X₁₅₁₀

NAPOLEON CROSSDIFFERENCE

X₁₅₁₀ = X₁₃₇-cross conjugate of X₁₄₃

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

X₁₅₁₀ (NAPOLEON CROSSDIFFERENCE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {30, 511}, {110, 1291}

• Related Centers: X₉₃₀ (ANTICOMPLEMENT OF X₁₃₇), X₃₀ (EULER INFINITY POINT), X₅₁₁ (ISOGONAL CONJUGATE OF X₉₈), X₁₁₀ (FOCUS OF KIEPERT PARABOLA), X₁₂₉₁ (COLLINGS TRANSFORM OF X₅₀)

Trilinears: \(\csc(B + \pi /6)\csc(C - \pi /6) - \csc(C + \pi /6)\csc(B - \pi /6) : :\)

Barycentrics: \(a^{2} (b^{2} - c^{2}) (a^{4} + b^{4} + c^{4} - 2 a^{2} b^{2} - 2 a^{2} c^{2} - b^{2} c^{2}) : :\)

▸ Alternate Coordinates1 form

• Trilinears: \((3 \sin^{2} A - \cos^{2} A)(\cos B \sin C - \sin B \cos C) : :\)

▸ Incident Central Lines2 lines

{30, 511}, {110, 1291}

▸ Functional Relations10 relations

• X₁₅₁₀ = isogonal conjugate of X₉₃₀

• X₁₅₁₀ = X_i-Ceva conjugate of X_j for these (i,j): (2, 39018), (4, 137), (110, 1493)

• X₁₅₁₀ = X₁₃₇-cross conjugate of X₁₄₃

• X₁₅₁₀ = crosspoint of X_i and X_j for these (i,j): (4, 933), (110, 1173)

• X₁₅₁₀ = crosssum of X_i and X_j for these (i,j): (140, 523), (512, 570)

• X₁₅₁₀ = crossdifference of every pair of points on line X₆X₁₇

• X₁₅₁₀ = X₅₂₃-of-orthic-triangle

• X₁₅₁₀ = perspector of hyperbola {{A, B, C,X₂,X₆₁,X₆₂}}

• X₁₅₁₀ = infinite point of normal to hyperbola {{A, B, C,X₄,X₁₅}} at X₁₅ and normal to hyperbola {{A, B, C,X₄,X₁₆}} at X₁₆

• X₁₅₁₀ = barycentric square root of X₃₉₀₁₈