Question 8
Let $P$ be a point lying inside triangle $ABC$. Let $Q$ be a point on the segment $AB$, and let $R$ be a point on the segment $AC$ such that both circles ($BPQ$) and ($CPR$) are tangent to line $AP$. Through $B$ and $C$ we draw the lines passing through the center of the circle ($BPC$), and through $Q$ and $R$ we draw the lines passing through the center of the circle ($PQR$). Prove that there exists a circle tangent to the four drawn lines.