Sentence:  For any two circles (spheres) there always exists at least one "midcircle" (<WSD>"midsphere</WSD>") 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 midcircle (<WSD>midsphere</WSD>) is the center of similitude of two given circles (spheres)....Clearly the midcircle (<WSD>midsphere</WSD>) 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:  (geometry) The sphere, between two given spheres, by which each of the latter may be inverted into the other.