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Math Help - complex numbers - Equilateral triangle

  1. #1
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    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

    i cant thin of a equation about an Equilateral triangle. please help
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  2. #2
    MHF Contributor
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    Quote Originally Posted by stud_02 View Post
    a,b,c are complex numbers.
    if abc is an Equilateral triangle,show

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

    i cant thin of a equation about an Equilateral triangle. please help
    Hi

    Have a look to this link

    http://www.mathhelpforum.com/math-he...x-numbers.html
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  3. #3
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    thanks.is there any other way to prove it without vectors
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  4. #4
    Senior Member pankaj's Avatar
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    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

     <br />
(a-b)(\bar{a}-\bar{b})=(b-c)(\bar{b}-\bar{c})=(c-a)(\bar{c}-\bar{a})=k^2<br />

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

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

    (3)  <br />
\bar{c}-\bar{a}=\frac{k^2}{c-a}<br />

    Adding (1),(2) and (3)

     <br />
0=\frac{k^2}{a-b}+\frac{k^2}{b-c}+\frac{k^2}{c-a}<br />

     <br />
\frac{1}{a-b}+\frac{1}{b-c}+\frac{1}{c-a}=0<br />

    Taking LCM

     <br />
a^2+b^2+c^2=ab+bc+ca<br />
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