Semicircles are constructed on AB, AC, and BC. A circle is tangent to all three semicircles. Find the radius of the circle.
Let O be the cemter of the red circle and r = its radius
Let N = the center od the right semi-circle
BN^2 + BO^2 = ON^2
1^2 + (2 -r)^2 = (1 + r)^2
1 + 4 - 4r +r^2 = r^2 + 2r + 1
5 - 4r = 2r + 1
4 = 6r
r = 4/6 = 2/3
+673 
