Feuerbach Triangle

Author

Eric W. Weisstein
November 8, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/FeuerbachTriangle.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/FeuerbachTriangle.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Figure

t=Triangle[{{0,0},{1,0},{.2,.7}}];
Show[Graphics[{
t,
{Dashing[{.01}],Sides[t,2]},
{Red,FeuerbachTriangle[t]},
{Red,NinePointCircle[t]},
{Blue,Excircles[t]},
{Purple,Incircle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[ExcentralTriangle[t]],{"(Subscript[J, A])","(Subscript[J, B])","(Subscript[J, C])"}}],
Append[#,Red]&/@Transpose[{VertexPoints[FeuerbachTriangle[t]],{"(Subscript[F, A])","(Subscript[F, B])","(Subscript[F, C])"}}],
{
{FeuerbachPoint[t],"F",Purple},
{Incenter[t],"I",Purple}
}
],{PointSize[.01],LabelOffset->.06}
]
}],AspectRatio->Automatic
]

Vertices

Trilinears[FeuerbachTriangle]//TraditionalForm

Area

area=Together//@DistanceTrilinears[Area[FeuerbachTriangle]Δ/Area[]/.{((Cos[(1/2) (-x_+y_)])^2) :>(1/2)+(1/2) Cos[x] Cos[y]+(1/2) Sin[x] Sin[y],
((Sin[(1/2) (-x_+y_)])^2) :>(1/2)-(1/2) Cos[x] Cos[y]-(1/2) Sin[x] Sin[y]}]

FullSimplify[area,TriangleQ[]]

Factor[(((((a-b))^2) (a+b)+((a^2)-3 a b+(b^2)) c-(a+b) (c^2)-(c^3)) ((a-b) (((a+b))^2)-((a^2)+3 a b+(b^2)) c-(a-b) (c^2)+(c^3)) ((a^3)+(a^2) (b+c)-(((b-c))^2) (b+c)-a ((b^2)-3 b c+(c^2))))]

Circle Center and Radius

rs=(Together//@(ToSidelengths[HalfAngleFormulas[Distance[#,Trilinears[NinePointCenter]]^2&/@TrilinearVertexMatrix[FeuerbachTriangle]]/.Cos[x_]:>TrigExpand[Cos[x]]])/.Sqrt[x_]Sqrt[y_]:>FullSimplify[Sqrt[x y],TriangleQ[]]);

Refine[(Sqrt[Factor[rs]]),TriangleQ[]]

Circle[NinePointCircle]

Centers

(pts={#,KimberlingLookup[#[FeuerbachTriangle]]}&/@Take[KimberlingCenters,100]);//Timing

{#[[1]],Name[#[[1]]],X[#[[2,1]]],Name[X[#[[2,1]]]]}&/@Cases[pts,_?(Last[#]=!={}&)]//TableForm