a) \(H F \parallel A B\) \(\Rightarrow \frac{H F}{A B} = \frac{C F}{C A} \Rightarrow \frac{H F}{C F} = \frac{A B}{A C}\)

\(\Rightarrow \frac{H F}{C F} . \frac{A B^{2}}{A C^{2}} = \frac{A B^{3}}{A C^{3}} \Rightarrow \frac{H F}{C F} . \frac{B H . B C}{C H . B C} = \frac{A B^{3}}{A C^{3}}\)

\(\Rightarrow \frac{H F . B H}{C F . C H} = \frac{A B^{3}}{A C^{3}} \Rightarrow \frac{H F . B H}{C H} . \frac{1}{C F} = \frac{A B^{3}}{A C^{3}} \left(\right. 1 \left.\right)\)

Ta có: \(H F \parallel A B\)\(\Rightarrow \angle C H F = \angle C B A\)

Xét \(\Delta B E H\) và \(\Delta H F C :\) Ta có: \(\left{\right. \angle B E H = \angle H F C = 90 \\ \angle C H F = \angle C B A\)

\(\Rightarrow \Delta B E H sim \Delta H F C \left(\right. g - g \left.\right) \Rightarrow \frac{B E}{B H} = \frac{H F}{H C} \Rightarrow B E . H C = H F . B H\)

\(\Rightarrow B E = \frac{H F . B H}{H C} \left(\right. 2 \left.\right)\)

Từ (1) và (2) \(\Rightarrow \frac{B E}{C F} = \frac{A B^{3}}{A C^{3}}\)