From:elkies@osgood.harvard.edu (Noam Elkies)Newsgroups:sci.mathSubject:Is there a five-circle theorem?Keywords:Euclidean geometry, five circlesDate:18 Jan 90 22:43:10 GMTOrganization:Harvard Math Department

Karl Rubin (rubin@function.mps.ohio-state.edu) has observed the following result experimentally; we could find neither a general proof nor a reference, and wondered whether sci.math readers could do any better: Let L_0,L_1,L_2,L_3,L_4 be five lines in general position on the Euclidean plane---think of the subscripts mod 5 and draw L_i as the consecutive lines of a (not necessarily regular!) pentagram. Let C_i be the circle inscribed about the triangle formed by L_i, L_{i-2}, and L_{i+2}. Then C_{i-1} and C_{i+1} meet at the intersection of L_{i-1} and L_{i+1}, and again at some other point P_i (which we take to be the same point if the two circles are tangent there). Show that these five points P_i are concyclic. --Noam D. Elkies (elkies@zariski.harvard.edu) Dept. of Mathematics, Harvard University

From:ara@zurich.ai.mit.edu (Allan Adler)Newsgroups:sci.math,sci.math.symbolicSubject:Re: Elkies-Rubin (was: deciding geometric conjectures)Date:5 Feb 92 05:54:53 GMTOrganization:M.I.T. Artificial Intelligence Lab.

I haven't really thought about this, but the statement seems to be very much in the spirit of some theorems due to Steiner,Wallace, de Lonchamps, Miquel and Bath which are described in an obituary notice of Frederick Bath written by W.L.Edge (Proc. Edinburgh Math. Soc (1983) vol.26, pp.279-281. I quote from the discussion of Bath's Theorem on p.280: "One must now speak of Bath's Theorem. A plane quadrilateral affords, by omitting each of its sides in turn, four triangles; their four circumcircles have a common point P (Wallace's Theorem) and their four circumcentres are on a circle C (Steiner). Five coplanar lines, no three concurrent, afford, by omitting each in turn, five quadrilaterals; the five points P are on a circle \Gamma (the Miquel circle) and the five circles C have a common point Q (de Longchamps). Bath's theorem is that Q is on \Gamma." Edge remarks that Bath proved it by projecting from a figure in four dimensional space (F.Bath "On circles determined by five lines in a plane," Proc. Cambridge Phil. Soc. vol.35 (1939) 518-519). Even if these results do not imply the Elkies-Rubin result, it seems natural to look at the techniques used there for a synthetic proof. Allan Adler ara@altdorf.ai.mit.edu

Newsgroups:sci.math,sci.math.symbolicFrom:edgar@function.mps.ohio-state.edu (Gerald Edgar)Subject:Re: Elkies-Rubin (was: deciding geometric conjectures)Organization:The Ohio State University, Dept. of Math.Date:Wed, 5 Feb 1992 15:34:50 GMT

Here is the statement, from sci.math in January, 1990. >Let L_0,L_1,L_2,L_3,L_4 be five lines in general position on the Euclidean >plane---think of the subscripts mod 5 and draw L_i as the consecutive lines >of a (not necessarily regular!) pentagram. Let C_i be the circle inscribed >about the triangle formed by L_i, L_{i-2}, and L_{i+2}. Then C_{i-1} and >C_{i+1} meet at the intersection of L_{i-1} and L_{i+1}, and again at some >other point P_i (which we take to be the same point if the two circles are >tangent there). Show that these five points P_i are concyclic. And here is Bath's Theorem, as reported by Allan Adler: "One must now speak of Bath's Theorem. A plane quadrilateral affords, by "omitting each of its sides in turn, four triangles; their four circumcircles "have a common point P (Wallace's Theorem) and their four circumcentres are on "a circle C (Steiner). Five coplanar lines, no three concurrent, afford, "by omitting each in turn, five quadrilaterals; the five points P are on a "circle \Gamma (the Miquel circle) and the five circles C have a common "point Q (de Longchamps). Bath's theorem is that Q is on \Gamma. A little thought shows (using the fact that no three lines are concurrent) that the five points P_i constructed in the first, are the same five points P constructed in Bath's theorem, so Bath's theorem does the trick. In fact, there is even more information there... This is a good example showing how useful the network can be. -- Gerald A. Edgar Internet: edgar@mps.ohio-state.edu Department of Mathematics Bitnet: EDGAR@OHSTPY The Ohio State University telephone: 614-292-0395 (Office) Columbus, OH 43210 -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)

From:djoyce@black.clarku.edu (David E. Joyce)Newsgroups:sci.math,sci.math.symbolicSubject:Re: Elkies-Rubin (was: deciding geometric conjectures)Date:13 Feb 92 09:13:05 GMTOrganization:Clark University (Worcester, MA)

The statement, from sci.math in January, 1990, starting the thread: >Let L_0,L_1,L_2,L_3,L_4 be five lines in general position on the Euclidean >plane---think of the subscripts mod 5 and draw L_i as the consecutive lines >of a (not necessarily regular!) pentagram. Let C_i be the circle inscribed >about the triangle formed by L_i, L_{i-2}, and L_{i+2}. Then C_{i-1} and >C_{i+1} meet at the intersection of L_{i-1} and L_{i+1}, and again at some >other point P_i (which we take to be the same point if the two circles are >tangent there). Show that these five points P_i are concyclic. >And here is Bath's Theorem, as reported by Allan Adler: > >"One must now speak of Bath's Theorem. A plane quadrilateral affords, by >"omitting each of its sides in turn, four triangles; their four circumcircles >"have a common point P (Wallace's Theorem) and their four circumcentres are on >"a circle C (Steiner). Five coplanar lines, no three concurrent, afford, >"by omitting each in turn, five quadrilaterals; the five points P are on a >"circle \Gamma (the Miquel circle) and the five circles C have a common >"point Q (de Longchamps). Bath's theorem is that Q is on \Gamma. Gerald Edgar noted: > >A little thought shows (using the fact that no three lines are concurrent) >that the five points P_i constructed in the first, are the same five points >P constructed in Bath's theorem, so Bath's theorem does the trick. >In fact, there is even more information there... Frank Morley and F. V. Morley wrote a book _Inversive_Geometry_, Ginn & Co., Boston, 1933, that includes a description of the complete 4-line and Wallace's theorem, the 5-line and the Miquel circle, and the 6-line and the Clifford point. They complete their book with a chapter on the n-line which describes Clifford's chain: The theorem of Clifford is as follows. Two lines, say 1 and 2, have a common point 12. Three lines 1, 2, 3 have three common points 12, 23, 31 which lie on a circle 123. Four lines have four circumcircles 123 which are on a point 1234. Five lines give five such points which lie on a circle 12345. An so on. We thus get the complete n-line, the lines 1,2,...,n meeting in (n choose 2) points 12,... these lying on (n choose 3) circles 123,... these meeting in (n choose 4) points 1234,... ending with a point or a circle 123...n as n is even or odd. Regarding the lines as circles on the point infinity, we have the Clifford configuration /Gamma n of 2^(n-1) points and and 2^(n-1) circles, each point on n circles and each circle on n points. >-- >David E. Joyce Dept. Math. & Comp. Sci. >Internet: djoyce@black.clarku.edu Clark University >BITnet: djoyce@clarku Worcester, MA 01610-1477 -- David E. Joyce Dept. Math. & Comp. Sci. Internet: djoyce@black.clarku.edu Clark University BITnet: djoyce@clarku Worcester, MA 01610-1477