[ Triangle Inequality ]  [ Greatest or Least Length ]

Points $M$ and $N$ are located on the same side of line $l$. Using a compass and a straightedge, construct a point $K$ on line $l$ such that the sum $M K+N K$ is the smallest.
Consider the image of one of the given points under reflection about the line $l$ and apply the triangle inequality.

## Solution

Let $N_{1}$ be the point symmetric to $N$ with respect to the line $l$. Then for any point $K$ on the line $l$,

$$
M K+N K=M K+N_{1} K \geqslant M N_{1} .
$$

Equality is achieved when $K$ is the intersection point of the lines $l$ and $M N_{1}$.

From this statement, it follows that the tangent to the ellipse forms equal angles with the focal radii drawn to the point of tangency. (See D. Hilbert and S. Cohn-Vossen. Geometry and the Imagination. Chapter 1.)

![](https://cdn.mathpix.com/cropped/2024_05_06_cd0469c76668f3ff0930g-19.jpg?height=380&width=734&top_left_y=96&top_left_x=662)

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