To explain how, I'll need to explain how I constructed it in the first place. I started with these two circles. The radius of the outer circle is R. The radius of the inner circle is R/2. They are both centered at (0, 0).
Well, if two circles are tangent, then a line drawn through the centers of both will pass through the point of tangency. So if you have a circle, and you know what point you want to be tangent, then you can draw a line through the center of that circle and through that point. Then any circle that is centered on that line and that goes through that point will be tangent to the first circle. And if you have two circles like that, and you want to find one circle that is tangent to both, then you can draw two lines. Where they intersect will be the center of the third circle.
So I'll start with the red circles. I drew lines through the centers of the red circles and the points where the red circles intersect the inner circle. Where they intersect is the center of the circular arc that connect the two arcs. Note that each red line comes close to where the other red circle intersects the outer circle, but not quite. Also, each red line comes close to where the blue circles intersect the inner circle, but again, not quite.
So this time, I started not with the outer circle and inner circle, but rather the outer circle and one of the green circles.
R, which is approximately .5176R.