Question

Cho \(\frac{a+b}{b+c}\) = \(\frac{c+d}{d+a}\) ( Với c+d ≠ 0 , b+c ≠ 0 , d+a ≠ 0 )

Chứng minh a = c hoặc a + b + c + d = 0

Answer 1

https://i.imgur.com/z4bn8DU.jpg

Answer 2

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

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

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

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

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

Nếu \(a+b+c+d\ne0.\)

\(\Rightarrow c+d=d+a\)

\(\Rightarrow c=a\left(đpcm1\right).\)

Nếu \(a+b+c+d=0\) thì hợp với đề.

\(\Rightarrow a+b+c+d=0\left(đpcm2\right).\)

Chúc bạn học tốt!