Cut The Kont offers a proof of the Pythagorean Theorem based on a converging geometric series of similar right triangles. The second image on that page (linked) is the most relevant for this question.
It is already know that altitude $h=\frac{ab}{c}$ and that $b_{1}=b\frac{a^{2}}{c^{2}}$. To finish the proof as per this example, I would have to get $\frac{b}{b_{1}}=\frac{b^{2}}{c^{2}}$. Then, by setting $x=\frac{b^{2}}{c^{2}} < 1$ the formula for a geometric series can be applied so that:
$\displaystyle \begin{align} b=b_1+b_2+b_3+\ldots b&=b_1(1+x+x^2+\ldots)\\ &=b\frac{a^2}{c^2}\cdot\frac{1}{1-b^2/c^2}\\ &=b\frac{a^2}{c^2-b^2} \end{align}$
and $\frac{a^{2}}{c^{2}-b^{2}}=1$
My only issue with this proof is that I cannot get $\frac{b}{b_{1}}=\frac{b^{2}}{c^{2}}$. I have been able to get $\frac{b}{b_{1}}=\frac{c^{2}}{b^{2}}$, which does not work with the aforementioned method. Using the image provided, how, if possible, can this be done?