Wow!! This one put relation geometry with algebra!

[Hasib]

A triangle satisfy the following equation: $a^2+b^2+c^2=ab+bc+ca$
prove that, the triangle's angle are the same angle=60

\[\clubsuit\]

[Hasib]

It's a nice problem!\[\clubsuit\clubsuit\]

[Moon]

Yes, it is nice, and quite old problem (I guess it appeared in BdMO, TWICE!!!)
Just use the famous trick, the square of a real number is always $\geq 0$.
What are you waiting for?? Just make a square.

[Hasib]

Yeah! Now i can do it. Bt while i see this in BdMO, i tried so hard. Bt, couldn't do it.

Oh! Please don't post the solution. Because, it's a good problem for the begginers of BdMO!\[\clubsuit\]

[Muhiminul Maruf]

its a very ^3 -1 nice problem

[mugdho.snigdho]

I solved thus:(a+b+c)(a^2 +b^2 +c^2 -ab-bc-ca)=a^3 +b^3 +c^3-3abc=.5(a+b+c)((a-b)^2 +(b-c)^2 +(c-a)^2)=o. so,a=b=c[since, a+b+c not equals to o]

[Moon]

^Hey, Musician Borthers, please learn LaTeX to write equations. The basic stuff is very easy, for example you could just write your equation between two dollar signs, and it would look as nice as the following..
$(a+b+c)(a^2 +b^2 +c^2 -ab-bc-ca)=a^3 +b^3 +c^3-3abc$
$=\frac{1}{2}(a+b+c)((a-b)^2 +(b-c)^2 +(c-a)^2)=0$

[Tahmid Hasan]

এইখানে লাঈক দেয়ার উপাই থাকলে মুন ভাই এর reply তে ১টা likeদিতাম