Question 110
Let $O$ be the circumcenter of a triangle $ABC$. A circle $k$ is tangent to the lines $BC$, $CA$, $AB$ at points $D, E, F$, respectively, such that $A$ is on the other side of the line $BC$ with respect to the circle. Suppose the circle $k$ is equal to the circumcircle of the triangle (that is, has the same radius). Prove that lines $OD$ and $EF$ are perpendicular.