X₁₉₂₀

CROSSSUM OF PU(12)

X₁₉₂₀ = perspector of Gemini triangle 31 and cross-triangle of ABC and Gemini triangle 31

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

X₁₉₂₀ (CROSSSUM OF PU(12)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {76, 85}, {2, 561}

• Related Centers: X₈₉₃ (-CROSS CONJUGATE OF X₂₉₂), X₇₆ (3rd BROCARD POINT), X₈₅ (ISOTOMIC CONJUGATE OF X₉), X₂ (CENTROID), X₅₆₁ (ISOGONAL CONJUGATE OF 4th POWER POINT)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = b^{3}c^{3}(a^{2} + bc)\)

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

Search Value (6, 9, 13): \(4.450646987674\)

▸ Incident Central Lines10 lines

{2, 561}, {37, 1221}, {75, 982}, {76, 85}, {210, 668}, {310, 321}, {334, 1581}, {1002, 1611}, {1215, 1237}, {1240, 1441}

▸ Functional Relations5 relations

• X₁₉₂₀ = isotomic conjugate of X₈₉₃

• X₁₉₂₀ = complement of polar conjugate of isogonal conjugate of X₂₃₁₉₂

• X₁₉₂₀ = perspector of Gemini triangle 31 and cross-triangle of ABC and Gemini triangle 31

• X₁₉₂₀ = trilinear pole of perspectrix of ABC and Gemini triangle 32

• X₁₉₂₀ = X_i-Ceva conjugaute of X_j for these (i,j): (334, 1921), (1240, 76)