A circle centered at B is tangent to AC.

Hint:  Use Heron's formula to calculate the area, then use this formula  (h = 2*A / a) to find the radius.

 

The altitude from point B to line segment AC is the radius of a circle.

Here is another way, (though Heron's formula is probably easier.

 

Let X be the point of intersection of the circle and the tangent

Let r be the radius

 

Now you have triangles BCX  and  BAX

 

The angles at X are right angles so you can use pythagoras's theorum to find XC  and  XA  in terms of r

added they equal 12

 

solve

(it is a bit tricky)