X₁₂₃

X₁₀₈-OF-MEDIAL-TRIANGLE

X₁₂₃ = complement of X₁₀₈

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

X₁₂₃ (X₁₀₈-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)

• Central Lines: {2, 108}

• Related Centers: X₂ (CENTROID), X₁₀₈ (Ψ(CIRCUMCENTER, INCENTER))

Trilinears: \((\csc A)(\sec B - \sec C)[(\sec A)(\sin^{2} B - \sin^{2} C) + \sin C \tan C - \sin B \tan B] : :\)

Barycentrics: \((\sec B - \sec C)[(\sec A)(\sin^{2} B - \sin^{2} C) + \sin C \tan C - \sin B \tan B]\)

Search Value (6, 9, 13): \(3.984933555176\)

▸ Alternate Coordinates1 form

• Barycentrics: \((a - b - c)(b - c)^{2}(a^{2} - b^{2} - c^{2})(a^{4} - b^{4} - c^{4} + 2a^{2}bc - 2ab^{2}c - 2abc^{2} + 2b^{2}c^{2}) : :\)

X₁₂₃ lies on the nine-point circle and hese lines: 2,108 3,119 10,117 113,960

▸ Functional Relations10 relations

• X₁₂₃ = isogonal conjugate of X₁₅₃₈₅

• X₁₂₃ = complement of X₁₀₈

• X₁₂₃ = complementary conjugate of X₅₂₁

• X₁₂₃ = X₂-Ceva conjugate of X₆₅₈₈

• X₁₂₃ = X₄-Ceva conjugate of X₅₂₁

• X₁₂₃ = Spieker-radical-circle-inverse of X₃₄₄₅₅

• X₁₂₃ = X₁₀₈-of-medial triangle

• X₁₂₃ = Stevanovic-circle-inverse of X₃₈₉₇₂

• X₁₂₃ = crossdifference of every pair of points on line X₁₄₁₅X₂₄₄₃

• X₁₂₃ = trilinear product X_i*X_j for these {i,j}: {3436, 7004}, {6332, 6588}