Newsgroups: sci.math From: boshuck@triples.math.mcgill.ca (William Boshuck) Subject: Re: Your favourite proofs Date: Thu, 14 Jul 1994 20:10:28 GMT Hi, I just remembered a really cute proof of the following fact: Any finite collection L of points in the plane satisfying (*) a line passing through any two points in L passes through a third point in L is collinear. Proof: Suppose not. Among all triples (p,q,r) of non-collinear points in L, pick one which minimizes the distance between r and the line pq, and let p' be a point witnessing (*) for the line pq. Without loss, p' lies between p and q on pq. Then the distance between p' and (at least) one of the lines rp, rq is smaller than the distance between r and pq. Contradiction. cheers, ============================================================================== Newsgroups: sci.math From: stephen@mont.cs.missouri.edu (Stephen Montgomery-Smith) Subject: Re: Your favourite proofs Date: 15 Jul 94 05:16:02 GMT In <1994Jul14.201028.8961@sifon.cc.mcgill.ca> boshuck@triples.math.mcgill.ca (Wi lliam Boshuck) writes: >Hi, >I just remembered a really cute proof of the following fact: Any >finite collection L of points in the plane satisfying > (*) a line passing through any two points in L passes > through a third point in L >is collinear. >Proof: [ the short proof is ommitted] Apparently this was an unsolved problem for quite a while. Stephen ============================================================================== Newsgroups: sci.math Subject: Re: Your favourite proofs From: kubo@zariski.harvard.edu (Tal Kubo) Date: 15 Jul 94 03:32:25 EDT In article stephen@mont.cs.missouri.edu (Stephen Montgomery-Smith) writes: >boshuck@triples.math.mcgill.ca (William Boshuck) writes: > >>[...] Any finite collection L of points in the plane satisfying > >> (*) a line passing through any two points in L passes >> through a third point in L >> >>is collinear. > >>Proof: [ the short proof is ommitted] > >Apparently this was an unsolved problem for quite a while. Both this problem and the previous one (N red, N blue points) are projectively invariant assertions but the short proofs use metric concepts. Maybe "purity of method" was the stumbling block? (Speaking of which, are there projective geometry solutions to either problem? The one above even has a *combinatorial* solution but it's not simple.) ============================================================================= Newsgroups: sci.math From: edgar@math.ohio-state.edu (Gerald Edgar) Subject: Re: Your favourite proofs Date: 15 Jul 1994 09:21:58 -0400 In <1994Jul14.201028.8961@sifon.cc.mcgill.ca> boshuck@triples.math.mcgill.ca (Wi lliam Boshuck) wrote: >Hi, > >I just remembered a really cute proof of the following fact: Any >finite collection L of points in the plane satisfying > > (*) a line passing through any two points in L passes > through a third point in L > >is collinear. > >Proof: Suppose not. Among all triples (p,q,r) of non-collinear points >in L, pick one which minimizes the distance between r and the line pq, >and let p' be a point witnessing (*) for the line pq. Without loss, p' >lies between p and q on pq. Then the distance between p' and (at >least) one of the lines rp, rq is smaller than the distance between r >and pq. Contradiction. > As Stephen M-S notes, this proof uses apparenly extraneous metric properties. Is the theorem true in other geometries? Say the two-dimensional rational plane? [Yes, embed it in the real plane...another extraneous construction.] How about the two-complex-dimensional "plane", where line means of course one-complex-dimensional line?? How about planes over finite fields? -- Gerald A. Edgar Internet: edgar@math.