An easy problem

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From the point $C$ to the circle $k$ are drawn the tangents $CA$ and $CB$. On the arc $AB$ internal to the triangle $ABC$ is chosen a point $D$. The line $CD$ intersects again $k$ at the point $E$. Through $D$ is drawn a line parallel to $AC$, that intersects $AB$ at the point $F$. The line $EF$ intersects $AC$ at the point $M$. Prove that $M$ is the middle of the segment $AC$.

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Don't loose your time. It follows from: topic43754.html