Lucas Inner Triangle

Author

Eric W. Weisstein
November 9, 2005

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

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/LucasInnerTriangle.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,LucasInnerTriangle[t]},
{Green,LucasInnerCircle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"B","C","A"}}],
Append[#,Blue]&/@Transpose[{Center/@LucasCircles[t],{"(Subscript[L, B])","(Subscript[L, C])","(Subscript[L, A])"}}],
Transpose[{VertexPoints[LucasInnerTriangle[t]],{"(Subscript[I, B])","(Subscript[I, C])","(Subscript[I, A])"},{Red,LabelOffset->#}&/@(.035{{-1,-1},{1,-1},{0,1}})}]
],LabelOffset->.03,PointSize[.022]]
}],AspectRatio->Automatic,PlotRange->All]

Area

area=Factor[Together//@ToSidelengths[Area[LucasInnerTriangle]]/.Sqrt[x_]Sqrt[y_]:>FullSimplify[Sqrt[x y],TriangleQ[]]/.Sqrt[x_]:>FullSimplify[Sqrt[x],TriangleQ[]]]

area/. (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)]):>4Δ

t=Triangle[{{0,0},{1,0},{.3,.7}}];

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

Factor[7 (a^4)-14 (a^2) (b^2)+7 (b^4)-14 (a^2) (c^2)-14 (b^2) (c^2)+7 (c^4)]/.(a-b-c) (a+b-c) (a-b+c) (a+b+c):>-(4Area)^2

ToSidelengths[((a^2) (b^2) (a-b-c) (a+b-c) (c^2) (a-b+c) (a+b+c) (-112 (Area^2)-16 (a^2) Δ-16 (b^2) Δ-16 (c^2) Δ))/(4 (-4 (a^4)+8 (a^2) (b^2)-4 (b^4)+8 (a^2) (c^2)+8 (b^2) (c^2)-4 (c^4)+12 (a^2) Δ+8 (b^2) Δ+8 (c^2) Δ) (-4 (a^4)+8 (a^2) (b^2)-4 (b^4)+8 (a^2) (c^2)+8 (b^2) (c^2)-4 (c^4)+8 (a^2) Δ+12 (b^2) Δ+8 (c^2) Δ) (-4 (a^4)+8 (a^2) (b^2)-4 (b^4)+8 (a^2) (c^2)+8 (b^2) (c^2)-4 (c^4)+8 (a^2) Δ+8 (b^2) Δ+12 (c^2) Δ))/.Δ:>Area]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

Factor[-4 (a^4)+8 (a^2) (b^2)-4 (b^4)+8 (a^2) (c^2)+8 (b^2) (c^2)-4 (c^4)]/.(a-b-c) (a+b-c) (a-b+c) (a+b+c):>-(4Area)^2

ToSidelengths[((a^2) (b^2) (a-b-c) (a+b-c) (c^2) (a-b+c) (a+b+c) (-112 (Area^2)-16 (a^2) Δ-16 (b^2) Δ-16 (c^2) Δ))/(4 (64 (Area^2)+12 (a^2) Δ+8 (b^2) Δ+8 (c^2) Δ) (64 (Area^2)+8 (a^2) Δ+12 (b^2) Δ+8 (c^2) Δ) (64 (Area^2)+8 (a^2) Δ+8 (b^2) Δ+12 (c^2) Δ))/.Δ:>Area]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

Factor[12 (a^2) Δ+8 (b^2) Δ+8 (c^2) Δ]

ToSidelengths[(-16 (a^2) (b^2)  (c^2)  (-112 Area-16 ((a^2)+ (b^2)+ (c^2))))/(4 (64 Area+12 (a^2)+8 (b^2) +8 (c^2) ) (64 Area+8 (a^2) +12 (b^2) +8 (c^2) ) (64 Area+8 (a^2) +8 (b^2) +12 (c^2) ))/.Δ:>Area]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

