Sentence:  For any two circles (spheres) there always exists at least one <WSD>"midcircle</WSD>" ("midsphere") which inverts the two given circles (spheres) into each other. in the older literature this is also known as the circle (sphere) of "antisimilitude"....The center of a <WSD>midcircle</WSD> (midsphere) is the center of similitude of two given circles (spheres)....Clearly the <WSD>midcircle</WSD> (midsphere) of two equal circles (spheres) is the midline (midplane) between the two, which is partial justification for calling inversion in a circle or sphere "reflection" in a circle or sphere. 
Meaning of the word marked with <WSD> tags:  (inversive geometry) A reference generalised circle through which two given circles are inverses of each other.