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The problem provided involves a circle with radius 1 and a square inscribed within it, centered at \( O \). The goal is to find the expression \( \overline{AB} - \overline{BC} + \overline{CD} - \overline{DA} \).

Given that the square is inscribed within a circle with a radius of 1, each side of the square lies on the circle, and the vertices of the square are equidistant from each other. The diagonal of the square is equal to the diameter of the circle, which is 2 (since the radius is 1).

### Steps to solve:
1. **Determine the side length of the square**: 
   - The diagonal \( d \) of the square can be calculated using the Pythagorean theorem in one of its right triangles:
     \[
     d^2 = s^2 + s^2 = 2s^2
     \]
   - Given \( d = 2 \) (since it's the circle's diameter), we solve:
     \[
     2s^2 = 4 \implies s^2 = 2 \implies s = \sqrt{2}
     \]
   - Therefore, the side length \( s \) of the square is \( \sqrt{2} \).

2. **Substitute the side lengths into the expression**:
   - \( \overline{AB} = \overline{BC} = \overline{CD} = \overline{DA} = \sqrt{2} \)
   - The expression becomes:
     \[
     \sqrt{2} - \sqrt{2} + \sqrt{2} - \sqrt{2}
     \]
   - Which simplifies to:
     \[
     0
     \]

### Conclusion:
The value of \( \overline{AB} - \overline{BC} + \overline{CD} - \overline{DA} \) is \( 0 \). This result makes sense geometrically since the square is symmetrically inscribed in the circle and opposite sides cancel out when subtracted and added in this alternating fashion.

Would you like to learn more details about any specific part of this solution, or do you have any other questions?

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1. **What is the geometric significance of the inscribed square's symmetry?**
2. **How would the result change if the polygon inscribed in the circle wasn't a square?**
3. **Can we apply similar calculations for a regular polygon with more sides inscribed in the circle?**
4. **What is the area of the circle and the square, and how do they compare?**
5. **Why does the expression involving the sides of the square sum up to zero?**

**Tip:** When solving geometric problems, always consider the symmetry and properties of the shapes involved, as they often simplify calculations and provide intuitive understanding.

Duotas apskritimas, kurio spindulys lygus 1, ir į kurį įbrėžtas kvadratas ABCD. O – kvadrato įstrižainių susikirtimo taškas. Raskite |ĀB - B̃C + C̄D - ḊA|.

Learn how to solve a geometric expression involving the sides of an inscribed square in a circle with radius 1. This problem covers key geometry and vector concepts, using the Pythagorean theorem and the properties of inscribed figures.

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