If $a=b$, then clearly $P$ is the midpoint of a chord, so we're done. So suppose that $a\ne b$. 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 $\mathrm{QR}$ 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 ${}^{\circ }$ round, $\mathrm{QP}$ will have length $b$, and $\mathrm{PR}$ will have length $a$, in other words, the segments will have switched. So now $|\mathrm{QP}|>|\mathrm{PR}|$, when they started the other way round. But as we turn the chord, the lengths of the segments change continuously, so to switch from $\mathrm{QP}$ being shorter to $\mathrm{QP}$ being longer, we must have had $|\mathrm{QP}|=|\mathrm{QR}|$ at some point. But then $P$ will be the midpoint of this chord.