Incircle and Excircles of A Triangle - Other Incircle Properties

Other Incircle Properties

Suppose the tangency points of the incircle divide the sides into lengths of x and y, y and z, and z and x. Then the incircle has the radius

and the area of the triangle is

If the altitudes from sides of lengths a, b, and c are h_a, h_b, and h_c then the inradius r is one-third of the harmonic mean of these altitudes, i.e.

The distance d between the circumcenter and the incenter is given by

where r is the incircle radius and R is the circumcircle radius. Thus the incircle radius is no larger than half the circumcircle radius (Euler's triangle inequality).

The product of the incircle radius and the circumcircle radius of a triangle with sides a, b, and c is

Some relations among the sides, incircle radius, and circumcircle radius are:

Any line through a triangle that splits both the triangle's area and its perimeter in half goes through the triangle's incenter (the center of its incircle). There are either one, two, or three of these for any given triangle.

The distance from the incenter to the centroid is less than one third the length of the longest median of the triangle.

Denoting the distance from the incenter to the Euler line as d, the length of the longest median as v, the length of the longest side as u, and the semiperimeter as s, the following inequalities hold:

Denoting the center of the incircle of triangle ABC as I, we have