MathNet 2019-part-02 — 184 problems (EN–VI) Question 147

On the side $\overline{AB}$ of the cyclic quadrilateral $ABCD$ there is a point $X$ such that the diagonals $\overline{BD}$ and $\overline{AC}$ bisect the segments $\overline{CX}$ and $\overline{DX}$, respectively.
Find the smallest possible value of $|AB| : |CD|$. (Belarus)