Editing Incircle and excircles (section)

==Related constructions==

===Nine-point circle and Feuerbach point===
{{Main|Nine-point circle}}

[[File:Circ9pnt3.svg|right|thumb|250px|The nine-point circle is tangent to the incircle and excircles]]

In [[geometry]], the '''nine-point circle''' is a [[circle]] that can be constructed for any given [[triangle]]. It is so named because it passes through nine significant [[concyclic points]] defined from the triangle. These nine [[point (geometry)|points]] are:<ref>{{harvtxt|Altshiller-Court|1925|pp=103–110}}</ref><ref>{{harvtxt|Kay|1969|pp=18,245}}</ref>

* The [[midpoint]] of each side of the triangle
* The [[perpendicular|foot]] of each [[altitude (triangle)|altitude]]
* The midpoint of the [[line segment]] from each [[vertex (geometry)|vertex]] of the triangle to the [[orthocenter]] (where the three altitudes meet; these line segments lie on their respective altitudes).

In 1822, Karl Feuerbach discovered that any triangle's nine-point circle is externally [[tangent circles|tangent]] to that triangle's three excircles and internally tangent to its incircle; this result is known as [[Feuerbach's theorem]]. He proved that:<ref>{{citation |ref={{harvid|Feuerbach|1822}} |last1=Feuerbach |first1=Karl Wilhelm |author1-link=Karl Wilhelm Feuerbach |last2=Buzengeiger |first2=Carl Heribert Ignatz |year=1822 |title=Eigenschaften einiger merkwürdigen Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten Linien und Figuren. Eine analytisch-trigonometrische Abhandlung |publisher=Wiessner |location=Nürnberg |edition=Monograph |language=de |url=https://gdz.sub.uni-goettingen.de/id/PPN512512426}}.</ref>
:... the circle which passes through the feet of the altitudes of a triangle is tangent to all four circles which in turn are tangent to the three sides of the triangle ... {{harv|Feuerbach|1822}}

The [[triangle center]] at which the incircle and the nine-point circle touch is called the [[Feuerbach point]].

===Incentral and excentral triangles===
The points of intersection of the interior angle bisectors of <math>\triangle ABC</math> with the segments <math>BC</math>, <math>CA</math>, and <math>AB</math> are the vertices of the '''incentral triangle'''. Trilinear coordinates for the vertices of the incentral triangle <math>\triangle A'B'C'</math> are given by{{Citation needed|date=May 2020}}
:<math display=block>\begin{array}{ccccccc}
  A' &=& 0 &:& 1 &:& 1 \\[2pt]
  B' &=& 1 &:& 0 &:& 1 \\[2pt]
  C' &=& 1 &:& 1 &:& 0
\end{array}</math>

The '''excentral triangle''' of a reference triangle has vertices at the centers of the reference triangle's excircles. Its sides are on the external angle bisectors of the reference triangle (see figure at [[#top|top of page]]). Trilinear coordinates for the vertices of the excentral triangle <math>\triangle A'B'C'</math> are given by{{Citation needed|date=May 2020}}
:<math display=block>\begin{array}{ccrcrcr}
  A' &=& -1 &:& 1 &:& 1\\[2pt]
  B' &=& 1 &:& -1 &:& 1 \\[2pt]
  C' &=& 1 &:& 1 &:& -1
\end{array}</math>