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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)?

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It must be the equilateral triangle with one vertex at a corner of the square, and the other two vertices on the sides of the square not adjacent to that corner. It should be easy to compute all the lengths and angles of the resulting figure, and then to decide whether they are compass-and-straightedge constructible. – Gerry Myerson Aug 25 '11 at 6:13
How to prove that? – DKhanh Aug 25 '11 at 6:25

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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.

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