X₄₁₇

X₃-CEVA CONJUGATE OF X₁₈₅

X₄₁₇ = isogonal conjugate of the polar conjugate of X₆₅₀₉

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

X₄₁₇ (X₃-CEVA CONJUGATE OF X₁₈₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {1624, 2883}

• Related Centers: X₃ (CIRCUMCENTER), X₁₈₅ (NAGEL POINT OF THE ORTHIC TRIANGLE), X₂ (CENTROID), X₁₆₂₄ (TCC-PERSPECTOR OF X₁₆₂), X₂₈₈₃ (COMPLEMENTARY CONJUGATE OF X₄)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = (\cos A)(\sec^{2} B + \sec^{2} C)\)

Barycentrics: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = (\sin 2A)(\sec^{2} B + \sec^{2} C)\)

Search Value (6, 9, 13): \(4.693494549184\)

▸ Alternate Coordinates1 form

• Barycentrics: \(a (a + b) \cdot (a + c) \cdot (a^{7} b^{2} - a^{6} b^{3} - 2 a^{5} b^{4} + 2 a^{4} b^{5} + a^{3} b^{6} - a^{2} b^{7} + a^{7} b c - a^{6} b^{2} c - a^{5} b^{3} c + 2 a^{4} b^{4} c - a^{3} b^{5} c - a^{2} b^{6} c + a b^{7} c + a^{7} c^{2} - a^{6} b c^{2} + a^{5} b^{2} c^{2} - a^{4} b^{3} c^{2} - a^{3} b^{4} c^{2} + a^{2} b^{5} c^{2} - a b^{6} c^{2} + b^{7} c^{2} - a^{6} c^{3} - a^{5} b c^{3} - a^{4} b^{2} c^{3} + 2 a^{3} b^{3} c^{3} + a^{2} b^{4} c^{3} - a b^{5} c^{3} + b^{6} c^{3} - 2 a^{5} c^{4} + 2 a^{4} b c^{4} - a^{3} b^{2} c^{4} + a^{2} b^{3} c^{4} + 2 a b^{4} c^{4} - 2 b^{5} c^{4} + 2 a^{4} c^{5} - a^{3} b c^{5} + a^{2} b^{2} c^{5} - a b^{3} c^{5} - 2 b^{4} c^{5} + a^{3} c^{6} - a^{2} b c^{6} - a b^{2} c^{6} + b^{3} c^{6} - a^{2} c^{7} + a b c^{7} + b^{2} c^{7}) : :\)

▸ Incident Central Lines28 lines

{2, 3}, {185, 6509}, {216, 51030}, {389, 13409}, {577, 14379}, {1092, 23606}, {1624, 2883}, {1942, 4558}, {2055, 43574}, {2972, 5562}, {3164, 45255}, {3917, 31388}, {5907, 44436}, {6389, 15653}, {6760, 14152}, {8763, 22341}, {9729, 46832}, {9730, 42441}, {10575, 40948}, {10607, 35602}, {11459, 14059}, {12359, 35442}, {13367, 20775}, {15043, 30258}, {15958, 19210}, {16035, 19166}, {22401, 47406}, {26900, 37526}

▸ Functional Relations9 relations

• X₄₁₇ = isogonal conjugate of the polar conjugate of X₆₅₀₉

• X₄₁₇ = X_i-Ceva conjugate of X_j for these (i,j): {3, 185}, {4558, 32320}

• X₄₁₇ = X_i-isoconjugate of X_j for these (i,j): {4, 821}, {158, 1105}, {775, 1093}, {801, 6520}, {6521, 41890}

• X₄₁₇ = X_i-Dao conjugate of X_j for these (i,j): {264, 13567}, {801, 37867}, {821, 36033}, {1093, 2883}, {1105, 1147}, {3269, 14618}

• X₄₁₇ = crosspoint of X₃ and X₁₀₉₂

• X₄₁₇ = crosssum of X₄ and X₁₀₉₃

• X₄₁₇ = barycentric product X_i*X_j for these {i,j}: {3, 6509}, {63, 820}, {185, 394}, {255, 6508}, {577, 41005}, {774, 6507}, {800, 3964}, {1092, 13567}, {4100, 17858}, {5562, 19180}, {10607, 45199}, {14379, 45200}

• X₄₁₇ = barycentric quotient X_i/X_j for these {i,j}: {48, 821}, {185, 2052}, {577, 1105}, {774, 6521}, {800, 1093}, {820, 92}, {1092, 801}, {1624, 15352}, {3964, 40830}, {4100, 775}, {6509, 264}, {16035, 8794}, {19180, 8795}, {23606, 41890}, {41005, 18027}

• X₄₁₇ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 3, 26897}, {3, 6617, 1593}, {3, 6638, 20}, {3, 15781, 3520}, {3, 26876, 34003}, {577, 14379, 43652}, {3524, 26876, 3}, {15717, 26874, 3}