Xét \(ΔAB'H'\) và \(ΔABH\):
\(\widehat{AB'H'}=\widehat{ABH}(H'B'//HB)\)
\(\widehat{H'AB'}=\widehat{HAB}\) (đối đỉnh)
\(→ΔAB'H'\backsim ΔABH(g-g)\)
\(→\dfrac{AB'}{A'H'}=\dfrac{AB}{AH}↔\dfrac{AB'}{AB}=\dfrac{A'H'}{AH}\)
Xét \(ΔAB'C'\) và \(ΔABC\):
\(\widehat{AB'C'}=\widehat{ABC}(B'C'//BC)\)
\(\widehat{B'AC'}=\widehat{BAC}\) (đối đỉnh)
\(→ΔAB'C'\backsim ΔABC(g-g)\)
\(→\dfrac{AB'}{B'C'}=\dfrac{AB}{BC}↔\dfrac{AB'}{AB}=\dfrac{B'C'}{BC}\)
mà \(\dfrac{AB'}{AB}=\dfrac{AH'}{AH}\)
\(→\dfrac{AH'}{AH}=\dfrac{B'C'}{BC}\)