X₃₅₄₅

EULER LINE INTERCEPT OF LINE X₁₁X_1056

X₃₅₄₅ = midpoint of X₄ and X₃₅₂₄

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

X₃₅₄₅ (EULER LINE INTERCEPT OF LINE X₁₁X_1056) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 3524}, {2, 3} (Euler line)

• Related Centers: X₄ (ORTHOCENTER), X₃₅₂₄ (EULER LINE INTERCEPT OF LINE X), X₂ (CENTROID), X₃ (CIRCUMCENTER)

Trilinears: \(3 \cos B \cos C + 2 \sin B \sin C : :\)

Barycentrics: \(a^{4} + 4a^{2}(b^{2} + c^{2}) - 5(b^{2} - c^{2})^{2} : :\)

Search Value (6, 9, 13): \(-0.139293941983\)

▸ Linear Combinations1 combination

• X₃₅₄₅ = \(5\) X₂ - \(2\) X₃ = \(4\) X₃ + \(5\) X₄

▸ Incident Central Lines15 lines

{2, 3}, {11, 1056}, {12, 1058}, {69, 1568}, {114, 671}, {230, 1285}, {262, 538}, {355, 3241}, {371, 1328}, {372, 1327}, {551, 944}, {1131, 3312}, {1132, 3311}, {1352, 1992}, {3058, 3085}

▸ Functional Relations10 relations

• X₃₅₄₅ = midpoint of X₄ and X₃₅₂₄

• X₃₅₄₅ = reflection of X_i in X_j for these (i,j): (376, 3524), (3524, 2)

• X₃₅₄₅ = complement of X₁₀₃₀₄

• X₃₅₄₅ = anticomplement of X₅₀₅₄

• X₃₅₄₅ = circumcircle-inverse of X₃₇₉₅₃

• X₃₅₄₅ = orthocentroidal-circle inverse of X₃₇₆

• X₃₅₄₅ = circle-O(PU(5))-inverse of X₃₈₂

• X₃₅₄₅ = trisector nearest X₂ of segment X₂X₄

• X₃₅₄₅ = trisector nearest X₃₈₁ of segment X₂X₃₈₁

• X₃₅₄₅ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): (2, 3, 15702), (2, 4, 376), (2, 5, 5071), (2, 20, 549), (2, 381, 4), (2, 382, 15715), (2, 384, 33224), (2, 11737, 3544), (3, 4, 33703), (3, 5, 5056), (3, 381, 3845), (4, 5, 3090), (5, 381, 2), (5, 5068, 3544)

Notes

As a point on the Euler line, X₃₅₄₅ has Shinagawa coefficients \((2, 3)\).