Reflection Triangle

Author

Eric W. Weisstein
November 19, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[{
t,
{Red,ReflectionTriangle[t]},
{Blue,lines=Line/@Transpose[Vertices/@{t,ReflectionTriangle[t]}],
MapIndexed[EqualSideTicksAboutMidpoint[#1,#2[[1]],.02,.015]&,lines],
PerpSquare[#,.04,1]&/@Transpose[{lines,Sides[t]}]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[ReflectionTriangle[t]],{"(SuperStar[A])","(SuperStar[B])","(SuperStar[C])"}}],
{
{Orthocenter[t],"H",{Blue,LabelOffset->{.06,-.05}}}
}
],LabelOffset->.04]
}],AspectRatio->Automatic]

Perspector

(1/(4a b Cos[A]Cos[B]Cos[C]))Perspector[ReflectionTriangle,ReferenceTriangle]

IncidentCenters[Take[KimberlingCenters,10],{Sec[A],Sec[B],Sec[C]}]//Timing

CenterName[X[4]]

Trilinear Vertex Matrix

TrilinearVertexMatrix[ReflectionTriangle]//TraditionalForm

Area

Factor[Refine[ToAngles[Factor[Together//@ToSidelengths[Area[ReflectionTriangle]]]],Area>0]]

Factor[ToSidelengths[4-(((((Distance[Circumcenter,Orthocenter])/Circumradius)))^2)]]Area

Side Lengths

Refine[Sqrt[Together//@(ToSidelengths[SideLengths[ReflectionTriangle]]^2)],TriangleQ[]]

Points

Centroid

g=-Symmetrize[ToSidelengths[Centroid[ReflectionTriangle]]]

ToAngles[%]

Factor[ToSidelengths[a (S^2-SA^2+2 SB SC)]]

IncidentCenters[KimberlingCenters,g]//Timing

Circumcenter

o=-Symmetrize[ToSidelengths[Circumcenter[ReflectionTriangle]]]

IncidentCenters[KimberlingCenters,o]//Timing

Name[X[195]]

CenterFunction[X[195]]

Orthocenter

h=Symmetrize[ToSidelengths[Orthocenter[ReflectionTriangle]]]

IncidentCenters[KimberlingCenters,h]//Timing

Perspective with

t=RandomTriangle[50];
t2=ReflectionTriangle[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

Checks

Together//@ToSidelengths[PerspectiveQ[AntipedalTriangle[X[1]],Triangle[ReflectionTriangle]]]

Factor/@(Together//@ToSidelengths[PerspectiveQ[AntipedalTriangle[X[5]],Triangle[ReflectionTriangle]]/.f_Cos:>TrigExpand[f]]/.Sqrt[x_]Sqrt[y_]:>FullSimplify[Sqrt[x y],TriangleQ[]])

Together//@ToSidelengths[PerspectiveQ[AntipedalTriangle[X[20]],Triangle[ReflectionTriangle]]]

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

Derivation

Reflection@@@Transpose[Through[{Vertices,Sides}[Triangle[ReferenceTriangle]]]]

-ToAngles[Factor[ToSidelengths[%]]]/{a,b,c}

Centers

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

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