# 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.

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.

Thus $$|MO -NO| \leq MN$$, with equality when ...
• Thank you for the solution! In the case where MN is parallel to $\alpha$, I said there is no maximum value for |MO−NO|, as the difference tends to be equal to MN as the point O heads more and more away from MN, in the direction of (MN, but never actually reaches that value. A much simpler solution than I anticipated! Commented Dec 15, 2019 at 13:34