Isodynamic Points

Author

Eric W. Weisstein
December 25, 2004

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

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

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

Graphic

Cevians[Incenter]

Cartesian[#[t]]&/@Cevians[Incenter]

AngleBisectors[t]

t=Triangle[{{0.,0.},{2.,-.3},{1.,.7}}];
Show[Graphics[{
t,
Transpose[{{Red,Green,Blue},ApolloniusCircles[t]}],
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Purple]&/@Transpose[{IsodynamicPoints[t],{"S","S'"}}]
],
LabelOffset->.12
],
AngleBisectorsExterior[t],
Cartesian[#[t]]&/@Cevians[Incenter]
}],
AspectRatio->Automatic,
PlotRange->{{-2,4},{-1.5,3.5}}]

tri={{0.,0.},{1.,.7},{2.,-.3}};
iso=IsodynamicPoints[tri];
iso1=iso[[1]];
iso2=iso[[2]];
pts={iso[[1]],"(Subscript[I, 1])",iso[[2]],"(Subscript[I, 2])"};

Show[Graphics[{
	{Thickness[.01],Triangle[tri]},
			PointLabels[tri,pts,.03,{.2,.2}],
	AngleBisectorsExterior[tri],AngleBisectors[tri],
  {Dashing[{.02,.02}],IsodynamicCircles[tri]}
}], AspectRatio->Automatic,PlotRange->{{-2,4},{-1.5,3.5}}]

Inverses

t=Triangle[{{0,0},{1,0},{.09,.5}}];
Show[Graphics[{
t,
{Red,Circumcircle[t]},
{Blue,Line[Coordinates/@{IsodynamicPoint[2][t],Circumcenter[t]}]
},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"A","B","C"}}],
{
{IsodynamicPoint[1][t],"S",Blue},
{IsodynamicPoint[2][t],"S'",Blue},
{Circumcenter[t],"O",Red}
}
],{PointSize[.03],LabelOffset->.03}]
}],AspectRatio->Automatic,PlotRange->All]

d=Det[DistanceTrilinears/@{IsodynamicPoint[1],IsodynamicPoint[2],Circumcenter}]

CollinearQ[{IsodynamicPoint[1],IsodynamicPoint[2],Circumcenter}]//Timing

d=Distance[Circumcenter[t],#]&/@(IsodynamicPoint[#][t]&/@{1,2})

{Times@@d,Circumradius[t]^2}

IsodynamicPoint[#][t]&/@{1,2}

InversePoint[t,IsodynamicPoint[#][t]]&/@{1,2}