CR-MO 2018 G1

Let $O$ be the center of the circle circumscribed to $\vartriangle ABC$ , and let $P$ be any point on $BC$ ( $P \ne B$ and $P \ne C$ ). Suppose that the circle circumscribed to $\vartriangle BPO$ intersects $AB$ at $R$ ( $R \ne A$ and $R \ne B$ ) and that the circle circumscribed to $\vartriangle COP$ intersects $CA$ at point $Q$ ( $Q \ne C$ and $Q \ne A$ ).

1. Show that $\vartriangle PQR \sim \vartriangle ABC$ and that $O$ is orthocenter of $\vartriangle PQR$ .
2. Show that the circles circumscribed to the triangles $\vartriangle BPO$ , $\vartriangle COP$ , and $\vartriangle PQR$ all have the same radius.