If-a-b-c-b-c-a-c-a-b-1-and-a-b-c-0-then-prove-that-1-a-1-b-1-c-

Question Number 211485 by MATHEMATICSAM last updated on 10/Sep/24

If \frac{a - b}{c} + \frac{b - c}{a} + \frac{c + a}{b} = 1 and

a - b + c \neq 0 then prove that

\frac{1}{a} = \frac{1}{b} + \frac{1}{c} .

Answered by Frix last updated on 10/Sep/24

\frac{a - b}{c} + \frac{b - c}{a} + \frac{c + a}{b} = 1

(a - b + c) (a b + a c - b c) = 0

a - b + c \neq 0 \Rightarrow a b + a c - b c = 0

\frac{a b}{a b c} + \frac{a c}{a b c} - \frac{b c}{a b c} = 0

\frac{1}{c} + \frac{1}{b} - \frac{1}{a} = 0

\frac{1}{a} = \frac{1}{b} + \frac{1}{c}

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