Geometry Problem with Tangent Circles Amd Trapezoids

Question

I think the drawing is very self explanatory besides the fact $MN$ is the distance from the touching point of the circles to the common tangent line.

I need to show $MN=\frac{2rR}{R+r}$ which I don't know how to do. The only progress I did is to show the following triangle is a right triangle

and I could perahps use that $MN$ is the geometric mean of the projections of the sides over the tangent line, which I found no way to do.

Thanks in advance!

Answer 1

Let angle ECD be $\theta$

Then $$\cos\theta=\frac{R-r}{R+r}=\frac{MN-r}{r}$$

From this the result follows immediately