The midcircle of two given circles is the circle which would invert the circles into each other. Dixon (1991) gives constructions for the midcircle for four of the five possible configurations. In the case of the two intersecting circles, there are two midcircles.
The midcircle is in the same coaxal system as the two given circles.
The center of the midcircle(s) is one or both of the centers of similitude.
1. If one circle is inside the other, the unique midcircle has center at the internal similitude center.
2. If the circles are disjoint, the unique midcircle has center at the external similitude center.
3. If the circles intersect in two points, there are two midcircles, one centered at each center of similitude.
4. If the circles intersect in a single point, the unique midcircle has center at the external similitude center (since the midcircle that would be centered at internal similitude center degenerates to a point).
If the given circles intersect, their two midcircles bisect their angles and are orthogonal to each other.