Subsets of the circle not contained in a semi-circle

Question

I'm reading a paper (Bullett and Sentenac, "Ordered orbits of the shift...", Ergodic Theory and Dynamical Systems), and have found that a proposition (Proposition 1) is (1) slightly incorrect (I have found a counterexample to the proposition that is stated); and (2) implicitly uses a lemma that is not stated in the paper.

The following is the statement of the implicit lemma:

Lemma Let $A$ be a subset of a circle $C$ such that

• $A$ is not contained in any closed semi-circle of $C$; and
• $A$ does not consist exactly of two pairs of diametrically opposite points.

Then there exist three points $a,b,c$ in $A$ that do not lie in any closed semi-circle of $C$ (i.e. there is a 3-point witness to the fact that $A$ does not lie in a semi-circle).

I have a proof, but it's less clean than I would like (with a couple of cases). I feel this should either be a consequence of a well known theorem, or there should be a very simple clean proof. Any suggestions?