US-EGMO-TST 2020 Problem 2

Let $ABC$ be a triangle and let $P$ be a point not lying on any of the three lines $AB$ , $BC$ , or $CA$ . Distinct points $D$ , $E$ , and $F$ lie on lines $BC$ , $AC$ , and $AB$ , respectively, such that $\overline{DE}\parallel \overline{CP}$ and $\overline{DF}\parallel \overline{BP}$ . Show that there exists a point $Q$ on the circumcircle of $\triangle AEF$ such that $\triangle BAQ$ is similar to $\triangle PAC$ .

Andrew Gu