D-Triangle

Author

Eric W. Weisstein
November 11, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/D-Triangle.nb.

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

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

Figure

t=Triangle[{{0,0},{1,0},{.2,.5}}];
Show[Graphics[{
t,
{Red,DTriangle[t]},
{Green,OrthocentroidalCircle[t],
Cartesian[(Line@@(Trilinears/@{Centroid,Orthocenter}))[t]]
},
{Blue,Medians[t]},
EqualSideTicksAboutMidpoints[t,.02],
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[MedialTriangle[t]],{"(Subscript[M, A])","(Subscript[M, B])","(Subscript[M, C])"}}],
Append[#,Red]&/@Transpose[{VertexPoints[DTriangle[t]],{"(Subscript[D, A])","(Subscript[D, B])","(Subscript[D, C])"}}],
{
{Centroid[t],"G",{Blue,LabelOffset->{.03,-.02}}},
{Orthocenter[t],"H",{Green,LabelOffset->{-.03,.02}}}
}
],LabelOffset->.035]
}],AspectRatio->Automatic,PlotRange->All]

t=Triangle[{{0,0},{1,0},{.2,.5}}];
Show[Graphics[{
t,
{Blue,ApolloniusCircles[t]},
{Red,DTriangle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[DTriangle[t]],{"(Subscript[D, A])","(Subscript[D, B])","(Subscript[D, C])"}}]
],LabelOffset->.04]
}],AspectRatio->Automatic,PlotRange->{{-1,1.1},{-.2,1}}]

Trilinear Vertex Matrix

TrilinearVertexMatrix[DTriangle]//TraditionalForm

Angles

{a1,a2,a3}=TrilinearVertexMatrix[ReferenceTriangle];
{d1,d2,d3}=TrilinearVertexMatrix[DTriangle];

Simplify[ToAngles[Simplify[Together//@ToSidelengths[{Angle[{a2,d1,a3}],Angle[{a3,a1,a2}]}],TriangleQ[]]],TriangleQ[]]/.ArcCos[-Cos[x_]]:>Pi-x

Simplify[ToAngles[Simplify[Together//@ToSidelengths[{Angle[{a3,d2,a1}],Angle[{a1,a2,a3}]}],TriangleQ[]]],TriangleQ[]]/.ArcCos[-Cos[x_]]:>Pi-x

Simplify[ToAngles[Simplify[Together//@ToSidelengths[{Angle[{a1,d3,a2}],Angle[{a2,a3,a1}]}],TriangleQ[]]],TriangleQ[]]/.ArcCos[-Cos[x_]]:>Pi-x

Area

Map[Factor,Simplify[Together//@ToSidelengths[Area[DTriangle]],TriangleQ[]],3]

Side Lengths

Map[Factor,Together//@ToSidelengths[SideLengths[DTriangle]],4]

Map[Factor,Simplify[%,TriangleQ[]],3]

Invsersely Similar to the Medial Triangle

SimilarQ[MedialTriangle,DTriangle]

Circumcircle is the Orthocentroidal Circle

Symmetrize/@Factor[ToSidelengths[Midpoint[{Centroid,Orthocenter}]]]

ToAngles[%11]

IncidentCenters[Take[KimberlingCenters,500],%11]//Timing

CenterName[X[381]]

Orthocentroidal Circle

Derivation

2a b c IsogonalConjugate/@TrilinearVertexMatrix[BrocardTriangle[2]]

InputForm[%]

Centers

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

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

moses={{3,381},{6,6},{15,13},{16,14},{23,111},{111,2},{187,115},{511,542},{512,690},{1296,4},{2502,1648},{2780,1499},{2854,524}};

l==moses

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