1998 OIM Problems/Problem 2

Problem

The circle inscribed in the triangle $ABC$ is tangent to the sides $BC$, $CA$, and $AB$ at the points $D$, $E$, and $F$, respectively. $AD$ cuts the circle at a second point $Q$. Show that the line $EQ$ passes through the midpoint of $AF$ if, and only if, $AC = BC$.

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

Solution

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See also

https://www.oma.org.ar/enunciados/ibe13.htm