X₄₃₀

EULER X₈₁-5th-SUBSTITUTION POINT

X₄₃₀ = tangential-to-orthic similarity image of X₁₉₉

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

X₄₃₀ (EULER X₈₁-5th-SUBSTITUTION POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {118, 136}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₁₁₈ (MIDPOINT OF X₄ AND X₁₀₁), X₁₃₆ (INTERSECTION OF LINE X)

Trilinears: \((v + w)\sec A : (w + u)\sec B : (u + v)\sec C\), where \(u : v : w\) = X₈₁; e.g., \(u = u(a,b,c) = 1/(b + c)\)

Barycentrics: \((v + w)\tan A : (w + u)\tan B : (u + v)\tan C\)

Search Value (6, 9, 13): \(-10.356504391821\)

▸ Alternate Coordinates1 form

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

▸ Incident Central Lines38 lines

{2, 3}, {11, 40956}, {33, 1865}, {53, 3192}, {55, 50036}, {115, 40984}, {118, 136}, {125, 20622}, {210, 594}, {226, 1893}, {343, 48902}, {661, 23615}, {1172, 44097}, {1211, 24703}, {1213, 1839}, {1230, 3702}, {1827, 1856}, {1848, 2969}, {1855, 38930}, {1859, 42069}, {1867, 2901}, {1897, 17982}, {2245, 7082}, {2328, 32431}, {2333, 40586}, {2970, 21665}, {3190, 24045}, {5130, 17156}, {5139, 5513}, {5155, 39594}, {8735, 40976}, {8754, 42070}, {15496, 46835}, {17056, 17605}, {21243, 48940}, {34335, 34338}, {38360, 50329}, {40954, 44546}

▸ Functional Relations18 relations

• X₄₃₀ = orthocentroidal-circle-inverse of X₁₈₈₉

• X₄₃₀ = polar-circle-inverse of X₅₁₉₆

• X₄₃₀ = polar conjugate of X₃₂₀₁₄

• X₄₃₀ = isogonal conjugate of the isotomic conjugate of X₄₄₁₄₃

• X₄₃₀ = polar conjugate of the isotomic conjugate of X₁₂₁₃

• X₄₃₀ = polar conjugate of the isogonal conjugate of X₂₀₉₇₀

• X₄₃₀ = orthic-isogonal conjugate of X₁₈₃₉

• X₄₃₀ = X_i-Ceva conjugate of X_j for these (i,j): {4, 1839}, {1897, 2501}, {26705, 523}, {44143, 1213}

• X₄₃₀ = X₂₀₉₇₀-cross conjugate of X₁₂₁₃

• X₄₃₀ = X_i-isoconjugate of X_j for these (i,j): {3, 40438}, {48, 32014}, {63, 1171}, {81, 1796}, {656, 6578}, {905, 4629}, {1126, 1444}, {1255, 1790}, {1268, 1437}, {1459, 4596}, {4558, 47947}, {4575, 4608}, {4592, 50344}, {4632, 22383}, {7254, 37212}, {17206, 28615}

• X₄₃₀ = X_i-Dao conjugate of X_j for these (i,j): {69, 1125}, {136, 4608}, {1171, 3162}, {1213, 17206}, {1249, 32014}, {1444, 3647}, {1796, 40586}, {3120, 4025}, {4064, 21709}, {5139, 50344}, {6578, 40596}, {15419, 35076}, {36103, 40438}

• X₄₃₀ = crosspoint of X₄ and X₁₈₂₆

• X₄₃₀ = crosssum of X₃ and X₁₇₉₀

• X₄₃₀ = crossdifference of every pair of points on line {647, 7254}

• X₄₃₀ = tangential-to-orthic similarity image of X₁₉₉

• X₄₃₀ = barycentric product X_i*X_j for these {i,j}: {4, 1213}, {6, 44143}, {10, 1839}, {19, 4647}, {25, 1230}, {27, 8013}, {92, 1962}, {158, 3958}, {225, 3686}, {264, 20970}, {278, 4046}, {281, 3649}, {286, 21816}, {321, 2355}, {393, 41014}, {594, 31900}, {648, 6367}, {1100, 41013}, {1125, 1826}, {1269, 2333}, {1783, 30591}, {1824, 4359}, {1880, 3702}, {1897, 4988}, {2052, 22080}, {2501, 4427}, {3683, 40149}, {4103, 46542}, {4115, 7649}, {4983, 6335}, {6331, 8663}, {6531, 51417}, {7140, 8025}, {14618, 35327}, {24006, 35342}

• X₄₃₀ = barycentric quotient X_i/X_j for these {i,j}: {4, 32014}, {19, 40438}, {25, 1171}, {42, 1796}, {112, 6578}, {1100, 1444}, {1125, 17206}, {1213, 69}, {1230, 305}, {1783, 4596}, {1824, 1255}, {1826, 1268}, {1839, 86}, {1897, 4632}, {1962, 63}, {2308, 1790}, {2333, 1126}, {2355, 81}, {2489, 50344}, {2501, 4608}, {3649, 348}, {3683, 1812}, {3686, 332}, {3958, 326}, {4046, 345}, {4115, 4561}, {4427, 4563}, {4647, 304}, {4977, 15419}, {4983, 905}, {4988, 4025}, {4990, 15411}, {6367, 525}, {7140, 6539}, {8013, 306}, {8040, 4001}, {8663, 647}, {8750, 4629}, {20970, 3}, {21816, 72}, {22080, 394}, {23201, 18604}, {30591, 15413}, {31900, 1509}, {35327, 4558}, {35342, 4592}, {41013, 32018}, {41014, 3926}, {44143, 76}, {50512, 7254}, {51417, 6393}

• X₄₃₀ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 4, 1889}, {4, 24, 7546}, {4, 403, 15762}, {4, 429, 407}, {4, 451, 31902}, {4, 469, 427}, {4, 3542, 7534}, {4, 4207, 25}, {4, 4213, 27}, {4, 6353, 6994}, {4, 7414, 46467}, {4, 7513, 1885}, {4, 7563, 23047}, {4, 14004, 428}, {27, 4213, 468}, {429, 37398, 4205}, {1824, 1826, 7140}, {3136, 33329, 440}, {6818, 7377, 37439}, {11323, 37385, 13615}, {28044, 37385, 25}, {52286, 52287, 37226}

Notes

As a point on the Euler line, X₄₃₀ has Shinagawa coefficients \((F,2\$bc\$ + E + F)\).