if a b c are in H.P then prove that (a/(b+c)),(b/(c+a)),(c/(a+b)) arw also in H.P.


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Question Number 40236 by scientist last updated on 17/Jul/18

$i f a b c a r e i n H . P t h e n p r o v e t h a t \frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b} a r w$ $a l s o i n H . P .$
	Answered by tanmay.chaudhury50@gmail.com last updated on 17/Jul/18

	$a, b, c H . P$ $s o \frac{1}{a}, \frac{1}{b}, \frac{1}{c} i n A . P$ $\frac{2}{b} = \frac{1}{a} + \frac{1}{c}$ $t o p r o v e \frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b} i n H . P$ $i f w e c a n p r o v e r e c i p r o c a l a r e i n A . P$ $\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c} a r e i n A . P$ $n o w \frac{1}{a}, \frac{1}{b}, \frac{1}{c} a r e i n A . P$ $m u l t i p l y a + b + c w i t h e a c h t e r m . . r e s u l t s a r e$ $i n A . P$ $\frac{a + b + c}{a}, \frac{a + b + c}{b}, \frac{a + b + c}{c} i n A . P$ $1 + \frac{b + c}{a}, 1 + \frac{a + c}{b}, 1 + \frac{a + b}{c} i n A . P$ $\frac{b + c}{a}, \frac{a + c}{b}, \frac{a + b}{c} i n A . P$ $\frac{a}{b + c}, \frac{b}{a + c}, \frac{c}{a + b} a r e i n H . P p r o v e d$
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