Question

cho \(\dfrac{a+b}{b+c}=\dfrac{c+d}{d+a}\)

chứng minh:rằng: a=c hoặc a+b+c+d=0

Answer 1

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

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

\(\Rightarrow\dfrac{a+b}{c+d}+1=\dfrac{b+c}{d+a}+1\)

\(Hay:\dfrac{a+b+c+d}{c+d}=\dfrac{b+c+d+a}{d+a}\)

Nếu: \(a+b+c+d\ne0\Rightarrow c+d=d+a\Rightarrow c=a\left(dpcm\right)\)

Nếu: \(a+b+c+d=0\left(dpcm\right)\)

Answer 2

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

\(\Rightarrow\dfrac{a+b}{b+c}+1=\dfrac{c+d}{d+a}\)

Hay \(\dfrac{a+b+c+d}{c+d}=\dfrac{b+c+d+a}{d+a}\)

Nếu \(a+b+c+d\ne0\) thì: \(c+d=d+a\Rightarrow c=a\)

Nếu \(a+b+c+d=0\) thì hợp với đề.

\(\rightarrowđpcm\)