Problem 1.5. Two spheres of radius $r$ are externally tangent. Three other spheres of radius $R$ are externally tangent to each other, pair by pair. Each of these three spheres is also externally tangent to the first two. Find the relationship between $R$ and $r$.
Solution Problem 1.5 The centers of the three spheres of radius $R, O_{1}, O_{2}$, and $O_{3}$, are the vertices of an equilateral triangle
![](https://cdn.mathpix.com/cropped/2024_04_26_23b9750f7dc8395e0b8dg-3.jpg?height=656&width=1448&top_left_y=800&top_left_x=360)

with side $2 R$. The point of tangency, $T$, of the two spheres of radius $r$ is the center of that triangle and, therefore, is two-thirds of the height from the vertices. The height of the triangle is $h=\frac{2 R \sqrt{3}}{2}=R \sqrt{3}$ and two-thirds of $h$ is $\frac{2 R}{\sqrt{3}}$.

If we call $Q_{1}, Q_{2}$ the centers of the circles of radius $r$, the triangle $O_{1} T Q_{1}$ is a right triangle at $T$ and its sides are: $\frac{2 R}{\sqrt{3}}, r$ and $R+r$. Applying the Pythagorean theorem, we have:

$$
\left(\frac{2 R}{\sqrt{3}}\right)^{2}+r^{2}=(R+r)^{2}
$$

and simplifying results in: $R=6 r$.