Tangential Triangle

Author

Eric W. Weisstein
November 11, 2005

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

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

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

Figure

t=Triangle[{{0.,0.},{1.,.2},{.5,1.}}];
Show[Graphics[{
{
Red,
tt=TangentialTriangle[t]
},
{
Blue,
Circumcircle[t],
PerpSquare[#,.07,1]&/@Transpose[{pl=PedalLines[tt,Circumcenter[t]],Sides[tt]}],
Dashing[{.02}],
pl
},
{
Green,
Anticevians[t,SymmedianPoint[t]]
},
t,
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[tt],{"(Subscript[T, A])","(Subscript[T, B])","(Subscript[T, C])"}}],
{
{Circumcenter[t],"O",Blue},
{SymmedianPoint[t],"K",Green}
}
],{PointSize[.025],LabelOffset->.06}]
}], AspectRatio->Automatic,PlotRange->All]

Vertices

AnticevianTrilinears[SymmedianPoint]//MatrixForm//TraditionalForm

Side Lengths

Timing[abc=FullSimplify[DistanceTrilinears[AnticevianTriangleSideLengths[SymmedianPoint]],TriangleQ[]]]

FullSimplify[abc,AcuteQ[]]

Sum

Timing[sum=FullSimplify[Plus@@abc,TriangleQ[]]]

FullSimplify[LawOfCosines[FullSimplify[sum,AcuteQ[]]]]

Product

Timing[prod=FullSimplify[Times@@abc,TriangleQ[]]]

LawOfCosines[((8 (a^5) (b^5) (c^5))/(((((a^2)-(b^2)+(c^2)))^2) (((-(a^2)+(b^2)+(c^2)))^2) ((((a^2)+(b^2)-(c^2)))^2)))]

Area

<<PlaneGeometry`

FullSimplify[LawOfCosines[AnticevianTriangleArea[SymmedianPoint]]Δ/TriangleArea[],TriangleQ[]]

Tangential triangle and original are Perspective from the Gergonne Point of A

Show[Graphics[{
{
Blue,
Line/@Transpose[Vertices/@{t,TangentialTriangle[t]}],
PointLabels[t,bluepts,.03,.07]
},
{
Red,
TangentialTriangle[t],PointLabels[t,redpts,.03,.07]
},
t,
PointLabels[t,pts,.03,.07]
},AspectRatio->Automatic,PlotRange->All]
]

GergonnePoint[TangentialTriangle[t]]==SymmedianPoint[t]

Side Extensions

tri={{0.,0.},{.9,.2},{-.05,1.2}};
p=TangentialTrianglePoints[tri]/. Point->Identity;
t=TangentialTriangle[tri][[1]];
pts={
		tri[[1]],"(Subscript[A, 1])",tri[[2]],"(Subscript[A, 2])",tri[[3]],"(Subscript[A, 3])",
		t[[1]],"(Subscript[T, 1])",t[[2]],"(Subscript[T, 2])",t[[3]],"(Subscript[T, 3])",
		p[[1]],"(Subscript[L, 1])",p[[2]],"(Subscript[L, 3])",p[[3]],"(Subscript[L, 2])"
		};

Show[Graphics[{
Triangle[tri],
  {Dashing[{.02,.02}],Circumcircle[tri],TangentialTriangleLines[tri]},
  {Thickness[.008],TangentialTriangle[tri]},
			PointLabels[tri,pts,.02,{.12,.01}],
			Line[LongestLine[p]]
}], AspectRatio->Automatic,PlotRange->All]

Centers

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

From Moses

moses={{1,3},{2,154},{3,26},{4,155},{5,156},{6,157},{7,6},{8,1498},{9,159},{11,110},{12,1614},{21,2917},{30,1154},{35,2937},{36,2070},{37,160},{55,22},{56,24},{57,25},{60,1601},{63,161},{65,4},{79,195},{80,399},{104,2931},{142,206},{145,64},{173,1486},{174,55},{175,1152},{176,1151},{177,1},{181,1629},{200,1619},{202,1605},{203,1606},{218,2353},{222,2351},{226,184},{234,31},{241,237},{244,1624},{258,197},{259,1631},{260,2915},{269,1609},{279,3053},{284,2934},{354,2},{388,1181},{390,1350},{479,1611},{481,372},{482,371},{497,394},{513,523},{514,512},{516,511},{517,30},{518,1503},{522,924},{523,1510},{527,2393},{528,2781},{553,51},{555,2176},{650,669},{900,526},{926,804},{934,2079},{942,5},{943,2918},{946,1147},{1086,1576},{1088,1613},{1111,1625},{1122,393},{1155,23},{1156,2930},{1317,74},{1319,186},{1320,2935},{1323,187},{1354,2383},{1357,107},{1358,112},{1359,1299},{1361,1300},{1362,98},{1364,925},{1365,933},{1385,1658},{1387,1511},{1401,2052},{1439,2165},{1476,2929},{1486,1485},{1488,198},{1638,351},{1836,1993},{2089,56},{2091,19},{2099,378},{2310,1634},{2346,2916},{2446,1114},{2447,1113},{2801,542},{2802,2777},{2808,2782},{2809,2794},{2810,2790},{2820,2799},{2821,2797},{2832,2848},{3021,1297},{3022,99},{3023,1303},{3024,930},{3025,476},{3027,1298},{3028,1141}};

True only for acute

With[{t=Triangle[{{0,0},{1,0},{.3,.5}}]},{X[1][TangentialTriangle[t]],X[3][t]}]

With[{t=Triangle[{{0,0},{1,0},{1.3,.5}}]},{X[1][TangentialTriangle[t]],X[3][t]}]

Symmetrize[ToAngles[Simplify[Together//@ToSidelengths[ctr]/.Sqrt[s_]:>Simplify[Sqrt[s],TriangleQ[]&&AcuteQ[]],TriangleQ[]&&AcuteQ[]]]]

Trilinears[X[3]]

Symmetrize[ToAngles[Simplify[Together//@ToSidelengths[ctr]/.Sqrt[s_]:>Simplify[Sqrt[s],TriangleQ[]],TriangleQ[]]]]

Symmetrize[Simplify[%15,TriangleQ[]]]

Symmetrize[ToAngles[Symmetrize[ToSidelengths[Symmetrize[Simplify[%21,AcuteQ[]&&TriangleQ[]]]]]]]

Symmetrize[Simplify[%21,ObtuseQ[]&&TriangleQ[]]]