I think the area of GAD isn't equal to the area of GEC

You're absolutely right! I mistakenly assumed the areas of \(\Delta GAD\) and \(\Delta GEC\) are equal, but since they are only similar (not congruent), their areas are proportional to the square of the ratio of their corresponding sides, not equal.

Let me rework the calculation for the area of triangle \(GAD\).

### 4. Correctly Calculating the Area of Triangle \(GAD\):

We know that \(\Delta AGD \sim \Delta GEC\), and the ratio of the areas of similar triangles is the square of the ratio of their corresponding sides.

In parallelogram \(ABCD\), since \(AB \parallel CD\), and \(AG = GC\) (because diagonals of a parallelogram bisect each other), the ratio of the corresponding sides of \(\Delta AGD\) and \(\Delta GEC\) is 1:1.

However, since \(BE/CE = 2/3\), we need to find the correct base for \(\Delta GAD\). Let's analyze this carefully:
- The ratio of sides between triangles \(GAD\) and \(GEC\) is determined by how the triangles share portions of the parallelogram's diagonal and sides. Since \(BE\) and \(CE\) relate directly to the height or base of triangles, and the area formula for triangles depends on both.



Explore the calculation of triangle areas within parallelograms, focusing on the relationship between similar triangles and their proportional areas. This problem addresses the specifics of triangles GAD and GEC within parallelogram ABCD. Suitable for high school geometry students.

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