Equilateral Triangle ProofDate: 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/ |
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