2006 Canadian MO Problems/Problem 5

Problem

The vertices of a right triangle $ABC$ inscribed in a circle divide the circumference into three arcs. The right angle is at $A$ , so that the opposite arc $BC$ is a semicircle while arc $AB$ and arc $AC$ are supplementary. To each of the three arcs, we draw a tangent such that its point of tangency is the midpoint of that portion of the tangent intercepted by the extended lines $AB$ and $AC$ . More precisely, the point $D$ on arc $BC$ is the midpoint of the segment joining the points $D^\prime$ and $D^\prime^\prime$ (Error compiling LaTeX. Unknown error_msg) where the tangent at $D$ intersects the extended lines $AB$ and $AC$ . Similarly for $E$ on arc $AC$ and $F$ on arc $AB$ . Prove that triangle $DEF$ is equilateral.