Yff Triangles

Author

Eric W. Weisstein
November 8, 2005

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

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

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

Figures

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[GraphicsArray[{
Graphics[{
t,
{Blue,YffTriangle[1][t]},
{Red,cev=Cartesian[#[t]]&/@Cevians[YffPoint[1]]},
EqualSideTicks[#,1,.02]&/@Transpose[{Coordinates/@EndPoint/@cev,RotateLeft[Vertices[t]]}],PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[YffTriangle[1][t]],{"A'","B'","C'"}}],
{
{YffPoint[1][t],"U",Red}
}
],LabelOffset->.04,PointSize[.035]]
},AspectRatio->Automatic,PlotRange->All],
Graphics[{
t,
{Blue,YffTriangle[2][t]},
{Red,cev=Cartesian[#[t]]&/@Cevians[YffPoint[2]]},
EqualSideTicks[#,2,.02]&/@Transpose[{Coordinates/@EndPoint/@cev,RotateRight[Vertices[t]]}],
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[YffTriangle[2][t]],{"A''","B''","C''"}}],
{
{YffPoint[2][t],"U'",Red}
}
],LabelOffset->.04,PointSize[.035]]
},AspectRatio->Automatic,PlotRange->All]
}],ImageSize->500]

Check Coincident of Cevians

First Yff Point

Cartesian[#[t]]&/@(Intersections@@@Partition[N[Cevians[YffPoint[1]]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]],20],2,1,1])

Equal@@(Rest[#]/First[#]&/@(Intersections@@@Partition[Cevians[YffPoint[1]],2,1,1]))

Second Yff Point

Cartesian[#[t]]&/@(Intersections@@@Partition[N[Cevians[YffPoint[2]]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]],20],2,1,1])

Equal@@(Rest[#]/First[#]&/@(Intersections@@@Partition[Cevians[YffPoint[2]],2,1,1]))

Area

u=YffU=Module[{pp=a+b+c,qq=a b+b c+c a,rr=a b c,W,Q,V},W=Sqrt[27 (8 qq^3+4 pp^3 rr-36 pp qq rr+108 rr^2-pp^2 qq^2)];
Q=pp^3-9 pp qq+54 rr;
V=(W+Q)^(1/3);
pp/6-(6 qq-pp^2)/(6 V)+V/6]

ToSidelengths[Area[Evaluate[YffTriangle[1]]]-((u^3)/(2Circumradius))]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

Simplify[Together//@ToSidelengths[Area[Evaluate[YffTriangle[1]]]-((u^3)/(2Circumradius))],TriangleQ[]]

Centers

First

(pts={#,KimberlingLookup[#[YffTriangle[1]]]}&/@Take[KimberlingCenters,100]);//Timing

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

Second

(pts={#,KimberlingLookup[#[YffTriangle[2]]]}&/@Take[KimberlingCenters,100]);//Timing

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