In right triangle ABC, the length of side AC is 8, the length of side BC is 6, and angle C is 90 degrees. The circumcircle of triangle ABC is drawn. The angle bisector of ACB (the right angle) meets the circumcircle at point M. Find the length of CM.
I've labeled by myself that the intersection of CM and BA is X. What I get by using the angle bisector theorem is that AX=40/7 and BX=30/7, which is confirmed since AB=10 from the right triangle ABC. By using the Power of a Point theorem, we know that CX*CM=AX*BX=1200/49.