Yff Central Triangle

Author

Eric W. Weisstein
November 8, 2005

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

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

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

Graphic

t=Triangle[{{0,0},{1,0},{.3,.5}}];
g1=Show[Graphics[{
t,
{Blue,
i=Isoscelizers[t,{.5,.6,.4}]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{Point/@Flatten[Coordinates/@i,1],{"(Subscript[I, AB])","(Subscript[I, AC])","(Subscript[I, BC])","(Subscript[I, BA])","(Subscript[I, CA])","(Subscript[I, CB])"}}]
],LabelOffset->.035,PointSize[.025]]
}],AspectRatio->Automatic,PlotRange->All]

t=Triangle[{{0,0},{1,0},{.3,.5}}];
g2=Show[Graphics[{
t,
{Blue,
i=Isoscelizers[t,YffCentralTriangleLengths[Vertices[t]][[1]]]
},
{Thickness[.01],Red,y=YffCentralTriangle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[y],{"A'","B'","C'"}}],
Append[#,Blue]&/@Transpose[{Point/@Flatten[Coordinates/@i,1],{"(Subscript[I, AB])","(Subscript[I, AC])","(Subscript[I, BC])","(Subscript[I, BA])","(Subscript[I, CA])","(Subscript[I, CB])"}}]
],LabelOffset->.035,PointSize[.025]]
}],AspectRatio->Automatic,PlotRange->All]

Show[GraphicsArray[{g1,g2}]]

is=Flatten[Coordinates/@i,1];

SideLengths[Triangle[{Vertices[y][[1]],is[[3]],is[[6]]}]]

SideLengths[Triangle[{is[[2]],Vertices[y][[2]],is[[5]]}]]

SideLengths[Triangle[{is[[1]],is[[4]],Vertices[y][[3]]}]]

SideLengths[Evaluate[y]]

Relation to Extangents Triangle

t==ExtangentsTriangle[YffCentralTriangle[t]]//Chop

t=Triangle[{{0,0},{1,0},{.3,.5}}];
g2=Show[Graphics[{
t,
{Blue,
i=Isoscelizers[t,YffCentralTriangleLengths[Vertices[t]][[1]]]
},
{Thickness[.01],Red,y=YffCentralTriangle[t]},
{Purple,Excircles[y]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[y],{"A'","B'","C'"}}],
Append[#,Blue]&/@Transpose[{Point/@Flatten[Coordinates/@i,1],{"(Subscript[I, AB])","(Subscript[I, AC])","(Subscript[I, BC])","(Subscript[I, BA])","(Subscript[I, CA])","(Subscript[I, CB])"}}]
],LabelOffset->.035,PointSize[.025]]
}],AspectRatio->Automatic,PlotRange->All]

Trilinear Vertex Matrix

Trilinears[YffCentralTriangle]//MatrixForm//TraditionalForm

Side Lengths

Area

area=Together//@Area[YffCentralTriangle]

area2=FullSimplify[area Δ/Area[],TriangleQ[]&&AcuteQ[]]

area2=-(4 a b c Δ Cos[(A/2)] Cos[(B/2)] Cos[(C/2)] ((((Cos[(A/2)]-Cos[(B/2)]))^2) (Cos[(A/2)]+Cos[(B/2)])-(((Cos[(A/2)]+Cos[(B/2)]))^2) Cos[(C/2)]-(Cos[(A/2)]+Cos[(B/2)]) ((Cos[(C/2)])^2)+((Cos[(C/2)])^3)))/((a+b+b Cos[A]+2 c Cos[(A/2)] Cos[(B/2)]+a Cos[B]+2 (b Cos[(A/2)]+a Cos[(B/2)]) Cos[(C/2)]) (2 c ((Cos[(A/2)])^3) (c Cos[(B/2)]+b Cos[(C/2)])+a Cos[(C/2)] (Cos[(B/2)]+Cos[(C/2)]) (b+c+c Cos[B]+2 a Cos[(B/2)] Cos[(C/2)]+b Cos[C])+Cos[(A/2)] (c Cos[(B/2)]+b Cos[(C/2)]) (a+b+c+c Cos[B]+4 a Cos[(B/2)] Cos[(C/2)]+(a+b) Cos[C])+2 ((Cos[(A/2)])^2) ((c^2) (1+Cos[B])+(a+2 b) c Cos[(B/2)] Cos[(C/2)]+(1/2) b (b+c) (1+Cos[C]))))

Together//@DistanceTrilinears[area2/.Cos[((x_)/2)]->((Sqrt[1+Cos[x]])/(Sqrt[2]))/.Sin[((x_)/2)]->((Sin[x])/((Sqrt[2]) (Sqrt[1+Cos[x]])))]

Centers

(pts={#,KimberlingLookup[#[YffCentralTriangle]]}&/@Take[KimberlingCenters,100]);//Timing

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