X₁₈₁₇

INVERSE MIMOSA TRANSFORM OF X₁₂₄₉

X₁₈₁₇ = isogonal conjugate of X₁₉₀₃

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

X₁₈₁₇ (INVERSE MIMOSA TRANSFORM OF X₁₂₄₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {57, 77}

• Related Centers: X₁₉₀₃ (ZOSMA TRANSFORM OF X₁₂₄₉), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₅₇ (ISOGONAL CONJUGATE OF X₉), X₇₇ (ISOGONAL CONJUGATE OF X₃₃)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = (\cos A)/(y + z)\),\(x : y : z\) = X₁₂₄₉

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)

Search Value (6, 9, 13): \(69.376468782008\)

▸ Incident Central Lines14 lines

{2, 3}, {40, 1819}, {57, 77}, {58, 937}, {63, 610}, {100, 306}, {110, 972}, {189, 333}, {196, 347}, {198, 329}, {572, 1730}, {662, 1812}, {1172, 1214}, {1396, 1465}

▸ Functional Relations3 relations

• X₁₈₁₇ = isogonal conjugate of X₁₉₀₃

• X₁₈₁₇ = X_i-Ceva conjugate of X_j for these (i,j): (333, 81), (1444, 21)

• X₁₈₁₇ = cevapoint of X_i and X_j for these (i,j): (3, 610), (40, 198)

Notes

As a point on the Euler line, X₁₈₁₇ has Shinagawa coefficients \((4EF^{2}S^{2} - \$aS_{A}\$(S^{2} - 4F^{2})S^{2} + 3\$bcS_{B}S_{C}\$S^{2} + 4\$abS_{C}\$FS^{2} - 2\$bc(S_{B})^{2}(S_{C})^{2}\$, [S^{2} - 4(E + F)F]ES^{2} + \$bc\$(S^{2} - 4F^{2})S^{2} - 3\$bcS_{B}S_{C}\$S^{2} - 4\$abS_{C}\$FS^{2} + 2\$bc(S_{B})^{2}(S_{C})^{2}\$)\).