The shortest distance between any two distinct points is the line segment joining them.How can I see why this is true? On a euclidean plane, the shortest distance between any two distinct points is the line segment joining them. How can I see why this is true? 

Let $\gamma(s)$ be a continuous curve in the plane with end-points $\gamma(0) = a$ and $\gamma(1) = b$. Using the Euler-Lagrange equations, the only stationary solution is $\gamma(s) = bs + (1 - s)a$, which is a line connecting the two end-points. See also this.
