Có : \(\dfrac{a}{c}=\dfrac{b}{d}\Leftrightarrow\dfrac{a^2}{c^2}=\dfrac{b^2}{d^2}\)
Áp dụng tính chất dãy tỉ số ...... :
\(\dfrac{a^2}{c^2}=\dfrac{b^2}{d^2}=\dfrac{a^2+b^2}{c^2+d^2}\)
\(\Leftrightarrow\dfrac{a^2}{c^2}=\dfrac{a^2+b^2}{c^2+d^2}\)
\(\Leftrightarrow\dfrac{a}{c}.\dfrac{b}{d}=\dfrac{a^2+b^2}{c^2+d^2}\)
Ta có: \(\dfrac{a}{b}=\dfrac{c}{d}\)
\(\Rightarrow\dfrac{a}{c}=\dfrac{b}{d}\)
\(\Rightarrow\left(\dfrac{a}{c}^2\right)=\left(\dfrac{b}{d}\right)^2=\dfrac{ab}{cd}\)
\(\Rightarrow\dfrac{a^2}{c^2}=\dfrac{b^2}{d^2}=\dfrac{ab}{cd}\)
Áp dụng tính chất của dãy tỉ số bằng nhau ta có:
\(\dfrac{ab}{cd}=\dfrac{a^2}{c^2}=\dfrac{b^2}{d^2}=\dfrac{a^2+b^2}{c^2+d^2}\)
\(\Rightarrow\dfrac{a^2+b^2}{c^2+d^2}=\dfrac{ab}{cd}\left(đpcm\right)\)