Tangents and parallel lines

[ins-]

Let quadrilateral ABCD with intersection points of the diagonals P is inscribed in a circle k. M is the intersection point of the tangent to k through C and a line through P, parallel to BC. N is the intersection point of the tangent to k through D and a line through P, parallel to AD. Prove that CM=DN.

[Ilya Nek]

$NDPC$ and $DPCM$ is inscribed ($\angle ACD=\angle PND$ and $\angle BDC=\angle PMC$), so easy to prove that triangle $CDN$ and $CDM$ are equal.