I drew a diagram and labeled some points. is the centre of the circle and D, E and F are points where the sides of the triangle are tangent to the circle. Thus angles CAB, PDB and BEP are right angles. Suppose the radius of the circle is r and the side length of CA is s. By the symmetry of the diagram E is the midpoint of BC. Notice also that the length of AD is r.
I used Pythagoras Theorem 3 times. Once for the triangle ABC to find the length of BC and hence BE in terms of s. A second for triangle BEP to find the length of BP, and a third time for triangle BPD to write an equation involving s and r. Solve for r.
