Question

cho  \(\frac{a+b}{c+d}=\frac{b+c}{d+a}\)trong đó a+b+c+d khác 0 . CM:  a=c

Answer 1

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

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

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

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

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

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

xét a< c =>a.(a+b+d)<c(c+b+d)

xét a>c =>a.(a+b+d)>c(c+b+d)

=>a=c

=>đpcm