X₅₀₀

ORTHOCENTER OF THE INCENTRAL TRIANGLE

X₅₀₀ is the orthocenter of the incentral triangle.

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

X₅₀₀ (ORTHOCENTER OF THE INCENTRAL TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Incentral Triangle △I_aI_bI_c — vertices I_a, I_b, I_c (Cevian triangle of Incenter X₁ (angle bisector feet))

• Concurrence Lines: I_a–X₅₀₀, I_b–X₅₀₀, I_c–X₅₀₀ concurring at X₅₀₀

Trilinears: \(a(b^{2} +c^{2} - a^{2} + bc)[2abc + (b + c)(a^{2} - (b - c)^{2})] : :\)

Search Value (6, 9, 13): \(3.085237628787\)

▸ Incident Central Lines3 lines

{1, 30}, {3, 6}, {651, 943}

▸ Functional Relations4 relations

• X₅₀₀ = inverse-in-Brocard-circle of X₅₈₂

• X₅₀₀ = crosspoint of X₁ and X₃₅

• X₅₀₀ = crosssum of X₁ and X₇₉

• X₅₀₀ = X₁-Ceva conjugate of X₉₄₂