a/b = c/d => 1 - a/b = 1 - c/d

              => b/b - a/b = d/d - c/d

              => (b - a)/b = (d - c)/d

\(\frac{a}{b}=\frac{c}{d}\)

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

\(\Rightarrow1+\frac{-a}{b}=1+\frac{-c}{d}\)

\(\Rightarrow\frac{b-a}{b}=\frac{d-c}{d}\left(dpcm\right)\)

\(\frac{a}{b}:\frac{c}{d}=\frac{a}{b}.\frac{c}{d}\Rightarrow\frac{a}{b}=\frac{a}{b}.\frac{c}{d}.\frac{c}{d}\Rightarrow\frac{a}{b}:\frac{a}{b}=\frac{c}{d}.\frac{c}{d}\)

\(\Rightarrow1=\left(\frac{c}{d}\right)^2\Rightarrow\frac{c}{d}=1\text{ hoặc }\frac{c}{d}=-1\)

ta có: \(\frac{a}{b}:\frac{c}{d}=\frac{a}{b}.\frac{c}{d}\Leftrightarrow\frac{a}{b}.\frac{d}{c}=\frac{a}{b}.\frac{c}{d}\Leftrightarrow\frac{a.d}{b.c}=\frac{a.c}{bd}\Leftrightarrow\frac{d}{c}=\frac{c}{d}\Leftrightarrow d^2=c^2\)

suy ra d=c hoặc d=-c

suy ra \(\frac{c}{d}=\frac{c}{c}=1\) hoặc \(\frac{c}{d}=\frac{c}{-c}=-1\)

thì a.b=c.d