X₃₁₂₂

DANNEELS PERSPECTOR FOR X₄₂

X₃₁₂₂ = X₃₁₂₄-cross conjugate of X₃₁₂₅

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

X₃₁₂₂ (DANNEELS PERSPECTOR FOR X₄₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {11, 244}, {86, 256}

• Related Centers: X₃₁₂₄ (DANNEELS PERSPECTOR FOR X₇₆), X₃₁₂₅ (DANNEELS PERSPECTOR FOR X₈₁), X₁₁ (FEUERBACH POINT), X₂₄₄ (-LINE CONJUGATE OF X₁₀₀), X₈₆ (CEVAPOINT OF INCENTER AND CENTROID), X₂₅₆ (1st SHARYGIN POINT)

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

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

Search Value (6, 9, 13): \(0.901351548407\)

▸ Incident Central Lines12 lines

{6, 2054}, {11, 244}, {86, 256}, {190, 291}, {351, 865}, {674, 3009}, {1015, 1960}, {1400, 1918}, {1929, 2640}, {2092, 2667}, {2110, 2277}, {2642, 2643}

▸ Functional Relations8 relations

• X₃₁₂₂ = X_i-Ceva conjugate of X_j for these (i,j): (10, 661), (42, 512), (58, 649), (244, 3125), (256, 513), (310, 514), (1015, 3121), (1400, 798), (1474, 1919)

• X₃₁₂₂ = X₃₁₂₄-cross conjugate of X₃₁₂₅

• X₃₁₂₂ = crosspoint of X_i and X_j for these (i,j): (10, 661), (37, 513), (42, 512), (58, 649), (244, 1015), (310, 514), (667, 1402)

• X₃₁₂₂ = crosssum of X_i and X_j for these (i,j): (10, 190), (58, 662), (81, 100), (86, 99), (101, 1918), (306, 1331), (314, 668), (333, 643), (645, 1043), (765, 1016), (1332, 1792)

• X₃₁₂₂ = isogonal conjugate of X₄₆₀₀

• X₃₁₂₂ = intersection of tangents to Steiner inellipse at X₁₀₈₄ and X₁₀₈₆

• X₃₁₂₂ = crosspoint wrt medial triangle of X₁₀₈₄ and X₁₀₈₆

• X₃₁₂₂ = crossdifference of every pair of points on the line X₉₉X₁₀₁