New, extremely simple golden ratio construction with two identical circles and line. Is there any prior art?

Slightly more general answer.

Let $R$ and $B$ be the lengths of the red and blue lines respectively. If the radius of the circles is $r$, then we have the equations $$R=2r$$ since $R$ is the diameter of one of the circles, and $$B+r=\sqrt{r^2+(2r)^2}=r\sqrt5$$ since $B+r$ is the hypotenuse of a right triangle with legs of length $r$ and $2r$.

Hence $$\frac{R}{B}=\frac{2r}{r\sqrt5-r}=\frac{2}{\sqrt5-1}=\frac{2\left(\sqrt{5}+1\right)}{\left(\sqrt{5}-1\right)\left(\sqrt{5}+1\right)}=\frac{2\left(\sqrt{5}+1\right)}{4}=\frac{\sqrt{5}+1}{2}=\varphi$$

Here's the proof: $$6^2+3^2=36+9=45$$ $$\frac{6}{\sqrt{45}-3}=\frac{2}{\sqrt{5}-1}$$ This was simple enough to just write down. For the last step we have: $$\frac{\sqrt{45}}{3}=\sqrt{x}$$ $$\frac{45}{9}=5$$ NB That was relatively simple so I can't claim any credit, the OP had the idea which may be original.