MathNet 2000 — Math Olympiad Vault Question 12

Equilateral triangle $A B C$ is inscribed in a circle. Let $D$ be the midpoint of $A B$, and let $E$ be the midpoint of $A C$. The ray $\overrightarrow{D E}$ meets the circle at $P$. Prove that $D E^{2} = D P \cdot P E$.