Question

[Hoàng ngọc hà]

chứng minh rằng nếu a+b/c+d=b+c/d+a với a+b+c+d khác 0 thì a=c

Answer 1

[ST]

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

<=>\(\frac{a+b}{c+d}+1=\frac{b+c}{d+a}+1\)

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

<=> \(\frac{a+b+c+d}{c+d}-\frac{a+b+c+d}{d+a}=0\)

<=> \(\left(a+b+c+d\right)\left(\frac{1}{c+d}-\frac{1}{d+a}\right)=0\)

<=> \(\orbr{\begin{cases}a+b+c+d=0\\\frac{1}{c+d}-\frac{1}{d+a}=0\end{cases}\Leftrightarrow\orbr{\begin{cases}a+b+c+d=0\\c=a\end{cases}\left(đpcm\right)}}\)