(a+b+c)^{2} = a^{2}+2*a*b+2*a*c+b^{2}+2*b*c+c^{2}

so we have

a^{2}+2*a*b+2*a*c+b^{2}+2*b*c+c^{2} = a^{2}+b^{2}+c^{2}

Subtract a^{2}+b^{2}+c^{2} from both sides

2*a*b+2*a*c+2*b*c = 0

Divide all terms by 2*a*b*c and we are left with

1/c + 1/b + 1/a = 0

.