1. **Problem statement:** We have a circle with center $O$ and points $A$ and $B$ on the circle. The chord $AB$ is extended through $B$ to $C$ such that $|BC|$ equals the radius $r$ of the circle. The line $CO$ intersects the circle at points $D$ and $E$, with $D$ between $C$ and $O$. Given $\angle BCO = x$, we want to express $\angle AOE$ in terms of $x$. 2. **Key facts and notation:** - Radius $r = |OB| = |OA| = |OC|$ since $C$ lies on the extension and $|BC|=r$. - $O$ is the center, so $OA$ and $OB$ are radii. - $\angle BCO = x$ is given. - $E$ lies on the circle on line $CO$ beyond $O$. 3. **Step 1: Understand the geometry of points $B$, $C$, and $O$** Since $|BC|=r$ and $|OB|=r$, triangle $OBC$ is isosceles with $OB=BC=r$. 4. **Step 2: Find $\angle OBC$** In triangle $OBC$, angles at $B$ and $C$ satisfy: $$\angle OBC + \angle BCO + \angle COB = 180^\circ$$ We know $\angle BCO = x$ and $OB=BC$, so $\angle OBC = x$. 5. **Step 3: Find $\angle COB$** Using the triangle sum: $$\angle COB = 180^\circ - 2x$$ 6. **Step 4: Relate $\angle AOE$ to $\angle COB$** Points $A$ and $B$ lie on the circle, so $\angle AOB$ is the central angle subtending chord $AB$. Since $O$ is center, $\angle AOB$ is the angle at $O$ between radii $OA$ and $OB$. 7. **Step 5: Note that $E$ lies on line $CO$ beyond $O$** Since $E$ is on the circle and on line $CO$, $OE$ is a radius. 8. **Step 6: Express $\angle AOE$** $\angle AOE$ is the angle between radii $OA$ and $OE$. Since $OE$ lies on line $CO$, and $CO$ forms angle $\angle BCO = x$ with chord $BC$, the angle $\angle AOE$ relates to $\angle COB$. 9. **Step 7: Use the fact that $\angle AOB = \angle COB$** Because $A$ and $B$ lie on the circle and $O$ is center, $\angle AOB$ equals the central angle subtending chord $AB$. 10. **Step 8: Since $E$ lies on $CO$ beyond $O$, $\angle AOE = 180^\circ - \angle AOB$** Because $OE$ is opposite $OC$, the angle between $OA$ and $OE$ is the supplement of $\angle AOB$. 11. **Step 9: Substitute $\angle AOB = \angle COB = 180^\circ - 2x$** So, $$\angle AOE = 180^\circ - (180^\circ - 2x) = 2x$$ **Final answer:** $$\boxed{\angle AOE = 2x}$$