How many circles pass through 2 points but also tangent to a given circle? Given: a circle $O$. and $2$ points $A, B$ out of that circle.
How many new circles that are tangent to circle $O$ can we form which also pass through points $A,B$.
My Geometry is a somewhat rusty - I find hard time solving this because I am missing the new circle as there could be virtually infinite circles (with different radii and centers)  that pass through 3 points ($A$, $B$, another point on circle $O$) - how do I know how many of them are tangent? My guess is only $2$ but I can't show it. I understand that this forum demands one to show effort, but really I have nothing to write here. Any help would be appreciated.
edit: I understand that in some settings the answer is $0$. so let's add a new condition that the points $A,B$ are both below the circle $O$. so the line between them does not reach the given circle.
 A: 
Given a circle with radius $CE$ and two points $A$, $B$ outside the circle.
Join $AB$ and set up the perpendicular bisector at $D$.
On this line lie the centers of all circles passing through $A$, $B$,
and from each of these points lines can be drawn: to $A$, and through center $C$ to the opposite side of the given circle (or to center $C$ through the near side).
Let $F$ be the (unique) point on the perpendicular bisector such that$$FA=FCE$$The circle drawn with center $F$ and radius $FA$ is internally tangent to the given circle, since common point $E$ lies on the line through their centers but not between them.
Next let $G$ be the (unique) point equidistant from $A$ and point $H$ on the near side of the given circle. The circle with center $G$ and radius $GA$ is externally tangent to the given circle since now $H$ lies on the line through their centers and between them.
I have not shown how to construct points $F$ and $G$, but if it's clear that these points on the perpendicular bisector exist, then two and only two circles pass through points $A$, $B$ tangent to the given circle, one internally and one externally.
A: Draw a perpendicular bisector to line AB. It cuts the given circle at two points say T1 and T2 (not labeled that way in the sketch).
Then there are two circles, ABT1 sand ABT2 one of which makes external contact and the other makes internal contact. That is, the given circle can be placed once outside and once inside.
A straight line can be taken to be circle as special case if points A,B,T1 or A,B,T2 are collinear.


