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Teorema lui Pompeiu

 
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AndraS
Euclid


Joined: 22 Nov 2008
Posts: 10

PostPosted: Thu Dec 11, 2008 6:10 pm    Post subject: Teorema lui Pompeiu Reply with quote

Fie ABC un triunghi echilateral si P un punct din plan. Sa se arate ca segmentele [PA], [PB], [PC] pot fi laturile unui triunghi .
Una dintre metodele utilizate in rezolvare este cea cu numere complexe si tocmai pe aceea as dori sa o aflu. Smile
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Marius Mainea
Gauss


Joined: 26 May 2008
Posts: 1099
Location: Gaesti (Dambovita)

PostPosted: Thu Dec 11, 2008 9:06 pm    Post subject: Reply with quote

Fie P(z), A(a), B(b) ,C(c) cele patru puncte cu afixele lor.

Avem relatia : (z-a)(b-c)+(z-b)(c-a) +(z-c)(a-b)=0 (calcul)

Trecand la module |(z-a)(b-c)+(z-b)(c-a)|=|-(z-c)(a-b)| si folosind |x+y|\le |x|+|y|

obtinem

PA\cdot BC+PB\cdot CA\ge PC\cdot AB deci PA+PB\ge PC si analoagele.
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andrusca



Joined: 10 Oct 2010
Posts: 2

PostPosted: Sun Oct 10, 2010 4:21 pm    Post subject: Reply with quote

As dori daca poate sa ma ajute cineva in rezolvarea acestei probleme utilizand teorema lui pompeiu

Fie M un punct arbitrar in interiorul triunghiului ABC echilateral. Sa se arate ca [AB] ; [BC] ;[CM] pot fi laturile unui triunghi.

Multumesc anticipat
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