a) $\dfrac{AB}{AC} = \dfrac{AH}{CH}$
$\Rightarrow \dfrac{AB^2}{AC^2} = \dfrac{AH^2}{CH^2} = \dfrac{BH.CH}{CH^2} = \dfrac{BH}{CH}$
b) $\dfrac{AB^2}{AC^2} = \dfrac{BH}{CH}$
$\Rightarrow \dfrac{AB^4}{AC^4} = \dfrac{BH^2}{CH^2} = \dfrac{BE.AB}{CF.AC}$
$\Rightarrow \dfrac{AB^3}{AC^3} = \dfrac{BE}{CF}$
c) Ta có:
$AB.AC = AH.BC = 2S_{ABC}$
$\Rightarrow BC = \dfrac{AB.AC}{AH}$
Ta được:
$BC.BE.CF$
$= \dfrac{AB.AC}{AH}.BE.CF$
$= \dfrac{(AB.BE).(AC.CF)}{AH}$
$= \dfrac{BH^2.CH^2}{AH}$
$= \dfrac{AH^4}{AH} = AH^3$
d) $BC.HE.HF$
$= \dfrac{AB.AC}{AH}.HE.HF$
$= \dfrac{(AB.HE).(AC.HF)}{AH}$
$= \dfrac{BH.AH.CH.AH}{AH}$
$= \dfrac{AH^4}{AH} = AH^3$