Question

If $a+b+c\ne 0$ and $\frac{b+c}{a},\frac{c+a}{b},\frac{a+b}{c}$ are in A.P. prove that $\frac{1}{a},\frac{1}{b},\frac{1}{c}$ are also in A.P.

Answer 1

$a+b+c\ne 0,$

$\frac{b+c}{a}\frac{c+a}{b},\frac{a+b}{c}$ are in AP

$\frac{b+c}{a}+1,c+ab+1,a+bc+1$ also in AP

$\frac{a+b+c}{a},\frac{a+b+c}{b},\frac{a+b+c}{c}$ also in AP

Divide by $a+b+c$ gives

$\frac{1}{a},\frac{1}{b},\frac{1}{c}$ also in AP