Editing Barycentric coordinate system (section)

===Examples of special points===
In the homogeneous barycentric coordinate system defined with respect to a triangle <math>ABC</math>, the following statements about special points of <math>ABC</math> hold.

The three [[vertex (geometry)|vertices]] {{mvar|A}}, {{mvar|B}}, and {{mvar|C}} have coordinates<ref name=Scott>Scott, J. A. "Some examples of the use of areal coordinates in triangle geometry", ''[[Mathematical Gazette]]'' 83, November 1999, 472–477.</ref>

<math display=block>\begin{array}{rccccc}
  A = & 1 &:& 0 &:& 0 \\
  B = & 0 &:& 1 &:& 0 \\
  C = & 0 &:& 0 &:& 1
\end{array}</math>

The [[centroid]] has coordinates <math>1:1:1.</math><ref name=Scott/>

If {{mvar|a}}, {{mvar|b}}, {{mvar|c}} are the [[length|edge lengths]] <math>BC</math>, <math>CA</math>, <math>AB</math> respectively, <math>\alpha</math>, <math>\beta</math>, <math>\gamma</math> are the [[angle | angle measures]] <math>\angle CAB</math>, <math>\angle ABC</math>, and <math>\angle BCA</math> respectively, and {{mvar|s}} is the [[semiperimeter]] of <math>ABC</math>, then the following statements about special points of <math>ABC</math> hold in addition.

The [[circumcenter]] has coordinates<ref name=Scott/><ref name=Olympiad>{{cite web|last1=Schindler|first1=Max|last2=Chen|first2=Evan|title=Barycentric Coordinates in Olympiad Geometry|url=https://www.mit.edu/~evanchen/handouts/bary/bary-full.pdf|access-date=14 January 2016|date=July 13, 2012}}</ref><ref name=ck>Clark Kimberling's Encyclopedia of Triangles {{cite web |url=http://faculty.evansville.edu/ck6/encyclopedia/ETC.html |title=Encyclopedia of Triangle Centers |access-date=2012-06-02 |url-status=dead |archive-url=https://web.archive.org/web/20120419171900/http://faculty.evansville.edu/ck6/encyclopedia/ETC.html |archive-date=2012-04-19 }}</ref><ref name=":0">[http://mathworld.wolfram.com/BarycentricCoordinates.html Wolfram page on barycentric coordinates]</ref>

<math display=block>\begin{array}{rccccc}
  & \sin 2\alpha &:& \sin 2\beta &:& \sin 2\gamma \\[2pt]
  =& 1-\cot\beta\cot\gamma &:& 1-\cot\gamma\cot\alpha &:& 1-\cot\alpha\cot\beta \\[2pt]
  =& a^2(-a^2+b^2+c^2) &:& b^2(a^2-b^2+c^2) &:& c^2(a^2+b^2-c^2)
\end{array}</math>

The [[orthocenter]] has coordinates<ref name=Scott/><ref name=Olympiad/>

<math display=block>\begin{array}{rccccc}
  & \tan\alpha &:& \tan\beta &:& \tan\gamma \\[2pt]
  =& a\cos\beta\cos\gamma &:& b\cos\gamma\cos\alpha &:& c\cos\alpha\cos\beta \\[2pt]
  =& (a^2+b^2-c^2)(a^2-b^2+c^2) &:& (-a^2+b^2+c^2)(a^2+b^2-c^2) &:& (a^2-b^2+c^2)(-a^2+b^2+c^2)
\end{array}</math>

The [[incenter]] has coordinates <math>a:b:c=\sin \alpha:\sin \beta:\sin \gamma.</math><ref name=Olympiad/><ref name=NK>Dasari Naga, Vijay Krishna, "On the Feuerbach triangle",
''Forum Geometricorum'' 17 (2017), 289–300: p. 289. http://forumgeom.fau.edu/FG2017volume17/FG201731.pdf</ref>

The [[excenter]]s have coordinates<ref name=NK/>

<math display=block>\begin{array}{rrcrcr}
  J_A = & -a &:& b &:& c \\
  J_B = & a &:& -b &:& c \\ 
  J_C = & a &:& b &:& -c 
\end{array}</math>

The [[nine-point center]] has coordinates<ref name=Scott/><ref name=NK/>

<math display=block>\begin{array}{rccccc}
  & a\cos(\beta-\gamma) &:& b\cos(\gamma-\alpha) &:& c\cos(\alpha-\beta) \\[4pt]
  =& 1+\cot\beta\cot\gamma &:& 1+\cot\gamma\cot\alpha &:& 1+\cot\alpha\cot\beta \\[4pt]
  =& a^2(b^2+c^2) - (b^2-c^2)^2 &:& b^2(c^2+a^2) - (c^2-a^2)^2 &:& c^2(a^2+b^2) - (a^2-b^2)^2
\end{array}</math>

The [[Gergonne Point|Gergonne point]] has coordinates <math>(s-b)(s-c):(s-c)(s-a):(s-a)(s-b)</math>.

The [[Nagel point]] has coordinates <math>s-a:s-b:s-c</math>.

The [[symmedian point]] has coordinates <math>a^2:b^2:c^2</math>.<ref name=":0" />