### 2007 National, Junior 11

Posted:

**Sun Jan 27, 2013 1:06 am**If $a,b,c$ are the sides of a triangle such that $a^2+b^2+c^2=ab+bc+ca$. Prove that the triangle is equilateral.

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Posted: **Sun Jan 27, 2013 1:06 am**

If $a,b,c$ are the sides of a triangle such that $a^2+b^2+c^2=ab+bc+ca$. Prove that the triangle is equilateral.

Posted: **Sun Jan 27, 2013 1:19 pm**

It is also in mathematical quickies and গনিতের মজা মজার গণিত ।

the solution is like this.

\[a^{2}+b^{2}+c^{2}=ab+bc+ca\]

\[2a^{2}+2b^{2}+2c^{2}=2ab+2bc+2ca\]

\[(a-b)^{2}+(b-c)^{2}+(c-a)^{2}=0\]

\[a=b=c\]

the solution is like this.

\[a^{2}+b^{2}+c^{2}=ab+bc+ca\]

\[2a^{2}+2b^{2}+2c^{2}=2ab+2bc+2ca\]

\[(a-b)^{2}+(b-c)^{2}+(c-a)^{2}=0\]

\[a=b=c\]

Posted: **Sat Feb 23, 2019 10:16 pm**

This problem is also a problem of BdMO National 2008 Junior!Fahim Shahriar wrote: ↑Sun Jan 27, 2013 1:06 amIf $a,b,c$ are the sides of a triangle such that $a^2+b^2+c^2=ab+bc+ca$. Prove that the triangle is equilateral.

Question repeat!