
4. In a triangle $A B C$, points $D$ and $E$ are on segments $B C$ and $A C$ such that $B D=3 D C$ and $A E=4 E C$. Point $P$ is on line $E D$ such that $D$ is the midpoint of segment $E P$. Lines $A P$ and $B C$ intersect at point $S$. Find the ratio $B S / S D$.


Solution. Let $F$ denote the midpoint of the segment $A E$. Then it follows that $D F$ is parallel to $A P$. Therefore, in triangle $A S C$ we have $C D / S D=C F / F A=3 / 2$. But $D C=B D / 3=$ $(B S+S D) / 3$. Therefore $B S / S D=7 / 2$.
