X₅₇₀

ORTHOGONAL OF X₅₂

X₅₇₀ = inverse-in-Brocard-circle of X₅₇₁

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

X₅₇₀ (ORTHOGONAL OF X₅₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Medial Triangle △M_aM_bM_c — vertices M_a, M_b, M_c (Formed by the midpoints of sides BC, CA, AB)

• Central Lines: {2, 311}

• Related Centers: X₂ (CENTROID), X₃₁₁ (ISOTOMIC CONJUGATE OF X₅₄)

Trilinears: \(\sin A + \sin 2A \cos(B - C) : :\) (Joe Goggins, 11/26/08)

Search Value (6, 9, 13): \(2.856483116793\)

▸ Alternate Coordinates1 form

• Trilinears: \((\sin A)(1 - 2 \sin^{2} B - 2 \sin^{2} C) : :\)

▸ Incident Central Lines6 lines

{2, 311}, {3, 6}, {53, 232}, {115, 128}, {140, 231}, {157, 184}

▸ Functional Relations13 relations

• X₅₇₀ = inverse-in-Brocard-circle of X₅₇₁

• X₅₇₀ = complement of X₃₁₁

• X₅₇₀ = crosspoint of X₂ and X₅₄

• X₅₇₀ = crosssum of X₅ and X₆

• X₅₇₀ = X₇₁-of-orthic-triangle if ABC is acute

• X₅₇₀ = {X₆,X₃₉}-harmonic conjugate of X₅₄₂₁

• X₅₇₀ = perspector of circumconic centered at X₁₂₀₉

• X₅₇₀ = center of circumconic that is locus of trilinear poles of lines passing through X₁₂₀₉

• X₅₇₀ = X₂-Ceva conjugate of X₁₂₀₉

• X₅₇₀ = polar conjugate of isogonal conjugate of X₂₃₁₉₅

• X₅₇₀ = polar conjugate of isotomic conjugate of X₁₂₁₆

• X₅₇₀ = X₆₃-isoconjugate of X₁₁₇₉

• X₅₇₀ = crossdifference of every pair of points on line X₅₂₃X₂₀₇₀ (the polar of X₅ wrt the circumcircle)