Circum-Medial Triangle

Author

Eric W. Weisstein
November 11, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/Circum-MedialTriangle.nb.

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

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.7}}];
p=Centroid[t];
Show[Graphics[{
t,
{Blue,Circumcircle[t]},
{Red,ct=CircumMedialTriangle[t],
Dashing[{.025}],
Line/@Transpose[Vertices/@{t,ct}]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[ct],{"A'","B'","C'"}}],
Append[#,Orange]&/@Transpose[{Midpoints[t],{"(Subscript[M, A])","(Subscript[M, B])","(Subscript[M, C])"}}],
{
{p,"G",Red},
{Incenter[t],"I"}
}
],LabelOffset->.04,PointSize[.03]]
}],AspectRatio->Automatic,ImageSize->200,
PlotRange->All]

Intersection

Intersections@@@Partition[AngleBisectors,2,1,1]

Trilinears[Incenter]

Trilinear Vertex Matrix

Factor[CircumcevianTriangle[Centroid][[1]]{(b c)^2,(c a)^2,(a b)^2}]//TraditionalForm

TrilinearVertexMatrix[CircumMedialTriangle]//TraditionalForm

Derivation

Factor[ {b c, a c,a b}^2Vertices[CircumcevianTriangle[Centroid]]]//TraditionalForm

Area

FullSimplify[DistanceTrilinears[Together//@Area[CircumcevianTriangle[Centroid]]]Δ/Area[]]

Centers

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