Ta có: \(\left(a+b+c-d\right)\left(a-b-c-d\right)=\left(a+b-c+d\right)\left(a-b+c+d\right)\)
\(\Rightarrow\frac{a+b+c-d}{a+b-c+d}=\frac{a-b+c+d}{a-b-c-d}\Leftrightarrow\frac{\left(a+b\right)+\left(c-d\right)}{\left(a+b\right)-\left(c-d\right)}=\frac{\left(a-b\right)+\left(c+d\right)}{\left(a-b\right)-\left(c+d\right)}.\)
Đặt \(A=a+b;B=c-d;C=a-b;D=c+d.\)Ta được:
\(\frac{A+B}{A-B}=\frac{C+D}{C-D}\Rightarrow\frac{A}{B}=\frac{C}{D}\Leftrightarrow\frac{a+b}{c-d}=\frac{a-b}{c+d}\Rightarrow\frac{a+b}{a-b}=\frac{c-d}{c+d}\)
Vậy ta được:
\(\left(a+b+c-d\right)\left(a-b-c-d\right)=\left(a+b-c+d\right)\left(a-b+c+d\right)\)
\(\Rightarrow\frac{a+b}{a-b}=\frac{c-d}{c+d}.\)
