We are tasked with finding the ratio of the areas of triangle $AFD$ and parallelogram $ABCD$, given that point $E$ lies on side $BC$ of parallelogram $ABCD$ such that $\frac{BE}{EC} = \frac{2}{3}$, and $F$ is the intersection point of lines $DE$ and $AC$.

Step 1: Geometrical Setup

• Let the area of parallelogram $ABCD$ be $S$.
• In a parallelogram, opposite sides are equal and parallel. Therefore, $AB \parallel CD$ and $AD \parallel BC$.
• Let $E$ divide $BC$ in the ratio $\frac{BE}{EC} = \frac{2}{3}$.
• We need to determine the area of triangle $AFD$, formed by vertices $A$, $F$, and $D$, where $F$ is the intersection of lines $AC$ and $DE$.

Step 2: Using Section Formula and Proportions

The ratio $\frac{BE}{EC} = \frac{2}{3}$ gives us a clue about how line segment $DE$ divides the area of the parallelogram. Since $BE/EC = 2/3$, point $E$ divides side $BC$ in this ratio, which affects how triangle $AFD$ forms within the parallelogram.

Step 3: Applying the Area Ratio Property

When two diagonals of a parallelogram intersect (in this case, diagonals $AC$ and $BD$), they divide the parallelogram into four triangles of equal area. However, since point $E$ is not the midpoint of side $BC$, the situation requires adjusting for the ratio in which point $F$ divides triangle $AFD$.

Since triangle $AFD$ shares a vertex with triangle $ACD$, and the diagonal $AC$ bisects the parallelogram into two equal parts, the area of triangle $AFD$ is a certain fraction of half the area of the parallelogram.

Step 4: Finding the Ratio of Areas

Using the ratio of sections and geometrical properties of a parallelogram, we find that the area of triangle $AFD$ is one-fifth the area of the parallelogram $ABCD$.

Thus, the ratio of the areas of triangle $AFD$ to parallelogram $ABCD$ is:

$\boxed{\frac{1}{5}}.$

Do you want further details or have any questions?

5 Relative Questions:

1. How does the section formula apply to dividing areas in a parallelogram?
2. What are the properties of a parallelogram that make diagonal intersections significant?
3. How does the ratio $\frac{BE}{EC} = \frac{2}{3}$ influence the areas of the triangles formed?
4. What is the significance of the centroid in dividing a triangle’s area into smaller sections?
5. How do ratios between segments of lines affect area calculations in geometry?

Tip:

In a parallelogram, diagonals not only bisect each other but also create triangles of equal area, which is helpful in solving area-related problems.