Three circles having radii 1, 2, and 3 are in pairs externally tangent to each other.

Three circles having radii 1, 2, and 3 are in pairs externally tangent to each other. Find the radius of the circle passing through the centers of these circles.

We connect the centers of the circle and determine the lengths of the sides of the formed triangle ABC.

The length of each side of the triangle is equal to the sum of the radii of the two tangent circles.

AC = 3 + 1 = 4 cm.

AB = 3 +2 = 5 cm.

BC = 2 + 1 = 3 cm.

In the triangle ABC, the Pythagorean theorem holds.

AB ^ 2 = AC ^ 2 + BC ^ 2.

25 = 16 + 9.

25 = 25.

Therefore, triangle ABC is rectangular, and segment AB is its hypotenuse.

Then the hypotenuse AB will lie on the diameter of the circle circumscribed about the triangle ABC.

Then R = AB / 2 = 5/2 = 2.5 cm.

Answer: The radius of the circle is 2.5 cm.