if four circles touch each other externally then points of contact are concyclic

Question

Question -

if four circles S1,S2,S3,S4 touch each other externally then points of contact A,B,C,D are concyclic...

Figure -

My proof -

First I invert about A and I get two parallel lines S1' and S2' and in between two circles S3' and S4' touching one another at C' and touching corresponding lines at D' and B' ....

now because line S1' and S2' are parallel and tangent to corresponding circles so D'B' must have to be transversal and therefore D'C'B' are COLLINEAR and inverting back we see that ABCD are concyclic ....

Is my proof correct ?

Answer 1

Introducing $S_1'$ and $S_2'$ was a good move. But the second figure should look more like the following:

You should state that the two circles are in a similar (or homothetic) position with respect to their touching point. This will then imply that the red line is actually going through the three special points.

Answer 2

Elementary solution will be connecting the points to their circle center to obtain 4 isosceles triangle. Checking the angle, opposite angle sums up to $180$ so they are concyclic.