Calculate largest rectangle in a rotated rectangle

Question

I\'m trying to find the best way to calculate the biggest (in area) rectangle which can be contained inside a rotated rectangle.

Some pictures should help (I hope) i

Answer 1

Trying not to break tradition putting the solution of the problem as a picture:)

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Edit: Third equations is wrong. The correct one is:

3.w * cos(α) * X + w * sin(α) * Y - w * w * sin(α) * cos(α) - w * h = 0

To solve the system of linear equations you can use Cramer rule, or Gauss method.