Equilateral Triangle Proof

Date: 11/05/2001 at 21:14:06
From: Chenelle Horton
Subject: Geometry

The question that I am trying to answer is as follows:

Let a, b, and c be the lengths of the sides of a triangle. Show that 
if a*a + b*b + c*c = bc + ca + ab, then the triangle is equilateral.

I have drawn a picture, but I don't know how to show the equation is 
true without numbers. Can you give me a hint as to how to start?

Thank you so much,
Chenelle

Date: 11/06/2001 at 08:08:06
From: Doctor Floor
Subject: Re: Geometry

Hi, Chenelle,

Thanks for your question.

When we have a^2 + b^2 + c^2 = bc + ca + ab we can derive

  a^2 + b^2 + c^2 - bc - ca - ab = 0

  2a^2 + 2b^2 + 2c^2 - 2bc - 2ca - 2ab = 0

and this is equivalent to 

  (a-b)^2 + (b-c)^2 + (c-a)^2 = 0

(just expand).

But the sum of three squares can only be equal to zero if each square 
itself is equal to zero. That gives a = b, b = c and c = a, and the 
triangle is equilateral.

If you need more help, just write back.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/