BCI Triangle

Author

Eric W. Weisstein
November 13, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[{
{Blue,TriangulationTriangles[t,Incenter]},
t,
{Green,Incircle/@(Triangle[FlattenAt[#,2]]&/@(Coordinates/@{Incenter[t],#}&/@Sides[t]))
},
{Red,bci=BCITriangle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[bci],{"A'","B'","C'"}}],
{
{Incenter[t],"I",Blue}
}
],PointSize[.025],LabelOffset->.03]

}],AspectRatio->Automatic,PlotRange->All]

Perspective

<<KimberlingCenters`

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[{
t,
{Green,Incircle/@(Triangle[FlattenAt[#,2]]&/@(Coordinates/@{Incenter[t],#}&/@Sides[t]))
},
{Red,bci=BCITriangle[t]},
{Blue,Line/@Transpose[Vertices/@{t,bci}]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[bci],{"A'","B'","C'"}}],
{
{X[1127][t],"(Subscript[X, 1127])",Blue}
}
],PointSize[.025],LabelOffset->.03]

}],AspectRatio->Automatic,PlotRange->All]

CenterFunction[X[1127]]

Trilinears

Trilinears[BCITriangle]//MatrixForm//TraditionalForm

Side Lengths

sides=SideLengths[BCITriangle]

FullSimplify[DistanceTrilinears[sides],TriangleQ[]]

Area

area=Factor[Together//@ToSidelengths[HalfAngleFormulas[Area[BCITriangle]]]]

Simplify[ToAngles[%],TriangleQ[]]

Factor[Refine[%26/. 1+Cos[x_]:>2Cos[x/2]^2,TriangleQ[]]]

Factor[(a^3)-(a^2) b-a (b^2)+(b^3)-(a^2) c-6 a b c-(b^2) c-a (c^2)-b (c^2)+(c^3)]

((a+b+c) ((a^2)-2 a b+(b^2)-2 a c-2 b c+(c^2))+8a b c(Cos[(A/2)] Cos[(B/2)]+Cos[(A/2)] Cos[(C/2)]+Cos[(B/2)] Cos[(C/2)]+2Cos[(A/2)] Cos[(B/2)] Cos[(C/2)]))//TraditionalForm

times

(Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)])/(8 (s+ b Cos[(A/2)]+a Cos[(B/2)]) (s+c Cos[(A/2)]+a Cos[(C/2)]) (s+c Cos[(B/2)]+b Cos[(C/2)]))//TraditionalForm

Centers

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

...crash...

(pts={#,KimberlingLookup[#[BCITriangle]]}&/@Take[KimberlingCenters,{1001,1500}]);//Timing

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

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