X₄₂₃

X₄-HIRST INVERSE OF X₂₇

X₄₂₃ is the X₄-hirst inverse of X₂₇.

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

X₄₂₃ (X₄-HIRST INVERSE OF X₂₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {107, 2700}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₁₀₇ (Ψ(SYMMEDIAN POINT, ORTHOCENTER)), X₂₇₀₀ (SR(X₉₈, X₁₀₁))

Trilinears: \((u^{2} - vw)\sec A : (v^{2} - wu)\sec B : (v^{2} - uv)\sec C\), where \(u : v : w\) = X₈₁; e.g., \(u = u(a,b,c) = 1/(b + c)\)

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

Search Value (6, 9, 13): \(1.100890707601\)

▸ Alternate Coordinates1 form

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

▸ Incident Central Lines16 lines

{2, 3}, {19, 2905}, {86, 18161}, {92, 31997}, {107, 2700}, {110, 48381}, {648, 8756}, {2322, 27958}, {2501, 4608}, {4600, 18020}, {6542, 17927}, {7119, 52412}, {7282, 27691}, {11363, 27954}, {23864, 44451}, {24559, 25533}

▸ Functional Relations11 relations

• X₄₂₃ = polar conjugate of X₁₁₅₉₉

• X₄₂₃ = polar conjugate of the isotomic conjugate of X₁₇₇₃₁

• X₄₂₃ = polar conjugate of the isogonal conjugate of X₁₃₂₆

• X₄₂₃ = X₁₃₂₆-cross conjugate of X₁₇₇₃₁

• X₄₂₃ = X_i-isoconjugate of X_j for these (i,j): {3, 9278}, {37, 17972}, {48, 11599}, {63, 2054}, {71, 1929}, {72, 17962}, {228, 6650}, {647, 37135}, {656, 2702}, {810, 35148}, {906, 18014}, {1332, 18001}, {1437, 6543}, {2200, 18032}, {3990, 17982}

• X₄₂₃ = X_i-Dao conjugate of X_j for these (i,j): {63, 39042}, {72, 39041}, {306, 41841}, {525, 35080}, {1249, 11599}, {1326, 20813}, {2054, 3162}, {2702, 40596}, {4466, 27929}, {5190, 18014}, {9278, 36103}, {17972, 40589}, {20546, 20825}, {35148, 39062}, {37135, 39052}

• X₄₂₃ = crosssum of X₃ and X₂₀₇₆₆

• X₄₂₃ = crossdifference of every pair of points on line {647, 22080}

• X₄₂₃ = barycentric product X_i*X_j for these {i,j}: {4, 17731}, {19, 52137}, {27, 6542}, {28, 20947}, {86, 17927}, {92, 1931}, {264, 1326}, {286, 1757}, {648, 2786}, {811, 9508}, {5029, 6331}, {6336, 31059}, {7649, 17934}, {17735, 44129}, {17943, 46107}

• X₄₂₃ = barycentric quotient X_i/X_j for these {i,j}: {4, 11599}, {19, 9278}, {25, 2054}, {27, 6650}, {28, 1929}, {58, 17972}, {112, 2702}, {162, 37135}, {286, 18032}, {648, 35148}, {1326, 3}, {1474, 17962}, {1757, 72}, {1826, 6543}, {1931, 63}, {2786, 525}, {2905, 39921}, {5029, 647}, {6541, 3695}, {6542, 306}, {7649, 18014}, {8747, 17982}, {9508, 656}, {17569, 40793}, {17731, 69}, {17735, 71}, {17927, 10}, {17934, 4561}, {17943, 1331}, {17976, 3682}, {18004, 4064}, {18266, 228}, {20693, 3949}, {20947, 20336}, {27929, 24459}, {28346, 51366}, {28602, 14429}, {31059, 3977}, {31905, 40725}, {52137, 304}

• X₄₂₃ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {4, 31904, 27}, {27, 28, 31913}, {27, 4248, 31912}, {27, 14013, 31914}, {27, 31912, 31915}, {28, 31909, 27}, {29, 14013, 27}, {422, 468, 415}, {1375, 3109, 448}, {4248, 31916, 31915}, {13739, 37448, 27}, {15149, 31905, 27}, {15149, 37168, 31905}, {31908, 31925, 27}, {31912, 31916, 27}

Notes

As a point on the Euler line, X₄₂₃ has Shinagawa coefficients \((2(E + F)F + 3\$bc\$F,-(E + F)^{2} - \$bc\$(E + F) + S^{2})\).