Question

[Bảo Ngọc]

cho a,b,c khác 0.

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

cmr a=b=c

Answer 1

[Đinh Đức Hùng]

\(\frac{ab}{a+b}=\frac{ac}{a+c}=\frac{bc}{b+c}\Rightarrow\frac{abc}{c\left(a+b\right)}=\frac{abc}{b\left(a+c\right)}=\frac{abc}{a\left(b+c\right)}\)

\(\Rightarrow c\left(a+b\right)=b\left(a+c\right)\Leftrightarrow ac+bc=ab+bc\Rightarrow ac=ab\Rightarrow c=b\) (1)

\(\Rightarrow b\left(a+c\right)=a\left(b+c\right)\Leftrightarrow ab+bc=ab+ac\Rightarrow bc=ac\Rightarrow b=a\) (2)

\(\Rightarrow c\left(a+b\right)=a\left(b+c\right)\Leftrightarrow ac+bc=ab+ac\Rightarrow bc=ab\Rightarrow c=a\) (3)

Từ (1) ; (2) ; (3) => \(a=b=c\) (ĐPCM)