# Thread: complex numbers - Equilateral triangle

1. ## complex numbers - Equilateral triangle

a,b,c are complex numbers.
if abc is an Equilateral triangle,show

$a^2 + b^2 + c^2 = ab + bc + ca$

2. Originally Posted by stud_02
a,b,c are complex numbers.
if abc is an Equilateral triangle,show

$a^2 + b^2 + c^2 = ab + bc + ca$

Hi

Have a look to this link

http://www.mathhelpforum.com/math-he...x-numbers.html

3. thanks.is there any other way to prove it without vectors

4. Let $|a-b|=|b-c|=|c-a|=k$ (since a,b and c are vertices of an equilateral triangle)

$|a-b|^2=|b-c|^2=|c-a|^2=k^2$

$
(a-b)(\bar{a}-\bar{b})=(b-c)(\bar{b}-\bar{c})=(c-a)(\bar{c}-\bar{a})=k^2
$

(1) $\bar{a}-\bar{b}=\frac{k^2}{a-b}$

(2) $\bar{b}-\bar{c}=\frac{k^2}{b-c}$

(3) $
\bar{c}-\bar{a}=\frac{k^2}{c-a}
$

$
0=\frac{k^2}{a-b}+\frac{k^2}{b-c}+\frac{k^2}{c-a}
$

$
\frac{1}{a-b}+\frac{1}{b-c}+\frac{1}{c-a}=0
$

Taking LCM

$
a^2+b^2+c^2=ab+bc+ca
$