X₅₉₅

2nd HATZIPOLAKIS-YIU POINT

Let O(A) be the circle tangent to line BC and to the circumcircle of triangle ABC at vertex A, and define O(B) and O(C) cyclically. X₅₉₅ is the radical center of O(A), O(B), O(C).

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

X₅₉₅ (2nd HATZIPOLAKIS-YIU POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 21}, {10, 82}

• Related Centers: X₅₉₆ (ISOGONAL CONJUGATE OF X₅₉₅), X₁ (INCENTER), X₂₁ (SCHIFFLER POINT), X₁₀ (SPIEKER CENTER), X₈₂ (ISOGONAL CONJUGATE OF X₃₈)

Trilinears: \(r + R (\cos 2A - 1) : :\)

Barycentrics: \(a^{2}(a^{2} + ab + ac - bc) : :\)

Search Value (6, 9, 13): \(0.211981313071\)

▸ Alternate Coordinates2 forms

• Trilinears: \(a(a^{2} + ab + ac - bc) : :\)

• Trilinears: \(sa^{2} - SR : sb^{2} - SR sc^{2} - SR : :\)

▸ Incident Central Lines13 lines

{1, 21}, {3, 995}, {10, 82}, {32, 101}, {35, 902}, {40, 602}, {46, 614}, {55, 386}, {56, 106}, {110, 849}, {171, 1125}, {387, 390}, {517, 580}

▸ Functional Relations4 relations

• X₅₉₅ = isogonal conjugate of X₅₉₆

• X₅₉₅ = crosssum of X₂₄₄ and X₅₂₃

• X₅₉₅ = Vu tangential transform of X₁

• X₅₉₅ = {X₁,X₃₁}-harmonic conjugate of X₅₈

Notes

See Antreas Hatzipolakis, Paul Yiu, Hyacinthos #2070

See Antreas Hatzipolakis and Paul Yiu, Hyacinthos 2070.