D and E are points on the sides CA and CB respectively of a triangle ABC right angled at C. Prove that `A E^2+B D^2=A B^2+D E^2`.

Explanation

To prove that \( AE^2 + BD^2 = AB^2 + DE^2 \) in triangle \( ABC \) right-angled at \( C \), where \( D \) and \( E \) are points on sides \( CA \) and \( CB \) respectively, we will use the Pythagorean theorem.

### Step-by-Step Solution:

1. **Identify the Right Triangle**:
We have triangle \( ABC \) with \( C \) as the right angle. Therefore, by the Pythagorean theorem:
\[
AB^2 = AC^2 + BC^2 \tag{1}
...