Inscribe an Equilateral Triangle in a Square


Determine a construction to inscribe an equilateral triangle in a square as in the upper figure at the right.   Make a GSP construction.

Is this the inscribed equilateral triangle that has the maximum proportion of the area of the square?




That is, consider (construct) any other inscribed equilateral triangle. Will it cover more, the same, or less of the area? For example, see the second figure at the right. Make a GSP construction.



Prove that the first example shows the maximum area of an equilateral triangle inscribed in a square.