Two externally tangent circles have a common tangent segment of $20 \mathrm{~cm}$ in length. What are the radii of these circles if the radius of the circle that can be placed between the two circles and their common tangent, touching all three, is as large as possible?
Let's use the notations from the diagram created based on the problem text. According to the Pythagorean theorem,

$$
E_{1} E_{2}^{2}=(R+r)^{2}-(R-r)^{2}=4 R \cdot r
$$

and thus

$$
E_{1} E_{2}=2 \sqrt{R r}
$$

According to the conditions, $E_{1} E_{2}=20(\mathrm{~cm})$, so $2 \sqrt{R r}=20$, which means

$$
R r=100
$$

Similarly to (1), $E_{1} E=2 \sqrt{r x}$ and $E E_{2}=2 \sqrt{R x}$. Since $E_{1} E_{2}=E_{1} E+E E_{2}$, we have $2 \sqrt{R r}=2 \sqrt{r x}+2 \sqrt{R x}$, from which $\sqrt{x}=\frac{10}{\sqrt{R}+\sqrt{r}}$. Since the sides are positive, squaring both sides gives an equivalent relationship:

$$
x=\frac{100}{R+r+20}=\frac{50}{\frac{R+r}{2}+10}
$$

According to (3), $x$ will be the largest when $\frac{R+r}{2}$ is the smallest. Since (2) states that $R r$ is constant, the inequality $\frac{R+r}{2} \geq \sqrt{R r}$ implies that $\frac{R+r}{2}$ is the smallest when $R=r=10$. Based on this, the maximum value of $x$ can also be determined: $x_{\max }=\frac{100}{40}=2.5(\mathrm{~cm})$

Based on the work of Gyula Pap (Debrecen, Fazekas M. Gymnasium, 1st year)

![](https://cdn.mathpix.com/cropped/2024_05_02_0609a67df66757d3b808g-1.jpg?height=678&width=859&top_left_y=1233&top_left_x=622)