John the shortest distance between two points on a flat surface is a straight line.
Show this is the case, presuming you mean any flat surface, and I'll gladly admit I'm wrong. Also, why it is relevant.
The triangle inequality, a <= b+c is true in flat space.
Consider the straight line AB and some curve c that goes from A to B and isn't equal to AB.
A set of C points will be used to approximate c.
Step 1
Pick a point C1 on c, the path A,C1, B is clearly longer than AB by the triangle inequality.
Step n
Pick another point Cn on the path c, then find the points Cr and Cs that are before and after Cn on the curve c.
The path Cr,Cn,Cs is longer or equal to the path Cr,Cs by the triangle inequality. So the approximation of c given by the n C points is still longer than AB.
So by induction any approximation of a curve from A to B by a finite number of points is longer than AB.
This can be extended to the length of the curve itself.
Consider the sequence T. The value of Tn is the length of a finite approximation of c by a set of points where no two successive points are more than 1/n distance apart. Also the approximation used for Tn+1 must contain all the points used for Tn.
The limit of the sequence will be greater than AB, and will be equal to the length of c.
Therefore the straight line AB is the shortest path between A and B.