Question

If $\frac{b+c}{a},\frac{c+a}{b},\frac{a+b}{c}$ are in A.P , show that $bc,ca,ab$ are in A.P

Answer 1

Given,

$\frac{b+c}{a},\frac{c+a}{b},\frac{a+b}{c}$ are in A.P.

$\Rightarrow \frac{1}{a},\frac{1}{b},\frac{1}{c}$ are also in A.P.

$\therefore \frac{1}{b}-\frac{1}{a}=\frac{1}{c}-\frac{1}{b}$

$\frac{a-b}{ab}=\frac{b-c}{bc}$

$\frac{a-b}{a}=\frac{b-c}{c}$

$c(a-b)=a(b-c)$

$ac-bc=ab-ac$

$2ac=ab+bc$

Therefore $ab,bc,ca$ are in A.P.