Fuhrmann Circle

Author

Eric W. Weisstein
August 16, 2005

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

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

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

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}]
},
{Red,FuhrmannCircle[t]},
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},
{Incenter[t],"I",Orange}
}
],LabelOffset->.03]
}],AspectRatio->Automatic]

tri={{0,0},{1,-.2},{.45,.6}};
ms[i_][tri_]:=MidArcSidePoints[tri][[i]];
ma[i_][tri_]:=MidArcAntiPoints[tri][[i]];
mp[i_][tri_]:=MidArcPoints[tri][[i]];
pts={Circumcenter,"O",NagelPoint,"Na",Orthocenter,"H",
		tri[[1]],"A",tri[[2]],"B",tri[[3]],"C",
		ms[1],"(Subscript[S, C])",ms[2],"(Subscript[S, A])",ms[3],"(Subscript[S, B])",
		ma[1],"(Subscript[F, C])",ma[2],"(Subscript[F, A])",ma[3],"(Subscript[F, B])",
		mp[1],"(Subscript[M, C])",mp[2],"(Subscript[M, A])",mp[3],"(Subscript[M, B])"
};

Show[Graphics[{
	Triangle[tri],
	PointLabels[tri,pts,.02,{.03,.05}],
	MidArcLines[tri],
	Circumcircle[tri],
	FuhrmannTriangle[tri],
	FuhrmannCircle[tri]
}],AspectRatio->Automatic]

tri={{0,0},{1,-.2},{.45,.6}};
ms[i_][tri_]:=MidArcSidePoints[tri][[i]];
ma[i_][tri_]:=MidArcAntiPoints[tri][[i]];
mp[i_][tri_]:=MidArcPoints[tri][[i]];
f[i_][tri_]:=FuhrmannPoints[tri][[i]];
pts={Circumcenter,"O",NagelPoint,"Na",Orthocenter,"H",
		tri[[1]],"A",tri[[2]],"B",tri[[3]],"C",
		FuhrmannCenter,"F",f[1],"(Subscript[F, A])",f[2],"(Subscript[F, B])",f[3],"(Subscript[F, C])",
		ms[1],"",ms[2],"",ms[3],"",
		ma[1],"(Subscript[F, C])",ma[2],"(Subscript[F, A])",ma[3],"(Subscript[F, B])",
		mp[1],"",mp[2],"",mp[3],""
};

Show[Graphics[{
	Triangle[tri],
	PointLabels[tri,pts,.02,{.03,.05}],
	MidArcLines[tri],
	Circumcircle[tri],
	FuhrmannTriangle[tri],
	FuhrmannCircle[tri],
			{Dashing[{.02,.02}],Altitudes[tri],Line[{Orthocenter[tri],NagelPoint[tri]}/. Point->Identity]}
}],AspectRatio->Automatic]

Equation

FullSimplify[Midpoint[{Orthocenter,NagelPoint}],TriangleQ[]]

Together//@DistanceTrilinears[Together//@Distance[Orthocenter,NagelPoint]/2]

FullSimplify[((Sqrt[a b c ((a^3)-(a^2) b-a (b^2)+(b^3)-(a^2) c+3 a b c-(b^2) c-a (c^2)-b (c^2)+(c^3))])/(Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])) R/Circumradius[],TriangleQ[]]

FullSimplify[DistanceTrilinears[CircleEquation[{((2 a (b+c) Sec[A]+(-a+b+c) (b Sec[B]+c Sec[C]))/a),((a (a-b+c) Sec[A]+2 b (a+c) Sec[B]+c (a-b+c) Sec[C])/b),(((a+b-c) (a Sec[A]+b Sec[B])+2 (a+b) c Sec[C])/c)},((Sqrt[a b c ((a^3)-(a^2) b-a (b^2)+(b^3)-(a^2) c+3 a b c-(b^2) c-a (c^2)-b (c^2)+(c^3))])/(Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]))]],TriangleQ[]]

Circle Function

LawOfCosines[FullSimplify[CircleFunction[((a-b-c) (a+b-c) (a-b+c) ((a^5) (α^2)+(a^4) α (b β+c γ)+a ((b^3)+(c^3)) α (b β+c γ)+(b-c) (b+c) (b β+c γ) ((b^2) β-(c^2) γ)-(a^3) ((b^2) α (α+β)-b c β γ+(c^2) α (α+γ))-(a^2) ((b^3) β (α+β)+(c^3) γ (α+γ))))]]]

IncidentCenters[KimberlingCenters,CyclicTrilinears[-((2 a Cos[A])/(a+b+c))]]

CenterName[X[48]]

CenterFunction[X[48]]

Center

Simplify[CircleCenter[CyclicTrilinears[-((2 a Cos[A])/(a+b+c))]]]

FullSimplify[((2 a (b+c) Sec[A]+(-a+b+c) (b Sec[B]+c Sec[C]))/a)]

IncidentCenters[KimberlingCenters,%]

CenterName[X[355]]

CenterFunction[X[355]]

Radius

Together//@DistanceTrilinears[Together//@Distance[Orthocenter,NagelPoint]/2]

FullSimplify[((Sqrt[a b c ((a^3)-(a^2) b-a (b^2)+(b^3)-(a^2) c+3 a b c-(b^2) c-a (c^2)-b (c^2)+(c^3))])/(Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])) R/Circumradius[],TriangleQ[]]

FullSimplify[Together//@ToSidelengths[CircleRadius[FuhrmannCircle]],TriangleQ[]]

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

Circumradius[]

Equal to O I

Factor/@FullSimplify[Sqrt[Factor[ToSidelengths[Distance[Circumcenter,Incenter]]^2]],TriangleQ[]]

FullSimplify[ToSidelengths[Subtract@@{%,CircleRadius[FuhrmannCircle]}],TriangleQ[]]

Parallel to Diameter

t=Triangle[{{0,0},{1,0},{.3,.5}}];
Show[Graphics[{
t,
{
Red,
FuhrmannCircle[t],
Line[Coordinates/@Through[{Orthocenter,NagelPoint}[t]]],
EqualSideTicksAboutMidpoint[Line[Coordinates/@Through[{Orthocenter,NagelPoint}[t]]],1,.015]
},
{
Blue,
Line[Coordinates/@Through[{Incenter,Circumcenter}[t]]]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{
{Orthocenter[t],"H",Red},
{NagelPoint[t],"Na",Red},
{X[355][t],"(Subscript[X, 355])",Red},
{Incenter[t],"I",Blue},
{Circumcenter[t],"O",Blue}
}],LabelOffset->.04]
}],AspectRatio->Automatic,PlotRange->All]

Simplify[ToSidelengths[ParallelQ@@TrilinearLine/@{Line[Circumcenter,Incenter],Line[NagelPoint,Orthocenter]}]]

Incident Points

IncidentCenters[KimberlingCenters,TrilinearEquation[FuhrmannCircle]]//Timing

(ctrs=IncidentCenters[KimberlingCenters,CircleEquation[FuhrmannCircle]])//Timing

{#,CenterName[#]}&/@ctrs//TableForm