\(\frac{1}{c}=\frac{1}{2}\left(\frac{1}{a}+\frac{1}{b}\right)\\ \Rightarrow2\cdot\frac{1}{c}=\frac{a}{ab}+\frac{b}{ab}\\ \frac{2}{c}=\frac{a+b}{ab}\\ \Rightarrow2ab=c\left(a+b\right)\\ ab+ab=ca+cb\\ ab-cb=ca-ab\\ b\left(a-c\right)=a\left(c-b\right)\\ \Rightarrow\frac{a}{b}=\frac{a-c}{c-b}\)
\(\frac{1}{c}=\frac{1}{2}.\left(\frac{1}{a}+\frac{1}{b}\right)\)
\(\Rightarrow\frac{1}{c}:\frac{1}{2}=\frac{1}{a}+\frac{1}{b}\)
\(\Rightarrow\frac{2}{c}=\frac{1}{a}+\frac{1}{b}\)
\(\Rightarrow\frac{2}{c}=\frac{a}{ab}+\frac{b}{ab}\)
\(\Rightarrow\frac{2}{c}=\frac{a+b}{ab}\)
\(\Rightarrow2ab=\left(a+b\right).c\)
\(\Rightarrow2ab=ac+bc\)
\(\Rightarrow ab+ab=ac+bc\)
\(\Rightarrow ab-bc=ac-ab\)
\(\Rightarrow b.\left(a-c\right)=a.\left(c-b\right)\)
\(\Rightarrow\frac{a}{b}=\frac{a-c}{c-b}\left(đpcm\right).\)
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