Miquel Point

Author

Eric W. Weisstein
December 14, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.2,.7}}];
x={.7,.6,.5};
Show[Graphics[{
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]

Point

MiquelPoint[{(Subscript[k, a]),(Subscript[k, b]),(Subscript[k, c])}]

{a ((a^2) ((Subscript[k, a])-1) (Subscript[k, a])+(b^2) (Subscript[k, a]) (Subscript[k, b])+(c^2) ((Subscript[k, a])-1) ((Subscript[k, c])-1)),b ((a^2) ((Subscript[k, a])-1) ((Subscript[k, b])-1)+(b^2) ((Subscript[k, b])-1) (Subscript[k, b])+(c^2) (Subscript[k, b]) (Subscript[k, c])),c ((a^2) (Subscript[k, a]) (Subscript[k, c])+(b^2) ((Subscript[k, b])-1) ((Subscript[k, c])-1)+(c^2) ((Subscript[k, c])-1) (Subscript[k, c]))}/.{(Subscript[k, a])-1->-Subsuperscript[k,a,′],(Subscript[k, b])-1->-Subsuperscript[k,b,′],(Subscript[k, c])-1->-Subsuperscript[k,c,′]}//Column//TraditionalForm

Special Cases

1/2

KimberlingLookup[MiquelPoint[{1,1,1}/2]]

Name[X[3]]

Symmetrize[ToAngles[Symmetrize[MiquelPoint[{1,1,1}/2]]]]

t=Triangle[{{0,0},{1,0},{.2,.7}}];
x={.5,.5,.5};
Show[Graphics[{
t,
{Red,#[t]&/@MiquelCircles[x]},
{Blue,Circumcircle[t]},
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]

1/3

KimberlingLookup[MiquelPoint[{1,1,1}/3]]

Symmetrize[ToAngles[Symmetrize[MiquelPoint[{1,1,1}/3]]]]

KimberlingLookup[MiquelPoint[{1,1,1}2/3]]

Symmetrize[ToAngles[Symmetrize[MiquelPoint[{1,1,1}2/3]]]]

1/4

KimberlingLookup[MiquelPoint[{1,1,1}/4]]

Symmetrize[ToAngles[Symmetrize[MiquelPoint[{1,1,1}/4]]]]

KimberlingLookup[MiquelPoint[{1,1,1}3/4]]

Symmetrize[ToAngles[Symmetrize[MiquelPoint[{1,1,1}3/4]]]]