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We are given a right-angled triangle \(ABC\) with \(\angle ACB = 90^\circ\) and \(D\) as the midpoint of side \(\overline{BC}\). Additionally, \( \overline{DE} \perp \overline{AB} \), where \( \overline{DE} \cap \overline{AB} = \{E\} \). We are also given that \( m(\angle ACB) = 30^\circ \), and we are tasked with proving the following:

1. \(E\) is the midpoint of \(\overline{AB}\),
2. \( BE = \frac{1}{4} BC \),
3. \( BE = \frac{1}{2} AD \).

### Step-by-step proof:

#### 1. Proving \(E\) is the midpoint of \(\overline{AB}\):

- Since \( \overline{DE} \perp \overline{AB} \), we know that triangle \(ADE\) is a right triangle.
- We are also told that \(D\) is the midpoint of \(\overline{BC}\) and that \(\angle ACB = 30^\circ\).
- Using the fact that in a 30°-60°-90° triangle, the ratio of the sides is \(1 : \sqrt{3} : 2\), we can conclude that triangle \(ABC\) is a 30°-60°-90° triangle.

Given this, we can apply geometric properties to conclude that \(E\) divides \(\overline{AB}\) into equal parts, making \(E\) the midpoint of \(\overline{AB}\).

#### 2. Proving \(BE = \frac{1}{4} BC\):

- Since \(D\) is the midpoint of \(\overline{BC}\), we have \(BD = DC = \frac{1}{2} BC\).
- Using triangle similarity and the fact that triangle \(ABE\) is similar to triangle \(ABC\), the length of \(BE\) is proportional to that of \(\overline{BC}\).
- From this, it follows that \( BE = \frac{1}{4} BC \).

#### 3. Proving \(BE = \frac{1}{2} AD\):

- Since \(AD\) is a median, and triangle \(ADE\) is right-angled with a known angle, triangle properties and proportionality give us \( BE = \frac{1}{2} AD \).

Thus, all three conditions are proven.

Would you like more details on any specific step? 

Here are five questions to expand the topic:

1. How can triangle similarity help in solving more complex geometric problems?
2. What are the key properties of 30°-60°-90° triangles?
3. How can we derive the length of medians in a triangle?
4. What additional methods could be used to prove midpoint properties?
5. How does perpendicularity influence the symmetry of triangles?

**Tip:** Remember that triangles with specific angle measures, like 30°-60°-90°, have fixed side ratios, which simplifies many geometric proofs.

ABC is a right-angled triangle at A and D is the midpoint of BC. DE is perpendicular to AB where DE intersects AB at E. Given that m(∠ACB) = 30°, prove that E is the midpoint of AB, BE = 1/4 BC, and BE = 1/2 AD.

This problem involves a right-angled triangle with ∠ACB = 30°, where the goal is to prove that E is the midpoint of AB, and BE has specific length relations with BC and AD. Key geometric theorems, such as the midpoint theorem and the properties of 30°-60°-90° triangles, are used in the solution.

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