Perimetre egale
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Claudiu Mindrila
- Fermat
- Posts: 520
- Joined: Mon Oct 01, 2007 2:25 pm
- Location: Targoviste
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Perimetre egale
Problema. Fie \( ABC \) un triunghi oarecare cu \( AD \) bisectoarea unghiului \( \angle BAC \). Sa se demonstreze ca daca triunghiurile \( ABD \) si \( ACD \) au perimetre egale atunci \( AB=AC \).
elev, clasa a X-a, C. N. "C-tin Carabella", Targoviste
\( AB+AD+BD=AC+AD+CD \Leftrightarrow AB+BD=AC+CD \)
Presupunem ca \( DB<DC \)sau \( DB>DC \) sau \( DB=DC \)
\( DB<DC \Leftrightarrow AC>AB \Leftrightarrow AB+BD<AC+CD \)
\( DB>DC \) analog
\( DB=DC \Leftrightarrow AC=AB \Leftrightarrow AB+BD=AC+CD \)
Presupunem ca \( DB<DC \)sau \( DB>DC \) sau \( DB=DC \)
\( DB<DC \Leftrightarrow AC>AB \Leftrightarrow AB+BD<AC+CD \)
\( DB>DC \) analog
\( DB=DC \Leftrightarrow AC=AB \Leftrightarrow AB+BD=AC+CD \)
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