Amir sent us this solution:
Draw any chord passing through P, and call its endpoints Q and
R. Let a be the length of the line segment PQ, and b the
length of PR.
If a=b, then clearly P is the midpoint of a chord, so we're
done. So suppose that a\neq 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^{\circ} 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.