Johnson Triangle

Author

Eric W. Weisstein
March 20, 2006

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

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

©2006 Wolfram Research, Inc. except for portions noted otherwise

Trilinear Vertex Matrix

TrilinearVertexMatrix[JohnsonTriangle]//TraditionalForm

Area

Refine[ToAngles[Factor[ToSidelengths[Area[JohnsonTriangle]/.c_Cos:>TrigExpand[c]]/.Sqrt[s_]Sqrt[t_]:>Simplify[Sqrt[s t],TriangleQ[]]]],Area>0]

Side Lengths

Simplify[Together//@(ToSidelengths[SideLengths[JohnsonTriangle]/.c_Cos:>TrigExpand[c]]/.Sqrt[s_]Sqrt[t_]:>Simplify[Sqrt[s t],TriangleQ[]]),TriangleQ[]]

Figure

t=Triangle[{{0,0},{1,0},{.7,.8}}];
ctrs=Center/@JohnsonCircles[t];
Show[Graphics[{
t,
{Green,Circumcircle[t]},
{Blue,JohnsonCircles[t]},
{Red,JohnsonTriangle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{ctrs,{"(Subscript[J, A])","(Subscript[J, B])","(Subscript[J, C])"}}],
{
{Circumcenter[t],"O",{Green}}
}
],LabelOffset->.06]
}],AspectRatio->Automatic]

Perspector

Simplify[PerspectiveQ[ReferenceTriangle,JohnsonTriangle],TriangleQ[]]

persp=Symmetrize[Simplify[Perspector[ReferenceTriangle,JohnsonTriangle],TriangleQ[]]]

KimberlingLookup[persp]

Name[X[5]]

t=Triangle[{{0,0},{1,0},{.7,.8}}];
ctrs=Center/@JohnsonCircles[t];
Show[Graphics[{
t,
{Green,Circumcircle[t]},
{Blue,JohnsonCircles[t]},
{Red,JohnsonTriangle[t]},
{Orange,Line/@Transpose[Map[Vertices,{JohnsonTriangle[t],t},{1}]]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{ctrs,{"(Subscript[J, A])","(Subscript[J, B])","(Subscript[J, C])"}}],
{
{Circumcenter[t],"O",{Green}},
{X[5][t],"Ni",Orange}
}
],LabelOffset->.06]
}],AspectRatio->Automatic]

Homothetic

t=Triangle[{{0,0},{1,0},{.7,.8}}];
rl=RadicalLine@@@Subsets[JohnsonCircles[t],{2}];
Show[Graphics[{
t,
{Blue,JohnsonCircles[t]},
{Red,rl},
{Purple,jt=JohnsonTriangle[t],EqualSideTicksAboutMidpoints[jt,.03]},
PointLabels[Join[
Append[#,Purple]&/@Transpose[{VertexPoints[JohnsonTriangle[t]],{"(Subscript[J, A])","(Subscript[J, B])","(Subscript[J, C])"}}],
Append[#,Red]&/@Transpose[{VertexPoints[MedialTriangle[JohnsonTriangle[t]]],{"(Subscript[M, A])","(Subscript[M, B])","(Subscript[M, C])"}}],
Transpose[{VertexPoints[t],{"A","B","C"}}],
{
{Orthocenter[t],"H",{Blue}}
}
],LabelOffset->.06]
}],AspectRatio->Automatic]

Trilinears[Orthocenter]

FullSimplify[ToSidelengths[PerspectiveQ[Triangle[Midpoints[JohnsonTriangle]],Triangle[ReferenceTriangle]]]]

Symmetrize[ToAngles[Symmetrize[ToSidelengths[Perspector[Triangle[Midpoints[JohnsonTriangle]],Triangle[ReferenceTriangle]]]]]]