Given two circles (the thick green ones), the dashed red circle is the circle through the point D orthogonal to both. Then in general the red circle intersects each green circle in two points. For any two of those 4 points, the two tangent lines at these points, along with the radii of the red circle form a kite with two right angles. For each pair of equal sides of the kite, there is a circle with these segments as radii. Since the equal vertex angles of the kite are right angles, the two circles are orthogonal.
In this figure one of the circles is the red circle. If the two points on the red circle are chosen one from each green circle, the second circle is one of the blue circles. Since the blue and green circles are both orthogonal to the red circle at the same point, they are tangent.
There are 4 ways to choose one intersection point from each green circle, so there are 4 blue circles.
James King, 1/30/2004