Circle

Circle

A circle is a 2D geometrical shape defined as the set of all points in a plane that are equidistant from a given point called the center. 

Here are the key definitions related to the parts of a circle:

1. Radius (r): The distance from the center of the circle to any point on its circumference.

2. Diameter (d): A straight line passing through the center, connecting two points on the circumference; it is twice the radius. This is the longest chord in a circle.

3. Circumference: The total length around the circle, calculated as \(2\pi r\) or \(\pi d\).

4. Area of Circle: The space inside the circle, given by \(\pi r^2\).

5. Center: The exact midpoint inside the circle.

6. Chord: A straight line connecting two points on the circumference. It can vary in length from zero (if it's just a point) to the diameter.

7. Arcs: Segments of the circumference between two points. They can be minor arcs (less than 180 degrees) or major arcs (more than 180 degrees).

Additionally:

8. Tangent Line: A straight line touching the circle at exactly one point and not crossing through its interior.

Cyclic Quadrilateral

A cyclic quadrilateral is a quadrilateral whose all four vertices lie on the circumference of a single circle. This means that it can be inscribed inside a circle. Hence, a quadrilateral inside a circle with four vertices on its circumference. 

Properties of a Cyclic Quadrilateral:

1. Opposite Angles are Supplementary: 

   - The sum of opposite angles in a cyclic quadrilateral is always \(180^\circ\). 

2. Exterior Angle Property:

   - The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.