Question

chứng minh rằng : (a+b+c+d)(a-b-c+d)=(a-b+c-d)(a+b-c-d) thì a/c=b/d

Answer 1

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

\(=-\left(b+c\right)^2+\left(a+d\right)^2\)   ( 1 )

\(\left(a+b-c-d\right)\left(a-b+c-d\right)=\left(b-c\right)^2-\left(a-d\right)^2\)    ( 2 )

Từ ( 1 ) và ( 2 ) suy ra 

\(b^2+2bc+c^2-a^2-2ad-d^2=a^2-2ad+d^2-b^2+2bc-c^2\)

\(4ad=4ac\Rightarrow ad=bc\)

\(\Rightarrow\)\(\frac{a}{c}=\frac{b}{d}\)( đpcm )