10.006. Two circles with radii $R=3$ cm and $r=1$ cm touch each other externally. Find the distance from the point of tangency of the circles to their common tangents.
Solution.

Let $F M=x$ (Fig. 10.6). From $\triangle O_{1} T O \sim \Delta F M O$ we have $\frac{R-r}{x}=\frac{R+r}{r},$ hence $x=\frac{r(R-r)}{R+r},$ and $E F=x+r=\frac{r(R-r)}{R+r}+r=\frac{2 R r}{R+r}=$ $=\frac{6}{4}=\frac{3}{2}$ cm. If, however, the common tangent of the circles is considered to be the line $E F$, then the required distance is zero.

Answer: 0 and $\frac{3}{2}$ cm.

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-559.jpg?height=352&width=404&top_left_y=94&top_left_x=174)

Fig. 10.7

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-559.jpg?height=408&width=330&top_left_y=98&top_left_x=815)

Fig. 10.8