MathNet 2009-part-03 — 175 problems (EN–VI) Question 102

Let $A$, $B$, $C$ be nodes of the lattice $\mathbb{Z} \times \mathbb{Z}$ such that inside the triangle $ABC$ lies a unique node $P$ of the lattice. Denote $E := AP \cap BC$. Determine $\max \frac{AP}{PE}$, over all such configurations.