\(\dfrac{1}{c}=\dfrac{1}{2}\left(\dfrac{1}{a}+\dfrac{1}{b}\right)\\ \Rightarrow\dfrac{1}{c}=\dfrac{1}{2}\left(\dfrac{a}{ab}+\dfrac{b}{ab}\right)\\ \Rightarrow\dfrac{1}{c}=\dfrac{1}{2}.\dfrac{a+b}{ab}\\ \Rightarrow\dfrac{1}{c}=\dfrac{a+b}{2ab}\\ \Rightarrow\left(a+b\right).c=2ab\\ \Rightarrow ac+bc=ab+ab\\ \Rightarrow ac-ab=ab-bc\\ \Rightarrow a\left(c-b\right)=b\left(a-c\right)\\ \Rightarrow\dfrac{a}{b}=\dfrac{a-c}{c-b}\)
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