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13 points on a circle: spoiler

Q: ??? 13 points in the plane are such that given any three some fourth one lies on the same circle as those three. Show that all 13 are cocircular.
Julian Gilbey, 13 Jun 1996

A: Partial solution: Label the points A..M. Call the fourth point the friend of the triple. There are 13C3 = 286 triples, so some point, wlog A, is the friend of at least 286/13 = 22 triples. These 22 triples involve 66 points (counting with multiplicity) so some point, wlog B, is in at least ceil(66/12) = 6 of them. These 6 triples involve 12 other points (counting with multiplicity), so some point, wlog C, is in at least ceil(12/11) = 2 of them. Thus we have two triples, wlog BCD and BCE, whose friend is A, and hence ABCDE are cocircular. What next?


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