Prove that the area of a triangle DEF is correct.

Question

There's any triangle ABC. First player 1 has to set D on AB so that in the end the triangle DEF has the highest possible area. Second player 2 has to set E on BC so that in the end the triangle has the lowest possible area. And third player 1 has to set F on CA so that in the end the triangle has the highest possible area. Question: What is the area of DEF? Task 2: Prove that the answer for the question is correct. I think the area of DEF is one quarter of the area of ABC but I have not found a way to mathematically prove that this answer is correct.