ELMO SL 2012 G2

In triangle $ABC$ , $P$ is a point on altitude $AD$ . $Q,R$ are the feet of the perpendiculars from $P$ to $AB,AC$ , and $QP,RP$ meet $BC$ at $S$ and $T$ respectively. the circumcircles of $BQS$ and $CRT$ meet $QR$ at $X,Y$ .

a) Prove $SX,TY, AD$ are concurrent at a point $Z$ .

b) Prove $Z$ is on $QR$ iff $Z=H$ , where $H$ is the orthocenter of $ABC$ .

Ray Li.