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If the complex a,b,c satisfy equations they are real

Source: Romanian TST 2001

January 16, 2011
trigonometrycomplex numbersalgebra proposedalgebra

Problem Statement

Show that if a,b,ca,b,c are complex numbers that such that (a+b)(a+c)=b(b+c)(b+a)=c(c+a)(c+b)=a (a+b)(a+c)=b \qquad (b+c)(b+a)=c \qquad (c+a)(c+b)=a then a,b,ca,b,c are real numbers.
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