Ancient Chinese method to calculating $\pi$

Question

I'm trying to understand the following passage from Boyer's and Merzbach's History of Mathematics:

The question I have is: how does the author derive that $w^2=2rv$?

Answer 1

Typing out Sam's solution from the comment section:

$w^2=v^2+\frac{s^2}4$, $v=r−u$, and $u^2=r^2−\frac{s^2}4$. Substituting $v$ and $\frac{s^2}4$ in the first equation from information in the latter two equations gives us $$w^2=(r−u)^2+r^2−u^2=2r^2−2ru=2(r−u)=2rv$$ so $w^2=2rv$.