Fuhrmann Triangle

Author

Eric W. Weisstein
November 11, 2005

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

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

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

Fuhrmann Triangle

Figure

t=Triangle[{{0,0},{1,0},{.2,.5}}];
Show[Graphics[{
t,
{Green,Circumcircle[t]},
{
Blue,
CircumcircleMidArcTriangle[t]
},
{
Red,
FuhrmannTriangle[t]
},
{
Purple,
{Dashing[{.02}],Line[{Coordinates[Circumcenter[t]],#}]&/@Vertices[MedialTriangle[t]]},
l=Line/@Transpose[Vertices/@Through[{CircumcircleMidArcTriangle,FuhrmannTriangle}[t]]],
MapIndexed[EqualSideTicksAboutMidpoint[#1,#2[[1]],.012,.04]&,l],
PerpSquare[#,.03,.3]&/@Transpose[{Sides[t],l}]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Purple]&/@Transpose[{VertexPoints[MedialTriangle[t]],{"(Subscript[M, A])","(Subscript[M, B])","(Subscript[M, C])"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[CircumcircleMidArcTriangle[t]],{"(Subscript[(Overscript[M, ⌢]), A])","(Subscript[(Overscript[M, ⌢]), B])","(Subscript[(Overscript[M, ⌢]), C])"}}],
Append[#,Red]&/@Transpose[{VertexPoints[FuhrmannTriangle[t]],{"(Subscript[F, A])","(Subscript[F, B])","(Subscript[F, C])"}}],
{{Circumcenter[t],"O",Purple}}
],LabelOffset->.03]
}],AspectRatio->Automatic]

Trilinear Vertex Matrix

TrilinearVertexMatrix[FuhrmannTriangle]//TraditionalForm

Concurrent in Circumcenter

Map[Symmetrize,ToAngles[Map[Symmetrize,Factor[Intersections@@@Partition[Transpose[TrilinearVertexMatrix/@{CircumcircleMidArcTriangle,FuhrmannTriangle}],2,1,1]],{2}]
],{2}]

Trilinears[Circumcenter]

Center of Circumcircle

Factor[Together//@CircleEquation[TrilinearVertexMatrix[FuhrmannTriangle]]]

eqn=((a^5) (α^2)-(a^3) (b^2) (α^2)-(a^3) (c^2) (α^2)+(a^4) b α β-(a^3) (b^2) α β-(a^2) (b^3) α β+a (b^4) α β+a b (c^3) α β-(a^2) (b^3) (β^2)+(b^5) (β^2)-(b^3) (c^2) (β^2)+(a^4) c α γ+a (b^3) c α γ-(a^3) (c^2) α γ-(a^2) (c^3) α γ+a (c^4) α γ+(a^3) b c β γ+(b^4) c β γ-(b^3) (c^2) β γ-(b^2) (c^3) β γ+b (c^4) β γ-(a^2) (c^3) (γ^2)-(b^2) (c^3) (γ^2)+(c^5) (γ^2));

Map[Factor,Together//@ToSidelengths[Circumcircle[eqn]],{2}]

Factor[Simplify[Together//@ToSidelengths[TrigExpand[CenterFunction[FuhrmannCenter]]],TriangleQ[]]]

(Sqrt[Factor[(((a^4) b c-(a^3) (b^2) c-(a^2) (b^3) c+a (b^4) c-(a^3) b (c^2)+3 (a^2) (b^2) (c^2)-a (b^3) (c^2)-(a^2) b (c^3)-a (b^2) (c^3)+a b (c^4))/(-(a-b-c) (a+b-c) (a-b+c) (a+b+c)))]])

CircleRadius[FuhrmannCircle]

Derivation

Simplify[Together//@ToSidelengths[Solve[
{CircleEquation[Circumcircle]==0,PerpendicularLine[TrilinearVertexMatrix[MedialTriangle][[1]],α]==0,
2Area[]==(a α+b β+c γ)},{α,β,γ}]],TriangleQ[]]

Map[Factor,%117,{3}]

TrilinearVertexMatrix[CircumcircleMidArcTriangle]:={{-a ,b+c,b+c},{a+c,-b,a+c},{a+b,a+b,-c}}

midarcpts=TrilinearVertexMatrix[CircumcircleMidArcTriangle]

midpts=TrilinearVertexMatrix[MedialTriangle]

Map[Symmetrize,Reflection@@@Transpose[{midarcpts,midpts}],{2}]

TrilinearVertexMatrix[FuhrmannTriangle]:={{a,((-(a^2)+b c+(c^2))/b),((-(a^2)+(b^2)+b c)/c)},{((-(b^2)+a c+(c^2))/a),b,(((a^2)-(b^2)+a c)/c)},{((a b+(b^2)-(c^2))/a),(((a^2)+a b-(c^2))/b),c}}

TrilinearVertexMatrix[FuhrmannTriangle] := 
  {{a, (-a^2 + b*c + c^2)/b, (-a^2 + b^2 + b*c)/c}, 
   {(-b^2 + a*c + c^2)/a, b, (a^2 - b^2 + a*c)/c}, 
   {(a*b + b^2 - c^2)/a, (a^2 + a*b - c^2)/b, c}}

Area

Factor[Area[FuhrmannTriangle]]

area=Factor[FullSimplify[Factor[ToSidelengths[Area[FuhrmannTriangle]]],TriangleQ[]]]

Factor[FullSimplify[area(R/(ToSidelengths[Circumradius])),TriangleQ[]]]

Factor[FullSimplify[area((ToSidelengths[Inradius])/r),TriangleQ[]]]

Factor[-FullSimplify[ToSidelengths[Distance[Incenter,Circumcenter]^2(((a+b+c))/(4Circumradius))],TriangleQ[]]]

Sidelengths

Sqrt[Factor[Simplify[SideLengths[FuhrmannTriangle],Area>0]^2]]

Factor[Simplify[Factor[ToSidelengths[(((b+c-a)(a+b+c)((Distance[Circumcenter,Incenter])^2))/(b c))]]]]

Points

Circumcenter

cc=Symmetrize[Circumcenter[FuhrmannTriangle]]

IncidentCenters[KimberlingCenters,cc]//Timing

Name[X[355]]

Incenter

Incenter[FuhrmannTriangle]

Nine-Point Center

NinePointCenter[FuhrmannTriangle]

Orthocenter

oc=-Symmetrize[Factor[ToSidelengths[Orthocenter[FuhrmannTriangle]]]]

IncidentCenters[KimberlingCenters,oc]//Timing

Name[X[1]]

Symmedian Point

sp=SymmedianPoint[FuhrmannTriangle]

Centers

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

l=Map[First,DeleteCases[pts,_?(Last[#]==={}&)],{2}]

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