Four houses are located at the vertices of a convex quadrilateral. Where should a well be dug so that the sum of the distances from it to the four houses is the smallest?

#
Suppose that the desired point does not lie on one of the diagonals and use the triangle inequality.

## Solution

Let the houses be located at the vertices of a convex quadrilateral $A B C D$, the diagonals $A C$ and $B D$ of which intersect at point $O$. Suppose that the well is located at some point $M$, different from $O$. Then

$$
A M + C M \geqslant A C, \quad B M + D M \geqslant B D
$$

and at least one of these inequalities is strict. Therefore,

$$
A M + C M + B M + D M > A C + B D = A O + C O + B O + D O,
$$

i.e., the sum of the distances to the vertices of the quadrilateral $A B C D$ from point $O$ is less than from any other point.

## Answer

At the intersection of the diagonals of the quadrilateral.