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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.

Learn how to calculate the area of triangle AEG within square ABCD with a step-by-step solution. This problem involves coordinate geometry and triangle properties, focusing on the area calculation of a degenerate triangle. Suitable for students in Grades 10-12.

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