Friendship Among Triangle Centers

Abstract

If we erect on the sides of a scalene triangle three squares, then at the vertices of the triangle we find new triangles, the flanks. 

 

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We study pairs of triangle centers $X$ and $Y$ such that the triangle of $X$s in the three flanks is perspective with $ABC$ at $Y$ , and vice versa. These centers $X$ and $Y$ we call friends. Some examples of friendship among triangle centers are given.