The following problem has been given to me and I have not been able to solve it at first glance completely (only that there is a unique solution).
Given two points A and B located on either side of a line MN determine with the ruler and compass a point of MN whose difference of the distances to points A and B is maximum.
Suppose $A,B$ is not symmetric, otherwise the problem is meaningless. And when the distances of $A,B$ to the line $MN$ is the same, the maximum does not exist since the difference of distances will keep growing as the chosen point tends to infinity. Now we rule out those two cases. Reflect the point $B$ along the line $MN$ to get a point $B'$(this is possible with ruler and compass). Suppose the line $B'A$ intersect with $MN$ at $K$. Then $|BK-AK|$ attains its maximum.