Let's look at the figure below.
Construct the common tangent line (blue). Denote the touching points by $A_1$ and $A_2$. Then drop perpendiculars from $C_1$ and from $C_2$ to the blue line. Then take the line connecting $L_1$ and $A_1$ and the line connecting $L_2$ and $A_2$. These two lines will meet on $C$ at $X$.
Certain triangles and the small circles on the figure are homothetic images of each other. The homothetic center is $H$. As a result the triangles $L_1A_1D_1$ and $MA_2D_2$ and the triangles $L_1A_1C_1$ and $MA_2C_2$ are similar triangles.
Comparing the angles of these triangles with the corresponding angles of the big triangle $L_1XL_2$ will show that, say, $L_1A_1D_1$ and $L_1XL_2$ are similar. It turns out that the angle at $X$ is a right angle. (Why are the angles at $D_2$ and $L_2$ equal?) That is, the big black triangle is a Thales triangle of the red circle. This is why is $X$ on the red circle.
This proves that doing the construction in the revers order we will find $A_1$ and $A_2$ on the tangent line.