X₃₅₂₂

EULER LINE INTERCEPT OF LINE X₈X_165

Other recent results on representations of points on the Euler line are given in

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

X₃₅₂₂ (EULER LINE INTERCEPT OF LINE X₈X_165) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {20, 3091}, {2, 3} (Euler line)

• Related Centers: X₂₀ (DE LONGCHAMPS POINT), X₃₀₉₁ (INTERSECTION OF LINES X), X₂ (CENTROID), X₃ (CIRCUMCENTER)

Trilinears: \(3 \cos A - \cos B \cos C : :\)

Barycentrics: \(7 a^{4} - b^{4} - c^{4} - 6 a^{2} b^{2} - 6 a^{2} c^{2} + 2 b^{2} c^{2} : :\)

Search Value (6, 9, 13): \(9.274159355221\)

▸ Alternate Coordinates2 forms

• Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = \sec A - 4 \sec B \sec C\)

• Trilinears: \(4 \sec B \sec C - 3 \sin B \sin C : :\)

▸ Linear Combinations1 combination

• X₃₅₂₂ = \(3\) X₂ - \(8\) X₃

▸ Incident Central Lines19 lines

{2, 3}, {8, 165}, {40, 145}, {56, 390}, {78, 144}, {84, 3219}, {97, 3346}, {185, 2979}, {193, 1350}, {323, 1181}, {489, 1270}, {490, 1271}, {590, 1131}, {615, 1132}, {1038, 3100}, {1155, 3486}, {1193, 1742}, {1204, 3098}, {2646, 3474}

▸ Functional Relations10 relations

• X₃₅₂₂ = midpoint of X₂₀ and X₃₀₉₁

• X₃₅₂₂ = reflection of X_i in X_j for these (i,j): (4, 1656), (631, 3), (3091, 631)

• X₃₅₂₂ = complement of X₁₇₅₇₈

• X₃₅₂₂ = anticomplement of X₃₀₉₁

• X₃₅₂₂ = crosspoint of X₂₂₂₃₅ and X₂₂₂₃₇

• X₃₅₂₂ = crosssum of X₂₂₂₃₆ and X₂₂₂₃₈

• X₃₅₂₂ = circumcircle-inverse of X₃₇₉₄₄

• X₃₅₂₂ = Thomson-isogonal conjugate of X₅₆₄₄

• X₃₅₂₂ = homothetic center of 2nd Conway triangle and cross-triangle of excentral and 2nd circumperp triangles

• X₃₅₂₂ = {X₂,X₃}-harmonic conjugate of X₁₅₇₁₇

Notes

As a point on the Euler line, X₃₅₂₂ has Shinagawa coefficients \((3, -4)\).

X₃₅₂₂ is one of many points on the Euler line that have a certain convenient representation, received from Peter Moses, February 8, 2010. Specifically, the appearance of (t,k) in the following list means that

X_k =\(f(A,B,C) : f(B,C,A): f(C,A,B)\), where \(f(A,B,C) = t \cos A + \cos B \cos C\):

(-4,3528), (-3,3522), (-5/2, 548), (-2,376), (-3/2,550), (-5/4,3534), (-1,20), (-3/4,1657), (-2/3,3529), (-1/3,3146), (-1/4,382), (-1/5,3543), (0,4), (1/6,546), (1/4,381), (1/3,3091), (2/5, 3545), (4/9,3544), (1/2,5), (2/3,3090), (7/10,547), (3/4,1656), (1,2), (7/6,632), (6/5,3533), (5/4,3526), (4/3,3525), (3/2,140), (2,631), (5/2,549), (3,3523), (7/2,3530), (4,3525), (infinity,3)

Peter J. C. Moses and Clark Kimberling, “Perspective isoconjugate triangle pairs, Hofstadter pairs, and crosssums on the nine-point circle,” Forum Geometricorum 11 (2011) 83-93. Click here to download a pdf.