X₃₁₉₇

X₄₀-CEVA CONJUGATE OF X₅₅

X₃₁₉₇ = isogonal conjugate of the polar conjugate of X₃₁₇₆

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

X₃₁₉₇ (X₄₀-CEVA CONJUGATE OF X₅₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {6, 19}, {48, 55}

• Related Centers: X₄₀ (BEVAN POINT), X₅₅ (INSIMILICENTER), X₆ (SYMMEDIAN POINT), X₁₉ (CLAWSON POINT), X₄₈ (CROSSPOINT OF X₁ AND X₆₃)

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

Search Value (6, 9, 13): \(-3.901018142390\)

▸ Linear Combinations1 combination

• X₃₁₉₇ = \(3\) X₁₅₄ - \(2\) X₁₈₆₂₁

X₃₁₉₇ lies on the cubics K179 and K750.

▸ Incident Central Lines96 lines

{3, 1630}, {6, 19}, {9, 6001}, {22, 11445}, {24, 11460}, {25, 3611}, {37, 1854}, {40, 219}, {48, 55}, {64, 71}, {101, 1604}, {109, 15905}, {155, 8141}, {159, 20468}, {184, 11406}, {218, 2270}, {222, 18725}, {268, 12330}, {281, 5776}, {394, 3101}, {517, 3211}, {910, 2911}, {958, 15656}, {966, 20306}, {1030, 3207}, {1181, 6197}, {1190, 2266}, {1191, 21770}, {1376, 10174}, {1407, 26934}, {1503, 2550}, {1615, 7964}, {1802, 12335}, {1853, 3925}, {1903, 7079}, {1993, 9536}, {2093, 18594}, {2183, 7355}, {2259, 11051}, {2289, 10310}, {2301, 4254}, {2343, 8602}, {3196, 5036}, {3198, 3990}, {3779, 9924}, {4258, 11434}, {5415, 17819}, {5416, 17820}, {5928, 30686}, {6237, 17834}, {6252, 17840}, {6253, 17845}, {6255, 17846}, {6404, 17843}, {6759, 10306}, {6769, 22153}, {7291, 23144}, {7688, 10606}, {7719, 9119}, {7724, 17835}, {7957, 7973}, {8251, 17814}, {8539, 17813}, {8802, 17833}, {8804, 12779}, {9537, 11441}, {9816, 17825}, {10119, 17847}, {10319, 17811}, {10636, 17826}, {10637, 17827}, {10902, 17821}, {11206, 17784}, {11428, 17809}, {11435, 17810}, {11471, 15811}, {12417, 17836}, {12661, 17838}, {12662, 17839}, {12663, 17842}, {13041, 17841}, {13042, 17844}, {13567, 18921}, {15509, 24310}, {17837, 22840}, {18405, 18406}, {18451, 18453}, {18598, 22132}, {19132, 19133}, {19180, 19181}, {19430, 19432}, {19431, 19433}, {21778, 21779}, {26908, 26909}, {26952, 26953}, {26957, 26958}

▸ Functional Relations9 relations

• X₃₁₉₇ = isogonal conjugate of the polar conjugate of X₃₁₇₆

• X₃₁₉₇ = X_i-Ceva conjugate of X_j for these (i,j): {40, 55}, {219, 6}, {610, 198}

• X₃₁₉₇ = X_i-isoconjugate of X_j for these (i,j): {2, 3345}, {57, 1034}, {63, 7149}, {75, 7152}, {81, 8806}, {85, 7037}, {189, 3342}, {333, 8811}, {348, 7007}, {8064, 17896}

• X₃₁₉₇ = crosspoint of X₁₀₀ and X₂₃₉₈₄

• X₃₁₉₇ = crosssum of X_i and X_j for these (i,j): {9, 6769}, {514, 16596}, {650, 3318}

• X₃₁₉₇ = crossdifference of every pair of points on line {521, 14837}

• X₃₁₉₇ = barycentric product X_i*X_j for these {i,j}: {1, 1490}, {3, 3176}, {8, 1035}, {40, 3341}, {55, 5932}, {65, 13614}, {72, 8885}, {78, 207}, {109, 14302}

• X₃₁₉₇ = barycentric quotient X_i/X_j for these {i,j}: {25, 7149}, {31, 3345}, {32, 7152}, {42, 8806}, {55, 1034}, {207, 273}, {1035, 7}, {1402, 8811}, {1490, 75}, {2175, 7037}, {2187, 3342}, {2212, 7007}, {3176, 264}, {3341, 309}, {5932, 6063}, {8885, 286}, {13614, 314}

• {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {6, 21767, 221}, {19, 2261, 2264}, {19, 19350, 6}, {48, 2272, 1436}, {55, 10536, 154}, {607, 1409, 6}, {610, 1498, 7152}, {2187, 2357, 55}, {5584, 6254, 64}, {10536, 11190, 55}