Two circles with radii $r_1$ and $r_2$ respectively, touch each other externally. Let $R$ be the radius of a circle that touches these two circles as well as their common tangent line. Which of the following relations is true?

The length of the direct common tangent segment between two externally touching circles with radii $a$ and $b$ is given by $2\sqrt{ab}$.
Let the two large circles have radii $r_1$ and $r_2$, and the small circle nested between them and the tangent line have radius $R$.

1. The tangent segment length between the circle of radius $r_1$ and the circle of radius $r_2$ is:

$$L=2\sqrt{r_1{r}_2}$$

2. This segment is divided into two smaller tangent segments by the contact point of the third circle of radius $R$:

* Between $r_1$ and $R$: $L_1=2\sqrt{r_{1}R}$
* Between $r_2$ and $R$: $L_2=2\sqrt{r_{2}R}$

Since the total length is the sum of its parts ($L=L_1+L_2$):

$$2\sqrt{r_1{r}_2}=2\sqrt{r_{1}R}+2\sqrt{r_{2}R}$$

Divide the entire equation by $2\sqrt{r_1{r}_{2}R}$:

$$\frac{\sqrt{r_1{r}_2}}{\sqrt{r_1{r}_{2}R}}=\frac{\sqrt{r_{1}R}}{\sqrt{r_1{r}_{2}R}}+\frac{\sqrt{r_{2}R}}{\sqrt{r_1{r}_{2}R}}$$

$$\frac{1}{\sqrt{R}}=\frac{1}{\sqrt{r_2}}+\frac{1}{\sqrt{r_1}}$$