Editing Incircle and excircles (section)

====Other properties====
If the [[altitude (triangle)|altitudes]] from sides of lengths <math>a</math>, <math>b</math>, and <math>c</math> are <math>h_a</math>, <math>h_b</math>, and <math>h_c</math>, then the inradius <math>r</math> is one-third of the [[harmonic mean]] of these altitudes; that is,<ref>{{harvtxt|Kay|1969|p=203}}</ref>
:<math display=block> r = \frac{1}{\dfrac{1}{h_a} + \dfrac{1}{h_b} + \dfrac{1}{h_c}}.</math>

The product of the incircle radius <math>r</math> and the [[circumcircle]] radius <math>R</math> of a triangle with sides <math>a</math>, <math>b</math>, and <math>c</math> is{{sfn|Johnson|1929|p=189, #298(d)}}
:<math display=block>rR = \frac{abc}{2(a + b + c)}.</math>

Some relations among the sides, incircle radius, and circumcircle radius are:<ref name=Bell/>
:<math display=block>\begin{align}
     ab + bc + ca &=  s^2 +  (4R + r)r, \\
  a^2 + b^2 + c^2 &= 2s^2 - 2(4R + r)r.
\end{align}</math>

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.<ref>Kodokostas, Dimitrios, "Triangle Equalizers", ''Mathematics Magazine'' 83, April 2010, pp. 141-146.</ref>

The incircle radius is no greater than one-ninth the sum of the altitudes.<ref>Posamentier, Alfred S., and Lehmann, Ingmar. ''[[The Secrets of Triangles]]'', Prometheus Books, 2012.</ref>{{rp|289}}

The squared distance from the incenter <math>I</math> to the [[circumcenter]] <math>O</math> is given by<ref name=Franzsen>{{cite journal
|last=Franzsen
|first=William N.
|journal=Forum Geometricorum
|mr=2877263
|pages=231–236
|title=The distance from the incenter to the Euler line
|volume=11
|year=2011
|url=http://forumgeom.fau.edu/FG2011volume11/FG201126.pdf
|access-date=2012-05-09
|url-status=dead
|archive-url=https://web.archive.org/web/20201205220605/http://forumgeom.fau.edu/FG2011volume11/FG201126.pdf
|archive-date=2020-12-05
}}.</ref>{{rp|232}}
:<math display=block>\overline{OI}^2 = R(R - 2r) = \frac{a\,b\,c\,}{a+b+c}\left [\frac{a\,b\,c\,}{(a+b-c)\,(a-b+c)\,(-a+b+c)}-1 \right ]</math>

and the distance from the incenter to the center <math>N</math> of the [[nine point circle]] is<ref name=Franzsen/>{{rp|232}}
:<math display=block>\overline{IN} = \tfrac12(R - 2r) < \tfrac12 R.</math>

The incenter lies in the [[medial triangle]] (whose vertices are the midpoints of the sides).<ref name=Franzsen/>{{rp|233, Lemma 1}}