Ta có:
\(\frac{a^2+b^2}{c^2+d^2}=\frac{a.b}{c.d}=\frac{a^2+b^2+a.b}{c^2+d^2+c.d}=\frac{a^2+a.b+b^2+a.b}{c^2+c.d+d^2+c.d}\)
\(\frac{a^2+b^2}{c^2+d^2}=\frac{a.b}{c.d}=\frac{a\left(a+b\right)+b\left(a+b\right)}{c\left(c+d\right)+d\left(c+d\right)}=\frac{\left(a+b\right)\left(a+b\right)}{\left(c+d\right)\left(c+d\right)}\)
\(\frac{\left(a+b\right)\left(a+b\right)}{\left(c+d\right)\left(c+d\right)}=\frac{a.b}{c.d}\Rightarrow\frac{c\left(a+b\right)}{a\left(c+d\right)}=\frac{b\left(c+d\right)}{d\left(a+b\right)}\)
\(\Rightarrow\frac{ca+cb}{ca+ad}=\frac{bc+bd}{ad+bd}=\frac{ca+bd}{ca-bd}=1\)
\(\Rightarrow ca+cb=ca+ad\)
\(\Rightarrow cb=ad\)
\(\Rightarrow\frac{a}{b}=\frac{c}{d}\left(đpcm\right)\)
