X₈₄₆

4th SHARYGIN POINT

This point solves a problem posed by Antreas Hatzipolakis in Hyacinthos #3001, June 11, 2001. For the construction as a Sharygin point, see the description at X₁₂₈₁.

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

X₈₄₆ (4th SHARYGIN POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 21}, {9, 43}

• Related Centers: X₁ (INCENTER), X₂₁ (SCHIFFLER POINT), X₉ (MITTENPUNKT), X₄₃ (-CEVA CONJUGATE OF X₁)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = - a^{2} + b^{2} + c^{2} + bc + ca + ab\)

Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = a f(a,b,c)\)

Search Value (6, 9, 13): \(2.519912547907\)

▸ Incident Central Lines11 lines

{1, 21}, {2, 1054}, {6, 1051}, {9, 43}, {35, 228}, {37, 171}, {55, 984}, {100, 756}, {333, 740}, {405, 986}, {982, 1001}

▸ Functional Relations6 relations

• X₈₄₆ = X_i-Ceva conjugate of X_j for these (i,j): (37, 1), (171, 43)

• X₈₄₆ = crossdifference of every pair of points on line X₆₆₁X₄₃₆₇

• X₈₄₆ = homothetic center of the excentral triangle and the antipedal triangle, wrt the incentral triangle, of X₁

• X₈₄₆ = X₂₇₅-of-excentral-triangle

• X₈₄₆ = excentral polar conjugate of X₃

• X₈₄₆ = perspector of 1st Sharygin triangle and unary cofactor triangle of 2nd Sharygin triangle