Question

If a, b, c are in H.P., show that $\frac{a}{b+c},\frac{b}{c+a},\frac{c}{a+b}$ are also in H.P.

Answer 1

Given that a, b, c are in H.P.
$\Rightarrow \frac{1}{a},\frac{1}{b},\frac{1}{c}$ are in A.P.
$\Rightarrow \frac{a+b+c}{a},\frac{a+b+c}{b},\frac{a+b+c}{c}$ are in A.P.
$\Rightarrow 1+\frac{b+c}{a},1+\frac{c+a}{b},1+\frac{a+b}{c}$ are in A.P.
$\Rightarrow \frac{b+c}{a},\frac{c+a}{b},\frac{a+b}{c}$ are in A.P.
$\Rightarrow \frac{a}{b+c},\frac{b}{c+a},\frac{c}{a+b}$ are in H.P.