The center of a Triangle's Incircle. It can be found as the intersection of Angle Bisectors, and it is the interior point for which distances to the sidelines are equal. Its Trilinear Coordinates are 1:1:1. The distance between the incenter and Circumcenter is .

The incenter lies on the Euler Line only for an Isosceles Triangle. It does, however, lie on the Soddy Line. For an Equilateral Triangle, the Circumcenter , Centroid , Nine-Point Center , Orthocenter , and de Longchamps Point all coincide with .

The incenter and Excenters of a Triangle are an Orthocentric System. The Power of the incenter with respect to the Circumcircle is

(Johnson 1929, p. 190). If the incenters of the Triangles , , and are , , and , then is equal and parallel to , where are the Feet of the Altitudes and are the incenters of the Triangles. Furthermore, , , , are the reflections of with respect to the sides of the Triangle (Johnson 1929, p. 193).

If four points are on a Circle (i.e., they are Concyclic), the incenters of the four Triangles form a Rectangle whose sides are parallel to the lines connecting the middle points of opposite arcs. Furthermore, the connectors pass through the center of the Rectangle (Fuhrmann 1890, p. 50; Johnson 1929, pp. 254-255). More generally, the 16 incenters and excenters of the Triangles whose Vertices are four points on a Circle, are the intersections of two sets of four Parallel lines which are mutually Perpendicular (Johnson 1929, p. 255).

**References**

Carr, G. S. *Formulas and Theorems in Pure Mathematics, 2nd ed.* New York: Chelsea, p. 622, 1970.

Dixon, R. *Mathographics.* New York: Dover, p. 58, 1991.

Fuhrmann, W. *Synthetische Beweise Planimetrischer Sätze.* Berlin, 1890.

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.

Kimberling, C. ``Central Points and Central Lines in the Plane of a Triangle.'' *Math. Mag.* **67**, 163-187, 1994.

Kimberling, C. ``Incenter.'' http://cedar.evansville.edu/~ck6/tcenters/class/incenter.html.

© 1996-9

1999-05-26