Extouch Triangle

Author

Eric W. Weisstein
March 7, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Graphic

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[{
{
t,
Dashing[{.02}],Sides[t,1]
},
{
Red,
j=Excircles[t]
},
{
Blue,
ex=ExtouchTriangle[t],
{Dashing[{.02}],
Line/@Transpose[Vertices/@{ex,t}]
}
},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"(Subscript[A, 1])","(Subscript[A, 2])","(Subscript[A, 3])"}}],
Append[#,Blue]&/@Transpose[{Point/@Vertices[ex],{"(Subscript[T, 1])","(Subscript[T, 2])","(Subscript[T, 3])"}}],
Append[#,Red]&/@Transpose[{Point/@Coordinates/@j,{"(Subscript[J, 1])","(Subscript[J, 2])","(Subscript[J, 3])"}}],
{
{NagelPoint[t],"Na",Purple}
}
],{PointSize[.025],LabelOffset->.1}]
}],AspectRatio->Automatic]

Vertices

Vertices[ExtouchTriangle]//TraditionalForm

Side Lengths

ToAngles[Factor//@Expand//@Simplify[SideLengths[ExtouchTriangle],Area>0]]//Simplify

ReplaceAll@@@Transpose[{{(Sqrt[(a^2)-((Area^2)/(b c))]),(Sqrt[(b^2)-((Area^2)/(a c))]),(Sqrt[-((Area^2)/(a b))+(c^2)])},Thread[Area->{b c Sin[A],c a Sin[B],a b Sin[C]}]}]

Area

area=FullSimplify[CevianTriangleArea[NagelPoint]Δ/TriangleArea[],TriangleQ[]]

area r^2/Inradius[]^2 s/Semiperimeter[]

In Terms of Other Triangles

Pedal Triangle of Bevan Point

Map[Symmetrize,Triangle[ExtouchTriangle],{3}]

Map[Symmetrize,(Together//@ToSidelengths[PedalTriangle[X[40]]])/.Triangle[x_]:>Triangle[x {a,b,c}],{3}]

Centroid

Midpoints[Triangle[v_List?MatrixQ]]:=Midpoint/@Partition[RotateLeft@v,2,1,1]/;Dimensions[v]=={3,3}

m=Map[Symmetrize,Together//@Midpoints[Triangle[ExtouchTriangle]],{2}]

medians=Transpose[{m,Vertices[Triangle[ExtouchTriangle]]}]

Map[Symmetrize,Together//@Intersections@@@Map[Line@@#&,Partition[medians,2,1,1],{2}],{3}]

Equal@@%

IncidentCenters[{X[210]},%219[[1]]]

Centers

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