I'm trying to formally show that if two circles $\Gamma$ and $\Gamma'$ are tangent at $A$, then $\Gamma'$ lies either entirely inside or entirely outside $\Gamma$.
The one particular case I'm struggling with is when $\Gamma$ and $\Gamma'$ lie on the same side of the tangent line $l$ at $A$. I know that $O$, $O'$ and $A$ are collinear. I suppose the radius $r$ of $\Gamma'$ is smaller than radius $s$ of $\Gamma$, and I want to show that $\Gamma'$ is entirely inside $\Gamma$, except for $A$.
I know line $OO'$ meets $\Gamma'$ on the other side of $A$ at some point $C$. I've been able to show that $C$ is inside $\Gamma$. For if $C*O*O'$, then $CO<CO'=r<s$, so $C$ is inside $\Gamma$. If $O*C*O'$, then $CO<OO'<OA=s$, so again $C$ is inside $\Gamma$. But when I take an arbitrary point $B$ on $\Gamma'$, I have no idea how to show that $OB<s$, to prove that $B$ is inside $\Gamma$. Does anyone know a nice way to rigorously show this? Thank you.