X₁₉₃₁

CROSSDIFFERENCE OF PU(32)

X₁₉₃₁ = perspector of conic {{A, B, C,PU(31)}}

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

X₁₉₃₁ (CROSSDIFFERENCE OF PU(32)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 21}, {99, 239}

• Related Centers: X₁ (INCENTER), X₂₁ (SCHIFFLER POINT), X₉₉ (STEINER POINT), X₂₃₉ (-LINE CONJUGATE OF X₄₂)

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

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

Search Value (6, 9, 13): \(-4.354696641458\)

▸ Incident Central Lines12 lines

{1, 21}, {37, 757}, {44, 662}, {99, 239}, {172, 593}, {241, 1414}, {261, 894}, {661, 1019}, {1014, 1423}, {1326, 1757}, {1444, 1778}, {2641, 2642}

▸ Functional Relations8 relations

• X₁₉₃₁ = isogonal conjugate of X₉₂₇₈

• X₁₉₃₁ = complement of X₂₀₃₄₉

• X₁₉₃₁ = anticomplement of X₂₀₃₃₇

• X₁₉₃₁ = X₂-Ceva conjugate of X₃₉₀₄₂

• X₁₉₃₁ = X₁₉₂₉-Ceva conjugate of X₁₉₆₃

• X₁₉₃₁ = crosssum of X₁₇₅₇ and X₁₉₆₁

• X₁₉₃₁ = perspector of conic {{A, B, C,PU(31)}}

• X₁₉₃₁ = intersection of trilinear polars of P(31) and U(31)