OMCC 2007 Problem 6

Consider a circle $S$ , and a point $P$ outside it. The tangent lines from $P$ meet $S$ at $A$ and $B$ , respectively. Let $M$ be the midpoint of $AB$ . The perpendicular bisector of $AM$ meets $S$ in a point $C$ lying inside the triangle $ABP$ . $AC$ intersects $PM$ at $G$ , and $PM$ meets $S$ in a point $D$ lying outside the triangle $ABP$ . If $BD$ is parallel to $AC$ , show that $G$ is the centroid of the triangle $ABP$ .

Arnoldo Aguilar (El Salvador)