X₁₅₀₃

POINT ARKAB

As the isogonal conjugate of a point on the circumcircle, X₁₅₀₃ lies on the line at infinity.

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

X₁₅₀₃ (POINT ARKAB) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 6}, {3, 66}

• Related Centers: X₁₂₉₇ (COLLINGS TRANSFORM OF X₁₂₇), X₄ (ORTHOCENTER), X₆ (SYMMEDIAN POINT), X₃ (CIRCUMCENTER), X₆₆ (ISOGONAL CONJUGATE OF X₂₂)

Trilinears: \(\sin A \tan \omega - 2 \cos B \cos C : :\)

Barycentrics: \(S^{2} a^{2} - 2 S_{B} S_{C} S_{\omega} : :\)

▸ Alternate Coordinates1 form

• Trilinears: \(bc(b^{6} + c^{6} - 2a^{6} + a^{4}b^{2} +a^{4}c^{2} - b^{4}c^{2} - c^{4}b^{2}) : :\)

▸ Linear Combinations1 combination

• X₁₅₀₃ = X₄ - X₆

▸ Incident Central Lines30 lines

{2, 154}, {3, 66}, {4, 6}, {5, 182}, {11, 1428}, {20, 64}, {22, 161}, {30, 511}, {51, 428}, {67, 74}, {98, 230}, {110, 858}, {125, 468}, {147, 325}, {184, 427}, {221, 388}, {242, 1146}, {265, 1177}, {287, 297}, {376, 599}, {381, 597}, {382, 1351}, {383, 395}, {394, 1370}, {396, 1080}, {546, 575}, {576, 1353}, {611, 1478}, {613, 1479}, {946, 1386}

▸ Functional Relations14 relations

• X₁₅₀₃ = isogonal conjugate of X₁₂₉₇

• X₁₅₀₃ = isotomic conjugate of X₃₅₁₄₀

• X₁₅₀₃ = complementary conjugate of X₁₃₂

• X₁₅₀₃ = X_i-Ceva conjugate of X_j for these (i,j): (2, 23976), (4, 132), (287, 6), (297, 230), (685, 523)

• X₁₅₀₃ = cevapoint of X₂₀ and X₁₄₇

• X₁₅₀₃ = crosspoint of X₄ and X₉₈

• X₁₅₀₃ = crosssum of X₃ and X₅₁₁

• X₁₅₀₃ = crossdifference of every pair of points on line X₆X₅₂₀

• X₁₅₀₃ = X₄-Hirst inverse of X₁₂₄₉

• X₁₅₀₃ = intercept of the van Aubel line and the line at infinity

• X₁₅₀₃ = Cundy-Parry Phi transform of X₁₄₃₇₆

• X₁₅₀₃ = Cundy-Parry Psi transform of X₈₇₄₃

• X₁₅₀₃ = crosspoint of X₂₀ and X₁₄₇ wrt both the excentral and anticomplementary triangles

• X₁₅₀₃ = infinite point of tangents at X₄ and X₂₀ to Darboux cubic K004