X₂₆₄₆

SUM OF PU(80)

X₂₆₄₆ = midpoint of X₁ and X₃₅

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

X₂₆₄₆ (SUM OF PU(80)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 35}

• Related Centers: X₁ (INCENTER), X₃₅ ({X)

Trilinears: \(2 \cos A + \cos B + \cos C : :\)

Search Value (6, 9, 13): \(2.487499019788\)

▸ Incident Central Lines42 lines

{1, 3}, {2, 1837}, {6, 1732}, {8, 2320}, {11, 214}, {12, 515}, {20, 1836}, {21, 60}, {24, 1905}, {28, 1859}, {29, 243}, {33, 1900}, {37, 48}, {41, 1212}, {58, 2361}, {72, 993}, {73, 820}, {78, 210}, {104, 943}, {140, 1737}, {154, 968}, {212, 1468}, {224, 1001}, {284, 1731}, {355, 498}, {377, 497}, {405, 997}, {409, 662}, {501, 2360}, {518, 2330}, {550, 1770}, {650, 2649}, {851, 2654}, {991, 1456}, {1100, 2245}, {1104, 1193}, {1152, 2362}, {1201, 1279}, {1476, 2346}, {1831, 2355}, {1848, 1852}, {2638, 2658}

▸ Functional Relations7 relations

• X₂₆₄₆ = midpoint of X₁ and X₃₅

• X₂₆₄₆ = X₁-Ceva conjugate of X₂₆₅₀

• X₂₆₄₆ = crosspoint of X₁ and X₂₁

• X₂₆₄₆ = crosssum of X₁ and X₆₅

• X₂₆₄₆ = {X₁,X₃}-harmonic conjugate of X₆₅

• X₂₆₄₆ = homothetic center of intouch triangle and medial triangle of 2nd circumperp triangle

• X₂₆₄₆ = PU(80)-harmonic conjugate of X₆₅₀

Notes

In the place of a triangle ABC, let I = X₁ = incenter, and let

• A'B'C' = intouch triangle;

• M_a = midpoint of segment AI;

• L_a = M_aA', and define L_b and L_c cyclically;

• \(R_{a}\) = reflection of L_a in AI, and define B_b and \(R_{c}\) cyclically.

The lines \(R_{a}\), \(R_{b}\), \(R_{c}\) concur in X₂₆₄₆. (Angel Montesdeoca, October 9, 2021)