How to construct a square equal to a given triangle.

Solution 1:

Given a rectangle with side lengths $a,b$ we want to construct a square of the same area. Without loss of generality assume $a>b$.

The first step is to obtain the square's side length $m=\sqrt{ab}$. This is a classical construction; erect a semicircle on a segment of length $a+b$ and draw the perpendicular from a point $a$ from one end. The length of that perpendicular within the semicircle is $m$.

The actual cuts to the rectangle that transform it into a square are based off a 2016 paper on illustrating the Bolyai–Gerwien theorem. From one vertex draw a cut to the long opposite side, creating a right triangle with legs $m$ and $b$. On the long side incident to said vertex, erect a perpendicular cut at the point $a-m$ from the vertex that stops at the first cut (this second cut has length $m-b$). This creates three pieces, which rearrange into a square, as shown above.

If $a>4b$, the above construction will not work. In this case, repeatedly bisect the rectangle parallel to its short side and stack the halves on top of each other until $a\le4b$.