excirkel

Excirkel, also known as a circumcircle or circumscribed circle, is a circle that passes through all the vertices of a polygon. The concept of an excircle is closely related to that of a circumcircle, but while a circumcircle is defined for any polygon, excircles are specific to triangles.

For a given triangle, there are three excircles: the excircle opposite each vertex of the triangle. An excircle touches the side of the triangle opposite its vertex and the extensions of the other two sides. The center of an excircle, known as the excenter, is the intersection of the external angle bisectors of the two non-adjacent angles of the triangle.

The radius of an excircle is given by the formula r = A / s, where A is the area of the triangle, and s is the semiperimeter (half the perimeter) of the triangle. The excircles are useful in various fields such as geometry, trigonometry, and number theory, particularly in problems involving tangents and angle bisectors.

In the context of a circumcircle, the radius is given by the formula R = abc / (4 A), where a, b, and c are the side lengths of the triangle, and A is the area of the triangle. The circumcircle is significant because it allows for the application of properties related to cyclic quadrilaterals and the Inscribed Angle Theorem.

Both excircles and circumcircles play crucial roles in understanding the geometric properties of triangles and other polygons, providing insights into angles, sides, and areas.