Question

1.Chứng minh:

\(\dfrac{a+b}{a-b}=\dfrac{c+a}{c-a}\Rightarrow a^2=bc\)

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Answer 1

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

Mặt khác: \(\dfrac{a+b}{c+a}=\dfrac{a-b}{c-a}=\dfrac{\left(a+b\right)-\left(a-b\right)}{\left(c+a\right)-\left(c-a\right)}=\dfrac{2b}{2a}=\dfrac{b}{a}\left(2\right)\)

Từ (1) và (2) suy ra \(\dfrac{a}{c}=\dfrac{b}{a}\Rightarrow a^2=bc\)