X₅₄₆

MIDPOINT OF X₄ AND X₅

X₅₄₆ is the midpoint of X₄ and X₅.

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

X₅₄₆ (MIDPOINT OF X₄ AND X₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 5} (Euler line)

• Related Centers: X₄ (ORTHOCENTER), X₅ (NINE-POINT CENTER)

Trilinears: \(3 \cos(B - C) - 2 \cos A : :\)

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

Search Value (6, 9, 13): \(-2.561873834646\)

▸ Alternate Coordinates1 form

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

▸ Linear Combinations1 combination

• X₅₄₆ = X₃ - \(3\) X₅ = X₄ + X₅ = X₃ + \(3\) X₄

▸ Incident Central Lines6 lines

{2, 3}, {13, 398}, {14, 397}, {113, 137}, {156, 578}, {946, 952}

▸ Functional Relations14 relations

• X₅₄₆ = midpoint of X_i and X_j for these (i,j): (4, 5), (382, 550)

• X₅₄₆ = reflection of X_i in X_j for these (i,j): (140, 5), (548, 140)

• X₅₄₆ = complement of X₅₅₀

• X₅₄₆ = anticomplement of X₃₅₃₀

• X₅₄₆ = circumcircle-inverse of X₃₇₉₂₂

• X₅₄₆ = orthocentroidal-circle-inverse of X₃₈₂

• X₅₄₆ = X₁₃₈₅-of-orthic-triangle if ABC is acute

• X₅₄₆ = X₅-of-Euler-triangle

• X₅₄₆ = {X₃,X₅}-harmonic conjugate of X₃₆₂₈

• X₅₄₆ = X₅-of-Ehrmann-mid-triangle

• X₅₄₆ = Ehrmann-side-to-orthic similarity image of X₅

• X₅₄₆ = Johnson-to-Ehrmann-mid similarity image of X₅

• X₅₄₆ = homothetic center of Ehrmann mid-triangle and Euler triangle

• X₅₄₆ = radical center of de Longchamps circles of ABC and 1st and 2nd Ehrmann inscribed triangles

Notes

As a point on the Euler line, X₅₄₆ has Shinagawa coefficients \((1,5)\).

Let A'B'C', A″B″C″ be the outer and inner Vecten triangles, resp. Let (O_a) be the circle inscribed in square A’BA″C, and define (O_b) and (O_c) cyclically. X₅₄₆ is the radical center of circles (O_a), (O_b), (O_c). (Randy Hutson, December 2, 2017)

Let M_A denote the point in which the A-median meets side BC. On 11/05/03, Andrew Crane noted that X₅₄₆ is the radical center of circles (A), (B), (C), where (A) denotes the circle centered at A and passing through M_A, and (B) and (C) are defined cyclically.