Question

Cho\(\frac{a+b}{b+c}=\frac{c+d}{d+a}\left(c+d\ne0\right)\)

CMR: \(a+b+c+d=0\)hoặc \(a=c\)

Answer 1

\(\frac{a+b}{b+c}=\frac{c+d}{d+a}\left[c+d\ne0\right]\)

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

\(a^2+ad+ab+bd=bc+bd+c^2+cd\)

\(a^2+ad+ab=bc+c^2+cd\)

\(a^2+ad+ab-bc-c^2-cd=0\)

\(\left[a+c\right]\left[a-c\right]+d\left[a-c\right]+b\left[a-c\right]=0\)

\(\left[a-c\right]\left[a+b+c+d\right]=0\)

\(\orbr{\begin{cases}a-c=0\\a+b+c+d=0\end{cases}}\Leftrightarrow\orbr{\begin{cases}a=c\\a+b+c+d=0\end{cases}}\)