Question

a/b+c=b/c+a=c/a+b then prove that each ratio is equal to 1/2 or -1​

Answer 1

Answer:

Shown that, Each ratio $= \frac{1}{2}$ or $= -1$

Step-by-step explanation:

Given that, $\frac{a}{b+c} = \frac{b}{c+a} = \frac{c}{a+b}$

From the first pair:

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

$=>ca + a^2 = b^2 + bc$  [by cross multiplication]

$=>a^2 - b^2 + ca - bc =0$

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

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

∴ $a - b = 0, ie, a = b$, or $a + b + c = 0, ie. a + b = -c$

From the other pairs, we can derive:

$b - c = 0, ie, b = c$, or $a + b + c = 0, ie. b + c = -a$

and

$c - a = 0, ie, c = a$, or $a + b + c = 0, ie. c + a = -b$

∴ $\frac{a}{b + c}$ $=\frac{a}{a + a}$ $=\frac{a}{2a}$ $= \frac{1}{2}$

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

Similarly, $\frac{b}{c + a}$ $= \frac{1}{2}$ or $= -1$

and, $\frac{c}{a + b}$ $= \frac{1}{2}$ or $= -1$