Prove that any geodesic curve on $S^2$ is a great circle
I want to prove that a geodesic curve ($\gamma$ with $k_g=0$) is in fact a great circle. I know that any curve on a sphere has konstant normal curvature $k_n$. Futheremore we got the Definition of a great circle as the intersection of a plane that passes through the midpoint of the sphere
I already proved the $\tau=0$ and since $k^2=k_n^2+k_g^2 \;$ is constant $\gamma$ is a circle but I dont know how I show that $\gamma$ is a great circle.
Any help would be appreciated.