From: elkies@osgood.harvard.edu (Noam Elkies)
Newsgroups: sci.math
Subject: Is there a five-circle theorem?
Keywords: Euclidean geometry, five circles
Date: 18 Jan 90 22:43:10 GMT
Organization: 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.symbolic
Subject: Re: Elkies-Rubin (was: deciding geometric conjectures)
Date: 5 Feb 92 05:54:53 GMT
Organization: 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.symbolic
From: 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.symbolic
Subject: Re: Elkies-Rubin (was: deciding geometric conjectures)
Date: 13 Feb 92 09:13:05 GMT
Organization: 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