Every triangle has a square which is the maximum size square which can be inscribed in the triangle. For most triangles there is only one way to do so. For an equilateral triangle there are three ways to inscribe the square - one for each side.
The equilateral is not the only triangle with that property; there is one other triangle whose maximum inscribed square can be placed on all three sides. Determine the dimensions of that triangle! (Assume the shortest edge is 1 unit.)
(In reply to A guess and a question
by Steve Herman)
I am specifically looking for three distinct ways in this puzzle, but I won't discount such a right triangle as an honorable mention. It would be a non-trivial way to have the square in two distinct positions.