X548

MIDPOINT OF X5 AND X20

X548 is the midpoint of X5 and X20.
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 5, 20 }: Euler Line A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_20 = DE LONGCHAMPS POINT (0.3878, -0.1916) X_548 = MIDPOINT OF X(5) AND X(20) (0.4719, 0.0286) A B C X5 X20 X548 (a, b, c) = (5, 6, 7) | AB = 1

X548 (MIDPOINT OF X5 AND X20) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(- \cos(B - C) + 6 \cos A : :\)
Search Value (6, 9, 13): \(9.897108470477\)
▸ Alternate Coordinates2 forms

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

  • Trilinears: \(2 \sec A - 5 \sec B \sec C : 2 \sec B - 5 \sec C \sec A : 2 \sec C - 5 \sec A \sec B\)

▸ Linear Combinations1 combination

X548 lies on the Euler line. (Antreas Hatzipolakis, 1/20/00, Hyacinthos #201)

▸ Functional Relations5 relations

  • X548 = midpoint of Xi and Xj for these (i,j): (3, 550), (5, 20)

  • X548 = reflection of Xi in Xj for these (i,j): (140, 3), (546, 140)

  • X548 = {X2,X3}-harmonic conjugate of X15712

  • X548 = {X3,X5}-harmonic conjugate of X3530

  • X548 = radical center of circles centered at vertices of ABC with diameter equal to opposite side

Notes

As a point on the Euler line, X548 has Shinagawa coefficients \((5,-7)\).