excirkler

The term "excirkler" refers to a concept in geometry, specifically related to triangles. An excircle, also known as an escribed circle, is a circle that is tangent to one side of a triangle and to the extensions of the other two sides. Every triangle has three excircles, each associated with one of its sides.

For a given triangle ABC, there is an excircle opposite vertex A, tangent to side BC and to the extensions of sides AB and AC. Similarly, there are excircles opposite vertices B and C. The center of an excircle is called an excenter. The excenters are the intersection points of the external angle bisectors of two angles and the internal angle bisector of the remaining angle.

The radius of an excircle is called the exradius. The area of a triangle can be calculated using its exradii and semiperimeter. Specifically, if s is the semiperimeter and r_a, r_b, and r_c are the radii of the excircles opposite vertices A, B, and C respectively, then the area K of the triangle satisfies K = r_a(s-a) = r_b(s-b) = r_c(s-c), where a, b, and c are the lengths of the sides opposite vertices A, B, and C. The excircles play a role in various geometric theorems and formulas related to triangles.