Question 192
Let $ABCD$ be a cyclic quadrilateral with $O$ is circumcenter and $AC$ meets $BD$ at $I$. Suppose that rays $DA$, $CD$ meet at $E$ and rays $BA$, $CD$ meet at $F$. The Gauss line of $ABCD$ meets $AB$, $BC$, $CD$, $DA$ at points $M$, $N$, $P$, $Q$ respectively. Prove that the circle of diameter $OI$ is tangent to two circles $(ENQ)$, $(FMP)$.