2008 OIM Problems/Problem 5

Problem

Let $ABC$ be a triangle, and $X, Y, Z$ be points on the sides $BC$, $AC$, $AB$ respectively. Let $A'$, $B'$, and $C'$ be the circumcenters corresponding to the triangles $AZY, BXZ,$ and $CYX$. Show that

\[(A'B'C') \ge \frac{(ABC)}{4}\]

and that the equality is fulfilled if and only if the lines $AA',BB',CC'$ have a point in common.

Observation: for any triangle $RST$, we denote its area by $(RST)$.

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

Solution

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See also

OIM Problems and Solutions