Indcircles

Indcircles are a type of geometric construction related to triangles. Specifically, an indcircle is a circle that is tangent to one side of a triangle and to the extensions of the other two sides. This is in contrast to the incircle, which is tangent to all three sides of the triangle internally. Indcircles are also known as excircles.

Every triangle has three indcircles, with each indcircle being associated with one of the triangle's sides. The center of an indcircle is called an excenter. The excenter is located at the intersection of the bisector of one interior angle of the triangle and the bisectors of the two exterior angles at the other two vertices. The radius of an indcircle is called the exradius.

The formulas for the exradius of a triangle with side lengths a, b, and c, and area A are often expressed in relation to the semi-perimeter s = (a+b+c)/2. If r_a is the exradius opposite side a, then A = r_a(s-a). Similarly, A = r_b(s-b) and A = r_c(s-c). These relationships are fundamental in triangle geometry and have applications in various mathematical fields. The concept of indcircles extends the understanding of tangent circles to triangles beyond the well-known incircle and circumcircle.