3 $[\underline{\text { Tangents to Spheres }}]$

A plane is drawn through the center of a sphere of radius $R$. Three equal spheres touch the given sphere, the plane, and each other. Find the radii of the spheres.
Let spheres of radius $r$ touch a plane passing through the center $O$ of a sphere of radius $R$ at points $A, B$, and $C$. Then $O$ is the center of the equilateral triangle $ABC$ with side $2r$. Therefore,

$$
O A = O B = O C = \frac{\frac{2 \varphi \sqrt{3}}{3}}{3}
$$

Consider a plane passing through point $O$ and a line passing through point $A$ and the center $O_1$ of the corresponding sphere. If the spheres touch the given sphere externally, we obtain a circle of radius $r$ with center at point $O_1$, touching the circle of radius $R$ externally, and also touching the extension of the diameter of this circle at point $A$. From the right triangle $OAO_1$, we find that

$$
O A = \sqrt{O O_{1}^{2} - O_{1} A^{2}} = \sqrt{(R + r)^{2} - r^{2}} = \sqrt{R^{2} + 2 r R}
$$

From the equation

$$
\sqrt{R^{2} + 2 r R} = \frac{2 r \sqrt{3}}{3}
$$

we find that

$$
r = \frac{(\sqrt{21} + 3) \cdot R}{4}
$$

If the spheres touch the given sphere internally, then, reasoning similarly, we obtain the equation

$$
\sqrt{R^{2} - 2 r R} = \frac{2 r \sqrt{3}}{3}
$$

Then

$$
r = \frac{(\sqrt{21} - 3)}{4}
$$

## Answer

Page: $\leq<\underline{41} \underline{42} \underline{43} 44 \underline{45} \underline{46} \underline{47} \geq \geq$ [Total problems: 12605]

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