For a generilized triangle on a manifold, (distance can be regarded as geodesic length)it is well known that for Eucilidean Geometry，the following is true:

Consider a triangle $ABC$, $D$ is the midpoint of $BC$, then $AD\leq \frac{1}{2}(AB+AC)$

what about some other cases in manifolds. according to my knowledge, it is also true in spheres with dimension n. However I did not konw general cases.

Any advice will be appreaciated.

shortestpath between them? I ask because two points can be connected by a geodesic that is not shortest, and one can define triangles formed by such geodesics. – Joseph O'Rourke Feb 22 '12 at 17:06