Third Morley Triangle

Author

Eric W. Weisstein
November 15, 2005

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

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

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

Equilateral

EquilateralQ[MorleysTriangle[3][t=RandomTriangle[20]]]

Simplify[#^2&/@EquilateralQ[CircumtangentialTriangle]/.C->π-A-B]

Figure

t=Triangle[{{0,0},{1,0},{.3,.7}}];
mt=MorleysTriangle[3][t];
Show[Graphics[{
t,
{Blue,
Line/@({MorleysTriangle[3][t][[1,#[[1]]]],
ReferenceTriangle[t][[1,#[[2]]]]}&/@Select[Tuples[Range[3],2],UnsameQ@@#&])},
{Red,mt},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[mt],{"A'","B'","C'"}}]
],LabelOffset->.04,PointSize[.025]]
}],AspectRatio->Automatic,PlotRange->All]

Trilinear Vertex Matrix

TrilinearVertexMatrix[MorleysTriangle[3]]//TraditionalForm

Side Lengths

SideLengths[Evaluate[MorleysTriangle[3][t]]]

(8 Circumradius Sin[A/3 - 4 Pi/3] Sin[B/3 - 4 Pi/3] Sin[C/3 - 4 Pi/3])/.TriangleRules/.Thread[{A,B,C}->Angles[t]]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

Area

Simplify[Area[Evaluate[Table[8 Circumradius Sin[A/3 - 4 Pi/3] Sin[B/3 - 4 Pi/3] Sin[C/3 - 4 Pi/3],{3}]]],Circumradius>0&&TriangleQ[]]

Perspectors

tris=DeleteCases[Triangles,MorleysTriangle[3]];

pts=DeleteCases[{#,k=KimberlingLookup[Perspector[MorleysTriangle[3],#]]}&/@Take[tris,All],_?(Last[#]==={}&)]

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

Centers

(pts={#,KimberlingLookup[#[MorleysTriangle[3]]]}&/@Take[KimberlingCenters,10]);//Timing

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

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