\(\frac{a^2+b^2}{c^2+d^2}=\frac{ab}{cd}.\)
\(\Rightarrow\left(a^2+b^2\right).cd=ab.\left(c^2+d^2\right)\)
\(\Rightarrow a^2cd+b^2cd=abc^2+abd^2\)
\(\Rightarrow a^2cd+b^2cd-abc^2-abd^2=0\)
\(\Rightarrow\left(a^2cd-abc^2\right)+\left(b^2cd-abd^2\right)=0\)
\(\Rightarrow ac.\left(ad-bc\right)+bd.\left(bc-ad\right)=0\)
\(\Rightarrow ac.\left(ad-bc\right)-bd.\left(ad-bc\right)=0\)
\(\Rightarrow\left(ad-bc\right).\left(ac-bd\right)=0\)
\(\Rightarrow\left[{}\begin{matrix}ad-bc=0\\ac-bd=0\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}ad=bc\\ac=bd\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}\frac{a}{b}=\frac{c}{d}\left(đpcm\right).\\\frac{a}{b}=\frac{d}{c}\end{matrix}\right.\)
Vậy \(\frac{a}{b}=\frac{c}{d}.\)
Chúc bạn học tốt!
