So this may be a question that is born of my inability to properly express my intentions to Google.

In a fully connected non-directional graph such that any three points can be properly represented in a triangle in Euclidean geometry it would seem that a greedy first algorithm which chooses the shortest edge would find the shortest path from a starting point to visit all other points.

I can't find a condition under which this is false, but one of my neighbors (another programmer) insists that I am wrong, though still cannot come up with a condition where I am wrong.