I have seen the standard variational proof that great circles are the geodesics on the $2$-sphere. Do you know a purely geometric proof of this fact, not involving calculus of variations or differential geometry?
It seems like it could be possible to provide a more elementary proof. If you disagree, please explain why.
Let us assume that existence of geodesic it trivial; yet assume that is it obvious that a geodesic can not have corners.
If you agree, then draw a great circle $\Gamma$ through two points of geodesic $\gamma$. If $\gamma\not\subset\Gamma$, reflect the part of $\gamma$ which lies on one side from $\Gamma$. You get a new geodesic, say $\gamma'$, and it has corners, a contradiction.