X₃₁₅₄

X₄₇₇-CEVA CONJUGATE OF X₅₂₃

For a construction see Antreas Hatzipolakis and Kadir Altintas, euclid 1791.

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

X₃₁₅₄ (X₄₇₇-CEVA CONJUGATE OF X₅₂₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {125, 523}

• Related Centers: X₄₇₇ (TIXIER ANTIPODE), X₅₂₃ (ISOGONAL CONJUGATE OF X₁₁₀), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₁₂₅ (CENTER OF JERABEK HYPERBOLA)

Trilinears: \(u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z)\), where \(x : y : z\) = X₄₇₇ and \(u : v : w\) = X₅₂₃

Barycentrics: \((b^{2} - c^{2})^{2} (-3 a^{8} + 7 a^{6} b^{2} - 3 a^{4} b^{4} - 3 a^{2} b^{6} + 2 b^{8} + 7 a^{6} c^{2} - 13 a^{4} b^{2} c^{2} + 7 a^{2} b^{4} c^{2} - b^{6} c^{2} - 3 a^{4} c^{4} + 7 a^{2} b^{2} c^{4} - 2 b^{4} c^{4} - 3 a^{2} c^{6} - b^{2} c^{6} + 2 c^{8}) : :\)

Search Value (6, 9, 13): \(3.723792188231\)

▸ Incident Central Lines34 lines

{2, 3}, {110, 31945}, {115, 9209}, {122, 16221}, {125, 523}, {136, 13611}, {265, 14934}, {476, 15059}, {477, 14644}, {525, 16280}, {1112, 12052}, {1503, 16319}, {1553, 36518}, {2452, 26869}, {3233, 5972}, {3448, 14611}, {5627, 41522}, {6723, 22104}, {8901, 15470}, {9140, 14480}, {10113, 38610}, {10733, 38701}, {14220, 15453}, {15025, 38678}, {15061, 20957}, {15359, 22264}, {16168, 20304}, {16178, 35968}, {17702, 31379}, {18319, 21315}, {20417, 32417}, {23515, 25641}, {30714, 31378}, {34128, 38609}

▸ Functional Relations17 relations

• X₃₁₅₄ = 3 X₂ + X₁₇₅₁₁, 3 X₁₂₅ - X₆₀₇₀, X₄₇₆ - 5 X₁₅₀₅₉, X₄₇₇ + 3 X₁₄₆₄₄, X₁₅₅₃ - 3 X₃₆₅₁₈, 5 X₁₆₅₆ - X₃₆₁₉₃, 5 X₃₀₉₁ - X₃₆₁₇₂, 3 X₃₂₅₈ + X₆₀₇₀, 2 X₃₂₅₈ + X₁₂₀₇₉, 3 X₅₆₂₇ + 2 X₄₁₅₂₂, 2 X₆₀₇₀ - 3 X₁₂₀₇₉, 3 X₉₁₄₀ + X₁₄₄₈₀, X₁₀₇₃₃ + 3 X₃₈₇₀₁, 2 X₁₂₀₆₈ + X₁₇₅₁₁, X₁₃₄₇₁ + 3 X₁₄₈₉₃, 3 X₁₄₆₄₄ - X₃₄₁₅₀, 11 X₁₅₀₂₅ + X₃₈₆₇₈, 3 X₁₅₀₆₁ + X₂₀₉₅₇, 2 X₁₆₃₄₀ + X₃₆₁₆₉, X₁₈₃₁₉ - 3 X₂₁₃₁₅, 3 X₂₃₅₁₅ - X₂₅₆₄₁, X₃₀₇₁₄ - 3 X₃₁₃₇₈, 5 X₃₀₇₄₅ - X₃₆₁₈₈, 3 X₃₄₁₂₈ - X₃₈₆₀₉.

• X₃₁₅₄ = anticomplement of X₁₂₀₆₈

• X₃₁₅₄ = complement of X₇₄₇₁

• X₃₁₅₄ = midpoint of X_i and X_j for these {i,j}: {3, 36184}, {4, 36164}, {5, 16340}, {125, 3258}, {265, 14934}, {477, 34150}, {3448, 14611}, {7471, 17511}, {10113, 38610}

• X₃₁₅₄ = reflection of X_i in X_j for these {i,j}: {110, 31945}, {1112, 12052}, {3233, 5972}, {7471, 12068}, {12079, 125}, {22104, 6723}, {36169, 5}

• X₃₁₅₄ = nine-point-circle-inverse of X₃₁₃₄

• X₃₁₅₄ = polar-circle-inverse of X₄₂₄₀

• X₃₁₅₄ = orthoptic-circle-of-Steiner-inellipe-inverse of X₇₄₂₂

• X₃₁₅₄ = X_i-complementary conjugate of X_j for these (i,j): {656, 42424}, {15453, 10}, {32710, 8062}

• X₃₁₅₄ = X₄₇₇-Ceva conjugate of X₅₂₃

• X₃₁₅₄ = crosssum of X₁₁₀ and X₁₅₀₅₅

• X₃₁₅₄ = crossdifference of every pair of point on the line X₆₄₇X₂₄₂₀

• X₃₁₅₄ = orthogonal projection of X₁₂₅ on Euler line

• X₃₁₅₄ = Euler line intercept of Simpson line of X₇₄

• X₃₁₅₄ = barycentric product X₃₃₈*X₁₅₀₃₅

• X₃₁₅₄ = barycentric quotient X₁₅₀₃₅/X₂₄₉

• X₃₁₅₄ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 858, 36170}, {2, 7471, 12068}, {2, 17511, 7471}, {2, 36166, 468}, {477, 14644, 34150}, {868, 36189, 14120}, {1312, 1313, 3134}, {1650, 35235, 3134}, {14120, 37987, 868}

Notes

As a point on the Euler line, X₃₁₅₄ has Shinagawa coefficients \((9(E - 2F)F - 2S^{2}, -(E + F)(E + 10 F) + 6S^{2})\).