Extouch triangle: Difference between revisions

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{{short description|Triangle formed from the points of tangency of a given triangle's excircles}}

[[Image:Extouch Triangle and Nagel Point.svg|right|frame|325px|The extouch triangle (red, ΔT_aT_bT_c) and the [[Nagel point]] (blue, N) of a triangle (black, ΔABC). The orange circles are the [[excircles]] of the triangle.]]

[[Image:Extouch Triangle and Nagel Point.svg|right|frame|

The '''extouch triangle''' of a triangle is formed by joining the points at which the three [[excircle]]s touch the triangle. The vertices of the extouch triangle are given in [[trilinear coordinates]] by:

{{legend-line|solid #333333|Arbitrary triangle {{math|△''ABC''}}}}

{{legend-line|solid orange|[[Excircle]]s, tangent to the sides of {{math|△''ABC''}} at {{mvar|T{{sub|A}}, T{{sub|B}}, T{{sub|C}}}}}}

{{legend-line|solid red|'''Extouch triangle''' {{math|△''T{{sub|A}}T{{sub|B}}T{{sub|C}}''}}}}

{{legend-line|solid #1e90ff|[[Splitter (geometry)|Splitters]] of the perimeter {{mvar|{{overline|AT}}{{sub|A}}, {{overline|BT}}{{sub|B}}, {{overline|CT}}{{sub|C}}}}; intersect at the [[Nagel point]] {{mvar|N}}}}]]

TheIn [[Euclidean geometry]], the '''extouch triangle''' of a [[triangle]] is formed by joining the points at which the three [[excircle]]s touch the triangle. The vertices of the extouch triangle are given in [[trilinear coordinates]] by:

*<math>T_A = 0 : \csc^2{\left( B/2 \right)} : \csc^2{\left( C/2 \right)}</math>

*<math>T_B = \csc^2{\left( A/2 \right)} : 0 : \csc^2{\left( C/2 \right)}</math>

*<math>T_C = \csc^2{\left( A/2 \right)} : \csc^2{\left( B/2 \right)} : 0</math>

==Coordinates==

Or, equivalently, where a,b,c are the lengths of the sides opposite angles A, B, C respectively,

The [[vertex (geometry)|vertices]] of the extouch triangle are given in [[trilinear coordinates]] by:

<math display=block>\begin{array}{rccccc}

*<math>T_A = 0 : \frac{a-b+c}{b} : \frac{a+b-c}{c}</math>

*<math>T_BT_A =& 0 &:& \csc^2 \frac{-a+b+cB}{a2} &:& 0 :\csc^2 \frac{a+b-cC}{c2}</math> \\

*<math>T_CT_B =& \csc^2 \frac{-a+b+cA}{a2} &:& 0 &:& \csc^2 \frac{a-b+cC}{b2} : 0</math>\\

T_C =& \csc^2 \frac{A}{2} &:& \csc^2 \frac{B}{2} &:& 0

\end{array}</math>

Or,or equivalently, where {{mvar|a, b, c}} are the lengths of the sides opposite angles {{mvar|A, B, C}} respectively,

The intersection of the lines connecting the vertices of the original triangle to the corresponding vertices of the extouch triangle is the [[Nagel point]]. This is shown in blue and labelled "N" in the diagram.

<math display=block>\begin{array}{rccccc}

==Area==

*<math> T_A =& 0 &:& \frac{a \, - \, b \, + \, c}{b} &:& \frac{a \, + \, b \, - \, c}{c}</math> \\

T_B =& \frac{-a \, + \, b \, + \, c}{a} &:& 0 &:& \frac{a \, + \, b \, - \, c}{c} \\

T_C =& \frac{-a \, + \, b \, + \, c}{a} &:& \frac{a \, - \, b \, + \, c}{b} &:& 0

\end{array}</math>

Also, with {{mvar| s}} denoting the [[semiperimeter]] of the triangle, the vertices of the extouch triangle are given in [[barycentric coordinates]] by:

The area of the extouch triangle, A_T, is given by:

<math display=block>\begin{array}{rccccc}

T_A =& 0 &:& s-b &:& s-c \\

T_B =& s-a &:& 0 &:& s-c \\

T_C =& s-a &:& s-b &:& 0

\end{array}</math>

==Related figures==

The intersectiontriangle's of[[Splitter (geometry)|splitters]] are the lines connecting the vertices of the original triangle to the corresponding vertices of the extouch triangle; they bisect the triangle's perimeter and meet isat the [[Nagel point]]. This is shown in blue and labelled "N" in the diagram.

The [[Mandart inellipse]] is tangent to the sides of the reference triangle at the three vertices of the extouch triangle.<ref>{{citation

| last = Juhász | first = Imre

| journal = Annales Mathematicae et Informaticae

| mr = 3005114

| pages = 37–46

| title = Control point based representation of inellipses of triangles

| url = http://ami.ektf.hu/uploads/papers/finalpdf/AMI_40_from37to46.pdf

| volume = 40

| year = 2012}}.</ref>

==Area==

The area of the extouch triangle, A<{{mvar|K{{sub>|T</sub>}}}}, is given by:

::<math>A_T= A \frac{2r^2s}{abc}</math>

::<math>A_TK_T= A K\frac{2r^2s}{abc}</math>

where A, r, s are the area, radius of the [[incircle]] and [[semiperimeter]] of the original triangle, and a, b, c are the side lengths of the original triangle.

where A,{{mvar|K}} and {{mvar|r, s}} are the area, and radius of the [[incircle]], and{{mvar|s}} is the [[semiperimeter]] of the original triangle, and {{mvar|a, b, c}} are the side lengths of the original triangle.

This is the same area as the [[intouch triangle]].

This is the same area as that of the [[intouch triangle]].<ref>Weisstein, Eric W. "Extouch Triangle." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ExtouchTriangle.html</ref>

==See also==

*[[Excircle]]

*[[Incircle]]

*[[Intouch triangle]]

*[[Cevian triangle]]

==External linksReferences==

{{reflist}}

* [http://mathworld.wolfram.com/ExtouchTriangle.html Extouch triangle at MathWorld]

[[Category:Circles]]

[[Category:TriangleObjects Geometrydefined for a triangle]]