X₁₉₁

X₁₀-CEVA CONJUGATE OF X₁

X₁₉₁ = X₁₀-Ceva conjugate of X₁

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

X₁₉₁ (X₁₀-CEVA CONJUGATE OF X₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 21}, {9, 46}

• Related Centers: X₁₀ (SPIEKER CENTER), X₁ (INCENTER), X₂₁ (SCHIFFLER POINT), X₉ (MITTENPUNKT), X₄₆ (-CEVA CONJUGATE OF X₁)

Trilinears: \((b + c - a)(bc + ca + ab) + b^{3} + c^{3} - a^{3} : :\)

Search Value (6, 9, 13): \(4.664025211173\)

▸ Alternate Coordinates1 form

• Trilinears: \(S_{A} + rR : S_{B} + rR : S_{C} + rR\)

▸ Linear Combinations1 combination

• X₁₉₁ = X₁ - \(2\) X₂₁

▸ Incident Central Lines10 lines

{1, 21}, {9, 46}, {10, 267}, {30, 40}, {35, 72}, {36, 960}, {109, 201}, {165, 1079}, {190, 1089}, {329, 498}

▸ Functional Relations15 relations

• X₁₉₁ = reflection of X_i in X_j for these (i,j): (1, 21), (79, 442)

• X₁₉₁ = isogonal conjugate of X₂₆₇

• X₁₉₁ = X₁₀-Ceva conjugate of X₁

• X₁₉₁ = crosspoint of X_i and X_j for these (i,j): (10, 502)

• X₁₉₁ = crosssum of X₅₈ and X₅₀₁

• X₁₉₁ = excentral-isogonal conjugate of X₃

• X₁₉₁ = X_i-aleph conjugate of X_j for these (i,j): (2, 2), (8, 20), (10, 191), (37, 1045), (188, 3), (366, 6)

• X₁₉₁ = X₆₄₃-beth conjugate of X₁₉₁

• X₁₉₁ = crossdifference of every pair of points on line X₆₆₁X₂₆₀₅

• X₁₉₁ = X₅₄-of-excentral-triangle

• X₁₉₁ = perspector of excentral triangle and Fuhrmann triangle

• X₁₉₁ = intersection of Euler lines of outer and inner Garcia triangles

• X₁₉₁ = {X₁,X₂₁}-harmonic conjugate of X₅₄₂₆

• X₁₉₁ = complement of X₁₄₄₅₀

• X₁₉₁ = anticomplement of X₁₁₂₆₃

Notes

Let I_a be the reflection of X₁ in the perpendicular bisector of BC, and define I_b and I_c cyclically. X₁₉₁ = X₂₁ of I_aI_bI_c. (Randy Hutson, December 2, 2017)

Let J_aJ_bJ_c be the excentral triangle. Let N_a be the nine-point center of BCJ_a, and define N_b and N_c cyclically. The lines J_aN_a, J_bN_b, J_cN_c concur in X₁₉₁; c.f. X₅₅₀₆. (Randy Hutson, December 2, 2017)