The figure above shows a straight cone of base radius $R$ in which a sphere is inscribed that intersects the side of the cone in a circle of radius $r$. Point $C$ is the center of the base of the cone and tangent to the sphere. Points $A$ and $B$ are located on the same generatrix of the cone, where $A$ belongs to the circle of radius $r$ and $B$ to the circle of radius $R$.
Based on this hypothetical situation, judge the following items. The radius of the sphere is $\dfrac{R\sqrt r}{\sqrt{2R-r}}$
(S: True)
I try:
$CB = CA = R$
$\triangle VAO \sim \triangle VCB \sim \triangle VTA$
$\dfrac{VO+re}{VT}=\dfrac{r}{R}=\dfrac{VA}{VA+R}$
$\dfrac{VO}{VA+R}=\dfrac{re}{R}=\dfrac{VA}{VO+re}$
$\dfrac{r}{re}=\dfrac{VA}{VO}=\dfrac{VT}{VA}$