Miquel Triangle

Author

Eric W. Weisstein
December 14, 2005

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

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

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

Figure

Cartesian[MiquelTriangle[x][t]]

t=Triangle[{{0,0},{1,0},{.2,.7}}];
x={.7,.6,.5};
Show[Graphics[{
t,
{Blue,Cartesian[MiquelTriangle[x][t]]},
{Red,#[t]&/@MiquelCircles[x]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{Cartesian[#[t]]&/@SideRatioPoints[x],{"(Subscript[P, A])","(Subscript[P, B])","(Subscript[P, C])"}}],
{{Cartesian[MiquelPoint[x][t]],"M",Blue}}
],LabelOffset->.04,PointSize[.025]]
}],AspectRatio->Automatic,PlotRange->All]

Side Lengths

Collect[#,{a^2,b^2,c^2},Factor]&/@(FullSimplify[SideLengths[Evaluate[MiquelTriangle[{ka,kb,kc}]]],TriangleQ[]&&Area>0]^2)

Area

Factor[Area[Evaluate[MiquelTriangle[{ka,kb,kc}]]]]

Special Cases

2#&/@MiquelTriangle[{1,1,1}/2]

Symmetrize/@TrilinearVertexMatrix[MedialTriangle]