What is the length the side of of a square inscribed in a triangle? 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.
