$M$ is the centre of the circumcentre of the cyclic quadrilateral $ABCD$. Let $AB \cap DC = {N}$ and $(N, B, C) \cap (N, A, D) = P$ ($P \not\equiv N$). Prove that $MP \perp NP$.

Notation: $(X, Y, Z)$ denotes the circumcircle of $\triangle XYZ$.

If you are wondering, this is adapted from a recent competition.

I have noticed that $AMCP$ and $BMDP$ is cyclic quadrilaterals. But I don't know if that is going to help with this problem.


This is part of a larger geometric diagram which is often referred to as the "big diagram". Proofs for many properties, including the one you just stated, are given here: http://yufeizhao.com/olympiad/cyclic_quad.pdf


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