Orthoptic Circle of theSteiner Inellipse

Author

Eric W. Weisstein
October 23, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.5}}];
Show[Graphics[{
t,
{Red,OrthopticCircleOfTheSteinerInellipse[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{{Centroid[t],"G",Red}}
],
LabelOffset->.03
]
}],AspectRatio->Automatic,PlotRange->All]

Derivation

Equation

eqn=Numerator[Factor[ToSidelengths[TrilinearEquation[OrthopticCircleOfTheSteinerInellipse]]]]

Circle Function

ToAngles[Simplify[CircleFunction[eqn]]]

CircleFunction[OrthopticCircleOfTheSteinerInellipse]

Center

-Symmetrize[Center[eqn]]

Trilinears[Centroid]

Radius

CircleRadius[OrthopticCircleOfTheSteinerInellipse]

Incident Points

(pts=IncidentCenters[KimberlingCenters,OrthopticCircleOfTheSteinerInellipse])//Timing

Length[pts]

{#,Name[#]}&/@pts//ColumnForm

Orthogonal Circles

t=Triangle[{{0,0},{1,0},{-.1,.7}}];
circs={OrthopticCircleOfTheSteinerInellipse,PolarCircle,StevanovicCircle};
centers={"G","(Subscript[X, 351])","(Subscript[X, 650])"};
colors={Red,Green,Blue};
Show[Graphics[{
t,
Transpose[{colors,Through[circs[t]]}],
PointLabels[Join[
Transpose[{Center[#[t]]&/@circs,centers,colors}],
Transpose[{VertexPoints[t],{"A","B","C"}}]
],LabelOffset->.07
]
}],AspectRatio->Automatic,PlotRange->{{-.7,1.1},{-.7,1}}]