First Lemoine Circle

Author

Eric W. Weisstein
February 9, 2005

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

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

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

Code

ps={{(((a^2)+(c^2))/a),b,0},{0,(((a^2)+(b^2))/b),c},{a,0,(((b^2)+(c^2))/c)}};
qs={{(((a^2)+(b^2))/a),0,c},{a,(((b^2)+(c^2))/b),0},{0,b,(((a^2)+(c^2))/c)}};
LemoineHexagon1:=Flatten[RotateRight[Reverse[Transpose[{qs,ps}]]],1];LemoineHexagon2:=Flatten[Transpose[{qs,ps}],1];

Graphic

t=Triangle[{{0,0},{2.6,-.4},{1.3,1.4}}];
h=Flatten[Coordinates/@Parallels[t,SymmedianPoint[t]],1];
Show[Graphics[{
t,
{
Violet,
FirstLemoineCircle[t],
Parallels[t,SymmedianPoint[t]]
},
{
Blue,
l=Line[Coordinates/@Through[{SymmedianPoint,Circumcenter}[t]]],
EqualSideTicksAboutMidpoint[l,1,.05]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[Cartesian[Triangle[ps][t]]],{"(Subscript[P, A])","(Subscript[P, B])","(Subscript[P, C])"}}],
Append[#,Red]&/@Transpose[{VertexPoints[Cartesian[Triangle[qs][t]]],{"(Subscript[Q, A])","(Subscript[Q, B])","(Subscript[Q, C])"}}],
{{SymmedianPoint[t],"K",Blue}},
{{Circumcenter[t],"O",Blue}},
{{X[182][t],"(Subscript[X, 182])",Red}}
],LabelOffset->{-.15,0}]
}],AspectRatio->Automatic,PlotRange->All]

Equation

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

FullSimplify[CircleFunction[eqn]]

-((b*c*(b^2 + c^2))/(a^2 + b^2 + c^2)^2)

CircleEquation[FirstLemoineCircle]

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

CenterName[X[141]]

Incident Points

(ctrs=IncidentCenters[KimberlingCenters,FirstLemoineCircle])//Timing

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

Center

Symmetrize[Together//@ToSidelengths[CircleCenter[CircleEquation[FirstLemoineCircle]][[1]]]]

Symmetrize[Together//@ToSidelengths[Midpoint[{SymmedianPoint,Circumcenter}][[1]]]]

IncidentCenters[KimberlingCenters,ctr]

CenterName[X[182]]

Symmetrize[Simplify[Together//@ToSidelengths[TrigExpand[CenterFunction[X[182]]]],TriangleQ[]]]

Radius

FullSimplify[Together//@DistanceTrilinears[CircleRadius[FirstLemoineCircle]],TriangleQ[]]

R=((a b c Sqrt[(b^2) (c^2)+(a^2) (b^2)+(c^2) (a^2)])/(((a^2)+(b^2)+(c^2)) (Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])))

Map[Factor,Simplify[Circumradius[]Sec[BrocardAngle[]]/2,TriangleQ[]],{2}]

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

Side Ratios

q=qs[[3,1]]

p=ps[[1,1]]

Simplify[Together//@Distance@@@Partition[{{0,1,0},p,q,{0,0,1}},2,1],TriangleQ[]]

(((a^2)+(b^2)+(c^2))/a)%