Find the center of a Circle tangent to two circles with a known radius

Question

In this image, I need to find the center point of a tangent circle to two other circles with a known radius.

The information I can gather is: $rT$, $rA$, $rB$, $(Ax,Ay)$, $(Bx,By)$,

Not given is: $(TAx,TAy)$, $(TBx,TBy)$, $(Tx,Ty)$,

I searched for answers on this site, but couldn't find any before posting. Any insight would be much appreciated.

Answer 1

$(T_x,T_y)$ is a distance of $rT+rA$ from the center of $A.$

and $rT+rB$ from the center of $B.$

Solve this system of equations:

$(x-A_x)^2 + (y-A_y)^2 = (rT + rA)^2\\ (x-B_x)^2 + (y-B_y)^2 = (rT + rB)^2$