Let $C$ be the trace of a simple closed curve in a regular surface $S$. Let $p\in S-C$. Assume that $p$ is close enough to $C$ that a normal ball about $p$ intersects $C$. Let $q\in C$ be the point of $C$ closest to $p$. Then a minimizing geodesic from $p$ to $q$ must intersect $C$ orthogonally.
The statement is intuitively clear but I don't know how to prove it. Should I parameterize $C$? How can I show the orthogonality?