Contact Triangle

Author

Eric W. Weisstein
November 11, 2005

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

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

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

Illustration

t=Triangle[{{0,0},{1,0},{.2,.7}}];
ct=ContactTriangle[t];

Show[Graphics[{
t,
{
Blue,
Incircle[t],
p=PedalLines[t,Incenter[t]],
PerpSquare[#,.03,2]&/@Transpose[{Sides[t],p}]
},
{
Red,
ContactTriangle[t]
},
{
Green,
Cevians[t,GergonnePoint[t]]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[ct],{"(Subscript[C, A])","(Subscript[C, B])","(Subscript[C, C])"}}],
{
{Incenter[t],"I",Blue},
{GergonnePoint[t],"Ge",Green}
}
],{PointSize[.025],LabelOffset->.04}]
}],
AspectRatio->Automatic,PlotRange->All]

Nested Contact Triangles Converge to an Isosceles Triangle

t=Triangle[{{0,0},{1,0},{.2,.6}}];
colors={Red,Orange,Green,Blue,Violet};
es=NestList[ContactTriangle,t,4];
points=Transpose[Flatten/@{VertexPoints/@es,Transpose[Table[#<>StringJoin@@Table["'",{n}],{n,0,4}]&/@{"(Subscript[C, A])","(Subscript[C, B])","(Subscript[C, C])"}],
Table[colors[[i]],{i,5},{3}]
}];

Show[Graphics[{
Transpose[{colors,es}],
PointLabels[
points,
{PointSize[.02],LabelOffset->.03}
]
}],
AspectRatio->Automatic,PlotRange->All]

Gergonne Point = Symmedian Point of Contact Triangle

GergonnePoint[t]==SymmedianPoint[ContactTriangle[t]]

Vertices

CevianTrilinears[GergonnePoint]//MatrixForm//TraditionalForm

a b c CevianTrilinears[GergonnePoint]//MatrixForm//TraditionalForm

{{0,Sec[B/2]^2,Sec[C/2]^2},
{Sec[A/2]^2,0,Sec[C/2]^2},
{Sec[A/2]^2,Sec[B/2]^2,0}}//MatrixForm//TraditionalForm

Side Lengths

abc=FullSimplify[Together//@DistanceTrilinears[CevianTriangleSideLengths[GergonnePoint]],TriangleQ[]]

FullSimplify[LawOfCosines[ExpandAll[{(((Sqrt[ (a+b-c)  (a-b+c)]) (-a+b+c))/(2 Sqrt[b c])),(((a-b+c) (Sqrt[ (a+b-c)  (-a+b+c)]))/(2 Sqrt[a c])),(((a+b-c) (Sqrt[ (a-b+c) (-a+b+c)]))/(2 Sqrt[a b]))}]],TriangleQ[]]

s={(-a+b+c) Cos[(A/2)],(a-b+c) Cos[(B/2)],(a+b-c) Cos[(C/2)]}

Area

area=FullSimplify[Area[CevianTriangle[GergonnePoint]] Δ/Area,TriangleQ[]]

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

ToSidelengths[((Area^2)/(2 Circumradius Semiperimeter))(Δ/Area)]

Centers

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

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

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

l==moses

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