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Minimal Sum with Cyclic Octagon

Source: 2012 Baltic Way, Problem 12

November 22, 2012
geometry unsolvedgeometry

Problem Statement

Let P0P_0, P1P_1, …\dots, P8=P0P_8 = P_0 be successive points on a circle and QQ be a point inside the polygon P0P1⋯P7P_0 P_1 \dotsb P_7 such that ∠Pi−1QPi=45∘\angle P_{i - 1} QP_i = 45^\circ for i=1i = 1, …\dots, 8. Prove that the sum ∑i=18Pi−1Pi2\sum_{i = 1}^8 P_{i - 1} P_i^2 is minimal if and only if QQ is the centre of the circle.
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