\[\begin{array}{l} {\text{а)}} \hfill \\ {\text{Пусть }}\angle OBN = \alpha ,{\text{ }}K = OB \cap CN. \hfill \\ \angle OBC = \alpha \Rightarrow \angle KCB = \frac{\pi }{2} - \alpha \Rightarrow \angle MCN = \alpha \Rightarrow \hfill \\ \angle MON = 2\alpha \Rightarrow \angle MNO = \frac{\pi }{2} - \alpha \Rightarrow \angle ANM = \alpha \Rightarrow MN\parallel OB{\text{, ч}}{\text{.т}}{\text{.д}}{\text{.}} \hfill \\ {\text{б)}} \hfill \\ {\text{Пусть }}BC = y,{\text{ }}AM = x.{\text{ Тогда }}BN = y,{\text{ }}MC = 3x,{\text{ }}R = OM = \frac{{3x}}{2}. \hfill \\ \vartriangle AMN \sim \vartriangle AOB \Rightarrow \frac{{AN}}{{AB}} = \frac{{AM}}{{AO}} \Leftrightarrow \frac{{AN}}{{AN + y}} = \frac{x}{{x + \frac{{3x}}{2}}} \Leftrightarrow AN = \frac{2}{3}y. \hfill \\ \vartriangle AMN \sim \vartriangle AOB,{\text{ }}k = \frac{{MN}}{{OB}} = \frac{x}{{x + \frac{{3x}}{2}}} = \frac{2}{5}. \hfill \\ \frac{{MN}}{{CN}} = \operatorname{tg} \angle MCN \Leftrightarrow MN = 4\operatorname{tg} \alpha . \hfill \\ \frac{{ON}}{{BN}} = \operatorname{tg} \alpha \Leftrightarrow \operatorname{tg} \alpha = \frac{{3x}}{{2y}}. \hfill \\ AM \cdot AC = A{N^2} \Leftrightarrow x \cdot 4x = {\left( {\frac{2}{3}y} \right)^2} \Leftrightarrow y = 3x. \hfill \\ y = 3x \Rightarrow \operatorname{tg} \alpha = \frac{1}{2} \Rightarrow MN = 2 \Rightarrow OB = 2 \cdot \frac{5}{2} = 5. \hfill \\ BOMN{\text{ - трапеция}} \Rightarrow \hfill \\ S = \frac{{MN + OB}}{2} \cdot NK = \frac{{5 + 2}}{2} \cdot 2 = 7. \hfill \\ \end{array}\]

б) 7