Cho tam giác $ABC$ thỏa mãn: $a+b+c=2(a\cos A+b\cos B+c\cos C)$.
Chứng minh tam giác $ABC$ đều
Ta có: $a=b\cos C+c\cos B, b=c\cos A+a\cos C,c=a\cos B+b\cos A$
nên điều kiện đã cho tương đương với:
$(b\cos C+c\cos B)+(c\cos A+a\cos C)+(a\cos B+b\cos A)$
$=2(a\cos A+b\cos B+c\cos C)$
$ \Leftrightarrow (a\cos A+b\cos B-a\cos B-b\cos A)+(b\cos B+c\cos C$
$-b\cos C-c\cos B)+(c\cos C+a\cos A-c\cos A-a\cos C)=0$
$ \Leftrightarrow (a-b)(\cos A-\cos B)+(b-c)(\cos B-\cos C)+(c-a)(\cos C-\cos A)=0$
Ta chứng minh: $(a-b)(\cos A-\cos B) \leq  0$, dấu "=" xảy ra khi $a=b$
Xét $a=b$ thì bất phương trình đúng.
Xét $a>b$ thì $A>B \Rightarrow  \cos A<\cos B \Rightarrow  (a-b)(\cos A-\cos B)<0$
Xét $a<b$ thì $A<B \Rightarrow  \cos A>\cos B \Rightarrow  (a-b)(\cos A-\cos B)<0$
Tương tự thì $(b-c)(\cos B-\cos C) \leq  0, (c-a)(\cos C-\cos A) \leq  0$
Do đó, dấu đẳng thức đồng thời xảy ra nên $a=b=c$.
Vậy tam giác $ABC$ đều.

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