Question

CMR nếu \(\dfrac{a+b}{a-b}=\dfrac{c+a}{c-a}\)thì a²= bc

Answer 1

Ta có : \(\dfrac{a+b}{a-b}=\dfrac{c+a}{c-a}\)

\(\Rightarrow\left(a+b\right)\left(c-a\right)=\left(a-b\right)\left(c+a\right)\)

\(\Rightarrow ac-a^2+bc-ab=ac+a^2-bc-ba\)

\(\Rightarrow ac-a^2+bc-ab-\left(ac+a^2-bc-ab\right)=0\)

\(\Rightarrow ac-a^2+bc-ab-ac-a^2+bc+ab=0\)

\(\Rightarrow\left(ac-ac\right)+\left(-ab+ab\right)-a^2-a^2+bc+bc=0\)\(\Rightarrow-2a^2+2bc=0\)

\(\Rightarrow-2a^2=-2bc\)

\(\Rightarrow a^2=bc\left(đpcm\right)\)