X₁₅₇₅

EXSIMILICENTER OF SPIEKER AND (1/2)-MOSES CIRCLES

The Moses circle, M, is introduced at X₁₀₁₅; the (1/2)-Moses circle is concentric to M with half the radius of M. The insimilicenter of the Spieker and (1/2)-Moses circles is X₁₁₀₇.

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

X₁₅₇₅ (EXSIMILICENTER OF SPIEKER AND (1/2)-MOSES CIRCLES) 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, 350}

• Related Centers: X₂ (CENTROID), X₃₅₀ (-HIRST INVERSE OF X₇₅)

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

Search Value (6, 9, 13): \(-8.379650098317\)

▸ Linear Combinations1 combination

• X₁₅₇₅ = \((R \tan \omega \sin 2\omega )\) X₁₀ - \(r\) X₃₉ = \(R\) X₁ + \(R\) X₈ - \((2r \cot \omega \csc 2\omega )\) X₃₉

▸ Incident Central Lines13 lines

{2, 37}, {6, 43}, {10, 39}, {42, 1100}, {44, 513}, {71, 992}, {172, 404}, {239, 292}, {291, 518}, {519, 1015}, {574, 993}, {1009, 1104}, {1125, 1500}

▸ Functional Relations10 relations

• X₁₅₇₅ = isotomic conjugate of X₃₂₀₂₀

• X₁₅₇₅ = complement of X₃₅₀

• X₁₅₇₅ = anticomplement of X₂₀₅₃₀

• X₁₅₇₅ = complementary conjugate of X₂₀₅₄₂

• X₁₅₇₅ = X_i-Ceva conjugate of X_j for these (i,j): (239, 518), (292, 37)

• X₁₅₇₅ = cevapoint of X₄₃ and X₂₁₀₈

• X₁₅₇₅ = crosspoint of X₂ and X₂₉₁

• X₁₅₇₅ = crosssum of X_i and X_j for these (i,j): (1, 1575), (6, 238)

• X₁₅₇₅ = crossdifference of every pair of points on line X₁X₆₆₇

• X₁₅₇₅ = {X₁₀,X₃₉}-harmonic conjugate of X₁₁₀₇