Shortest Distance in Euclidean Geometry: Proven or Definition?

[aaaa202]

In euclidean geometry I believe the shortest distance is the perpendicular one. Can this be proven or is it a definition?

 

[Whovian]

Depends on your definition of a line. If it's a function with a constant slope, it can be proven with calculus of variations, I think. I don't think in Euclidian Geometry a line's rigorously defined.

EDIT: Never mind, I though you were asking if it could be proven that the shortest distance between two points is a line. See scurty's answer for a better reply

 

[scurty]

In euclidean geometry I believe the shortest distance is the perpendicular one. Can this be proven or is it a definition?

Not enough clarification here. Do you mean the shortest distance from a point not on a line to the line itself? I imagine it can be solved by considering the perpendicular line and another line emanating from the point. You will have a right triangle and the other line not perpendicular is the hypotenuse of the triangle, so therefore longer.

 

[HallsofIvy]

The length of the hypotenuse, c, of a right triangle, with legs a and b, satisfies $c^2= a^2+ b^2$ and so the hypotenuse is loner than either leg. Do you see why that means that the perpendicular line (one of the legs) is shorte than any othe line? (Ahh- that's essentially what scurty said.)