AB is line segment of length 24 cm. C is its midpoint. On AB, AC and BC semicircles are described. Find the radius of the circle which touches all the three semicircles.

Let the required radius be r cm.
O₁O₃ = Radius of smaller semicircle + r = 24/4 + r = 6 + r
O₁C = Radius of smaller semicircle = 24/4 = 6 cm
In right triangle O₁O₃C:
O₁O₃² = O₁C² + O₃C²
or O₃C² = O₁O₃² – O₁C²
  = 36 + r² + 12r – 36
  = r² + 12r 
or O₃C = (r² + 12r)^1/2
Also,
DC = DO₃ + O₃C
or 24/2 = 12 = r + (r² + 12r)^1/2
or 12 – r = (r² + 12r)^1/2
Squaring both sides
144 + r² – 24r = r² + 12r
or 36r = 144
or r = 144/36 = 4
Hence, radius of the circle which touches all three semicircles is 4 cm.