Circles $C_1$ and $C_2$ touch externally at a point $M$ and also touch a circle $C$ internally at $L_1$and $L_2$ respectively. Let $X$ be a point of intersection of $C$ with the common tangent to $C_1$ and $C_2$ at $M$. Line X $L_1$ meets $C_1$ again at $A_1$ and X $L_2$ meets $C_2$ again at $A_2$.
Prove $A_1A_2$ is a common tangent to $C_1$ and $C_2$.
All help welcome. Really struggling with this one.
We must show that line $A_1A_2$ is perpendicular to both $C_1A_1$ and $C_2A_2$.
By the tangent secant theorem we have $XA_1\cdot XL_1=XM^2=XA_2\cdot XL_2$. It follows that triangles $XA_1A_2$ and $XL_2L_1$ are similar, so that (see diagram below for angle names): $$ \angle XA_1A_2=\angle XL_2L_1={\pi\over2}-\alpha+\gamma, \quad \angle XA_2A_1=\angle XL_1L_2={\pi\over2}-\alpha+\beta. $$ In addition, notice that: $$\beta+\gamma=\angle L_1XL_2={1\over2}\angle L_1CL_2=\alpha.$$ We have then: $$ \angle C_1A_1A_2=\pi-\angle C_1A_1L_1-\angle XA_1A_2= \pi-\beta-\left({\pi\over2}-\alpha+\gamma\right)={\pi\over2}, $$ as it was to be proved. A similar reasoning can be made for $\angle C_2A_2A_1$.
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.