X₃₅₇₆

INTERSECTION OF LINES X₁X_3 AND X₂X_515

Let P_a be the parabola with focus A and directrix BC, and define P_b and P_c cyclically. X₃₅₇₆ is the centroid of the six points of tangency of lines from X₁ to P_a, P_b, and P_c. (Randy Hutson, December 2, 2017)

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

X₃₅₇₆ (INTERSECTION OF LINES X₁X_3 AND X₂X_515) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 165}

• Related Centers: X₁ (INCENTER), X₁₆₅ (CENTROID OF THE EXCENTRAL TRIANGLE)

Trilinears: \((3a - b - c)(a - b + c)(a + b - c) - 4abc\)

Search Value (6, 9, 13): \(5.085011432617\)

▸ Alternate Coordinates2 forms

• Trilinears: \(3 \cos A + \cos B + \cos C - 1 : :\) (Peter Moses, March 10, 2011)

• Trilinears: \(r + 2 R \cos A : :\)

▸ Linear Combinations1 combination

• X₃₅₇₆ = X₁ + \(2\) X₃

▸ Incident Central Lines53 lines

{1, 3}, {2, 515}, {4, 1125}, {8, 3523}, {9, 48}, {10, 631}, {11, 3523}, {20, 946}, {21, 84}, {30, 1699}, {58, 602}, {77, 102}, {78, 947}, {106, 1292}, {140, 355}, {154, 392}, {187, 1572}, {198, 374}, {200, 956}, {223, 1455}, {284, 2257}, {376, 516}, {380, 1108}, {405, 1490}, {519, 3158}, {549, 952}, {573, 1449}, {580, 1468}, {581, 1193}, {595, 601}, {936, 958}, {950, 3086}, {953, 1308}, {960, 1071}, {962, 3522}, {963, 2910}, {970, 1051}, {991, 995}, {1001, 1012}, {1151, 1702}, {1152, 1703}, {1210, 3486}, {1212, 3207}, {1350, 1386}, {1435, 1870}, {1511, 2948}, {1732, 2364}, {1766, 3247}, {1829, 3515}, {1902, 3516}, {2202, 3100}, {2320, 3306}, {2718, 2742}

▸ Functional Relations18 relations

• X₃₅₇₆ = midpoint of X₁ and X₁₆₅

• X₃₅₇₆ = reflection of X_i in X_j for these (i,j): (40, 165),(165, 3)

• X₃₅₇₆ = isogonal conjugate of X₃₅₇₇

• X₃₅₇₆ = X₂₃₂₀-Ceva conjugate of X₁

• X₃₅₇₆ = crosssum of X₂ and X₂₀₉₃

• X₃₅₇₆ = crossdifference of every pair points on the line X₆₅₀X₁₇₆₉

• X₃₅₇₆ = X₂-of-hexyl-triangle

• X₃₅₇₆ = anticomplement of X₁₀₁₇₅

• X₃₅₇₆ = X₂-of-2nd-circumperp-triangle

• X₃₅₇₆ = X₃₈₁-of-excentral-triangle

• X₃₅₇₆ = {X₁,X₃}-harmonic conjugate of X₄₀

• X₃₅₇₆ = homothetic center of hexyl triangle and medial triangle of 2nd circumperp triangle

• X₃₅₇₆ = homothetic center of Johnson triangle and cross-triangle of Aquila and anti-Aquila triangles

• X₃₅₇₆ = Thomson-isogonal conjugate of X₉

• X₃₅₇₆ = orthocenter of cross-triangle of ABC and anti-Aquila triangle

• X₃₅₇₆ = insimilicenter of circumcircles of ABC and anti-Aquila triangle; the exsimilicenter is X₁

• X₃₅₇₆ = trisector nearest X₃ of segment X₁X₃

• X₃₅₇₆ = endo-homothetic center of Ehrmann side-triangle and orthic triangle; the homothetic center is X₃₈₁

Notes

X₃₅₇₆ occurs in Hyacinthos #18095 (Quang Tuan Bui, August 2, 2009). Following a note by Seiichi Kirikami [see X₁₆₅], Peter Moses, October 19, 2010, found that if DEF is the pedal triangle of X₃₅₇₆, then |FB| + |CE| = |DC| + |AF| = |EA| + |BD|.

Let A'B'C' be the hexyl triangle. Let A_b = BC∩C'A', A_c = BC∩A'B', and define B_c, B_a, C_a, C_b cyclically. Then X₃₅₇₆ is the centroid of AbAcBcBaCaCb. (Randy Hutson, December 2, 2017)

Let I be the incenter, X₁, and I_a, I_b, I_c be the excenters. Let O_a be the de Longchamps circle of triangle IbIcI, and define O_b and O_c cyclically. Then X₃₅₇₆ is the radical center of O_a, O_b, O_c. (Randy Hutson, December 2, 2017)