I have a problem that even my smartest colleagues were able to solve. This is to get the radius of the smallest circle in the drawing below. Using a computer program, I managed to get that lightning is 0.25 cm, but I can not prove this with calculations. I could not solve with the Descartes Circle Theorem, because the three circles are not tangent to each other at all possibilities. Sorry for my imperfect English, because it is my second language. :-)
$\epsilon_1=-1, \epsilon_2=2$ and $\epsilon_3=\epsilon_4=\frac{1}{r}$
$2((-1^2)+2^2+\frac{1}{r^2}+\frac{1}{r^2})=(-1+2+\frac{1}{r}+\frac{1}{r})^2 \; \rightarrow 2(1+4+\frac{2}{r^2})=(1+\frac{2}{r})^2$
$2+8+\frac{4}{r^2}=1+\frac{4}{r}+\frac{4}{r^2} \; \rightarrow \; \frac{4}{r}=9 \; \rightarrow \; r=\frac{4}{9}= \mathbf {0.\bar{4}\neq 0.25}$
Many thanks to anyone who can help.
$E$ has coordinates $(s,r)$
$s^2+(0.5-r)^2=(0.5+r)^2$, so $s^2=2r$
$s^2+r^2=(1-r)^2$ so $2r+r^2=1-2r+r^2$, and $r=1/4$