VTMO 2026 - Grade 9 - Semifinal - SET 08 - Self-study Question 4

Let the isosceles trapezoid $ABCD$, with $AB\parallel CD$ and $AB\lt CD$, be inscribed in a circle with centre $O$. Let $K$ be the intersection of its diagonals. Let $M,N$ be the circumcentres of triangles $KAB,KCD$, respectively, and let $P,Q$ be the circumcentres of $KAD,KBC$, respectively.

a) Prove that $MQNP$ is a parallelogram.

b) Prove that $KM=ON$.