Following Penrose' notation, I omit letters "c" for circles and "P" for points, to avoid cluttering. Here we see four (n=4) blue circles c1, c2, c3, c4 through O intersecting in P12, P13, P14, P23, P24, P34 which define the four unique yellow circle c123, c124, c134, c234. The construction of the yellow circles corresponds to applying the case n=3 successively to three (blue) circles each, i.e. to c1,c2,c3 to give c123, etc. The surprising fact is that the four yellow circles c123, c124, c134, c234 intersect in one unique point P1234. Sir R. Penrose states that this "is actually a direct consequence of an old theorem, known to the ancient Greek geometer Appollonius" (Greece, Perge, ca. 260 - 190 BC.)
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Taking O to infinity changes the above into
The four blue straight lines are now four circles with infinite radius intersecting at infinity O.
Soli Deo Gloria. Created with Cinderella by Eckhard Hitzer (Fukui).