X₄₃₆

EULER X₄₈-7th-SUBSTITUTION POINT

X₄₃₆ = orthocentroidal-circle-inverse of X₅₂₂₄₉

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

X₄₃₆ (EULER X₄₈-7th-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), {51, 107}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₅₁ (CENTROID OF ORTHIC TRIANGLE), X₁₀₇ (Ψ(SYMMEDIAN POINT, ORTHOCENTER))

Trilinears: \((u^{2} + vw)\sec A : (v^{2} + wu)\sec B : (w^{2} + uv)\sec C\), where \(u : v : w\) = X₄₈; e.g., \(u(A,B,C) = \sin 2A\)

Barycentrics: \((u^{2} + vw)\tan A : (v^{2} + wu)\tan B : (w^{2} + uv)\tan C\)

Search Value (6, 9, 13): \(0.375185984287\)

▸ Alternate Coordinates2 forms

• Barycentrics: \(\tan A - \tan \omega ' : :\), where ω’ is the Brocard angle of the orthic triangle

• Barycentrics: \((a^{2} + b^{2} - c^{2}) \cdot (a^{2} - b^{2} + c^{2}) \cdot (a^{8} - 2 a^{6} b^{2} + a^{4} b^{4} - 2 a^{6} c^{2} + 3 a^{4} b^{2} c^{2} - b^{6} c^{2} + a^{4} c^{4} + 2 b^{4} c^{4} - b^{2} c^{6}) : :\)

▸ Incident Central Lines56 lines

{2, 3}, {6, 1988}, {51, 107}, {53, 10192}, {54, 8794}, {110, 324}, {154, 33971}, {182, 15466}, {184, 2052}, {185, 51031}, {216, 40402}, {264, 9306}, {459, 18950}, {578, 1093}, {648, 34986}, {1075, 7592}, {1196, 6531}, {1199, 43840}, {1304, 32439}, {1495, 1629}, {1614, 44732}, {1941, 34148}, {1947, 7193}, {1948, 3955}, {1970, 27359}, {1994, 35360}, {3167, 9308}, {4994, 38848}, {5012, 46106}, {5890, 40664}, {5943, 36794}, {6344, 50463}, {6524, 11427}, {6525, 14853}, {6530, 23292}, {6747, 14165}, {6749, 45867}, {6761, 12022}, {8795, 26887}, {8884, 8887}, {10982, 45062}, {11245, 51358}, {11424, 14249}, {11455, 48364}, {14361, 14912}, {14363, 37505}, {14978, 18350}, {16657, 51385}, {19357, 41365}, {26880, 42329}, {27377, 41588}, {32002, 32223}, {35259, 41244}, {37894, 44132}, {38833, 39286}, {43650, 52147}

▸ Functional Relations11 relations

• X₄₃₆ = orthocentroidal-circle-inverse of X₅₂₂₄₉

• X₄₃₆ = polar conjugate of X₉₂₉₀

• X₄₃₆ = isogonal conjugate of the isotomic conjugate of X₉₂₉₁

• X₄₃₆ = polar conjugate of the isogonal conjugate of X₁₉₇₀

• X₄₃₆ = X_i-Ceva conjugate of X_j for these (i,j): {1987, 41204}, {21449, 1970}, {40402, 4}

• X₄₃₆ = X_i-isoconjugate of X_j for these (i,j): {3, 9251}, {48, 9290}, {656, 1303}

• X₄₃₆ = X_i-Dao conjugate of X_j for these (i,j): {1249, 9290}, {1303, 40596}, {9251, 36103}

• X₄₃₆ = crosssum of X₅₂₀ and X₄₁₂₁₉

• X₄₃₆ = barycentric product X_i*X_j for these {i,j}: {1, 9252}, {5, 21449}, {6, 9291}, {92, 1954}, {95, 27359}, {112, 42331}, {264, 1970}

• X₄₃₆ = barycentric quotient X_i/X_j for these {i,j}: {4, 9290}, {19, 9251}, {112, 1303}, {1954, 63}, {1970, 3}, {9252, 75}, {9291, 76}, {21449, 95}, {27359, 5}, {42331, 3267}

• X₄₃₆ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 4, 52249}, {4, 3079, 7714}, {4, 52299, 6619}, {107, 275, 51}, {184, 2052, 41204}, {418, 41202, 8613}, {1495, 42400, 1629}, {1585, 1586, 52247}, {4240, 30506, 13595}, {6524, 11427, 41371}, {8887, 10282, 8884}

Notes

As a point on the Euler line, X₄₃₆ has Shinagawa coefficients \((2F^{2},-(E + F)F + S^{2})\).