X₃₅₅₆

INTERSECTION OF LINES X₂₅X_65 AND X₃₁X_56

X₃₅₅₆ = X₆₃-Ceva conjugate of X₆

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

X₃₅₅₆ (INTERSECTION OF LINES X₂₅X_65 AND X₃₁X_56) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {31, 56}, {25, 65}

• Related Centers: X₆₃ (ISOGONAL CONJUGATE OF X₁₉), X₆ (SYMMEDIAN POINT), X₃₁ (2nd POWER POINT), X₅₆ (EXSIMILICENTER), X₂₅ (HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES), X₆₅ (ORTHOCENTER OF THE INTOUCH TRIANGLE)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a(-aS_{B}S_{C} + bS_{C}S_{A} + cS_{A}S_{B})\)

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

Search Value (6, 9, 13): \(-2.083711635565\)

▸ Incident Central Lines21 lines

{1, 159}, {3, 960}, {6, 1245}, {20, 1610}, {25, 65}, {31, 56}, {40, 197}, {55, 976}, {63, 1619}, {64, 71}, {73, 2187}, {100, 1265}, {161, 2099}, {222, 1660}, {859, 1780}, {958, 1503}, {1012, 2217}, {1398, 1456}, {1498, 3428}, {1593, 2182}, {2176, 2178}

▸ Functional Relations8 relations

• X₃₅₅₆ = isogonal conjugate of X₇₂₁₉

• X₃₅₅₆ = X₆₃-Ceva conjugate of X₆

• X₃₅₅₆ = crosssum of X_i and X_j for these (i,j): (123, 522), (513, 2968)

• X₃₅₅₆ = pole wrt circumcircle of line X₅₂₁X₆₅₆

• X₃₅₅₆ = X₁₈₈-of-the-tangential-triangle

• X₃₅₅₆ = polar conjugate of isotomic conjugate of X₂₂₁₁₉

• X₃₅₅₆ = crosspoint of X₁₀₉ and X₁₅₃₈₅

• X₃₅₅₆ = X₆₅-of-the-excentral-triangle-of-the-tangential-triangle