2022/037) If a, b, c are in H.P., how do you prove that $\frac{b+c-a}{a}$, $\frac{c+a-b}{b}$, $\frac{a+b-c}{c}$ are in A.P.​?

 a,b,c are in HP so $\frac{1}{a}$, \frac{1}{b}$, $\frac{1}{c}$ are in AP

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

multiply both LHS and RHS by a + b + c we get

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

Or 

$(\frac{a+b+c}{b} - 2) - (\frac{a+b+c}{a}-2)  = (\frac{a+b+c}{c} -2)  - (\frac{a+b+c}{b} -2) $

or $(\frac{a+c-b }{b}) - (\frac{b+c-a }{a})  = (\frac{a+b-c }{c})  - (\frac{c + a - b}{b} $)

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