Four, (Full marks 14 points) On a line, there are four points $A, B, C, D$ in sequence, with $AB: BC: CD = 2: 1: 3$. Circles $\odot O_{1}, \odot O_{2}$ are constructed with diameters $AC$ and $ED$ respectively, and the two circles intersect at points $E, F$ (as shown in the figure). Find the value of $ED: EA$.
Connect $E B, E C$, and draw $C G$ perpendicular to $E B$ intersecting $A E, B E$ at $G, H$.
$$
\because D E \perp B E, \therefore D E
$$
//CG, from the given conditions $A C$ :
$$
\begin{array}{l}
C D=3,3=A G: G E, \therefore \\
A G=G E .
\end{array}
$$
$\because C H: D E=B C: B D=1: 4$, and $C G: D E=A C$ :
$$
A D=1: 2 \text {, }
$$
$\therefore H$ is the midpoint of $G C$, so $E B$ is the perpendicular bisector of $C G$.

Also, $\angle A E C=90^{\circ}, \therefore \triangle G E C$ is an isosceles right triangle, thus $\angle E C G=45^{\circ}$.
Therefore, $E D: E A=2 C G: 2 E G=\sqrt{2}$.