\[\begin{array}{l} {\text{а)}} \hfill \\ \angle TCD = \angle TED = \frac{\pi }{2} - \angle ETD = \angle DTA \hfill \\ \angle HTC = \frac{\pi }{2} - \angle HCT = \frac{\pi }{2} - \angle DTA = \angle TDA. \hfill \\ {\text{б)}} \hfill \\ \left. \begin{array}{l} \angle CDT = \angle CTB \Rightarrow \vartriangle BCT \sim \vartriangle HDT \hfill \\ \vartriangle CHT \sim \vartriangle TDA \hfill \\ \end{array} \right\} \Rightarrow BCHT \sim HDAT \Rightarrow \hfill \\ \frac{{BC}}{{TH}} = \frac{{TH}}{{AD}} \Leftrightarrow T{H^2} = 2\sqrt 2 \cdot 4\sqrt 2 \Leftrightarrow TH = 4. \hfill \\ \end{array}\]

б) 4