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) ============================================================================== Newsgroups: sci.math Subject: Re: Your favourite proofs From: kubo@germain.harvard.edu (Tal Kubo) Date: 15 Jul 94 17:48:30 EDT In article <3062hm$bpb@math.mps.ohio-state.edu> edgar@math.ohio-state.edu (Gerald Edgar) writes: > >As Stephen M-S notes, this proof uses apparenly extraneous >metric properties. Is the theorem true in other geometries? >Say the two-dimensional rational plane? [Yes, embed it in the >real plane...another extraneous construction.] How about >the two-complex-dimensional "plane", where line means of course >one-complex-dimensional line?? How about planes over finite fields? It's true in all of the above. The result follows from a purely combinatorial theorem of Haim Hanani stating that if the N points are not collinear then they determine at least N distinct "lines". This doesn't rely on any geometry whatsoever, just pure incidence relations. ============================================================================== Newsgroups: sci.math From: geoff@math.ucla.edu (Geoffrey Mess) Subject: Re: Your favourite proofs Date: Sat, 16 Jul 94 02:01:04 GMT In article <1994Jul15.033225.33540@hulaw1.harvard.edu> writes: > In article > stephen@mont.cs.missouri.edu (Stephen Montgomery-Smith) writes: > >boshuck@triples.math.mcgill.ca (William Boshuck) writes: > > > >>[...] Any finite collection L of points in the plane satisfying > > > >> (*) a line passing through any two points in L passes > >> through a third point in L > >>is collinear. > > > >>Proof: [ the short proof is ommitted] Tal Kubo: > (Speaking of which, are there projective geometry solutions to either > problem? The one above even has a *combinatorial* solution but it's > not simple.) ??? Gerald Edgar: >As Stephen M-S notes, this proof uses apparenly extraneous >metric properties. Is the theorem true in other geometries? >Say the two-dimensional rational plane? [Yes, embed it in the >real plane...another extraneous construction.] How about >the two-complex-dimensional "plane", where line means of course >one-complex-dimensional line?? How about planes over finite fields? It's false in CP^2: consider the 9 inflection points of a nonsingular cubic. They lie on 12 lines each containing 3 points. It's also false in many planes over finite fields: if the characteristic is not 3 and the field F_q contains enough roots of unity then the Hessian configuration of 9 points and 12 lines reduces mod p to embed in the projective plane over F_q. Here q is a power of p. I think its enough that the field contains a cube root of unity regardless of the characteristic, but I'd have to think some more in characteristic 2. The Hessian example is valuable as motivation for the "extraneous" approach to the problem in the real plane. -- Geoffrey Mess Department of Mathematics, UCLA. geoff@math.ucla.edu ============================================================================== Newsgroups: sci.math From: hrubin@snap.stat.purdue.edu (Herman Rubin) Subject: Re: Your favourite proofs Date: Sat, 16 Jul 1994 20:23:37 GMT In article <3062hm$bpb@math.mps.ohio-state.edu> edgar@math.ohio-state.edu (Geral d Edgar) writes: >In <1994Jul14.201028.8961@sifon.cc.mcgill.ca> boshuck@triples.math.mcgill.ca (W illiam Boshuck) wrote: >>Hi, >> >>I just remembered a really cute proof of the following fact: Any >>finite collection L of points in the plane satisfying >> >> (*) a line passing through any two points in L passes >> through a third point in L >> >>is collinear. How about planes over finite fields? Since a plane over a finite field (affine or projective) is finite, the theorem is obviously false for those planes. -- Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 Phone: (317)494-6054 hrubin@stat.purdue.edu (Internet, bitnet) {purdue,pur-ee}!snap.stat!hrubin(UUCP) ============================================================================== Newsgroups: sci.math From: edgar@math.ohio-state.edu (Gerald Edgar) Subject: Re: Your favourite proofs Date: 16 Jul 1994 17:14:32 -0400 >>>I just remembered a really cute proof of the following fact: Any >>>finite collection L of points in the plane satisfying >>> >>> (*) a line passing through any two points in L passes >>> through a third point in L >>> >>>is collinear. > > How about planes over finite fields? > >Since a plane over a finite field (affine or projective) is finite, >the theorem is obviously false for those planes. Well... Oh, yeah.. I should have said fields of finite characteristic. That't it. -- Gerald A. Edgar Internet: edgar@math.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) ==============================================================================