Let's solve the problem step by step based on the image provided. Here are the tasks:

Problem:

We are given a parallelogram $ABCD$ with the following conditions:

• $AE = EF = FC$.
  We are tasked to prove:

1. $DE$ is parallel to $FB$.
2. $DE = FB$.
3. $DEBF$ is a parallelogram.

Solution:

1. Prove $DE \parallel FB$:

Since $ABCD$ is a parallelogram, the opposite sides $AB \parallel DC$ and $AD \parallel BC$.
Given $AE = EF = FC$, $E$ and $F$ divide $AC$ into three equal parts. Thus:

• $AE = EF = FC = \frac{1}{3}AC$.

Join $D$ to $E$ and $B$ to $F$.
In $\triangle ADC$, line segment $DE$ joins $D$ to $E$, a point on $AC$. Similarly, in $\triangle ABC$, line $FB$ joins $B$ to $F$, a point on $AC$.

Now, since $E$ and $F$ are proportionally dividing $AC$, and $D$ and $B$ are endpoints of parallel sides $AD \parallel BC$, it follows from the Basic Proportionality Theorem (Thales' Theorem) that: $DE \parallel FB.$

2. Prove $DE = FB$:

In parallelogram $ABCD$, opposite sides $AD = BC$ and $AB = DC$.
Since $AE = EF = FC$, $E$ and $F$ divide $AC$ equally.

From the Basic Proportionality Theorem applied earlier, not only are $DE \parallel FB$, but also: $\frac{DE}{FB} = 1.$ Thus, $DE = FB$.

3. Prove $DEBF$ is a parallelogram:

To prove $DEBF$ is a parallelogram, we need to show that:

1. Opposite sides are parallel ($DE \parallel FB$ and $DF \parallel EB$), or
2. Opposite sides are equal ($DE = FB$ and $DF = EB$).

From part (1), $DE \parallel FB$.
From part (2), $DE = FB$.

Now consider $DF$ and $EB$:

• Since $D$ and $B$ are endpoints of opposite sides $AD$ and $BC$ of parallelogram $ABCD$, and $E$ and $F$ divide $AC$ proportionally, it follows (by symmetry) that $DF = EB$ and $DF \parallel EB$.

Thus, all conditions for $DEBF$ to be a parallelogram are satisfied: $DEBF \text{ is a parallelogram.}$

Final Answer:

1. $DE \parallel FB$.
2. $DE = FB$.
3. $DEBF$ is a parallelogram.

Would you like me to provide a diagram or further details?

Here are 5 related questions to expand on the topic:

1. What is the Basic Proportionality Theorem, and how does it apply to parallelograms?
2. How can we prove diagonals of a parallelogram bisect each other geometrically?
3. What are the properties of a parallelogram derived from its definition?
4. How does dividing diagonals into equal parts affect other sides of a parallelogram?
5. Can we generalize this result for quadrilaterals other than parallelograms?

Tip: Always use symmetry and proportionality when working with parallelograms or midpoints!