Here is the problem:
We start with a triangle ABC with area 1. We choose a point (F) on side AB, then someone else chooses a point (G) on side BC. We then choose the last point (H) on side CA. Our goal is to mximize the area of the triangle FGH. What is the maximum area we can get no matter where the other person puts his point?
I am stuck with this problem. I think that the best place for point F is the midpoint of AB, but I'm not sure how to prove it or what to do next. I appreciate any help you can give me.