Let the three sides of a triangle be $a,b$ and $c$. If the equation
$$a^2+b^2+c^2=ab +bc+ac$$
holds true, then the triangle is an equilateral triangle.
How do we prove this? An answer or even the slightest hint will be appreciated.
Asked
Active
Viewed 133 times
1 Answers
12
Note that $$a^2 + b^2 + c^2 = ab + bc + ca \implies (a-b)^2 + (b-c)^2 + (c-a)^2 = 0$$ I trust you can finish it off from here.
-
Thank you.I of course would love to know how you figured it out that quick. – rahul May 16 '13 at 07:18
-
@rahul Experience. – May 16 '13 at 07:19
-
@rahul, times 2 on both sides. – Easy May 16 '13 at 07:19
-
@Easy,I understood that.What I meant was how it took him hardly a minute to have proved it,which, as he says,is experience. – rahul May 16 '13 at 07:25
-
@Easy you are right from experience and memory this type of thing we can solve easily. – iostream007 May 16 '13 at 11:17