ToSidelengths[area2=((a^2) (b^2)  (c^2)  (7Area+((a^2)+ (b^2)+ (c^2))))/((16 Area+3 (a^2)+2 (b^2)+2 (c^2)) (16 Area+2 (a^2)+3 (b^2)+2 (c^2)) (16 Area+2 (a^2)+2(b^2)+3 (c^2) ))]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

area2//TraditionalForm

Side Lengths

abc=Factor[Together//@ToSidelengths[SideLengths[LucasInnerTriangle]]/.Sqrt[x_]:>FullSimplify[Sqrt[x],TriangleQ[]]]

FullSimplify[abc^2,TriangleQ[]]

a b c/Sqrt[{(-Factor[12 (a^4)-42 (a^2) ((b^2)+(c^2))+5 (2 (b^4)-9 (b^2) (c^2)+2 (c^4))]+Factor[4Area(16 (a^2)+20 (b^2) +20 (c^2) )]),(Factor[-10 (a^4)+42 (a^2) (b^2)-12 (b^4)+45 (a^2) (c^2)+42 (b^2) (c^2)-10 (c^4)] +4Area Factor[20 (a^2)+16 (b^2) +20 (c^2) ]),(Factor[-10 (a^4)+45 (a^2) (b^2)-10 (b^4)+42 (a^2) (c^2)+42 (b^2) (c^2)-12 (c^4)]+4Area Factor[20 (a^2) +20 (b^2)+16 (c^2)])}]//TraditionalForm

SideLengths[Evaluate[LucasInnerTriangle[t]]]

ToSidelengths[Sqrt[{((a^2) (b^2) (c^2))/(-12 (a^4)+42 (b^2) (a^2)+42 (c^2) (a^2)-10 (b^4)-10 (c^4)+45 (b^2) (c^2)+16 Area (4 (a^2)+5 (b^2)+5 (c^2))),((a^2) (b^2) (c^2))/(-10 (a^4)+42 (b^2) (a^2)+45 (c^2) (a^2)-12 (b^4)-10 (c^4)+42 (b^2) (c^2)+16  Area(5 (a^2)+4 (b^2)+5 (c^2))),((a^2) (b^2) (c^2))/(-10 (a^4)+45 (b^2) (a^2)+42 (c^2) (a^2)-10 (b^4)-12 (c^4)+42 (b^2) (c^2)+16  Area (5 (a^2)+5 (b^2)+4 (c^2)))}]]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

Sqrt[{((a^2) (b^2) (c^2))/(-12 (a^4)+42 (b^2) (a^2)+42 (c^2) (a^2)-10 (b^4)-10 (c^4)+45 (b^2) (c^2)+16 Area (4 (a^2)+5 (b^2)+5 (c^2))),((a^2) (b^2) (c^2))/(-10 (a^4)+42 (b^2) (a^2)+45 (c^2) (a^2)-12 (b^4)-10 (c^4)+42 (b^2) (c^2)+16  Area(5 (a^2)+4 (b^2)+5 (c^2))),((a^2) (b^2) (c^2))/(-10 (a^4)+45 (b^2) (a^2)+42 (c^2) (a^2)-10 (b^4)-12 (c^4)+42 (b^2) (c^2)+16  Area (5 (a^2)+5 (b^2)+4 (c^2)))}]//InputForm

Perspective Triangles

t=Triangle[{{0,0},{1,0},{3/10,7/10}}]//N[#,20]&;

Reference Triangle

p=Factor[Perspector[LucasInnerTriangle,ReferenceTriangle]]

{Cartesian[p[t]],Cartesian[CyclicTrilinears[2 Cos[A]+Sin[A]][t]]}

Lucas Central Triangle

7 Cos[A]+4 Sin[A]

p=Factor[Perspector[LucasInnerTriangle,LucasCentralTriangle]];

{Cartesian[p[t]],Cartesian[CyclicTrilinears[ 7 Cos[A]+4 Sin[A]][t]]}

Lucas Tangents Triangle

p=Factor[Perspector[LucasInnerTriangle,LucasTangentsTriangle]];

{Cartesian[p[t]],Cartesian[CyclicTrilinears[5 Cos[A]+3 Sin[A]][t]]}

Centers

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

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

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