How can I find the maximum area possible of equilateral triangle that inside a square whose sides have length a.
And how does that triangle look like? Can we construct it (with compass and straightedge)?
The last paragraph at the Mathworld piece on equilateral triangles gives the answer, and cites Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 115 and 129-131, 1979.