If-a-b-c-b-c-a-a-c-b-then-find-a-b-c-

Question Number 214402 by golsendro last updated on 07/Dec/24

If \frac{a + b}{c} = \frac{b + c}{a} = \frac{a + c}{b} then find

\frac{a + b}{c} .

Answered by efronzo1 last updated on 07/Dec/24

Let \frac{a + b}{c} = \frac{b + c}{a} = \frac{a + c}{b} = m

then a + b = mc; b + c = ma; a + c = mb

\Rightarrow a + b + b + c + a + c = mc + ma + mb

2 a + 2 b + 2 c = m (a + b + c)

2 (a + b + c) = m (a + b + c)

m = 2 \Rightarrow \frac{a + b}{c} = 2

Answered by Rasheed.Sindhi last updated on 07/Dec/24

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

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

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

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

{\begin{cases} a + b + c = 0 \Rightarrow a + b = - c \Rightarrow \frac{a + b}{c} = \frac{- c}{c} = - 1 \\ a = b = c \Rightarrow \frac{a + b}{c} = \frac{a + a}{a} = \frac{2 a}{a} = 2 \end{cases}

\frac{a + b}{c} = - 1 o r 2

Answered by Rasheed.Sindhi last updated on 07/Dec/24

\frac{a + b}{c} = \frac{b + c}{a} = \frac{c + a}{b} = \frac{(a + b) + (b + c) + c + a)}{c + a + b}

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

\frac{a + b}{c} = 2