54. In a right triangle $ABC$ with hypotenuse $|AB|=c$, a circle is constructed with the height of the triangle $CD$ as its diameter. Tangents to this circle, passing through points $A$ and $B$, touch it at points $M$ and $N$ and intersect when extended at point $K$. Find $|MK|$.
54. Let $|K M|=$, $K N|=x| A D|=$, $y| D B|=$, $z$. Then $|C D|=$ $=\sqrt{y z}, y+z=c$. The radius of the inscribed circle in $\triangle A K B$ is $\frac{1}{2}|C D|=\frac{1}{2} \sqrt{y z}$. Express the area of triangle $A K B$ using Heron's formula and $s=p r$. We get the equation

$$
\sqrt{(x+y+z) x y z}=(x+y+z) \frac{1}{2} \sqrt{\overline{y z}}
$$

Considering that $y+z=c$, we find $x=c / 3$.