If a=b, then clearly P is the midpoint of a chord, so we're done. So suppose that a ¹ b. We may as well assume that a < b (otherwise just switch round Q and R). Imagine that the shape is made out of a metal frame, and that the chord QR is made from elastic, just looped round the frame at Q and at R, but fixed at P (so that it can rotate). Rotate the chord around P, and the elastic will stretch so that the line is always a chord of the shape. When it's gone 180^° round, QP will have length b, and PR will have length a, in other words, the segments will have switched. So now |QP| > |PR|, when they started the other way round. But as we turn the chord, the lengths of the segments change continuously, so to switch from QP being shorter to QP being longer, we must have had |QP|=|QR| at some point. But then P will be the midpoint of this chord.