What is the length of the side of a square inscribed in a triangle?
This was inspired by this Numberphile video which showed multiple ways to construct the square with a side on one side of an acute triangle and the other two corners touching the other two sides of the triangle: https://www.youtube.com/watch?v=9ptyprXFPX0
My question is this:
Given an acute triangle and choosing a particular side, what is the length of a square with one side on one side of an acute triangle and the other two corners touching the other two sides of the triangle?
Here is my answer for a particular description of the triangle. I am interested in other ways of looking at this.
If the triangle has a base of length $c$ and the surrounding angles have tangents of $a$ and $b$, then the side of the inscribed square with a side on that base is $\dfrac{cab}{ab+a+b} $.
Suppose a square with side $s$ lies on $AB$, intersect $AC$ and $BC$ at $P$ and $Q$ respectively. Notice that triangles $CPQ$ and $CAB$ are similar. We can use similarity:
$$ \frac{h_{C}-s}{h_{C}}=\frac{s}{c} $$
where $h_{C}$ is altitude through $C$.
Of course it depends on what is known and what is not.