We have this situation:
where E is the center of the circle passing through the points of contact of the direct common tangents. A teacher claims that $x=\frac{r_1 +r_2}{2}$ and that it's true for any such system of two circles. They also claim that E is also the intersection of the transverse common tangents but I think he made a mistake there as when we take E as the intersection of transverse common tangents, we get $AE=\frac{r_1}{r_1+r_2}AB$ and $BE=\frac{r_2}{r_1+r_2}AB$ and upon writing the similar triangle equations we get $x=\frac{2r_1r_2}{r_1 + r_2}$ which clearly isn't what he claimed before. How do I prove that $x=\frac{r_1+r_2}{2}$?
$F$ is a midpoint of $DC$ because $\triangle DEC$ is isosceles ($DE=CE$ - radii) and $EF$ is a height to the base. $E$ is a midpoint of $AB$ (proportional segments theorem). Thus, $x$ is a midline in trapezoid $ABCD$ and $x=\frac{r_1+r_2}{2}$