Đặt \(\frac{a}{b}=\frac{c}{d}=k\Rightarrow a=bk,c=dk\)
Ta có :
\(\frac{a\times b}{c\times d}=\frac{bk\times b}{dk\times d}=\frac{b^2\times k}{d^2\times k}=\frac{b^2}{d^2}\left(1\right)\)
\(\frac{\left(a+b\right)^2}{\left(c+d\right)^2}=\frac{\left(bk+b\right)^2}{\left(dk+d\right)^2}=\frac{\left(b\times\left(k+1\right)\right)^2}{\left(d\times\left(k+1\right)\right)^2}=\frac{b^2\times\left(k+1\right)^2}{d^{2\times}\left(k+1\right)^2}=\frac{b^2}{d^2}\left(2\right)\)
Từ (1) và (2) , ta có :\(\frac{a\times b}{c\times d}=\frac{\left(a+b\right)^2}{\left(c+d\right)^2}\)