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.

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Hence such a curve has the equation:

$y = m x + c$

which defines a straight line.

Sources

• 1963: Charles Fox: An Introduction to the Calculus of Variations (2nd ed.) ... (next): Chapter $\text I$. The First Variation: $1.1$. Introduction