a)
$A{{B}^{2}}=BH.BC$
$A{{C}^{2}}=CH.BC$
$\Rightarrow \dfrac{A{{B}^{2}}}{A{{C}^{2}}}=\dfrac{BH}{CH}\Rightarrow \dfrac{AB}{AC}=\sqrt{\dfrac{BH}{CH}}$
b)
$AD.AB=AE.AC=A{{H}^{2}}$
c)
$ADHE$ là hình chữ nhật
$\Rightarrow DE=AH$
$\Rightarrow D{{E}^{2}}=A{{H}^{2}}$
$\Rightarrow A{{D}^{2}}+A{{E}^{2}}=A{{H}^{2}}$