Three circles with radii $1 \mathrm{~cm}, 2 \mathrm{~cm}$, and $3 \mathrm{~cm}$ are pairwise externally tangent, as shown in the figure below.

Determine the radius of the circle that is externally tangent to the three circles.

![](https://cdn.mathpix.com/cropped/2024_05_01_a444074f1f8ae50444d5g-10.jpg?height=420&width=415&top_left_y=1302&top_left_x=1321)
By connecting the centers of the three circles, we obtain the triangle $\triangle A B C$ with sides $A B=3 \mathrm{~cm}, A C=4 \mathrm{~cm}$, and $B C=5 \mathrm{~cm}$. Since $3^{2}+4^{2}=5^{2}$, this triangle is a right triangle with hypotenuse $B C$.

![](https://cdn.mathpix.com/cropped/2024_05_01_a444074f1f8ae50444d5g-40.jpg?height=426&width=469&top_left_y=676&top_left_x=749)

Construct the rectangle $A B D C$ by making a copy $\triangle B C D$, congruent to the triangle $\triangle A B C$ and with a common side $B C$.

![](https://cdn.mathpix.com/cropped/2024_05_01_a444074f1f8ae50444d5g-40.jpg?height=418&width=466&top_left_y=1268&top_left_x=754)

Since $D C=A B=3$ and the circle centered at $C$ also has a radius of $3 \mathrm{~cm}$, we see that point $D$ lies on this circle.

By connecting point $D$ to each vertex of the triangle $\triangle A B C$ and extending these segments until they intersect the circles, we obtain the points $P_{1}, P_{2}$, and $P_{3}$.

![](https://cdn.mathpix.com/cropped/2024_05_01_a444074f1f8ae50444d5g-40.jpg?height=456&width=501&top_left_y=1973&top_left_x=736)

We have:

- $D P_{2}=D B+B P_{2}=C A+B P_{2}=4+2=6$.
- $D P_{1}=D A+A P_{1}=5+1=6$.
- $D P_{3}=D C+C P_{3}=3+3=6$.

Thus, $D P_{1}=D P_{2}=D P_{3}=6$. Therefore, if we consider the circle centered at $D$ with a radius of $6 \mathrm{~cm}$, we see that this circle passes through $P_{1}, P_{2}$, and $P_{3}$. Additionally, since the points $\left\{D, A, P_{1}\right\}$, $\left\{D, B, P_{2}\right\}$, and $\left\{D, C, P_{3}\right\}$ are collinear, it follows that the circle centered at $D$ with a radius of $6 \mathrm{~cm}$ is tangent to the three given circles with centers at $A, B$, and $C$.

![](https://cdn.mathpix.com/cropped/2024_05_01_a444074f1f8ae50444d5g-41.jpg?height=499&width=499&top_left_y=821&top_left_x=737)