Consider two circles c'(r), c''(r/2) with centers O, F respectively, the second having half the radius of the first. Show that: [1] The lines A'A'' joining two points on them respectively, such that OA' and OF are parallel and opposite directed intersect the line of centers OF at a point G, such that GO/GF=2 (this is the inner similitude center of the two circles). [2] Define H to be the symmetric of O with respect to F. Show that the circle with diameter HG belongs to the bundle of circles generated by c' and c''. [3] If c', c'' do not intersect, then the circle-bundle of circles (d) simultaneously orthogonal to c' and c'' is of intersecting type and defines two points H', H'' on line FO, such that H and G are harmonic conjugate with respect to H' and H''.
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