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a triangle area game


BMAD
Go to solution Solved by gavinksong,

Question

Given a triangle ABC with the unit area. The first player chooses a point X on the side [AB], than the second -- Y on [bC] side, and, finally, the first chooses a point Z on [AC] side. The first tries to obtain the greatest possible area of the XYZ triangle, the second -- the smallest. 
 
What area can the first obtain for sure and how?
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Actually, since the problem is asking what area the first can obtain for sure, suggesting that the answer should be independent of what the second player picks, my answer must actually be correct.

 

If the first player chooses X and Z, such that it forms a line parallel to BC, then the resulting area of XYZ is independent of where the second player places Y. The area of XYZ can be expressed as

(1-r)*r where r the scale factor between the similar triangles AXZ and ABC. This is maximized when r = 1/2, which leaves us with a guaranteed area of 1/4.

 

We know that this is the maximum guaranteed area. If the line XZ is not parallel to BC, then the second player may choose the endpoint closest to the line XZ, minimizing the "height" of the triangle XYZ with respect to the base XZ. The parallel line passing through this point Y crosses the triangle ABC, forming two smaller triangles. Since the line XZ is parallel to the base of one of these triangles, the reasoning in the previous paragraph applies to this XYZ with respect to one of these smaller triangles. However, since the outer triangle in question is smaller than the original, the maximum area of XYZ is strictly smaller than 1/4.

Edited by gavinksong
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If the line XZ is parallel to the BC side, then no matter what the second player picks, the area will be half the length of XZ times the distance between XZ and BC. This area is maximized and equal to 1/4 when X and Z are midpoints.

Edited by gavinksong
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If the line XZ is parallel to the BC side, then no matter what the second player picks, the area will be half the length of XZ times the distance between XZ and BC. This area is maximized and equal to 1/4 when X and Z are midpoints.

would player 2 pick a midpoint?

Edited by BMAD
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If the line XZ is parallel to the BC side, then no matter what the second player picks, the area will be half the length of XZ times the distance between XZ and BC. This area is maximized and equal to 1/4 when X and Z are midpoints.

would player 2 pick a midpoint?

 

 

My answer is independent of what the second player picks but...

 

If you don't follow my strategy, a smart second player will always pick an endpoint.

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My answer is independent of what the second player picks but...

If you don't follow my strategy, a smart second player will always pick an endpoint.

 

 

Ack. I've made a mistake. :/

To simplify the problem, I assumed that the first player essentially chooses a line, and then the second player chooses a point. I forgot that the second player does not know which point the first player will choose last.

Still, at least we know that 1/4 is a lower bound.

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