X₁₉₁₉

BICENTRIC DIFFERENCE OF PU(12)

X₁₉₁₉ = cevapoint of X₆₆₉ and X₁₉₂₄

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

X₁₉₁₉ (BICENTRIC DIFFERENCE OF PU(12)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {101, 765}, {669, 688}

• Related Centers: X₁₉₇₈ (TRILINEAR PRODUCT OF PU(41)), X₁₀₁ (Ψ(INCENTER, SYMMEDIAN POINT)), X₇₆₅ (CEVAPOINT OF X₁ AND X₁₀₀), X₆₆₉ (CROSSDIFFERENCE OF X₂ AND X₃₉), X₆₈₈ (CROSSDIFFERENCE OF X₆ AND X₇₆)

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

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

Search Value (6, 9, 13): \(0.909923085465\)

▸ Incident Central Lines4 lines

{101, 765}, {649, 834}, {667, 788}, {669, 688}

▸ Functional Relations10 relations

• X₁₉₁₉ = isogonal conjugate of X₁₉₇₈

• X₁₉₁₉ = complement of X₂₁₃₀₄

• X₁₉₁₉ = anticomplement of X₂₁₂₆₂

• X₁₉₁₉ = X_i-Ceva conjugate of X_j for these (i,j): (32, 1977), (692, 1333)

• X₁₉₁₉ = cevapoint of X₆₆₉ and X₁₉₂₄

• X₁₉₁₉ = crosspoint of X_i and X_j for these (i,j): (31, 101), (81, 932), (692, 1333)

• X₁₉₁₉ = crosssum of X_i and X_j for these (i,j): (75, 514), (321, 693), (646, 668), (850, 1230)

• X₁₉₁₉ = PU(12)-harmonic conjugate of X₁₉₁₈

• X₁₉₁₉ = barycentric product of PU(25)

• X₁₉₁₉ = trilinear product of PU(42)