Consider a straight river of width $L$ and two points $A$ and $B$ on either side of this river. We want to build a bridge perpendicular to the river. Where should we build it to minimize the travel distance from $A$ to $B$?
![](https://cdn.mathpix.com/cropped/2024_05_10_699beca3fc1a5ab13db9g-53.jpg?height=1286&width=938&top_left_y=282&top_left_x=474)
Let $\vec{v}$ be the vector perpendicular to the river, directed towards $A$ and of length $L$, and $B^{\prime}$ the image of $B$ by the translation of vector $\vec{v}$: if the bridge goes from $P$ to $Q$, the length of the path is $A P+P Q+Q B=A P+P B^{\prime}+L$. Therefore, $A P+P B^{\prime}$ must be as small as possible, so $P$ must be aligned with $A$ and $B^{\prime}$. $P$ is thus the intersection of $\left(A B^{\prime}\right)$ and the riverbank on the side of $A$.