Editing Incircle and excircles (section)

===Gergonne triangle and point===
[[File:Intouch Triangle and Gergonne Point.svg|right|frame|
{{legend-line|solid black|Triangle {{math|△''ABC''}}}}
{{legend-line|solid #728fce|Incircle ([[incenter]] at {{mvar|I}})}}
{{legend-line|solid red|Contact triangle {{math|△''T{{sub|A}}T{{sub|B}}T{{sub|C}}''}}}}
{{legend-line|solid #1dc404|Lines between opposite vertices of {{math|△''ABC''}} and {{math|△''T{{sub|A}}T{{sub|B}}T{{sub|C}}''}} (concur at Gergonne point {{mvar|G{{sub|e}}}})}}
]]

The '''Gergonne triangle''' (of <math>\triangle ABC</math>) is defined by the three touchpoints of the incircle on the three sides. The touchpoint opposite <math>A</math> is denoted <math>T_A</math>, etc.

This Gergonne triangle, <math>\triangle T_AT_BT_C</math>, is also known as the '''contact triangle''' or '''intouch triangle''' of <math>\triangle ABC</math>. Its area is
:<math display=block>K_T = K\frac{2r^2 s}{abc}</math>

where <math>K</math>, <math>r</math>, and <math>s</math> are the area, radius of the incircle, and semiperimeter of the original triangle, and <math>a</math>, <math>b</math>, and <math>c</math> are the side lengths of the original triangle. This is the same area as that of the [[extouch triangle]].<ref>
Weisstein, Eric W. "Contact Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ContactTriangle.html</ref>

The three lines <math>AT_A</math>, <math>BT_B</math>, and <math>CT_C</math> intersect in a single point called the '''Gergonne point''', denoted as <math>G_e</math> (or [[triangle center]] ''X''<sub>7</sub>). The Gergonne point lies in the open [[orthocentroidal disk]] punctured at its own center, and can be any point therein.<ref name=Bradley>Christopher J. Bradley and Geoff C. Smith, "The locations of triangle centers", ''[[Forum Geometricorum]]'' 6 (2006), 57–70. http://forumgeom.fau.edu/FG2006volume6/FG200607index.html</ref>

The Gergonne point of a triangle has a number of properties, including that it is the [[symmedian point]] of the Gergonne triangle.<ref>
{{cite journal
|last=Dekov
|first=Deko
|title=Computer-generated Mathematics : The Gergonne Point
|journal=Journal of Computer-generated Euclidean Geometry
|year=2009
|volume=1
|pages=1&ndash;14
|url=http://www.dekovsoft.com/j/2009/01/JCGEG200901.pdf
|url-status=dead
|archive-url=https://web.archive.org/web/20101105045604/http://www.dekovsoft.com/j/2009/01/JCGEG200901.pdf
|archive-date=2010-11-05
}}</ref>

[[Trilinear coordinates]] for the vertices of the intouch triangle are given by{{Citation needed|date=May 2020}}
:<math display=block>\begin{array}{ccccccc}
  T_A &=& 0 &:& \sec^2 \frac{B}{2} &:& \sec^2\frac{C}{2} \\[2pt]
  T_B &=& \sec^2 \frac{A}{2} &:& 0 &:& \sec^2\frac{C}{2} \\[2pt]
  T_C &=& \sec^2 \frac{A}{2} &:& \sec^2\frac{B}{2} &:& 0.
\end{array}</math>

Trilinear coordinates for the Gergonne point are given by{{Citation needed|date=May 2020}}
:<math display=block>\sec^2\tfrac{A}{2} : \sec^2\tfrac{B}{2} : \sec^2\tfrac{C}{2},</math>

or, equivalently, by the [[Law of Sines]],
:<math display=block>\frac{bc}{b + c - a} : \frac{ca}{c + a - b} : \frac{ab}{a + b - c}.</math>