If (a + b +c + d) (a -b- c+ d)

=(a + b- c- d) (a -b+c- d);

prove that : a : b =c : d.

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Solution

Given
(a + b +c + d) (a -b- c+ d) =(a + b - c - d) (a -b+c- d)

$\frac{(a + b + c + d)}{(a + b - c - d)}$ = $\frac{(a + b - c - d)}{(a - b - c + d)}$

Applying componendo and dividendo,

we get

=> $\frac{2 (a + b)}{2 (c + d)}$ = $\frac{2 (a - b)}{2 (c - d)}$

=> $\frac{(a + b)}{(c + d)}$ = $\frac{(a - b)}{(c - d)}$

=> $\frac{(a + b)}{(a - b)}$ = $\frac{(c + d)}{c - d}$

applying componendo and dividendo

=> $\frac{a + b + a - b}{a + b - a + b}$ = $\frac{c + d + c - d}{c + d - c + d}$

=> $\frac{2 a}{2 b}$ = $\frac{2 c}{2 d}$

=> $\frac{a}{b}$ = $\frac{c}{d}$

Hence prove
a:b = c:d

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Please Use the Components of Proportion like Componendo and dividends etc

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