The problem involves a square $ABCD$ with side length 1, where $BE = 2EC$ and $CF = FD$. The task is to find the area of triangle $AEG$.

Let's break it down step by step:

1. Find the coordinates of point $E$:

   • Given that $BE = 2EC$, let $EC = x$.
   • Since $BE + EC = 1$ (because $BC = 1$), we have $2x + x = 1$, which gives $3x = 1$, so $x = \frac{1}{3}$.
   • Therefore, $EC = \frac{1}{3}$ and $BE = \frac{2}{3}$.
   • Point $E$ is $\left(0, \frac{2}{3}\right)$.

2. Find the coordinates of point $F$:

   • Given that $CF = FD$, let $CF = y$.
   • Since $CF + FD = 1$ (because $CD = 1$), we have $y + y = 1$, which gives $2y = 1$, so $y = \frac{1}{2}$.
   • Therefore, $CF = \frac{1}{2}$ and $FD = \frac{1}{2}$.
   • Point $F$ is $\left(\frac{1}{2}, 1\right)$.

3. Find the coordinates of point $G$:

   • Point $G$ is the intersection of $AD$ and $EF$.
   • Line $AD$ is $x = 0$.
   • Line $EF$ has endpoints $E(0, \frac{2}{3})$ and $F(\frac{1}{2}, 1)$.
   • The equation of line $EF$ can be found using the slope formula: $m = \frac{1 - \frac{2}{3}}{\frac{1}{2} - 0} = \frac{\frac{1}{3}}{\frac{1}{2}} = \frac{1}{3} \times \frac{2}{1} = \frac{2}{3}$
   • The equation of the line is $y = \frac{2}{3}x + \frac{2}{3}$.

4. Find the coordinates of point $G$:

   • Since $G$ lies on $AD$, $x = 0$.
   • Substitute $x = 0$ into the equation of $EF$: $y = \frac{2}{3} \times 0 + \frac{2}{3} = \frac{2}{3}$
   • Therefore, point $G$ is $(0, \frac{2}{3})$.

5. Calculate the area of triangle $AEG$:

   • The coordinates of $A$ are $(0, 1)$.
   • The coordinates of $E$ are $(0, \frac{2}{3})$.
   • The coordinates of $G$ are $(0, \frac{2}{3})$.

Since points $E$ and $G$ are the same in this case, triangle $AEG$ is degenerate, meaning it has zero area.

Therefore, the area of triangle $AEG$ is $0$.

If you have any questions or need further details, please let me know. Here are five more questions you might want to ask:

1. How do you find the equation of a line given two points?
2. What is the slope-intercept form of a linear equation?
3. How can you determine if a triangle is degenerate?
4. How do you find the area of a triangle given its vertices?
5. What are some properties of triangles in a square?

Tip: Always verify if the given conditions make the points collinear, as this can simplify solving the problem.