X₅₆₆

HARMONIC OF X₅₀

X₅₆₆ = Brocard-circle-inverse of X₅₀

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

X₅₆₆ (HARMONIC OF X₅₀) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}, {2, 94}

• Related Centers: X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₂ (CENTROID), X₉₄ (ISOGONAL CONJUGATE OF X₅₀)

Trilinears: \(\sin A + \cos A \cot D/2 : :\), where cot \(D/2 = 4 \operatorname{area}/(a^{2} + b^{2} + c^{2} - 6R^{2})\), where \(R = abc/(4 \operatorname{area})\)

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

Search Value (6, 9, 13): \(-13.317949197267\)

▸ Alternate Coordinates2 forms

• Trilinears: \(a[16(b^{2} + c^{2})\sigma ^{2} - 3a^{2}b^{2}c^{2}]\) (M.Iliev, 5/13/07)

• Barycentrics: \((S_{B} + S_{C}) (3 R^{2} - S_{A} - S_{\omega}) : :\)

▸ Incident Central Lines71 lines

{2, 94}, {3, 6}, {4, 9221}, {5, 9220}, {45, 13006}, {53, 1594}, {67, 43718}, {140, 16310}, {141, 18375}, {160, 9973}, {184, 7669}, {230, 7495}, {231, 7749}, {232, 5094}, {233, 1879}, {237, 9971}, {248, 19151}, {393, 37119}, {549, 14836}, {599, 36212}, {631, 46262}, {858, 3815}, {940, 21478}, {1180, 5306}, {1576, 37457}, {1656, 9222}, {1990, 37118}, {2165, 2963}, {2453, 47213}, {2548, 14791}, {2549, 44468}, {3054, 39576}, {3055, 15355}, {3087, 35471}, {3148, 18374}, {3291, 37637}, {5012, 14060}, {5475, 7574}, {6240, 6748}, {6749, 10295}, {7507, 14576}, {7579, 7603}, {7668, 39906}, {7736, 16063}, {7763, 35549}, {8253, 8963}, {8962, 13846}, {9019, 37184}, {10311, 21284}, {11079, 14385}, {11672, 17416}, {13160, 18353}, {13371, 42459}, {14096, 18371}, {15122, 16303}, {16776, 37465}, {17277, 22377}, {17392, 26636}, {18281, 19656}, {18570, 38872}, {19127, 37183}, {23195, 34751}, {26216, 38323}, {30537, 34288}, {34828, 40681}, {35473, 39176}, {35503, 40065}, {36990, 44437}, {37196, 44519}, {40138, 47228}, {41237, 44388}

▸ Functional Relations18 relations

• X₅₆₆ = isogonal conjugate of X₇₅₇₈

• X₅₆₆ = complement of X₄₄₁₃₅

• X₅₆₆ = Brocard-circle-inverse of X₅₀

• X₅₆₆ = Schoutte-circle-inverse of X₁₄₈₀₅

• X₅₆₆ = complement of the isotomic conjugate of X₃₄₃₁

• X₅₆₆ = isogonal conjugate of the polar conjugate of X₇₅₇₇

• X₅₆₆ = polar conjugate of the isotomic conjugate of X₂₃₀₃₉

• X₅₆₆ = X₃₄₃₁-complementary conjugate of X₂₈₈₇

• X₅₆₆ = X₃₆₈₂₉-Ceva conjugate of X₁₈₁₁₇

• X₅₆₆ = X₁₈₁₁₇-cross conjugate of X₃₆₈₂₉

• X₅₆₆ = crosspoint of X₂ and X₃₄₃₁

• X₅₆₆ = crosssum of X_i and X_j for these (i,j): {2, 11004}, {6, 381}

• X₅₆₆ = crossdifference of every pair of points on line {523, 5926}

• X₅₆₆ = X_i-Dao conjugate of X_j for these (i,j): {3, 7578}, {566, 44135}

• X₅₆₆ = X₁-isoconjugate of X₇₅₇₈

• X₅₆₆ = barycentric product X_i*X_j for these {i,j}: {3, 7577}, {4, 23039}, {99, 18117}, {523, 36829}, {5562, 19177}

• X₅₆₆ = barycentric quotient X_i/X_j for these {i,j}: {6, 7578}, {7577, 264}, {18117, 523}, {19177, 8795}, {23039, 69}, {36829, 99}

• X₅₆₆ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {3, 6, 50}, {6, 39, 13337}, {6, 570, 13351}, {6, 8553, 2965}, {6, 11063, 32}, {6, 15109, 577}, {6, 18573, 3003}, {6, 18578, 568}, {6, 33886, 5007}, {6, 36751, 8553}, {15, 16, 14805}, {39, 216, 3003}, {39, 3003, 6}, {50, 41335, 6}, {160, 23635, 9973}, {216, 570, 6}, {216, 3003, 18573}, {371, 372, 567}, {570, 3003, 39}, {570, 18573, 13337}, {574, 5158, 5063}, {577, 14806, 15109}, {800, 5421, 6}, {1340, 1341, 2088}, {5063, 5158, 6}, {7772, 33872, 6}, {13337, 13351, 39}, {30260, 30261, 40280}, {40695, 40696, 2}