I was trying to print out the largest possible equilateral triangle on a standard sheet of paper (8.5 by 11 inches) and got sidetracked into the following question: what is the maximum possible side length of an equilateral triangle to fit in a rectangle of size $l $ by $ w$ ($l \le w$), and how would that equilateral triangle be placed in the rectangle?
I found the case of a square easily, but wasn't able to find the answer for a rectangle. I tried placing one vertex in a corner and the other two vertices on sides, but that gives me a solution not even on the rectangle.
Edit: I believe that when the ratio between $l$ and $w$ is less than a certain value, then a vertex is on a corner. Otherwise, I think the triangle will be set up so a base is on a side of the rectangle.