Question

Cho tỉ lệ thức a+b/b+c=c+d/d+a . CMR a=c hoặc a+b+c+d=0

Answer 1

Ta có : a+b/b+c = c+d/d+a 
=> (a+b)/(c+d)= (b+c)/(d+a) 
=> (a+b)/(c+d)+1=(b+c)/(d+a)+1 
hay: (a+b+c+d)/(c+d)=(b+c+d+a)/(d+a) 
- Nếu a+b+c+d khác 0 thì : c+d=d+a => c=a 
- Nếu a+b+c+d = 0 (điều phải chứng minh)

Answer 2

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

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

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

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

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

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

\(\implies\)\(\orbr{\begin{cases}a+b+c+d=0\\\frac{1}{c+d}-\frac{1}{d+a}=0\end{cases}}\)

\(\implies\) \(\orbr{\begin{cases}a+b+c+d=0\\\frac{1}{c+d}=\frac{1}{d+a}\end{cases}}\)

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

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