Incircle

The Inscribed Circle of a Triangle $\Delta ABC$. The center $I$ is called the Incenter and the Radius $r$ the Inradius. The points of intersection of the incircle with $T$ are the Vertices of the Pedal Triangle of $T$ with the Incenter as the Pedal Point (c.f. Tangential Triangle). This Triangle is called the Contact Triangle.

The Area $K$ of the Triangle $\Delta ABC$ is given by

$$K = \Delta AIC+\Delta CIB+\Delta AIB$$

$$= {\textstyle{1\over 2}}br+{\textstyle{1\over 2}}ar+{\textstyle{1\over 2}}cr={\textstyle{1\over 2}}(a+b+c)r=sr,$$

where $s$ is the Semiperimeter.

Using the incircle of a Triangle as the Inversion Center, the sides of the Triangle and its Circumcircle are carried into four equal Circles (Honsberger 1976, p. 21). Pedoe (1995, p. xiv) gives a Geometric Construction for the incircle.

See also Circumcircle, Congruent Incircles Point, Contact Triangle, Inradius, Triangle Transformation Principle

References

Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 11-13, 1967.

Honsberger, R. Mathematical Gems II. Washington, DC: Math. Assoc. Amer., 1976.

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 182-194, 1929.

Pedoe, D. Circles: A Mathematical View, rev. ed. Washington, DC: Math. Assoc. Amer., 1995.