2007 OIM Problems/Problem 2

Problem

Let $ABC$ be a triangle with incenter $I$ and $\Gamma$ be a circle with center $I$, of radius greater than that of the inscribed circle and that does not pass through any of the vertices. Let $X_1$ be the point of intersection of $\Gamma$ with the line $AB$ closest to $B$; $X_2$, $X_3$ the points of intersection of $\Gamma$ with the line $BC$ being $X_2$ closest to $B$; and $X_4$ the point of intersection of $\Gamma$ with the line $CA$ closest to $C$. Let $K$ be the point of intersection of the lines $X_1X_2$ and $X_3X_4$. Show that $AK$ intersects segment $X_2X_3$ at its midpoint.

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

Solution

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See also

OIM Problems and Solutions