\[\begin{array}{l} {\text{а)}} \hfill \\ {\text{Пусть }}\angle BAC = \alpha ,{\text{ }}\angle CAD = x. \hfill \\ AB = BC \Rightarrow \angle BCA = \alpha \Rightarrow \angle ABC = \pi - 2\alpha . \hfill \\ {\text{С другой стороны }}\angle ABC = \pi - \angle BAD{\text{, т}}{\text{.е}}{\text{.}} \hfill \\ \pi - 2\alpha = \pi - \left( {\alpha + x} \right) \Leftrightarrow \alpha = x. \hfill \\ {\text{б)}} \hfill \\ {\text{Пусть }}BE{\text{ - высота }}\vartriangle ABC. \hfill \\ BE = \sqrt {A{B^2} - A{E^2}} = \sqrt {\frac{{169}}{4} - 36} = \frac{5}{2}. \hfill \\ \sin \alpha = \sin \angle BAE = \frac{{BE}}{{AB}} = \frac{5}{{13}}. \hfill \\ {\text{Теорема синусов для }}\vartriangle ACD: \hfill \\ \frac{{CD}}{{\sin \alpha }} = 2R \Leftrightarrow CD = \frac{5}{{13}} \cdot 13 = 5. \hfill \\ \end{array}\]

б) 5