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 ebaines Posts: 10,593, Reputation: 5799 Expert #2 Feb 6, 2012, 09:07 AM
Given an equilateral triangle with legs of length L and unknown angle $\theta$ between each leg and the triangle's base, you can set up an equation that correlates the area of the reactangle to the point measured along the leg where the rectangle's corner touches - call this distance x from the base. The base of the triangle has length $2L \cos \theta$. The height of the rectangle is then $x \sin \theta$, and its width is $2L \cos \theta - 2xcos \theta$. So the area of the rectangle is:

$
A = (2L\cos \theta - 2x \cos \theta)(x \sin \theta)
$

Take the derivative of A with respect to x, and set to zero to find the value for x that gives a maximum for A. When you do this - what happens to the $\theta$ terms?