Hexyl Triangle

Author

Eric W. Weisstein
November 8, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[{
t,
{Dashing[{.02}],Sides[t,3]},
{Purple,Excircles[t]},
{Blue,HexylHexagon[t]},
{Red,ht=HexylTriangle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[ht],{"A'","B'","C'"}}],
Append[#,Blue]&/@Transpose[{Excenters[t],{"(Subscript[J, A])","(Subscript[J, B])","(Subscript[J, C])"}}]
],PointSize[.015],LabelOffset->.15]
}],AspectRatio->Automatic]

Trilinears

Trilinears[HexylTriangle]//MatrixForm//TraditionalForm

Side Lengths

FullSimplify[Together//@DistanceTrilinears[SideLengths[HexylTriangle]],TriangleQ[]]

FullSimplify[LawOfCosines[Together//@ExpandAll[%155]],TriangleQ[]]

Area

FullSimplify[Together//@ToSidelengths[Area[HexylTriangle]],TriangleQ[]]

Refine[ToAngles[%],Area>0]

Perspectors

Reference Triangle

persp=Factor[ToSidelengths[Perspector[ReferenceTriangle,HexylTriangle]]]

IncidentCenters[KimberlingCenters,persp]//Timing

Perspective with

t=RandomTriangle[50];
t2=HexylTriangle[t];

Anticevian Triangle

(ctrs=Select[KimberlingCenters,PerspectiveQ[t2,AnticevianTriangle[t,#]]&])//Timing

First/@ctrs

Length[ctrs]

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

Antipedal Triangle

(ctrs=Select[KimberlingCenters,PerspectiveQ[t2,AntipedalTriangle[t,#]]&])//Timing

First/@ctrs

Length[ctrs]

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

Cevian Triangles

(ctrs=Select[KimberlingCenters,PerspectiveQ[t2,CevianTriangle[t,#]]&])//Timing

First/@ctrs

Length[ctrs]

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

Pedal Triangle

(ctrs=Select[KimberlingCenters,PerspectiveQ[t2,PedalTriangle[t,#]]&])//Timing

First/@ctrs

Length[ctrs]

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

Triangle Points

Centroid

g=Symmetrize[ToSidelengths[Centroid[HexylTriangle]]]

IncidentCenters[KimberlingCenters,g]//Timing

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[{
t,
{Blue,HexylTriangle[t]},
{Red,Medians[HexylTriangle[t]]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[HexylTriangle[t]],{"A'","B'","C'"}}],
Append[#,Red]&/@Transpose[{Midpoints[HexylTriangle[t]],{"(Subscript[M, A'])","(Subscript[M, B'])","(Subscript[M, C'])"}}],
{
{Centroid[HexylTriangle[t]],"G'",Red}
}
],PointSize[.02],LabelOffset->.06]

}],AspectRatio->Automatic]

Centers

(pts={#,KimberlingLookup[#[HexylTriangle]]}&/@Take[KimberlingCenters,1000]);//Timing

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

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

(pts={#,KimberlingLookup[#[HexylTriangle]]}&/@Take[KimberlingCenters,{1001,2000}]);//Timing

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

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