\(\left\{{}\begin{matrix}BH.CH=AH^2\\CH-BH=14\end{matrix}\right.\) \(\Rightarrow BH\left(14+BH\right)=24^2\)
\(\Leftrightarrow BH^2+14BH-576=0\Rightarrow BH=18\)
\(\Rightarrow CH=32\Rightarrow BC=50;AB=\sqrt{BH.BC}=30;AC=40\)
\(\left\{{}\begin{matrix}\frac{BD}{AB}=\frac{DC}{AC}\\BD+DC=BC\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}BD=\frac{3}{4}DC\\BD+DC=50\end{matrix}\right.\) \(\Rightarrow DC=\frac{200}{7}\)
Kẻ \(DK\perp AC\Rightarrow DK//AB\Rightarrow\frac{DK}{AB}=\frac{DC}{BC}=\frac{4}{7}\)
\(\Rightarrow DK=\frac{4}{7}AB=\frac{120}{7}\)
Tam giác ADK vuông cân tại K (tam giác vuông có 1 góc \(\widehat{DAK}=45^0\))
\(\Rightarrow AD=DK\sqrt{2}=\frac{120\sqrt{2}}{7}\)
