Question

9. Lungimile laturilor unui triunghi sunt a, b,c. Dacă a² +b^2+c^2 = ab + ac + bc, atunci triunghiul este A. isoscel B. dreptunghic C. echilateral D. scalen​

Answer 1

$\displaystyle\\a^2+b^2+c^2=ab+bc+ca \Longleftrightarrow a^2+b^2+c^2-ab-bc-ca=0|\cdot2 \Longleftrightarrow\\\\a^2-2ab+b^2+b^2-2bc+c^2+a^2-2ac+c^2=0\Longleftrightarrow\\\\ (a-b)^2+(b-c)^2+(c-a)^2=0~si~astfel~obtinem~a-b=b-c=c-a=0\\\\de~unde~a=b=c~si~triunghiul~cu~lungimile~laturilor~a,b,c~este~echilateral.$

$\displaystyle---------------------------------------\\Nota:~Puteam~scrie~\sum_{cyc}a^2=\sum_{cyc}ab\Longleftrightarrow \sum_{cyc}a^2-\sum_{cyc}ab=0|\cdot 2\Longleftrightarrow\\\\\sum_{cyc}(a-b)^2=0~si~obtinem~a-b=b-c=c-a=0,~pentru~a~scrie~mai~putin.$