TH-TMO 2021 Problem 8

Let $P$ be a point inside an acute triangle $ABC$ . Let the lines $BP$ and $CP$ intersect the sides $AC$ and $AB$ at $D$ and $E$ , respectively. Let the circles with diameters $BD$ and $CE$ intersect at points $S$ and $T$ . Prove that if the points $A$ , $S$ , and $T$ are colinear, then $P$ lies on a median of $\triangle ABC$ .