Example $$ 2: A B C D $$ is a trapezium with $$ A B \| D C $$. $$ E $$ and $$ F $$ are points on non-parallel sides $$ A D $$ and $$ B C $$ respectively such that $$ E F $$ is parallel to $$ A B $$

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Certainly! Let's delve into the geometry of the given trapezium and explore the relationships and properties that arise from the configuration you've described.

## Given:
- **Trapezium $$ ABCD $$** with $$ AB \parallel DC $$.
- **Points $$ E $$ and $$ F $$** are located on the non-parallel sides $$ AD $$ and $$ BC $$ respectively.
- **Segment $$ EF $$** is drawn such that $$ EF \parallel AB $$ (and consequently $$ EF \parallel DC $$).

## Objective:
While a specific question wasn't provided, a common exploration in such configurations involves determining the ratio in which $$ EF $$ divides the legs $$ AD $$ and $$ BC $$, calculating the length of $$ EF $$, or finding areas related to the trapezium and the smaller trapezoids formed.

Let's explore these aspects step by step.

---

### 1. **Dividing Ratios of the Legs**

Since $$ EF \parallel AB \parallel DC $$, by the **Basic Proportionality Theorem (Thales' Theorem)**, $$ EF $$ divides the legs $$ AD $$ and $$ BC $$ proportionally.

- **Let’s denote:**
  - $$ AE = k \cdot AD $$
  - $$ BF = k \cdot BC $$

Here, $$ 0 < k < 1 $$ represents the ratio in which $$ EF $$ divides the legs.

**Proof:**

Consider triangles $$ AED $$ and $$ BFC $$:

- Since $$ EF \parallel AB $$, angles $$ \angle AED = \angle ABF $$ and $$ \angle EAD = \angle FBA $$ by corresponding angles.
- Therefore, triangles $$ AED $$ and $$ BFC $$ are **similar** by the **AA (Angle-Angle) Similarity Criterion**.

This similarity implies:

$$
\frac{AE}{AD} = \frac{BF}{BC} = k
$$

Thus, $$ E $$ and $$ F $$ divide $$ AD $$ and $$ BC $$ in the same ratio $$ k $$.

---

### 2. **Calculating the Length of $$ EF $$**

The length of $$ EF $$ can be determined using the concept of weighted averages based on the ratio $$ k $$.

**Formula:**

$$
EF = AB + k (DC - AB) = (1 - k) AB + k DC
$$

**Derivation:**

- When $$ k = 0 $$, $$ EF = AB $$.
- When $$ k = 1 $$, $$ EF = DC $$.
- For intermediate values of $$ k $$, $$ EF $$ lies proportionally between $$ AB $$ and $$ DC $$.

**Example:**

If $$ AB = 8 $$ cm, $$ DC = 12 $$ cm, and $$ k = \frac{1}{2} $$:

$$
EF = 8 + \frac{1}{2} (12 - 8) = 8 + 2 = 10 \text{ cm}
$$

---

### 3. **Area Considerations**

The areas of trapeziums or triangles formed within the main trapezium can also be analyzed.

**Area of Trapezium $$ ABCD $$:**

$$
\text{Area} = \frac{1}{2} (AB + DC) \times h
$$

where $$ h $$ is the height (distance between the parallel sides).

**Area of Trapezium $$ AEBF $$:**

Since $$ EF \parallel AB \parallel DC $$ and divides the legs proportionally:

$$
\text{Area} = \frac{1}{2} (AB + EF) \times k h
$$

Similarly, the area of trapezium $$ EFCD $$ would be:

$$
\text{Area} = \frac{1}{2} (EF + DC) \times (1 - k) h
$$

---

### 4. **Special Case: Midsegment**

When $$ k = \frac{1}{2} $$, $$ EF $$ is known as the **midsegment** (or **median**) of the trapezium.

**Properties of the Midsegment:**

- **Length:**

$$
EF = \frac{AB + DC}{2}
$$

- **Area:**

$$
\text{Area of } AEBF = \text{Area of } EFCD = \frac{1}{2} \times \text{Area of } ABCD
$$

---

### 5. **Visualization**

To better understand the configuration, consider the following diagram:

```
A___________B
 \         /
  \       /
   E____F
    \   /
     \ /
      D
```

- $$ AB \parallel EF \parallel DC $$
- $$ E $$ lies on $$ AD $$, $$ F $$ lies on $$ BC $$
- The ratios $$ \frac{AE}{AD} = \frac{BF}{BC} = k $$

---

### 6. **Applications**

Such configurations are fundamental in various geometric constructions and proofs, including:

- **Determining unknown lengths** in trapeziums.
- **Proving similarity** between different geometric figures.
- **Dividing figures** into regions of equal area or specific proportions.
- **Analyzing properties** related to parallel lines and transversals.

---

If you have a specific question or a particular aspect of this configuration you'd like to explore further—such as calculating specific lengths, areas, or proving a particular property—please provide additional details, and I'd be happy to assist you further!
In trapezium $$ ABCD $$ with $$ AB \parallel DC $$, points $$ E $$ and $$ F $$ on non-parallel sides $$ AD $$ and $$ BC $$ respectively, and $$ EF \parallel AB \parallel DC $$. The segment $$ EF $$ divides the legs $$ AD $$ and $$ BC $$ in the same ratio $$ k $$. The length of $$ EF $$ is calculated as $$ EF = (1 - k) \times AB + k \times DC $$. When $$ k = \frac{1}{2} $$, $$ EF $$ is the midsegment, equal to the average of $$ AB $$ and $$ DC $$.

Example is a trapezium with . and
are points on non-parallel sides and
respectively such that is parallel to

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