Find center of sphere from 3 points

I know the radius R of a sphere and three points on its surface. I also know that the three points are below the horizontal plane through the center of the sphere.

I need to find the equation of that sphere. Any ideas?

My Attempt

I can find the normal to the plane through A,B,C by using the cross product.

Similarly I can find the angle between the vectors AB and AC using the dot product.

With a bit of trig I think I can find the radius of the circle of intersection between the sphere and the plane. Assuming the angle between AB and AC is half of the angle between XB and XC where X is the center of the circle (not sure if this is right).

From here I am a bit stumped.

This problem comes from the rolling sphere method for determining lightning protection coverage.