Understanding Circles and Circumcircles in Geometry

In geometry, a circle is a set of points that are all the same distance from a central point, called the center. The circle is defined by the equation (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius.

A circumcircle is a circle that passes through all the vertices of a polygon. In other words, it is a circle that is inscribed within the polygon, meaning that it touches every side of the polygon at its vertices. The circumcircle of a polygon is unique, and it can be used to find the properties of the polygon, such as its area and perimeter.

For example, if you have a regular hexagon (a hexagon with all sides and angles equal), the circumcircle will pass through all the vertices of the hexagon and will have a radius equal to half the length of one side.