X₃₁₂₁

DANNEELS PERSPECTOR FOR X₃₇

X₃₁₂₁ = crossdifference of every pair of points on the line X₉₉X₁₀₀

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

X₃₁₂₁ (DANNEELS PERSPECTOR FOR X₃₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {100, 292}, {244, 665}

• Related Centers: X₁₀₀ (ANTICOMPLEMENT OF FEUERBACH POINT), X₂₉₂ (-HIRST INVERSE OF X₂₉₁), X₂₄₄ (-LINE CONJUGATE OF X₁₀₀), X₆₆₅ (CROSSDIFFERENCE OF X₂ AND X₁₁)

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

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

Search Value (6, 9, 13): \(0.570468631130\)

▸ Incident Central Lines6 lines

{81, 893}, {100, 292}, {244, 665}, {351, 865}, {890, 1977}, {1402, 2205}

▸ Functional Relations7 relations

• X₃₁₂₁ = isogonal conjugate of X₄₆₀₁

• X₃₁₂₁ = X_i-Ceva conjugate of X_j for these (i,j): (37, 512), (213, 798), (274, 513), (893, 649), (1015, 3122), (1333, 667), (1402, 669), (2203, 1980)

• X₃₁₂₁ = crosspoint of X_i and X_j for these (i,j): (37, 512), (42, 649), (213, 798), (274, 513), (667, 1333)

• X₃₁₂₁ = crosssum of X_i and X_j for these (i,j): (81, 99), (86, 190), (100, 213), (274, 799), (314, 645), (321, 668)

• X₃₁₂₁ = crossdifference of every pair of points on the line X₉₉X₁₀₀

• X₃₁₂₁ = intersection of tangents to Steiner inellipse at X₁₀₁₅ and X₁₀₈₄

• X₃₁₂₁ = crosspoint wrt medial triangle of X₁₀₁₅ and X₁₀₈₄