We’ve been given a triangle A B C with an area = 1. Now Marcus gets to choose a point

Bruno Pittman 2022-07-01 Answered
We’ve been given a triangle A B C with an area = 1. Now Marcus gets to choose a point X on the line B C, afterwards Ashley gets to choose a point Y on line A C and finally Marcus gets to choose a point Z on line AB. They can choose every point on their given line (Marcus: B C; Ashley: C A; Marcus: A B) except of A, B or C. Marcus tries to maximize the area of the new triangle X Y Z while Ashley wants to minimize the area of the new triangle. What is the final area of the triangle X Y Z if both people choose in the best possible way?
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Answers (1)

enfeinadag0
Answered 2022-07-02 Author has 16 answers
Your guess in the comment is (almost) correct. The first two points are the midpoints, and the last move makes no difference; the resulting area is 1 4 .

To prove this, consider Ashley's move. If her move is further away from A B than the first move, Marcus will make the last move arbitrarily close to A, whereas if it's closer to A B than the first move, Marcus will make the last move arbitrarily close to B. In either case, Ashley would have been better off making her move at the same distance from A B as the first move, so this is what she does. Then the last move makes no difference, and the area is given by
1 2 λ h ( 1 λ ) c
in terms of the length c of A B, the height h above A B and the fraction λ ( 0 , 1 ) at which the first move is placed along B C. The maximum is at λ = 1 2 ; the corresponding area is 1 8 h c, and this is 1 4 of the area 1 2 h c of triangle A B C.
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