X₂₅₄₈

INSIMILICENTER(MOSES CIRCLE, (X₄,2R))

The circle (X₄,2R) is identified at X₂₄₇₄.

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

X₂₅₄₈ (INSIMILICENTER(MOSES CIRCLE, (X₄,2R))) 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, 3785}

• Related Centers: X₂ (CENTROID), X₃₇₈₅ (com[INVERSE(nT(a2, a2 + b2))])

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = 2 \cos B \cos C + \sin(A + \omega ) \csc \omega\)

Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)

Search Value (6, 9, 13): \(0.913829324005\)

▸ Alternate Coordinates2 forms

• Trilinears: \(g(a,b,c) : g(b,c,a): g(c,a,b)\), where \(g(a,b,c) = bc[a^{4} - b^{4} - c^{4} + 2(a^{2}b^{2} + b^{2}c^{2} + c^{2}a^{2})]\) (Eric Danneels, 11/11/2004)

• Trilinears: \(a^{3}R + bcS : b^{3}R + caS : c^{3}R + abS\) (C. Lozada, 9/07/2013)

▸ Incident Central Lines29 lines

{2, 32}, {4, 39}, {5, 6}, {10, 1572}, {20, 574}, {115, 147}, {148, 1569}, {172, 499}, {187, 631}, {211, 263}, {217, 1899}, {230, 1656}, {316, 2021}, {388, 1015}, {427, 2207}, {497, 1500}, {498, 1914}, {516, 1571}, {625, 1692}, {1478, 2275}, {1479, 2276}, {1504, 1588}, {1505, 1587}, {1573, 2551}, {1574, 2550}, {2033, 2543}, {2034, 2542}, {2035, 2547}, {2036, 2546}

▸ Functional Relations3 relations

• X₂₅₄₈ = complement of X₃₇₈₅

• X₂₅₄₈ = anticomplement of X₇₈₁₅

• X₂₅₄₈ = {X₅,X₆}-harmonic conjugate of X₃₇₆₇