VTMO 2017 — Grade 9 — final Question 4

An isosceles trapezoid $ABCD$ with $AB \parallel CD$ and $AB \lt CD$ is cyclic with center $O$. Let $K = AC \cap BD$; $M , N , P , Q$ are circumcenters of $KAB , KCD , KAD , KBC$. Prove $MQNP$ is a parallelogram and $KM = ON$.