Four circles' intersections points cocircular?

Question

In the configuration below, four circles $C_i$, $i=1,2,3,4$, are tangent as shown, and each tangent to a surrounding circle $C_0$.

Q. Are the four circle intersections shown cocircular?

     

Answer 1

Inversion of this picture centered at the touch point between circles $1$ and $3$ transforms this picture into two parallel lines with three equal size circles in between them, touching both lines and each other (in the order $2$, $0$, $4$). Clearly in that picture these four points are on a circle (they are the points where the two outer circles touch the parallel lines, so they form a rectangle). Then they are in the original picture as well.

It is not relevant that circles $2$ and $4$ also touch some circle $0$. It is sufficient that they touch both $1$ an $3$