Sylvesterin–Gallain lause

On määrätty tasolla n kpl pisteitä (A, B, C ja P), jotka eivät kuulu samalle suoralle. Sylvesterin-Gallain lauseen mukaisesti tällöin on olemassa sellainen suora (m), joka kulkee täsmälleen kahden pisteen kautta.

Sylvesterin–Gallain lause on kuuluisa geometrian lause, joka kuuluu seuraavasti:

Jos on määrätty n pistettä tasossa eivätkä ne sijaitse samalla suoralla, on aina olemassa sellainen suora, joka kulkee täsmälleen kahden määrätyn tason pisteen kautta.[1]

Lauseen otaksui James Joseph Sylvester vuonna 1893 ja todisti Tibor Gallai vuonna 1944. Lauseesta on olemassa yleisempikin versio nimeltä Beckin lause. Sylvesterin–Gallain lause ei ole voimassa jos pisteitä on äärettömän monta. Eräs vastaesimerkki saadaan lattiisista .

Todistus

Sylvesterin–Gallain lauseen todistus: Oletetaan vastoin väitettä, että kaikki pisteet eivät ole samalla suoralla. Tarkastellaan niiden suorien joukkoa, jotka kulkevat ainakin kahden annetun pisteen kautta. Näiden joukossa on suora l ja piste A siten, että etäisyys ||A-l|| on positiivinen, mutta pienin mahdollinen annettujen pisteiden joukossa. Oletetaan, että suoralla l on enemmän kuin kaksi pistettä. Tällöin ainakin kaksi l:n pistettä, olkoot ne B ja C, ovat samalla puolella pisteen A ja p kautta kulkevaa suoraa, missä p on A:n projektio l:llä. Tällöin joko ||AB-C||<||A-l|| tai ||AC-B||<||A-l||, mikä on ristiriita A:n ja l:n minimaalisen etäisyyden kanssa.

Lähteet

  1. ↑ Martin Aigner & Günter M. Ziegler: Proofs from the book, 3. edition, s. 68. Springer, 2004. ISBN 9783540404606 Google book (preview). (englanniksi)
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