X₅₅₀

MIDPOINT OF X₃ AND X₂₀

X₅₅₀ is the midpoint of X₃ and X₂₀.

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

X₅₅₀ (MIDPOINT OF X₃ AND X₂₀) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 20} (Euler line)

• Related Centers: X₃ (CIRCUMCENTER), X₂₀ (DE LONGCHAMPS POINT)

Trilinears: \(- \cos(B - C) + 4 \cos A : :\)

Barycentrics: \(4 a^{4} - 3 a^{2} (b^{2} + c^{2}) - (b^{2} - c^{2})^{2} : :\)

Search Value (6, 9, 13): \(13.011854046758\)

▸ Alternate Coordinates5 forms

• Trilinears: \(2 \sec A - 3 \sec B \sec C : 2 \sec B - 3 \sec C \sec A : 2 \sec C - 3 \sec A \sec B\)

• Trilinears: \(6 \cos A + \cos B \cos C - 3 \sin B \sin C : :\)

• Trilinears: \(\cos A - 4 \cos B \cos C + 2 \sin B \sin C : :\)

• Trilinears: \(6 \cos B \cos C - 2 \sin B \sin C - \cos(B - C) : :\)

• Trilinears: \(5 \cos A + \cos(B - C) - \cos B \cos C - 3 \sin B \sin C : :\)

▸ Linear Combinations1 combination

• X₅₅₀ = \(3\) X₃ - X₄

▸ Incident Central Lines9 lines

{2, 3}, {15, 397}, {16, 398}, {35, 495}, {36, 496}, {40, 952}, {74, 930}, {156, 1092}, {165, 355}

▸ Functional Relations17 relations

• X₅₅₀ = midpoint of X₃ and X₂₀

• X₅₅₀ = reflection of X_i in X_j for these (i,j): (3, 548), (4, 140), (5, 3), (382, 546)

• X₅₅₀ = isogonal conjugate of X₁₆₈₃₅

• X₅₅₀ = trisector nearest X₂₀ of segment X₅X₂₀

• X₅₅₀ = complement of X₃₈₂

• X₅₅₀ = anticomplement of X₅₄₆

• X₅₅₀ = X₅-of-circumcevian-triangle-of-X₃₀

• X₅₅₀ = X₁₄₃-of-hexyl-triangle

• X₅₅₀ = intersection of tangents to Evans conic at X₁₇ and X₁₈

• X₅₅₀ = inverse-in-orthocentroidal-circle of X₃₈₅₁

• X₅₅₀ = radical center of reflections of nine-point circle in A, B, C

• X₅₅₀ = antipedal isogonal conjugate of X₅

• X₅₅₀ = radical trace of circumcircle and Trinh circle

• X₅₅₀ = {X₂,X₃₅₂₉}-harmonic conjugate of X₃₈₂

• X₅₅₀ = {X₃,X₄}-harmonic conjugate of X₁₄₀

• X₅₅₀ = {X₂,X₃}-harmonic conjugate of X₃₅₃₀

• X₅₅₀ = {X₂,X₂₀}-harmonic conjugate of X₃₅₂₉

Notes

As a point on the Euler line, X₅₅₀ has Shinagawa coefficients \((3,-5)\).

Let O_A be the circle centered at the A-vertex of the anticomplementary triangle and passing through A; define O_B and O_C cyclically. X₅₅₀ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)

Let O_A be the circle centered at the A-vertex of the anti-Euler triangle and passing through A; define O_B and O_C cyclically. X₅₅₀ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)