Circles with radii $R$ and $r$ touch each other externally. Their external common tangents, as well as the tangents parallel to the common internal tangent, form a symmetric trapezoid. Express the sides and angles of the trapezoid in terms of $R$ and $r$.
Using similar triangles and the Pythagorean theorem, with a bit of calculation, the following expressions are obtained for the sides.

The length of the legs is $\frac{(R+r)^{2}}{\sqrt{R r}}$, the shorter parallel side is $2 r \sqrt{\frac{r}{R}}$, the longer parallel side is $2 R \sqrt{\frac{R}{r}}$, and for the tangent of the angle between the longer base and the leg, $\operatorname{tg} \alpha=\frac{2 \sqrt{R r}}{R-r}$.

The solution is incomplete if it uses data other than $R$ and $r$ in the final result, for example, the arc of the angles.