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
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