X₃₅₇₉

INTERSECTION OF LINES X₁X_3 AND X₁₀X_30

X₃₅₇₉ = X₃₂₁₀-Ceva conjugate of X₃₇

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

X₃₅₇₉ (INTERSECTION OF LINES X₁X_3 AND X₁₀X_30) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 3}, {10, 30}

• Related Centers: X₁ (INCENTER), X₃ (CIRCUMCENTER), X₁₀ (SPIEKER CENTER), X₃₀ (EULER INFINITY POINT)

Trilinears: \(2a^{3} - b^{3} - c^{3} + (a^{2} + bc)(b + c) - 2a(b^{2} + c^{2} + bc) : :\)

Search Value (6, 9, 13): \(9.328390086566\)

▸ Alternate Coordinates1 form

• Trilinears: \(r - 3 R \cos A : :\)

▸ Linear Combinations1 combination

• X₃₅₇₉ = X₁ - \(3\) X₃

▸ Incident Central Lines42 lines

{1, 3}, {4, 2355}, {5, 516}, {8, 376}, {10, 30}, {12, 1770}, {20, 355}, {24, 1902}, {31, 582}, {37, 2160}, {42, 500}, {44, 573}, {45, 1766}, {71, 2173}, {72, 74}, {109, 227}, {140, 946}, {191, 210}, {267, 2941}, {378, 1829}, {381, 1698}, {392, 404}, {498, 1836}, {515, 550}, {518, 3098}, {548, 952}, {549, 1125}, {631, 962}, {759, 1293}, {896, 3214}, {901, 2687}, {944, 3522}, {971, 1158}, {1191, 1480}, {1216, 2807}, {1250, 2306}, {1656, 1699}, {1702, 3312}, {1703, 3311}, {1827, 1872}, {3085, 3474}, {3218, 3555}

▸ Functional Relations6 relations

• X₃₅₇₉ = midpoint of X_i and X_j for these (i,j): (3, 40), (20, 355)

• X₃₅₇₉ = reflection of X_i in X_j for these (i,j): (946, 140), (1385, 3)

• X₃₅₇₉ = X₃₂₁₀-Ceva conjugate of X₃₇

• X₃₅₇₉ = X₅-of-1st-circumperp-triangle

• X₃₅₇₉ = X₁₄₀-of-excentral-triangle

• X₃₅₇₉ = centroid of excenters and X₄₀

Notes

Let O_A be the circle centered at the A-vertex of the Aquila triangle and passing through A; define O_B and O_C cyclically. X₃₅₇₉ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 28, 2020)