Three circles touch each other externally. The distance between their centres is 5 cm, 6 cm and 7 cm. Find the radii of the circles.

This problem can be solved using the formula for the radii of three externally tangent circles, which is derived from the Descartes' Circle Theorem. The theorem states that if four circles are mutually tangent (externally or internally), their curvatures (k, which is 1/radius) satisfy a certain equation.

For three externally tangent circles, the fourth circle is considered to be a circle with infinite radius (or zero curvature). Therefore, the equation simplifies to:

k1 + k2 + k3 = 2 * sqrt(k1k2 + k2k3 + k3*k1)

Given the distances between the centers of the circles (5 cm, 6 cm, and 7 cm), we can find the radii of the circles by solving the above equation.

Let's denote the radii of the circles as r1, r2, and r3, and their curvatures as k1, k2, and k3 (where ki = 1/ri).

We know that the sum of the radii of any two circles is equal to the distance between their centers. Therefore, we have:

r1 + r2 = 5 cm r2 + r3 = 6 cm r3 + r1 = 7 cm

Adding these three equations, we get 2*(r1 + r2 + r3) = 18 cm, so r1 + r2 + r3 = 9 cm.

Subtracting the first equation from the third, we get r3 - r1 = 2 cm.

Subtracting the second equation from the third, we get r1 - r2 = 1 cm.

Solving these two equations simultaneously, we get r1 = 3 cm, r2 = 2 cm, and r3 = 4 cm.

So, the radii of the three circles are 3 cm, 2 cm, and 4 cm.