\(\dfrac{a+b}{a-b}=\dfrac{c+d}{c-d}\Rightarrow\dfrac{a+b}{c+d}=\dfrac{a-b}{c-d}\)
Áp dụng tính chất dãy tỉ số bằng nhau, ta có :
\(\dfrac{a+b}{c+d}=\dfrac{a-b}{c-d}=\dfrac{a+b+a-b}{c+d+c-d}=\dfrac{2a}{2c}=\dfrac{a}{c}\) \(\left(1\right)\)
\(\dfrac{a+b}{c+d}=\dfrac{a-b}{c-d}=\dfrac{a+b-a+b}{c+d-c+d}=\dfrac{2b}{2d}=\dfrac{b}{d}\) \(\left(2\right)\)
Từ \(\left(1\right),\left(2\right)\), ta có :
\(\dfrac{a}{c}=\dfrac{b}{d}\Rightarrow\dfrac{a}{b}=\dfrac{c}{d}\)
Ta có: \(\dfrac{a+b}{a-b}=\dfrac{c+d}{c-d}\)
\(\rightarrow\left(a+b\right)\left(c-d\right)=\left(a-b\right)\left(c+d\right)\)
\(\rightarrow ac-ad+bc-bd=ac+ad-bc-bd\)
\(\rightarrow-ad+bc=ad-bc\)
\(\rightarrow bc+bc=ad+ad\)
\(\rightarrow2bc=2ad\)
\(\rightarrow bc=ad\)
\(\rightarrow\dfrac{a}{b}=\dfrac{c}{d}\left(đpcm\right)\)
Chúc bạn học tốt!
