My question might be a bit odd, so I'll first put it in context. I'm a 17 student in high school, and local university organises, "for fun" (and to attract people to mathematics, etc.) thematic researches for students. With a group of friends, we are working on the idea of building a new theory of geometry with specifics rules. I won't tell you more about it, because I don't want to get spoiled :D
But it makes me wonder about "classic" plane geometry and do some researchs about fundamentals of Euclidean geometry. Especially about the idea of distance (between points). The most intuitive way to define distance between points is "the length of the shortest path between them" (let's call them A and B), with the shortest path happening to be the line segment AB.
I was wondering if the unicity of this shortest path is an axiom or something that can be proven. I guess if we imagine another shortest path as a combination of line segments, it can be proven to be longer than AB ad absurdum by considering them as triangles and using the fact that in a triangle sum of 2 lengths is always superior to the third one. But can we prove, for instance, that no curve is shorter than AB ?