Question: There are numbers which are square roots of negative numbers and they're are those which are logarithms of negative numbers(both are complex numbers).

Are there any numbers whose moduli are negative?
I mean, $|x|<0$

If yes, what are they?

If no, why?

Answer: 
Suppose $\omega\_1$ touches $K\_1K\_2$. 

Apply inversion with center at $B$ and $r=\sqrt{BK\_1\cdot BA\_1}$.

Then this inversion takes $K\_1$ to $A\_1$ and vice versa and thus it takes $\omega\_1$ to it self. Since line $K\_1K\_2$ is tangent to $\omega \_1$ it image, which is some circle through $B$ and $A\_1$ also touches $\omega \_1$. But then this is exactly $\Omega$. Now we know that $\Omega $ and $\omega \_2$ are tangent to each other, therefore there images are also tangent to each other. Say image of $\omega \_2$ is $\omega \_2'$ and we know that image of $\Omega$ is $K\_1K\_2$. Thus $K\_1K\_2$ touches $\omega \_2'$. 

Now since $A\_2$ is on $\Omega $ and on $A\_2K\_2$ it image is on image of this two, thus on $K\_1K\_2$ and $A\_2K\_2$, so the image of $A\_2$ is $K\_2$ (and vice versa) which means that $\omega \_2' = \omega \_2$. Q.E.D.
Suppose $\omega\_1$ touches $K\_1K\_2$. 

Homothety at $A\_1$ which takes $\omega \_1$ to $\Omega$ takes $K\_1$ to $B$ and $K\_1K\_2$ to (parallel) tangent $t$ to $\Omega$ at $B$. Homothety at $A\_2$ which takes $\Omega $ to $\omega \_2$ takes $B$ to $K\_2$ and $t$ to (parallel) tangent $\ell$ to $\omega \_2$ at $K\_2$. Since $\ell|| K\_1K\_2$ we have $\ell = K\_1K\_2$. Q.E.D.