Question

CMR nếu có (a+b+c+d)(a−b−c+d)=(a−b+c−d)(a+b−c−d) thì \(\dfrac{a}{b}=\dfrac{c}{d}\)

Answer 1

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

\(\Rightarrow a^2-ab-ac+ad+ab-b^2-bc+bd+ca-bc-c^2+cd+da-db-dc+d^2=a^2+ab-ac-ad-ba-b^2+bc+bd+ca+bc-c^2-cd-da-bd+cd+d^2\) \(\Rightarrow a^2-b^2-c^2+d^2+2ad-2bc=a^2-b^2-c^2+d^2-2ad+2bc\)

\(\Rightarrow2ad-2bc=2bc-2ad\)

\(\Rightarrow ad-bc=bc-ad\)

\(\Rightarrow2ad=2bc\)

\(\Rightarrow ad=bc\)

\(\Rightarrow\dfrac{a}{b}=\dfrac{c}{d}\)

\(\Rightarrowđpcm\)