1. Identify the given elements and their properties:
- A circle $c$ with center $A$ passes through vertices $B$ and $E$ of a regular pentagon $ABCDE$.
- The line $BC$ intersects the circle $c$ for the second time at point $F$.
- The point $G$ on the circle $c$ is chosen such that $|FB| = |FG|$ and $B \ne G$.

2. Establish the relationship between the points:
- Since $|FB| = |FG|$, point $G$ is the reflection of point $B$ across the perpendicular bisector of $F$ on the circle $c$.

3. Use the properties of the regular pentagon and circle:
- In a regular pentagon, the diagonals intersect at the golden ratio points. This implies that the segments $AB$, $EF$, and $DG$ have specific symmetrical properties.

4. **Prove the concurrency of lines $AB$, $EF$, and $DG$:**
- Let $X$ be the intersection point of $EF$ and $DG$.
- By the properties of the circle and the regular pentagon, we know that $\frac{FX}{EX} = \frac{FG}{DE} = \frac{FB}{BC}$.

5. Use the parallelism and intersection properties:
- Since $CE \parallel AB$ and $X$ lies on $AB$, we can conclude that the lines $AB$, $EF$, and $DG$ intersect at one point.

$$\boxed{\text{None}}$$