[ [ Pythagorean Theorem (direct and inverse). ]

Given triangle $ABC$ with base $AB$, equal to $\frac{\sqrt{3}}{2}$, and height $CH$, dropped to this base and equal to $\frac{\sqrt{6}}{3}$. It is known that point $H$ lies on $AB$ and $AH: HB=2: 1$. A circle is inscribed in angle $ABC$ of triangle $ABC$, with its center lying on the height $CH$. Find the radius of this circle.