X₁₉₅₉

CROSSDIFFERENCE OF PU(23)

X₁₉₅₉ = crosspoint of X₁ and X₁₅₈₁

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

X₁₉₅₉ (CROSSDIFFERENCE OF PU(23)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 21}, {2, 257}

• Related Centers: X₁₉₁₀ (TRILINEAR POLE OF PU(23)), X₁ (INCENTER), X₂₁ (SCHIFFLER POINT), X₂ (CENTROID), X₂₅₇ (ISOGONAL CONJUGATE OF X₁₇₂)

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

Barycentrics: \(\cos(A + \omega ) : :\)

Search Value (6, 9, 13): \(10.209747041531\)

▸ Alternate Coordinates3 forms

• Trilinears: \(b^{4} + c^{4} - a^{2}b^{2} - a^{2}c^{2} : :\) (M. Iliev, 5/13/2007)

• Trilinears: \(S_{A}\)\(S_{A}\) - \(S_{B}\)\(S_{C}\) : : (C. Lozada, 9/07/2013)

• Barycentrics: \(a^{2} \cos B \cos C - b c \cos^{2} A : :\)

▸ Incident Central Lines10 lines

{1, 21}, {2, 257}, {19, 326}, {48, 1760}, {92, 304}, {329, 1655}, {514, 661}, {1444, 1761}, {1762, 1812}, {1821, 1934}

▸ Functional Relations10 relations

• X₁₉₅₉ = isogonal conjugate of X₁₉₁₀

• X₁₉₅₉ = isotomic conjugate of X₁₈₂₁

• X₁₉₅₉ = X_i-Ceva conjugate of X_j for these (i,j): (2, 39040), (293, 1958), (1934, 38)

• X₁₉₅₉ = X_i-cross conjugate of X_j for these (i,j): (6, 2065), (114, 2), (230, 98), (1692, 1976), (1733, 1821)

• X₁₉₅₉ = crosspoint of X₁ and X₁₅₈₁

• X₁₉₅₉ = crosssum of X_i and X_j for these (i,j): (1, 1580), 240,1957)

• X₁₉₅₉ = polar conjugate of X₃₆₁₂₀

• X₁₉₅₉ = pole wrt polar circle of trilinear polar of X₃₆₁₂₀ (line X₁₉X₇₉₈)

• X₁₉₅₉ = X₆-isoconjugate of X₉₈

• X₁₉₅₉ = perspector of hyperbola {{A, B, C,X₇₅,X₆₆₂,PU(22)}}