MathNet 2006 — Part 05 Question 4

Cyclic quadrilateral $ABCD$ has side lengths $AB = 1$, $BC = 2$, $CD = 3$ and $DA = 4$. Points $P$ and $Q$ are the midpoints of $\overline{BC}$ and $\overline{DA}$. Compute $PQ^{2}$.