<s>[INST] Solve triangle point given base, point height and difference of sides I have the intuition that one should be able to calculate the position of the circle in the image below (or the equivalent, solve a and b).
We have the following information:
h and d is known as well as the difference between the lengths a and b.
Looking at the sine law:
$$ \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$
and cosine law:
$$c^2 = a^2 + b^2 - 2ab\cos(\gamma)\,$$
I don't see how we have information to use those formulas.
How do I solve the position of the dot?

 [/INST] Let $a=x,b-a=t$. Here, note that $b=x+t$ and that $b\lt a+d\Rightarrow t\lt d$.
Then, by the Pythagorean theorem for the biggest triangle, we have
$$\begin{align}&(x+t)^2=h^2+\left(\sqrt{x^2-h^2}+d\right)^2\\&\Rightarrow x^2+2tx+t^2=h^2+x^2-h^2+2d\sqrt{x^2-h^2}+d^2\\&\Rightarrow 2tx+t^2-d^2=2d\sqrt{x^2-h^2}\\&\Rightarrow (2tx+t^2-d^2)^2=4d^2(x^2-h^2)\\&\Rightarrow (4d^2-4t^2)x^2+(-4t^3+4td^2)x-4d^2h^2-t^4-d^4+2t^2d^2=0\\&\Rightarrow a=x=\frac{t^3-d^2t+d\sqrt{(d^2-t^2)(d^2+4h^2-t^2)}}{2(d^2-t^2)}\end{align}$$
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