Cricumcevian Triangle

Author

Eric W. Weisstein
January 21, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.7}}];
p=Point[{.4,.3}];
Show[Graphics[{
t,
{Blue,Circumcircle[t]},
{Red,ct=CircumcevianTriangle[t,p],
Dashing[{.02}],
Line/@Transpose[Vertices/@{t,ct}]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[ct],{"A'","B'","C'"}}],
{
{p,"P",Red}
}
],LabelOffset->.03,PointSize[.025]]
}],AspectRatio->Automatic]

Trilinear Vertex Matrix

CircumcevianTriangle[{α,β,γ}]//MatrixForm//TraditionalForm

Side Lengths

area=Simplify[SideLengths[Evaluate[CircumcevianTriangle[{α,β,γ}]]]]R/Circumradius[]

Simplify[area,TriangleQ[]]

Area

area=FullSimplify[Together//@Area[Evaluate[CircumcevianTriangle[{α,β,γ}]]] Δ/Area[],TriangleQ[]]

Simplify[LawOfCosines[-(a b c (((c α β+b α γ+a β γ))^3) Δ)/((a b (α^2)+((a^2)+(b^2)-(c^2)) α β+a b (β^2)) (a c (α^2)+((a^2)-(b^2)+(c^2)) α γ+a c (γ^2)) ((a^2-c^2-(b^2) ) β γ-b c ((β^2)+(γ^2))))]]

Table

t=Triangle[Table[Random[Real,{0,1},50],{3},{2}]];
kc=KimberlingCenters;

Length[k=Vertices[Cartesian[CircumcevianTriangle[#][t]]]&/@kc]//Timing

l={};
Do[
Print[l0={
i,
p=Triangles[[i]],
tri=Vertices[p[t]];
kc[[Flatten[Position[k,_?(#==tri&),{1}]]]]//Timing
}];
AppendTo[l,l0],
{i,Length[Triangles]}
]

Nicely Formatted

DisplayForm[Grid[{#[[-1,-1,1]],CenterName[#[[-1,-1,1]]],TriangleName[#[[2]]]}&/@Sort[Select[l,#[[-1,-1]]=!={}&],#1[[-1,-1,1,1]]<#2[[-1,-1,1,1]]&],RowLines->True,ColumnAlignments->Left]]