In a few days I will be giving a talk to (smart) high-school students on a topic which includes a brief overview on the notions of curvature and of gedesic lines. As an example, I will discuss flight paths and, more in general, shortest length arcs on the 2-dimensional sphere.
Here is my question: let $p,q$ be distinct (and, for simplicity, non-antipodal) points on the sphere $S^2$. Is there an elementary proof of the fact that the unique shortest path between $p$ and $q$ is the shortest arc on the unique great circle containing $p$ and $q$?
By ''elementary'' I mean a (possibly not completely formal, but somehow convinving) argument that could be understood by a high-school student. For example, did ancient Greeks know that the geodesics of the sphere are great circles, and, if so, how did they ``prove'' this fact?
Here is the best explanation that came into my mind so far. Let $\gamma\colon [0,1]\to S^2$ be any path, and let $\gamma(t_0)=a$. The best planar approximation of $\gamma$ around $a$ is given by the orthogonal projection of $\gamma$ on the tangent space $T_a S^2$. This projection is a straight line (that is, a length-miminizing path on $T_a S^2$) only if $\gamma$ is supported in a great circle containing $a$. This sounds quite convincing, and with a bit of differential geometry it is not difficult to turn this argument into a proof. However, I was wondering if I could do better...