X₂₃₁₀

X₂-ISOCONJUGATE OF X₁₂₆₂

X₂₃₁₀ = trilinear square of X₆₅₀

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

X₂₃₁₀ (X₂-ISOCONJUGATE OF X₁₂₆₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {31, 33}, {45, 55}

• Related Centers: X₃₁ (2nd POWER POINT), X₃₃ (PERSPECTOR OF THE ORTHIC AND INTANGENTS TRIANGLES), X₄₅ (-BETH CONJUGATE OF X₁), X₅₅ (INSIMILICENTER)

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

Search Value (6, 9, 13): \(2.803686543049\)

X₂₃₁₀ lies on the incentral inellipse (a.k.a. Hofstadter ellipse E(1/2)).

▸ Incident Central Lines27 lines

{1, 651}, {4, 774}, {9, 294}, {11, 244}, {31, 33}, {37, 2293}, {38, 497}, {42, 1864}, {45, 55}, {73, 1898}, {84, 1106}, {90, 255}, {390, 984}, {516, 1736}, {522, 1090}, {692, 2265}, {748, 1040}, {896, 1776}, {926, 2170}, {950, 2292}, {971, 1458}, {990, 1471}, {1400, 1827}, {1445, 1721}, {1857, 2181}, {2112, 2268}, {2201, 2312}

▸ Functional Relations8 relations

• X₂₃₁₀ = X_i-Ceva conjugate of X_j for these (i,j): (1, 650), (4, 661), (11, 2170), (33, 663), (84, 649), (90, 652), (104, 1635), (1088,514, (1098, 1021)

• X₂₃₁₀ = isogonal conjugate of X₇₀₄₅

• X₂₃₁₀ = crosspoint of X_i and X_j for these (i,j): (1,650, (9, 522), (11, 1146), (19, 513), (514, 1088), (1021, 1098)

• X₂₃₁₀ = crosssum of X_i and X_j for these (i,j): (1, 651), (57, 109), (59, 1262), (63, 100), (101, 1253), (255, 1813), (269, 934), (412, 653), (1020, 1254), (1106, 1461)

• X₂₃₁₀ = polar conjugate of isotomic conjugate of X₃₄₅₉₁

• X₂₃₁₀ = trilinear square of X₆₅₀

• X₂₃₁₀ = trilinear pole wrt incentral triangle of line X₁X₃

• X₂₃₁₀ = X₆₃-isoconjugate of X₇₁₂₈

Notes

Let A'B'C' be the Mandart-incircle triangle. Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₂₃₁₀.