Shortest Distance between Two Points is Straight Line

Theorem
The shortest distance between $2$ points is a straight line.

Proof
Let $s$ be the length of a curve between $2$ points $A$ and $B$.

The problem becomes one of finding the curve for which $\ds \int_a^B \rd s$ is a minimum.

Hence such a curve has the equation:
 * $y = m x + c$

which defines a straight line.