Pons Asinorum solution: why is this so hard? Background
You will recognize the image below as a variation on Euclid's famous Pons Asinorum (Donkey's bridge) problem. I ran across it on a friends facebook page except that specific values had been given for x and y. I could not solve it on my own, and I ended up looking up a solution online (there are several). I realized that the solutions I was finding were constructing equilateral triangles by knowing x and y and I would never get there because I was seeking a more general solution to the problem as shown below.
This lead me to two Questions:
1) How to express the angle α in terms of x and y when x and y are not known in advance?
2) I made a number of attempts at forming a system of equations from the rules for similar triangles, and I always ended up with inadequate systems (that is ones that could not be reduced to a single solution) What is it about the nature of this problem that so resists an algebraic solution using similar triangles?


Recall first of all that in a triangle with base $a$ and base angles $\beta$, $\gamma$, the altitude can be expressed as $a/(\cot\beta+\cot\gamma)$. This can be easily proved by the sine rule, for instance.
Consider now our triangle below and set $AB=l$. By the above rule we get then
$$
DH={l\over\cot(80°)+\cot(80°-x)}={l\over\tan(10°)+\tan(10°+x)},
\quad
EG={l\over\tan(10°)+\tan(10°+y)}.
$$
In addition we have
$$
EF=GH=l-EG\tan10°-DH\tan10°,
$$
and from triangle $DEF$ one gets, after some algebra:
$$
\tan(\alpha+10°+x)={EF\over DH-EG}={\tan^2 10°-\tan(10°+x)\tan(10°+y)\over\tan(10°+x)-\tan(10°+y)}.
$$

