▸ Functional Relations16 relations

• X₃₃₃ = isogonal conjugate of X₁₄₀₀

• X₃₃₃ = isotomic conjugate of X₂₂₆

• X₃₃₃ = X_i-Ceva conjugate of X_j for these (i,j): (261, 21), (274, 86)

• X₃₃₃ = cevapoint of X_i and X_j for these (i,j): (2, 63), (8, 9), (283, 284)

• X₃₃₃ = X_i-cross conjugate of X_j for these (i,j): (8, 314), (9, 21), (21, 86), (283, 332), (284, 29)

• X₃₃₃ = crosspoint of X₂₇₄ and X₃₁₄

• X₃₃₃ = crosssum of X₂₁₃ and X₁₄₀₂

• X₃₃₃ = crossdifference of every pair of points on line X₅₁₂X₈₁₀

• X₃₃₃ = trilinear pole of line X₅₂₂X₆₆₃ (the perspectrix of ABC and Gemini triangle 1)

• X₃₃₃ = polar conjugate of X₂₂₅

• X₃₃₃ = crosspoint of X₂ and X₆₃ wrt both the excentral and anticomplementary triangles

• X₃₃₃ = trilinear product X₂*X₂₁

• X₃₃₃ = trilinear product of Feuerbach hyperbola intercepts of line X₂X₆

• X₃₃₃ = perspector of Gemini triangle 2 and cross-triangle of ABC and Gemini triangle 2

• X₃₃₃ = barycentric product of vertices of Gemini triangle 28

• X₃₃₃ = perspector of ABC and cross-triangle of ABC and Gemini triangle 28

Notes

Let A₂₇B₂₇C₂₇ be Gemini triangle 27. Let A' be the perspector of conic {A, B, C,B₂₇,C₂₇}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₃₃₃. (Randy Hutson, January 15, 2019)