Intangents Triangle

Author

Eric W. Weisstein
November 8, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.4,.7}}];
Show[Graphics[{
t,
{Blue,Incircle[t]},
{Dashing[{.015}],Sides[t,2]},
{Red,Sides[it=IntangentsTriangle[t],2]
},
{Purple,Excircles[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[it],{"A'","B'","C'"}}],
Append[#,Purple]&/@Transpose[{Excenters[t],{"(Subscript[J, A])","(Subscript[J, B])","(Subscript[J, C])"}}],
{
{Incenter[t],"I",Blue}
}
],LabelOffset->.1,PointSize[.015]]
}],AspectRatio->Automatic]

Trilinear Vertex Matrix

Trilinears[IntangentsTriangle]//MatrixForm//TraditionalForm

Simplify[Factor[Together//@DistanceTrilinears[Trilinears[IntangentsTriangle]]] 2a b c/(a+b+c)]//MatrixForm//TraditionalForm

Side Lengths

Together//@DistanceTrilinears[SideLengths[IntangentsTriangle]]

Simplify[%]

Together//@%102

FullSimplify[{(√((((a^4) (((a-b-c))^3) (((a+b-c))^3) (((a-b+c))^3) (a+b+c))/((((a^4)-(b^4)+2 (b^2) (c^2)-(c^4)))^2))(a-b-c) (a+b-c) (a-b+c) (a+b+c))),√((a-b-c) (a+b-c) (a-b+c) (a+b+c)(((b^4) (((-a+b-c))^3) (((a+b-c))^3) (((-a+b+c))^3) (a+b+c))/(((-(a^4)+(b^4)+2 (a^2) (c^2)-(c^4)))^2))),√((a-b-c) (a+b-c) (a-b+c) (a+b+c)(((((a-b-c))^3) (((a+b-c))^3) (c^4) (((a-b+c))^3) (a+b+c))/((((a^4)-2 (a^2) (b^2)+(b^4)-(c^4)))^2)))},TriangleQ[]]

%27 Circumradius[]^2/R^2 r^2/Inradius[]^2  s/Semiperimeter[]/.Abs[x_]:>Abs[Factor[x]]

Simplify[LawOfCosines[%],TriangleQ[]]

Area

area=FullSimplify[DistanceTrilinears[Area[IntangentsTriangle] Δ/Area[]],TriangleQ[]]

Simplify[LawOfCosines[area] r^2/Inradius[]^2Δ^2/Area[]^2]

Centers

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

...crash...?

(pts={#,KimberlingLookup[#[IntangentsTriangle]]}&/@Take[KimberlingCenters,1000]);//Timing

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

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

(pts={#,KimberlingLookup[#[IntangentsTriangle]]}&/@Take[KimberlingCenters,2000]);//Timing

...crash...

(pts={#,KimberlingLookup[#[IntangentsTriangle]]}&/@Take[KimberlingCenters,{1001,1100}]);//Timing

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

(pts={#,KimberlingLookup[#[IntangentsTriangle]]}&/@Take[KimberlingCenters,{1101,1200}]);//Timing

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

(pts={#,KimberlingLookup[#[IntangentsTriangle]]}&/@Take[KimberlingCenters,{1201,1300}]);//Timing

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

(pts={#,KimberlingLookup[#[IntangentsTriangle]]}&/@Take[KimberlingCenters,{1301,1400}]);//Timing

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

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