Giải bài 4 trang 59 – SGK Toán lớp 8 tập 2

a) Ta có:  $\dfrac{AB'}{AB} = \dfrac{AC'}{AC}$
$\Rightarrow \dfrac{AB}{AB'} = \dfrac{AC}{AC'} \\ \Leftrightarrow \dfrac{AB}{AB'} - 1 = \dfrac{AC}{AC'} - 1 \\ \Leftrightarrow \dfrac{AB}{AB'} - \dfrac{AB'}{AB'} = \dfrac{AC}{AC'} - \dfrac{AC'}{AC'} \\ \Leftrightarrow \dfrac{AB - AB'}{AB'} = \dfrac{AC - AC'}{AC'} \\ \Leftrightarrow \dfrac{BB'}{AB'} = \dfrac{CC'}{AC'} \\ \Rightarrow \dfrac{AB'}{BB'} = \dfrac{AC'}{CC'} \,\, \text{(đpcm)}$
b) Ta có: $\dfrac{AB'}{AB} = \dfrac{AC'}{AC}$
Mà $AB' = AB - BB'; \, AC' = AC - CC' \\$
$\Rightarrow \dfrac{AB'}{AB} = \dfrac{AC'}{AC} \\ \Leftrightarrow \dfrac{AB - BB'}{AB} = \dfrac{AC - CC'}{AC} \\ \Leftrightarrow \dfrac{AB }{AB} - \dfrac{BB' }{AB} = \dfrac{AC }{AC} - \dfrac{CC' }{AC}\\ \Leftrightarrow 1 - \dfrac{BB' }{AB} = 1 - \dfrac{CC' }{AC}\\ \Rightarrow \dfrac{BB' }{AB} = \dfrac{CC' }{AC} \,\, \text{(đpcm)}$