Circumcircle Mid-Arc Triangle

Author

Eric W. Weisstein
November 11, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/CircumcircleMid-ArcTriangle.nb.

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

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

Figure

t=Triangle[{{0,0},{1,0},{.2,.5}}];
Show[Graphics[{
t,
{Green,Circumcircle[t]},
{
Red,
CircumcircleMidArcTriangle[t]
},
{
Blue,
l=Line/@Transpose[Vertices/@Through[{CircumcircleMidArcTriangle,FuhrmannTriangle}[t]]];
PerpSquare[#,.03,.3]&/@Transpose[{Sides[t],l}],
Line[Coordinates/@{Circumcenter[t],#}]&/@VertexPoints[CircumcircleMidArcTriangle[t]]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Blue]&/@Transpose[{VertexPoints[MedialTriangle[t]],{"(Subscript[M, A])","(Subscript[M, B])","(Subscript[M, C])"}}],
Append[#,Red]&/@Transpose[{VertexPoints[CircumcircleMidArcTriangle[t]],{"(Subscript[(Overscript[M, ⌢]), A])","(Subscript[(Overscript[M, ⌢]), B])","(Subscript[(Overscript[M, ⌢]), C])"}}],
{{Circumcenter[t],"O",Blue}}
],LabelOffset->.03]
}],AspectRatio->Automatic]

Trilinear Vertex Matrix

TrilinearVertexMatrix[CircumcircleMidArcTriangle]//TraditionalForm

Concurrence

ToAngles[Together/@Intersections@@@Partition[Transpose[TrilinearVertexMatrix/@{CircumcircleMidArcTriangle,MedialTriangle}],2,1,1]]/2

Trilinears[Circumcenter]

Centers

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