X₈₅₂

X₂-LINE CONJUGATE OF X₄

X₈₅₂ = X₂-line conjugate of X₄

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

X₈₅₂ (X₂-LINE CONJUGATE OF X₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {216, 373}

• Related Centers: X₂ (CENTROID), X₄ (ORTHOCENTER), X₃ (CIRCUMCENTER), X₂₁₆ (-CEVA CONJUGATE OF X₅₁), X₃₇₃ (CENTROID OF THE PEDAL TRIANGLE OF THE CENTROID)

Trilinears: \(\sec C \sin 2B \sin(C - A) + \sec B \sin 2C \sin(B - A) : :\)

Barycentrics: \(a^{2} (a^{2} - b^{2} - c^{2}) \cdot (a^{6} b^{2} - 2 a^{4} b^{4} + a^{2} b^{6} + a^{6} c^{2} + 2 a^{4} b^{2} c^{2} - a^{2} b^{4} c^{2} - 2 b^{6} c^{2} - 2 a^{4} c^{4} - a^{2} b^{2} c^{4} + 4 b^{4} c^{4} + a^{2} c^{6} - 2 b^{2} c^{6}) : :\)

Search Value (6, 9, 13): \(2.089399019079\)

▸ Incident Central Lines43 lines

{2, 3}, {51, 6509}, {216, 373}, {511, 2972}, {520, 647}, {577, 5651}, {895, 1942}, {1503, 1624}, {1576, 15139}, {3168, 46717}, {3580, 35442}, {4993, 43975}, {5437, 26900}, {5462, 42441}, {5640, 30258}, {6090, 15905}, {6389, 40981}, {6688, 32078}, {6760, 43574}, {7308, 26901}, {8798, 45187}, {9155, 14961}, {9306, 23606}, {9475, 14580}, {11064, 23181}, {11412, 14059}, {11793, 31388}, {14152, 43598}, {14915, 40948}, {14919, 14941}, {15526, 41586}, {17825, 26898}, {18437, 18911}, {20977, 41172}, {22264, 39201}, {26880, 43650}, {32205, 46025}, {32428, 46106}, {34828, 35222}, {37648, 42353}, {38999, 43952}, {42453, 43988}, {44715, 46788}

▸ Functional Relations9 relations

• X₈₅₂ = reflection of X₂₉₇₂ in X₄₄₄₃₆

• X₈₅₂ = X₂-line conjugate of X₄

• X₈₅₂ = circumcircle-inverse of X₃₇₉₂₆

• X₈₅₂ = isotomic conjugate of the polar conjugate of X₃₃₃₁

• X₈₅₂ = crossdifference of every pair of points on line {4, 647}

• X₈₅₂ = barycentric product X₆₉*X₃₃₃₁

• X₈₅₂ = X_i-isoconjugate of X_j for these (i,j): {92, 26717}, {525, 36139}, {14208, 32725}

• X₈₅₂ = barycentric quotient X_i/X_j for these {i,j}: {184, 26717}, {3331, 4}, {32676, 36139}, {33571, 2972}

• X₈₅₂ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 6638, 418}, {25, 6617, 426}, {51, 6509, 13409}, {401, 450, 41202}, {441, 468, 44888}, {441, 44892, 44891}, {441, 44894, 237}, {468, 44886, 237}, {468, 44888, 44891}, {1113, 1114, 37926}, {3284, 34147, 3292}, {6638, 38283, 2}, {44886, 44889, 468}, {44888, 44892, 468}, {44889, 44894, 44892}

Notes

As a point on the Euler line, X₈₅₂ has Shinagawa coefficients \(((2E - F)F - S^{2},(E + F)F + S^{2})\).