Chứng minh AH . HD = HB . HE; AD^2= DH . DB; AH . HE = DH^2

To prove the statements made in your notes, we'll go through them step by step based on properties of right triangles and rectangles.

1. **Prove \( AH \cdot HD = HB \cdot HE \)**:
- This is a result of the geometric mean in right triangles. If you have right triangle \( AHB \) and \( AHE \), you can use the properties of similar triangles or the implication of the altitude in the right triangles formed.

2. **Prove \( AD^2 = DH \cdot DB \)**:
- This can again be proven using the property of the altitude to the hypotenuse in right triangle \( ADB \). We can show that \( AH \) being the altitude gives us that \( AD^2 \) equals the product of segments \( DH \) and \( DB \) on the hypotenuse.

3. **Prove \( AH \cdot HE = DH^2 \)**:
- This can be derived from the same principles. In right triangle \( ADH \), applying the relationships from the earlier proved equations can yield this result.

4. **For triangle \( ABC \)** (assuming it's the same triangle structure):
- We continue applying triangle relationships. Referring to properties of similar triangles, consider triangle transformations that would yield the necessary relationships.

5. **Using relationships between segments**:
- Define new segments \( BM = EN \) to establish similarity. The equality of the segments allows you to derive properties of proportions typical in similar triangles.

Continuing the process of proving segment relationships and applying geometric means will establish all statements effectively. Illustrating the triangles and their relationships will also significantly aid in visual proof.

Make sure to validate each step through diagrams and ensure geometric principles remain consistent throughout.