[ Tangent to Spheres ] $[$ Tangent Spheres ]

A sphere of radius $R$ touches the plane $\alpha$. Consider all possible spheres of radius $r$ that touch the given sphere and the plane $\alpha$. Find the geometric loci of the centers of these spheres and the points of their tangency with the plane and the given sphere.
Let $O$ be the center of a given sphere of radius $R, A$ be the point of tangency of this sphere with a given plane $\alpha, Q$ be the center of an arbitrary sphere of radius $r$, tangent to the plane $\alpha$ at point $M$, and to the given sphere at point $N$. Since $O A \perp \alpha$ and $Q M \perp \alpha$, then $O A \| Q M$. Through the parallel lines $O A$ and $Q M$, we draw a plane. Point $N$ lies in this plane, since the line of centers of two tangent spheres passes through their point of tangency. Drop perpendiculars $Q B$ and $N C$ from points $Q$ and $N$ to the line $O A$. We then find:

$$
O B=O A-A B=O A-Q M=R-r
$$

$$
\begin{aligned}
& Q B=\sqrt{O Q^{2}-O B^{2}}=\sqrt{(R+r)^{2}-(R-r)^{2}}=2 \sqrt{R r}, \\
& C N=B Q \cdot \frac{N O}{O Q}=2 \sqrt{R r} \cdot \frac{R}{R+r}=\frac{2 R \sqrt{R r}}{R+r}, \\
& O C=O B \cdot \frac{N O}{Q O}=(R-r) \cdot \frac{R}{R+r}=\frac{R(R-r)}{R+r}, \\
& A C=O A-O C=R-\frac{R(R-r)}{R+r}=\frac{2 R r}{R+r} .
\end{aligned}
$$

From this, the following statements follow. Point $Q$ lies on a circle of radius $2 \sqrt{\frac{R r}{R}}$ with center at a fixed point $B$. The plane of this circle passes through point $B$ parallel to the plane $\alpha$ and is at a distance $r$ from the plane $\alpha$. Clearly, each point of this circle is the center of a sphere of radius $r$, tangent to the plane $\alpha$ and the given sphere of radius $R$. Point $M$ lies on a circle of radius $2 \sqrt{R r}$ with center at point $A$. Clearly, each point of this circle is the point of tangency of the plane $\alpha$ with a sphere of radius $r$, also tangent to the given sphere of radius $R$. Point $N$ lies on a circle of radius $\frac{2 R \sqrt{R r}}{R+r}$ with center at a fixed point $C$. The plane of this circle passes through point $C$ parallel to the plane $\alpha$ and

![](https://cdn.mathpix.com/cropped/2024_05_06_6bf60d3f257a54792c0cg-30.jpg?height=71&width=1887&top_left_y=1022&top_left_x=14)
tangency of the given sphere with some sphere of radius $r$, also tangent to the plane $\alpha$.

## Answer

Circles of radii $2 \sqrt{R r}, 2 \sqrt{R r}$, and $\frac{2 R \sqrt{R r}}{R+r}$.