+613 Semicircles are constructed on AB, AC, and BC. A circle is tangent to all three semicircles. Find the radius of the circle.
+130477 Let M be the midpoint of BC
BM = 1
Let O be the center of the red circle and r its radius
OB = (radius of large circle - r) = (2 - r)
OM = (1 + r)
Triangle OBM is right with OM the hypotenuse
So
OB^2 + BM^2 = OM^2
(2 - r)^2 + 1^2 = (1 + r)^2
r^2 - 4r + 4 + 1 = r^2 + 2r + 1
6r = 4
r = 4 / 6 = 2 / 3
