X₃₁₂₅

DANNEELS PERSPECTOR FOR X₈₁

X₃₁₂₅ = cevapoint of X₂₆₄₃ and X₃₁₂₄

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

X₃₁₂₅ (DANNEELS PERSPECTOR FOR X₈₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {115, 125}, {65, 213}

• Related Centers: X₁₁₅ (CENTER OF KIEPERT HYPERBOLA), X₁₂₅ (CENTER OF JERABEK HYPERBOLA), X₆₅ (ORTHOCENTER OF THE INTOUCH TRIANGLE), X₂₁₃ (-CEVA CONJUGATE OF X₄₂)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = (b + c)(b - c)^{2}\)

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)

Search Value (6, 9, 13): \(1.360418150352\)

▸ Incident Central Lines20 lines

{1, 1929}, {6, 1718}, {10, 762}, {37, 1018}, {38, 1573}, {65, 213}, {115, 125}, {244, 665}, {257, 274}, {335, 668}, {614, 1572}, {758, 2238}, {910, 2251}, {918, 1086}, {1333, 2160}, {1411, 1415}, {1575, 1739}, {2092, 2294}, {2642, 2643}, {3271, 4560}

▸ Functional Relations13 relations

• X₃₁₂₅ = isogonal conjugate of X₄₅₆₇

• X₃₁₂₅ = isotomic conjugate of X₄₆₀₁

• X₃₁₂₅ = X_i-Ceva conjugate of X_j for these (i,j): (28, 667), (37, 661), (65, 512), (81, 513), (244, 3122), (257, 514), (274, 2530), (321, 523), (1086, 3120), (2160, 649)

• X₃₁₂₅ = cevapoint of X₂₆₄₃ and X₃₁₂₄

• X₃₁₂₅ = X_i-cross conjugate of X_j for these (i,j): (2643, 3120), (3124, 3122)

• X₃₁₂₅ = crosspoint of X_i and X_j for these (i,j): (10, 514), (37, 661), (81, 513), (244, 1086), (321, 523), (649, 1400), (1880, 2501)

• X₃₁₂₅ = crosssum of X_i and X_j for these (i,j): (37, 100), (58, 101), (72, 906), (81, 662), (110, 1333), (190, 333), (643, 2287), (692, 2205), (765, 1252), (1331, 2327)

• X₃₁₂₅ = intersection of tangents to Steiner inellipse at X₁₁₅ and X₁₀₁₅

• X₃₁₂₅ = crosspoint wrt medial triangle of X₁₁₅ and X₁₀₁₅

• X₃₁₂₅ = trilinear product of extraversions of X₃₇

• X₃₁₂₅ = crossdifference of every pair of points on the line X₁₀₀X₁₁₀

• X₃₁₂₅ = polar conjugate of isotomic conjugate of X₁₈₂₁₀

• X₃₁₂₅ = X₆₃-isoconjugate of X₅₃₇₉