## X(546) (MIDPOINT OF X(4) AND X(5))

 Interactive Applet

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This applet was built with the free and multiplatform dynamic geometry software C.a.R..

 Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears           f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = 3 cos(B - C) - 2 cos A,
= cos A + 6 cos B cos C : cos B + 6 cos C cos A : cos C + 6 cos A cos B

Barycentrics    (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)

X(546) lies on the Euler line. (Antreas Hatzipolakis, 1/20/00, Hyacinthos #201)

Let MA denote the point in which the A-median meets side BC. On 11/05/03, Andrew Crane noted that X(546) is the radical center of circles (A), (B), (C), where (A) denotes the circle centered at A and passing through MA, and (B) and (C) are defined cyclically.

X(546) lies on these lines: 2,3    13,398    14,397    113,137    156,578    946,952

X(546) = midpoint of X(I) and X(J) for these (I,J): (4,5), (382,550)
X(546) = reflection of X(I) in X(J) for these (I,J): (140,5), (548,140)
X(546) = inverse-in-orthocentroidal-circle of X(382)
X(546) = complement of X(550)

This is a joint work of
Humberto José Bortolossi, Lis Ingrid Roque Lopes Custódio and Suely Machado Meireles Dias.