BALTIC-WAY 2012 Problem 12

Let $P_{0}, P_{1}, \ldots, P_{8}=P_{0}$ be successive points on a circle and $Q$ be a point inside the polygon $P_{0} P_{1} \ldots P_{7}$ such that $\angle P_{i-1} Q P_{i}=45^{\circ}$ for $i=1, \ldots, 8$. Prove that the sum

$$\sum_{i=1}^{8} P_{i-1} P_{i}{ }^{2}$$

is minimal if and only if $Q$ is the centre of the circle.