## Conics and similarities of circles III

File CirclesSimilar.html contains a short discussion on the locus of points which serve as similarity centers F of some similarity S mapping a circle (d) to another circle (d'). In file ConicsAndSimilarities.html it is shown that, for such a similarity, line EE' joining a point E on (d) with E'=S(E) on (d') envelopes a conic with one focus at F and tangent (under certain conditions) to the two given circles (d) and (d').
The example below is a case where the two circles (d) and (d') intersect and the conic is tangent only to one of them.

Points {V,W} are the homothety centers of the two circles (d) and (d'). Circle (f) has these two points as diametral points. F is always on this circle. In this configuration the two circles (d) and (d') intersect and the circle with diametral points {V,W} passes through the intersection points {M,N} of (d) and (d'). For the locations of F which are inside the intersection of the two circular discs the corresponding conic is an ellipse. For the other locations of F on circle (f) the corresponding conic is a hyperbola.
In the first case, in which F is on the arc MN of (f) lying inside the intersection of the two circular discs, the resulting ellipse may contact, one, both or none of the two circles (d) and (d').
A clue for the behavior of these ellipses regarding their contact with the circles may give the picture contained in the file DistanceFunction.html .