3 points in a unit square

Discrete Mathematics Level pending

You and a friend decide to play the following game...

You draw three distinct dots inside a unit square.

Then your friend draws the biggest rectangle (with edges parallel to those of the square) she can which contains exactly one of the three dots.

With the initial three dots, you try to minimize the size of the rectangle she is able to draw.

If you play optimally, what is the area of the largest rectangle she will be able to draw containing exactly one of the three dots?

Give your answer to 3 decimal places.

Assumption: The dots are allowed to share the same \(x\) or \(y\) coordinate, but not both.

Bonus question: What about if you start with five dots?

Inspired by this problem by Sharky Kesa! :0)

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