A plane $\alpha$ and two distinct points M and N outside of it are given. Find O∈ $\alpha$ such that $\mid MO-NO \mid$ is maximal.

Question

This is a problem that comes from a 9th grade math tournament.

We have the following picture:

The plane $\alpha$ and the distinct points M and N that do not lie on it are fixed. We want to find the point O on the plane $\alpha$ such that $\mid MO-NO \mid$ has maximum value, if such point exists. If not, we want to prove it doesn't exist.

I've tried solving a simpler variation of the problem in 2 dimensions where instead of a plane there is a line d and the points M, N & O are in the plane rather than in space, but I couldn't a solution for it either.

Another idea I had was to show that for any given point O a point P can be constructed for which $\mid MP-NP \mid$ > $\mid MO-NO\mid$, but I couldn't find any construction for such P.

Any idea is welcome. Thanks.

Answer 1

(Don't overthink it.)

Hint: Triangle inequality.

Thus $|MO -NO| \leq MN$, with equality when ...