77. A circle is circumscribed around a right triangle $A B C$ ( $\widehat{C}==90^{\circ}$ ). Let $C D$ be the altitude of the triangle. A circle centered at $D$ passes through the midpoint of the arc $\widetilde{A B}$ and intersects $A B$ at point $M$. Find $|C M|$, if $|A B|=c$.
77. If $K$ is the midpoint of $\breve{A B}$, and $O$ is the center of the circle, $|A B|=2 R=c$, then

$$
\begin{aligned}
& |C M|^{2}=|C D|^{2}+|D M|^{2}=|C D|^{2}+|D K|^{2}= \\
& =|A D| \cdot|D B|+R^{2}+|D O|^{2}=(|O A|-|D O|)(|O B|+|D O|)+R^{2}+ \\
& \quad+|D O|^{2}=(R-|D O|)(R+|D O|)+R^{2}+|D O|^{2}=2 R^{2}
\end{aligned}
$$

Thus, $|C M|=R \sqrt{2}=\frac{c}{2} \sqrt{2}$.