In "The penguin dictionary of curious and interesting geometry" David Wells mentions the following property of closed convex curves without a reference nor a proof.
"Given any closed convex curve, it is possible to find a point P, and three chords inclined at $\pi/3$, such that P is the mid-point of all three".
Can anyone redirect me to some literature or provide an argument for such a property?
Apparently this is a special case of a theorem by Steinhaus: "Given any closed convex planar curve, there always exist three concurrent chords bisecting each other and cutting at prescribed angle". The reference should be "On chords of convex curves", H. Steinhaus - Bull. Acad. Polon. Sci. Cl. III, 1957, which is apparently very hard to find online. Fun fact: the analogous (and more general) problem for simple closed curves has been conjectured but it is still supposedly open.