1. Consider a $1 \times 1$ grid of squares. Let $A, B, C, D$ be the vertices of this square, and let $E$ be the midpoint of segment $C D$. Furthermore, let $F$ be the point on segment $B C$ satisfying $B F=2 C F$, and let $P$ be the intersection of lines $A F$ and $B E$. Find $\frac{A P}{P F}$.
Answer: $\square$
Let line $B E$ hit line $D A$ at $Q$. It's clear that triangles $A Q P$ and $F B P$ are similar so
$$
\frac{A P}{P F}=\frac{A Q}{B F}=\frac{2 A D}{\frac{2}{3} B C}=3
$$