Two circles over the same wall: a genealogy of an unnamed circle

Each external common tangent of two disjoint circles, centres O1 and O2, meets each internal one in a point, and these four points lie on one circle — the circle on O1O2 as diameter, a circle the classical literature states but never names. Its named twin, the circle of similitude, is orthogonal to it. This essay walks the pair through a single draggable figure: from Thales' right angle, through Newton's mean-proportional relation and its inversive twin, to the parabola that Menaechmus first drew for the sake of two mean proportionals. The asymmetry of names — one circle with a title and a literature, one with a descriptive clause — is measured along the way, and turns out to be a fact about transmission, not about geometry.