I have the following triange construction: The values for $\alpha$, $d$, $a$ and $b$ are known:
I am trying to calculate the angle $\phi$, depending only on $\alpha$, $d$, $a$ and $b$. To archive this, I need to calculate $x$ first. I think, for given values $\alpha$, $d$, $a$ and $b$, there exists a unique solution for $x$. Is this correct?
I tried different approaches, for example the Law of cosines: $$\cos(\phi) = \frac{(a + x)^2 + (b+x)^2 - (d + a + b)^2}{2 (a + x)(b + x)}$$
or calculating the height $h$: $$h = \sin(\alpha) (a + x)$$ to later use the Pythagorean theorem.
Can you give me a hint, how to calculate $x$, such that I can calculate $\phi$ later? Thanks a lot!
HINT
By law of cosine we have
$$(b+x)^2=(a+x)^2+(d+a+b)^2-2(a+x)(d+a+b)\cos \alpha$$