Extangents Triangle

Author

Eric W. Weisstein
November 11, 2005

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

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

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

Figure

ClawsonPoint[t]

h

Line[{#,Coordinates@ClawsonPoint[t]}]&/@Vertices[ex]

t=Triangle[{{0,0},{1,0},{0.4,0.8}}];
h=OrthicTriangle[t];
Show[Graphics[{t,
{Blue,Excircles[t]},
{BlueViolet,ex=ExtangentsTriangle[t],
Line[{#,Coordinates@ClawsonPoint[t]}]&/@Vertices[ex]},
{Red,h},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[h],{"(Subscript[H, A])","(Subscript[H, B])","(Subscript[H, C])"}}],
Append[#,Blue]&/@Transpose[{Excenters[t],{"(Subscript[J, A])","(Subscript[J, B])","(Subscript[J, C])"}}],
Append[#,BlueViolet]&/@Transpose[{VertexPoints[ex],{"(Subscript[T, A])","(Subscript[T, B])","(Subscript[T, C])"}}],
{
{ClawsonPoint[t],"Cl",BlueViolet}
}
]]}],
AspectRatio->Automatic]

SimilarQ[t,ExtangentsTriangle[t]]

Trilinear Vertex Matrix

Trilinears[ExtangentsTriangle]//MatrixForm//TraditionalForm

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

Inradius

r=Inradius[Together//@DistanceTrilinears[SideLengths[ExtangentsTriangle]]]

FullSimplify[r,TriangleQ[]]

Area

area=FullSimplify[Together//@DistanceTrilinears[Area[ExtangentsTriangle]] Δ/Area[],TriangleQ[]]

Factor[LawOfCosines[area]]

Simplify[(-(a^2) b-a (b^2)-(a^2) c-2 a b c-(b^2) c-a (c^2)-b (c^2))]

Simplify[Subtract@@{a^3+b^3+c^3-(a+b)(a+c)(b+c),((a^3)-(a^2) b-a (b^2)+(b^3)-(a^2) c-2 a b c-(b^2) c-a (c^2)-b (c^2)+(c^3))}]

((((((a^3)+(b^3)+(c^3)-(a+b) (a+c) (b+c)))^2) Δ (secA) (secB) (secC))/(8 (a^2) (b^2) (c^2)))

Excircle Tangential Triangle has incenter coinciding with circumcenter of J1J2J3 and radius 2R+r (doesn't look at all like the printed figure???)

tri={{0,0},{1,0},{.2,.7}};
ctr=Excenters[tri];
Show[Graphics[{{Thickness[.01],Triangle[tri]},ExcircleTangentsTriangle[tri],
  Circumcircle[{ctr[[1,1]],ctr[[2,1]],ctr[[3,1]]}],
  Circumcenter[{ctr[[1,1]],ctr[[2,1]],ctr[[3,1]]}],
  Incircle[ExcircleTangents[tri]],
  Incenter[ExcircleTangents[tri]],
  Table[{{PointSize[.02],ctr[[i]]},Excircles[tri][[i]]},{i,1,3}],
  {PointSize[.02],ExcircleTangentPoints[tri]}},
  AspectRatio->Automatic]]

Centers

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

...crash...?

(pts={#,KimberlingLookup[#[ExtangentsTriangle]]}&/@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[#[ExtangentsTriangle]]}&/@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