Two circles are inscribed and circumscribed about a square ABCD, length of each side of the square is 32. P and Q are two points repectively on these circles, then `sigma(PA)^(2) + sigma(QA)^(2)` equals __________ .

Explanation

To solve the problem, we need to find the value of \( \sigma(PA^2) + \sigma(QA^2) \) where \( P \) and \( Q \) are points on the inscribed and circumscribed circles of a square \( ABCD \) with a side length of 32.

### Step-by-Step Solution:

1. **Identify the Geometry**:
- The square \( ABCD \) has a side length of 32.
- The inscribed circle (incircle) has a radius equal to half the side of the square, which is \( r = \frac{32}{2} = 16 \).
- The circumscribed circle (circumcircle) has a radius equal to the distance from the center of the square to a vertex, which is \( R = \frac{32\sqrt{2}}{2} = 16\sqrt{2} \).
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