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Teorema lui Thales

Teorema lui Thales. Baza geometriei Clasa VII-a. Grecu Ioan. Fie ABC un triunghi oarecare , B’ Є ﴾ AB ﴿ C’ Є ﴾ AC ﴿. A. B’. C’. C. B. B’C’ ║ BC. Dac ă :. A. B’. C’. B. C. Atunci :. AB ’ = AC’ AB AC. Demonstra ţie. Se traseaz ă : B’E ┴ AC C ’D ┴ AB

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Teorema lui Thales

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  1. Teorema lui Thales Baza geometriei Clasa VII-a Grecu Ioan

  2. Fie ABC un triunghi oarecare , B’ Є﴾AB﴿ C’ Є﴾AC﴿ A B’ C’ C B

  3. B’C’ ║ BC Dacă: A B’ C’ B C Atunci: AB’ = AC’ AB AC

  4. Demonstraţie Se trasează : B’E ┴ AC C’D ┴ AB BM ┴ B’C’ şi CN ┴ B’C’ A D ┌ E ┐ C’ M N B’ -------------------- ┘ └ B C

  5. AB’ · C’D 2 AB’ = = ﴾1﴿ BB’ · C’D BB’ TAB’C’TC’BB’ 2 AC’ · B’E TAB’C’TB’C’C 2 AC’ ﴾2﴿ = = CC’ CC’ · B’E 2

  6. În dreptunghiul BCNM=> BM ═ CN => = > T BB’C’ = T CC’B’ ﴾3﴿ BM · B’C’ CN · B’C’ = 2 2 AC’ AB’ Din (1), (2), (3) = > = = = > BB’ CC’ AB’ AC’ AC’ AB’ = = > = BB’ + AB’ AB CC’ + AC’ AC

  7. În dreptunghiul BCNM=> BM ═ CN =>=> = > T BB’C’ = T CC’B’ ﴾3﴿ BM · B’C’ CN · B’C’ = 2 2 A B’ C’ M N B C

  8. Din (1), (2), (3)= > AC’ AB’ = = = > BB’ CC’ AB’ AC’ = > = BB’ + AB’ CC’ + AC’ AB’ AC’ = > = AB AC

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