X₃₅₇₄

INTERSECTION OF LINES X₄X_54 AND X₅X_51

X₃₅₇₄ = midpoint of X₄ and X₅₄

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

X₃₅₇₄ (INTERSECTION OF LINES X₄X_54 AND X₅X_51) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 54}

• Related Centers: X₄ (ORTHOCENTER), X₅₄ (KOSNITA POINT)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = bc[a^{2}(b^{2} + c^{2}) - (b^{2} - c^{2})^{2}](2a^{6} + b^{6} + c^{6} - 3a^{4}b^{2} - 3a^{4}c^{2} - b^{2}c^{4} - b^{4}c^{2})\)

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)

Search Value (6, 9, 13): \(-5.932189231698\)

▸ Incident Central Lines14 lines

{4, 54}, {5, 51}, {25, 2917}, {113, 137}, {125, 389}, {130, 132}, {155, 195}, {185, 427}, {193, 576}, {235, 1843}, {1204, 3541}, {1839, 2182}, {1907, 2883}, {3519, 3527}

▸ Functional Relations7 relations

• X₃₅₇₄ = X₂₁-of-the-orthic-triangle if ABC is acute. Let H_aH_bH_c be the orthic triangle, which is the pedal (and cevian) triangle of the orthocenter, H = X₄ . The Euler lines of the triangles HH_bH_c, HH_cH_a, HH_aH_b concur in X₃₅₇₄. (Michel Garitte, July 7, 2009)

• X₃₅₇₄ = midpoint of X₄ and X₅₄

• X₃₅₇₄ = reflection of X₁₂₀₉ in X₅

• X₃₅₇₄ = anticomplement of X₃₂₃₄₈

• X₃₅₇₄ = X₄-Ceva conjugate of X₃₅₇₅

• X₃₅₇₄ = crosspoint of X₄ and X₅

• X₃₅₇₄ = crosssum of X₃ and X₅₄

Notes

Let A'B'C' be the cevian triangle of X₅. Let L_a be the reflection of line B'C' in line BC, and define L_b, L_c cyclically. Let A″ = L_b ∩ L_c, and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in the isogonal conjugate of X₃₅₇₄. (Randy Hutson, January 29, 2018)

Let H_aH_bH_c be the orthic triangle. X₃₅₇₄ is the radical center of the circles O(3,4) of triangles AH_bH_c, BH_cH_a, CH_aH_b. (Randy Hutson, July 31 2018)