X₅₄₇

MIDPOINT OF X₂ AND X₅

X₅₄₇ is the midpoint of X₂ and X₅.

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

X₅₄₇ (MIDPOINT OF X₂ AND X₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 5} (Euler line)

• Related Centers: X₂ (CENTROID), X₅ (NINE-POINT CENTER)

Trilinears: \(5 \cos(B - C) + 2 \cos A : :\)

Search Value (6, 9, 13): \(1.591120267062\)

▸ Alternate Coordinates1 form

• Trilinears: \(7 \cos A + 10 \cos B \cos C : 7 \cos B + 10 \cos C \cos A : 7 \cos C + 10 \cos A \cos B\)

▸ Linear Combinations1 combination

• X₅₄₇ = X₃ + \(3\) X₄

X₅₄₇ lies on the Euler line. (Antreas Hatzipolakis, 1/20/00, Hyacinthos #201)

▸ Incident Central Lines2 lines

{2, 3}, {551, 952}

▸ Functional Relations8 relations

• X₅₄₇ = midpoint of X_i and X_j for these (i,j): (2, 5), (381, 549)

• X₅₄₇ = reflection of X₁₄₀ in X₂

• X₅₄₇ = complement of X₅₄₉

• X₅₄₇ = circumcircle-inverse of X₃₇₉₂₃

• X₅₄₇ = center of the Vu pedal-centroidal circle of X₅

• X₅₄₇ = {X₃₅₂₃₁,X₃₅₂₃₂}-harmonic conjugate of X₃₇₉₅₂

• X₅₄₇ = pole of Fermat axis wrt conic {X₅,X₁₃,X₁₄,X₁₅,X₁₆}}

• X₅₄₇ = {X₃,X₄}-harmonic conjugate of X₃₆₂₇

Notes

As a point on the Euler line, X₅₄₇ has Shinagawa coefficients \((7,3)\).