X₁₂₄

X₁₀₉-OF-MEDIAL-TRIANGLE

X₁₂₄ = X₁₀₉-of-medial triangle

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

X₁₂₄ (X₁₀₉-OF-MEDIAL-TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Medial Triangle △M_aM_bM_c — vertices M_a, M_b, M_c (Formed by the midpoints of sides BC, CA, AB)

• Concurrence Lines: M_a–X₁₂₄, M_b–X₁₂₄, M_c–X₁₂₄ concurring at X₁₂₄

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

Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = (b + c - a)(b - c)^{2} [(b + c)(b^{2} + c^{2} - a^{2} - bc) + abc]\)

Search Value (6, 9, 13): \(3.863590110143\)

X₁₂₄ lies on the nine-point circle

▸ Incident Central Lines5 lines

{2, 109}, {4, 102}, {5, 117}, {10, 119}, {116, 928}

▸ Functional Relations16 relations

• X₁₂₄ = X₁₀₉-of-medial triangle

• X₁₂₄ = center of the rectangular hyperbola that passes through A, B, C, and X₅₈

• X₁₂₄ = midpoint of X₄ and X₁₀₂

• X₁₂₄ = reflection of X₁₁₇ in X₅

• X₁₂₄ = complementary conjugate of X₅₂₂

• X₁₂₄ = X₄-Ceva conjugate of X₅₂₂

• X₁₂₄ = complement of X₁₀₉

• X₁₂₄ = crosssum of circumcircle intercepts of line X₃X₁₀

• X₁₂₄ = orthopole of line X₃X₁₀

• X₁₂₄ = focus of Mandart parabola

• X₁₂₄ = anticenter of cyclic quadrilateral ABCX₁₀₂

• X₁₂₄ = inverse-in-orthoptic-circle-of-Steiner-inellipse of X₁₃₁₁

• X₁₂₄ = Spieker-radical-circle-inverse of X₃₄₄₅₈

• X₁₂₄ = X₂-Ceva conjugate of X₆₅₈₉

• X₁₂₄ = Kirikami-six-circles image of X₅₈

• X₁₂₄ = center of rectangular circumhyperbola that is isogonal conjugate of line X₃X₁₀