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gabriela_brb
Euclid


Joined: 07 Feb 2013
Posts: 12

PostPosted: Thu Oct 12, 2017 8:59 pm    Post subject: Romb Reply with quote

Va rog daca se poate un ajutor pt o problema de clasa 7.
In triunghiul ABC, cu AB=AC, m(<A)=40 grade, consideram punctul D pe AC astfel incat m(<ABD)=60 grade. Daca bisectoarea unghiului A intersecteaza dreapta BD in punctul E si prelungim AE cu segmentul ET congruent cu latura AB, aratati ca patrulaterul ABTC este romb.
Va multumesc mult !
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Laurian Filip
Site Admin


Joined: 25 Nov 2007
Posts: 476
Location: Bucuresti

PostPosted: Tue Oct 17, 2017 1:39 pm    Post subject: Reply with quote

Se observa ca m(\angle AED)=m(\angle ADE)=80^\circ si deci triunghiul AED este isoscel.

Fie AD=AE=x, CD=y. Atunci avem AB=AC=ET=x+y.

Notand AT \cap BC = \{ O \} si aplicand T. Menelaus in triunghiul AOC cu transversala B-E-D, obtinem EO=\frac{CD}{2}=\frac{y}{2}. Avem acum AO=OT=x+\frac{y}{2}. Pe de alta parte, AO este inaltime in triunghiul ABC isoscel, deci AO este mediana si implicit BO=CO.

Observam ca in patrulaterul ABTC, medianele se injumatatesc si sunt perpendiculare deci ABTC este romb.
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