Editing Incircle and excircles (section)

===Nagel triangle and Nagel point===
{{Main|Extouch triangle}}
[[File:Extouch Triangle and Nagel Point.svg|right|frame|
{{legend-line|solid black|[[Extended side]]s of triangle {{math|△''ABC''}}}}
{{legend-line|solid orange|Excircles of {{math|△''ABC''}} (tangent at {{mvar|T{{sub|A}}. T{{sub|B}}, T{{sub|C}}}})}}
{{legend-line|solid red|'''Nagel/Extouch triangle''' {{math|△''T{{sub|A}}T{{sub|B}}T{{sub|C}}''}}}}
{{legend-line|solid #728fce|[[Splitter (geometry)|Splitters]]: lines connecting opposite vertices of {{math|△''ABC''}} and {{math|△''T{{sub|A}}T{{sub|B}}T{{sub|C}}''}} (concur at '''Nagel point''' {{mvar|N}})}}
]]

The '''Nagel triangle''' or '''extouch triangle''' of <math>\triangle ABC</math> is denoted by the vertices <math>T_A</math>, <math>T_B</math>, and <math>T_C</math> that are the three points where the excircles touch the reference <math>\triangle ABC</math> and where <math>T_A</math> is opposite of <math>A</math>, etc. This <math>\triangle T_AT_BT_C</math> is also known as the '''extouch triangle''' of <math>\triangle ABC</math>. The [[circumcircle]] of the extouch <math>\triangle T_AT_BT_C</math> is called the '''Mandart circle'''
(cf. [[Mandart inellipse]]).

The three line segments <math>\overline{AT_A}</math>, <math>\overline{BT_B}</math> and <math>\overline{CT_C}</math> are called the [[splitter (geometry)|splitters]] of the triangle; they each bisect the perimeter of the triangle,{{Citation needed|date=May 2020}}
:<math display=block>\overline{AB} + \overline{BT_A} = \overline{AC} + \overline{CT_A} = \frac{1}{2}\left( \overline{AB} + \overline{BC} + \overline{AC} \right).</math>

The splitters intersect in a single point, the triangle's [[Nagel point]] <math>N_a</math> (or [[triangle center]] ''X''<sub>8</sub>).

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

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

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

The Nagel point is the [[isotomic conjugate]] of the Gergonne point.{{Citation needed|date=May 2020}}