First Morley Adjunct Triangle

Author

Eric W. Weisstein
November 15, 2005

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

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

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

Morley Triangle and Adjunct

Figure

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[{
t,
{Blue,
Rest/@MapIndexed[RotateLeft[#1,#2+2]&,Cevians[t,#]&/@VertexPoints[MorleyTriangle[1][t]]]
},
{DarkGreen,
VertexArcs[t,.1],
EqualAngleDivisionTicks[t,3,.1,.02,.07]
},
{Red,mt=MorleyTriangle[1][t]},
{DarkOrange,mat=MorleyAdjunctTriangle[1][t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[mt],{"A'","B'","C'"}}],
Append[#,DarkOrange]&/@Transpose[{VertexPoints[mat],{"A''","B''","C''"}}]
],LabelOffset->.03]
}],AspectRatio->Automatic,PlotRange->All]

Trilinear Vertex Matrix

TrilinearVertexMatrix[MorleyAdjunctTriangle[1]]//MatrixForm//TraditionalForm

Area

FullSimplify[Together//@TrigToExp[Area[MorleyAdjunctTriangle[1]]/.TriangleRules]]

FullSimplify[%,TriangleQ[]]

area=(a b c Δ(-2+Cos[((2 (A-B))/3)]+Cos[((2 (A+B))/3)]+8 Cos[(A/3)] Cos[(B/3)] Cos[(C/3)]+2 (Cos[((2 B)/3)]+Cos[((2 A)/3)] (1+2 Cos[((2 B)/3)])) Cos[((2 C)/3)]))/(16 (b Cos[(B/3)]+Cos[(A/3)] (a+2 c Cos[(B/3)])) (c Cos[(C/3)]+Cos[(B/3)] (b+2 a Cos[(C/3)])) (c Cos[(C/3)]+Cos[(A/3)] (a+2 b Cos[(C/3)])));

TraditionalForm[area]

Perspectors

tris=DeleteCases[Triangles,MorleyAdjunctTriangle[1]];

pts=DeleteCases[{#,k=KimberlingLookup[Perspector[MorleyAdjunctTriangle[1],#]]}&/@Take[tris,All],_?(Last[#]==={}&)]

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

Centers

(pts={#,KimberlingLookup[#[MorleyAdjunctTriangle[1]]]}&/@Take[KimberlingCenters,All]);//Timing

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