Question

If $(a+b+c+d)(a-b-c+d)=(a+b-c-d)(a-b+c-d),$
prove that $a:b=c:d$

Answer 1

prove that $a:b=c:d$
Given:$(a+b+c+d)(a-b-c+d)=(a+b-c-d)(a-b+c-d),$

Concept of componendo and dividendo to be use
If $\frac{a}{b}=\frac{c}{d}$ then $\frac{a+b}{a-b}=\frac{c+d}{c-d}$

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

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

$\Rightarrow \frac{2a+2b}{2c+2d}=\frac{2a-2b}{2c-2d}$

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

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

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

[By using componendo and dividendo]
$\Rightarrow \frac{2a}{2b}=\frac{2c}{2d}$

$\Rightarrow a:b=c:d$

Hence, proved.