In Figure, `Pa n dQ` are the midpoints of the sides `C Aa n dC B` respectively of ` A B C` right angled at C. Prove that `4(A Q^2+B P^2)=5A B^2` .

Explanation

To prove that \( 4(AQ^2 + BP^2) = 5AB^2 \) given that \( P \) and \( Q \) are the midpoints of sides \( CA \) and \( CB \) respectively in triangle \( ABC \) right-angled at \( C \), we can follow these steps:

### Step 1: Identify the triangles and apply the Pythagorean theorem
In triangle \( ACQ \):
- Since \( Q \) is the midpoint of \( CB \), we have \( CQ = \frac{1}{2}CB \).
- By the Pythagorean theorem, we can write:
\[
AQ^2 = AC^2 + CQ^2
...