Question 7
In an acute triangle $ABC$ with $AB\lt AC$, the angle bisector from $A$ meets $BC$ at $P$. The line through $P$ parallel to $AB$ meets $AC$ at $Q$. On this line let $R$ lie on the ray from $Q$ not containing $P$, with $QR=QA$. Let $S$ be the perpendicular projection of $R$ onto $BC$, and let $T$ be the intersection of $AC$ with the line through $P$ perpendicular to $AP$.
a. Prove that $Q$ is the circumcenter of $APR$.
b. Prove that triangles $STC$ and $APC$ are similar.
a. Prove that $Q$ is the circumcenter of $APR$.
b. Prove that triangles $STC$ and $APC$ are similar.