Here's a sketch. If the quadrilateral is $ABCD$, then let $E=AB\cap CD, F = AD\cap BC, G = AC\cap BD$ be the diagonal points. Then let $EG$ intersect $AB,CD$ at $PR$ resp. and $FG$ intersect $AD,BC$ at $S,Q$ resp. Using Brianchon we can show that there is an inellipse tangent to $ABCD$ with intouch points $P,Q,R,S$. Now, note that affine transforms preserve homotheties, but we can affine transform the inellipse into a incircle, then it's a well-known config to show that $ABCD$ is cyclic, so we're done.