TH-TMO 2018 Problem 1

In $\vartriangle ABC$ , the incircle is tangent to the sides $BC, CA, AB$ at $D, E, F$ respectively. Let $P$ and $Q$ be the midpoints of $DF$ and $DE$ respectively. Lines $P C$ and $DE$ intersect at $R$ , and lines $BQ$ and $DF$ intersect at $S$ . Prove that a) Points $B, C, P, Q$ lie on a circle. b) Points $P, Q, R, S$ lie on a circle.