Complete quadrilateral and a middle

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A quadrilateral $ABCD$ is inscribed in a circle $k$. $E$ is the intersection point of the lines $AB$ и $CD$ and $F$ is the intersection point of the lines $AD$ and $BC$. If $P$ is the second intersection point of the circumference of the triangle $CEF$ and $k$, and $M$ is the intersection point of the line $AP$ and the segment $EF$ prove that $M$ is the middle of $EF$. ($A$ is in a different half-plane with respect of the diagonal $BD$ from $E$ and $F$)

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You can see some solutions here:
http://www.artofproblemsolving.com/Foru ... 8&t=403357
Hope you like it.
Is it a well known statement?