Minimal path between two points via a line
There is a unique shortest path from to reflecting in the line . It is seen in the diagram that the angles of incidence and reflection ( and ) are equal for this shortest path. The variational priciple here gives the well-known law for the reflection of light (and billiard balls!).
It was known to the Greek mathematician Heron of the first Century AD and it appears as a problem in his book Catoptrica.