Question

What is the number of different squares which can be inscribed in a given equilateral triangle?

Answer 1

If we lay one side along the x-axis, and one vertex at (0,0), and let the side be of length 1, the triangles sides can be written as

y=0
y=√3x
y=√3(1-x)

Assuming the obvious position where one side of the square lies along one side of the triangle, from (a,0) to (2a,0), and

√3x = a
√3(1-x) = √3 - 2a = a
which gives us a square of side 1/√3.
There are 3 sides to the triangle, giving us 3 identical squares.

Now, any other inscribed square must have a vertex on each side of the triangle, so if we pick any point (a,0) we must have lines

y = m(x-a)+b
y = (-1/m)(x-a)+c
where the sides from (a,0) to the other two sides of the triangle are of equal length.

So, set up your equations of the intersecting lines, and the distances, and fire away. I have no idea whether the squares will be of the same size as above. If not, then there are an infinite number of different squares possible.

I haven't yet come up with a solution using just geometry.