circles

A semicircle with radius 2 and two semicircles with diameter 2 are drawn as shown.  The red circle is tangent to all three semicircles.  Find r.

r = 0.666666667     or     2/3

Let  the  distance  between the center of   one of the smaller semicircles and the  red  circle  =  1 + r

Let  radius of  one of the semicircles   = 1

Let  the  distance  form the  center of the small red circle to the diameter of the  large semicircle  = 2 - r

These form a right triangle such that

1^2 + (2-r)^2  = (1 + r)^2   simplify

1  +  4  - 4r  + r^2  =  1  + 2r  + r^2

4  = 6r

r = 4/6  = 2/3