Externally Tangent Circles Given two distinct points $A, B$, what is the set of possible tangency points between externally tangent circles $o_1; o_2$ such that $o_1$ is tangent to the line $AB$ at $A$, and $o_2$ is tangent to the line $AB$ at $B$.
Any hints on how to do this?
I think the picture look like this

Thanks

Hints: Start by showing that the line from the midpoint of $AB$ to the point of tangency is perpendicular to $O_1 O_2$.  
Then argue that the distance from the point of tangency to $AB$ is $\frac{1}{2} |AB|$, and hence the locus is a circle centered at the midpoint of $AB$ with radius $\frac{1}{2}|AB|$.
Illustrated here:

