Lemoine Hexagon

Author

Eric W. Weisstein
February 8, 2005

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

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/LemoineHexagon.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];

Figure

Self-Intersecting

t=Triangle[{{0,0},{2.6,-.4},{1.3,1.4}}];
g1=Show[Graphics[{
t,
{Cyan,Polygon/@(Append[#,Coordinates[SymmedianPoint[t]]]&/@Partition[Coordinates[Cartesian[#[t]]]&/@LemoineHexagon1,2])},
{Red,FirstLemoineCircle[t]},
{Blue,ClosedLine[Coordinates[Cartesian[#[t]]]&/@LemoineHexagon1]},
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}}
],LabelOffset->{-.15,0},PointSize[.03]]
}],AspectRatio->Automatic,PlotRange->All]

Simple

t=Triangle[{{0,0},{2.6,-.4},{1.3,1.4}}];
g2=Show[Graphics[{
t,
{Cyan,Polygon[Coordinates[Cartesian[#[t]]]&/@LemoineHexagon2]},
{Red,FirstLemoineCircle[t]},
{Blue,ClosedLine[Coordinates[Cartesian[#[t]]]&/@LemoineHexagon2]},
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])"}}]
],LabelOffset->{-.15,0},PointSize[.03]]
}],AspectRatio->Automatic,PlotRange->All]

Both

Show[GraphicsArray[{g1,g2}],ImageSize->500]

Perimeter

Self-Intersecting

FullSimplify[Distance@@@Partition[LemoineHexagon1,2,1,1],TriangleQ[]&&Area>0]

Factor[Plus@@%]

Non-Cyclic

FullSimplify[Distance@@@Partition[LemoineHexagon2,2,1,1],TriangleQ[]&&Area>0]

Factor[Plus@@%]

Area

Self-Intersecting

Area/@Triangle/@(Append[#,Trilinears[SymmedianPoint]]&/@Partition[LemoineHexagon1,2,1,1])

Factor[Plus@@%]//TraditionalForm

Simple

Area/@Triangle/@(Append[#,Trilinears[SymmedianPoint]]&/@Partition[LemoineHexagon2,2,1,1])

Factor[Plus@@%]//TraditionalForm

Derivation

ParallelLine[tri_List?VectorQ,mat_List?MatrixQ]:=ParallelLine[tri,LineEquation[mat]]

ParallelLine[ctr_,line_]:=ParallelLine[Trilinears[ctr],line]/;MemberQ[Centers,ctr]

Partition[TrilinearVertexMatrix[ReferenceTriangle],2,1,1]

ParallelLine[tri_List?VectorQ,eqn_]:=Module[{l,m,n},{l,m,n}=Coefficient[eqn,#]&/@{α,β,γ};
Det[{tri,{α,β,γ},{b n-c m,c l-a n,a m-b l}}]]

sides=Partition[TrilinearVertexMatrix[ReferenceTriangle],2,1,1];

sides

peqn=ParallelLine[SymmedianPoint,#]&/@sides

leqn1=LineEquation/@RotateRight[sides]

leqn2=LineEquation/@RotateLeft[sides]

soln=(Solve[Append[Thread[#==0],2Area[]==(a α+b β+c γ)],{α,β,γ}]&/@Transpose[{peqn,#}])&/@{leqn1,leqn2}

(qs=2 ((a^2)+(b^2)+(c^2)){α,β,γ}/.soln[[1]])//ColumnForm

{((((a^2)+(b^2)) (Sqrt[-(a^4)+2 (a^2) (b^2)-(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-(c^4)]))/a),0,c (Sqrt[-(a^4)+2 (a^2) (b^2)-(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-(c^4)])}

{
{((((a^2)+(b^2)) (Sqrt[-(a^4)+2 (a^2) (b^2)-(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-(c^4)]))/a),0,c (Sqrt[-(a^4)+2 (a^2) (b^2)-(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-(c^4)])},
{a (Sqrt[-(a^4)+2 (a^2) (b^2)-(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-(c^4)]),((((b^2)+(c^2)) (Sqrt[-(a^4)+2 (a^2) (b^2)-(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-(c^4)]))/b),0},
{0,b (Sqrt[-(a^4)+2 (a^2) (b^2)-(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-(c^4)]),((((a^2)+(c^2)) (Sqrt[-(a^4)+2 (a^2) (b^2)-(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-(c^4)]))/c)}
}/(Sqrt[-(a^4)+2 (a^2) (b^2)-(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-(c^4)])

qs={{(((a^2)+(b^2))/a),0,c},{a,(((b^2)+(c^2))/b),0},{0,b,(((a^2)+(c^2))/c)}};

(ps=2 ((a^2)+(b^2)+(c^2)){α,β,γ}/.soln[[2]])//ColumnForm

ps={
{0,((((a^2)+(b^2)) (Sqrt[-(a^4)+2 (a^2) (b^2)-(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-(c^4)]))/b),c (Sqrt[-(a^4)+2 (a^2) (b^2)-(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-(c^4)])},
{a (Sqrt[-(a^4)+2 (a^2) (b^2)-(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-(c^4)]),0,((((b^2)+(c^2)) (Sqrt[-(a^4)+2 (a^2) (b^2)-(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-(c^4)]))/c)},
{((((a^2)+(c^2)) (Sqrt[-(a^4)+2 (a^2) (b^2)-(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-(c^4)]))/a),b (Sqrt[-(a^4)+2 (a^2) (b^2)-(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-(c^4)]),0}
}/(Sqrt[-(a^4)+2 (a^2) (b^2)-(b^4)+2 (a^2) (c^2)+2 (b^2) (c^2)-(c^4)])

ps={{(((a^2)+(c^2))/a),b,0},{0,(((a^2)+(b^2))/b),c},{a,0,(((b^2)+(c^2))/c)}}