the chain is from the yellow circles.

What I want is a way to draw the next yellow circle given all the ones before it:

[![enter image description here](https://i.stack.imgur.com/NywYE.jpg)](https://i.stack.imgur.com/NywYE.jpg)

I know how to draw the first circle $(P\_0Q\_0X)$.

I know the (nameless in the picture) contact point of the $n-th$ circle with the $n+1-th$ one lies on a circle centered at $H = P\_0Q\_0 \cap BC$ passing through $C$.

I know lines $P\_nQ\_n$ all meet in $H$.

I know quads $P\_nP\_{n+1}Q\_{n+1}Q\_n$ are cyclic.

I still can't find a simple way to construct the next circle given the previous one.I know there are one or two inversions that can do the trick but I would prefer if we avoid the temptation of looking for invertions.

I can't prove but I know the circles $P\_nQ\_nC$ are tangent to $BC$ at $C$

EDIT: also don't just use the general apollonius solution unless you can make sure you can show some symetry from this particular problem. For example: because we know the contact points between two circles lie on a circle centered at $H$ passing through $C$ we don't need the full $CCC$ but we can use $PCC$ (of course you guys are suposed to show more simplifications)