\(\dfrac{a^2+b^2}{b^2+c^2}=\dfrac{a^2+ac}{ac+c^2}=\dfrac{a\left(a+c\right)}{c\left(a+c\right)}=\dfrac{a}{c}\left(đpcm\right)\)
Thay b2 = ac vào biểu thức trên, ta có:
\(\dfrac{a^2+ac}{ac+c^2}=\dfrac{a\left(a+c\right)}{c\left(a+c\right)}=\dfrac{a}{c}\)
\(\Rightarrow\dfrac{a^2+b^2}{b^2+c^2}=\dfrac{a}{c}\)
\(b^2=ac\Leftrightarrow\dfrac{a}{b}=\dfrac{b}{c}\)
Áp dụng t/c dtsbn:
\(\dfrac{a}{b}=\dfrac{b}{c}=\dfrac{a^2}{b^2}=\dfrac{b^2}{c^2}=\dfrac{a^2+b^2}{b^2+c^2}\left(1\right)\)
Ta có \(b^2=ac\Leftrightarrow\dfrac{ac}{c^2}=\dfrac{b^2}{c^2}\Leftrightarrow\dfrac{a}{c}=\dfrac{b^2}{c^2}\left(2\right)\)
\(\left(1\right)\left(2\right)\LeftrightarrowĐpcm\)