Two circles of radius two touch each other and each of them touches internally a circle of radius four. The radius of the circle which touches all the three circles is r, and then find the value of 3r.

To solve this problem, we can use the concept of Descartes' Theorem, which states that if four circles are mutually tangent, the radii of the circles satisfy a certain equation.

The theorem is given by:

1/r1 + 1/r2 + 1/r3 + 1/r4 = 1/2 * (1/r1 * 1/r2 + 1/r1 * 1/r3 + 1/r1 * 1/r4 + 1/r2 * 1/r3 + 1/r2 * 1/r4 + 1/r3 * 1/r4)

In this case, we have three circles with known radii (two with radius 2 and one with radius 4) and one circle with unknown radius r.

The radii of the two smaller circles are both 2, so their curvature (which is the reciprocal of the radius) is 1/2. The radius of the larger circle is 4, so its curvature is 1/4.

We can plug these values into Descartes' Theorem and solve for the curvature of the unknown circle, which is 1/r.

1/2 + 1/2 + 1/4 + 1/r = 1/2 * (1/2 * 1/2 + 1/2 * 1/4 + 1/2 * 1/r + 1/2 * 1/4 + 1/2 * 1/r + 1/4 * 1/r)

Solving this equation for 1/r gives us the curvature of the unknown circle.

1/r = 1 - 1/2 - 1/4 = 1/4

Therefore, the radius of the unknown circle is r = 4.

Finally, we find the value of 3r, which is 3*4 = 12.