Two circles touch each other at point X. Two common tangents of the circles meet at point P and none of the tangents passes through X. These tangents touch the larger circle at points B and C. If the radius of the larger circles 15 cm and CP = 20 cm, then what is the radius (in cm) of the smaller circle?

$$\eqalign{ & {\text{In }}\Delta {O^1}CP \cr & {O^1}P = \sqrt {{{20}^2} + {{15}^2}} = 25{\text{ cm}} \cr & OP = 25 - 15 - r = 10 - r \cr & {\text{In }}\Delta OPD\,\& \,\Delta {O^1}CP \cr & \frac{{OP}}{{{O^1}P}} = \frac{{OD}}{{{O^1}C}} \cr & \frac{{10 - r}}{{25}} = \frac{r}{{15}} \cr & 150 - 15r = 25r \cr & r = \frac{{150}}{{40}} \cr & r = 3.75{\text{ cm}} \cr} $$