Euclid proves in III.2 that circles are convex: that is, that every point on a chord is inside the circle.
To better understand his proof, I reproved it below, to which I request verification and critique.
Lemma: In triangle $\triangle OAB$, with point $P$ on segment $AB$, cevian $OP$ is less than $\max(OA, OB)$.
Lemma Proof: Since $\angle OPB$ is an exterior angle to $\triangle OPA$, then $\angle OPB > \angle A$ since an exterior angle is greater than an opposite interior angle (I.16). Assuming WLOG that $\angle A \geq \angle B$, we have $\angle OPB > \angle A \geq \angle B$ and therefore $OB > OP$ since in any triangle, the greater side is opposite the greater angle (I.19).
Main Proof: Consider circle $\bigcirc O$ with center $O$ and chord $AB$. No point $P$ on segment $AB$ can be external to $\bigcirc O$, since by the lemma, $OP$ is less than the radius of the circle $OA$.