Lucas Central Triangle

Author

Eric W. Weisstein
November 8, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[{
t,
{Blue,LucasCircles[t]},
{Red,LucasTriangle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"B","C","A"}}],
Append[#,Red]&/@Transpose[{Center/@LucasCircles[t],{"(Subscript[L, B])","(Subscript[L, C])","(Subscript[L, A])"}}]
],LabelOffset->.05,PointSize[.03]]
}],AspectRatio->Automatic,PlotRange->All]

Trilinear Vertex Matrix

TrilinearVertexMatrix[LucasCentralTriangle]//TraditionalForm

Side Lengths

t=Triangle[{{0,0},{1,0},{.3,.5}}];

ToSidelengths[SideLengths[LucasCentralTriangle]]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

Tangent circles formula

abc=Factor[{aa,bb,cc}/.Solve[Thread[CircleRadius/@Circles[LucasCircles]=={-aa+bb+cc,aa-bb+cc,aa+bb-cc}/2],{aa,bb,cc}]][[1]]

ToSidelengths[abc]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

{((2 a R (R (b^2)+a c b+(c^2) R))/((a c+2 b R) (a b+2 c R))),((2 b R (R (a^2)+b c a+(c^2) R))/((b c+2 a R) (a b+2 c R))),((2 c R (R (a^2)+b c a+(b^2) R))/((b c+2 a R) (a c+2 b R)))}

Simplify[Together//@ToSidelengths[SideLengths[LucasCentralTriangle]],TriangleQ[]]

Area

ToSidelengths[Area[LucasCentralTriangle]]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

ToSidelengths[Area[Evaluate[abc]]]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

FullSimplify[Together//@Area[Evaluate[abc]],TriangleQ[]&&Circumradius>0]

(a b c (Circumradius^2) √(3 (a^2) (b^2) (c^2)+4 a b c ((a^2)+(b^2)+(c^2)) Circumradius+4 Factor[((b^2) (c^2)+(a^2) ((b^2)+(c^2)))] (Circumradius^2)))/((b c+2 a Circumradius) (a c+2 b Circumradius) (a b+2 c Circumradius))

Perspective with

t=RandomTriangle[50];
t2=LucasCentralTriangle[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

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

Center/@Circles[LucasCircles]

CyclicTrilinears[-2 (a^4)+3 (b^2) (a^2)+3 (c^2) (a^2)-(b^4)-(c^4)+2 (b^2) (c^2)]

Factor[ToAngles[FullSimplify[ToSidelengths[((-2 (a^4)+3 (b^2) (a^2)+3 (c^2) (a^2)-(b^4)-(c^4)+2 (b^2) (c^2)+2 Area (3 (a^2)+(b^2)+(c^2)))/(2((a^2)+2 Area) ))],TriangleQ[]]]/.(Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]):>4Δ/.Δ:>S/2/.b c Cos[A]:>SA]

Cartesian[Triangle[{{a(2 S+SA),b SB,c SC},{a SA,b(2 S+SB),c SC},{a SA,b SB,c(2 S+SC)}}/.TriangleRules/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]][t]]

LucasCentralTriangle[t]

Centers

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

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