Question

[Mio HiHiHiHi]

Cho a,b,c\(\in\)Z:

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

Chứng minh rằng:\(\frac{a-c}{c}\)=\(\frac{c}{b-c}\)

Answer 1

[Pham Van Hung]

\(\frac{1}{a}+\frac{1}{b}=\frac{1}{c}\Rightarrow\frac{bc+ac}{abc}=\frac{ab}{abc}\Rightarrow bc+ac=ab\)

\(\Rightarrow ab-ac-bc=0\Rightarrow a\left(b-c\right)-c\left(b-c\right)=c^2\)

\(\Rightarrow\left(b-c\right)\left(a-c\right)=c^2\Rightarrow\frac{a-c}{c}=\frac{c}{b-c}\)