Polar Circle

Author

Eric W. Weisstein
October 25, 2005

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

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

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

Anticomplement

Factor[ToSidelengths[TrilinearEquation[PolarCircle]/.Thread[{α,β,γ}->Anticomplement[{α,β,γ}]]]]

Factor[ToSidelengths[TrilinearEquation[DeLongchampsCircle]]]

Equation

eqn=CircleEquation[Orthocenter,Sqrt[(1/2)(a^2+b^2+c^2)-4Circumradius[]^2]];

eqn=CircleEquation[Orthocenter,2Circumradius[] Sqrt[-Cos[A]Cos[B]Cos[C]]];

CircleEquation[PolarCircle]//Simplify

Center Function

Simplify[ExpandAll[DistanceTrilinears[CircleFunction[eqn]]],TriangleQ[]]

LawOfCosines[%]

Center

ctr=CircleCenter[eqn];

Simplify[ExpandAll[DistanceTrilinears[ctr]],TriangleQ[]]

Factor[FullSimplify[%19,TriangleQ[]]8Circumradius[]^2//FullSimplify]

((Sec[A]Sec[B]Sec[C])/(4a b c ))LawOfCosines[%]//FullSimplify

Factor[FullSimplify[DistanceTrilinears[CircleCenter[PolarCircle]],TriangleQ[]](2a b c)]

Radius

FullSimplify[DistanceTrilinears[CircleRadius[PolarCircle]],TriangleQ[]]

Simplify[LawOfCosines[%]R/Circumradius[],TriangleQ[]]

Incident Points

(ctrs=IncidentCenters[KimberlingCenters,PolarCircle,ObtuseOnly->True])//Timing

{#,CenterName[#]}&/@ctrs//TableForm

Figure

t=Triangle[{{0,0},{1,0},{-.1,.7}}];
Show[Graphics[{
t,
{Blue,PolarCircle[t]},
{Red,
LongestLine[Append[Coordinates[#],Coordinates@Orthocenter[t]]]&/@Altitudes[t]
},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{Feet[Altitudes[t]],{"(Subscript[H, A])","(Subscript[H, B])","(Subscript[H, C])"}}],
{{Orthocenter[t],"H",Red}}
],
LabelOffset->.05
]
}],AspectRatio->Automatic,PlotRange->All]

Orthogonality

Check

Simplify[Together//@DistanceTrilinears[First[OrthogonalQ[PolarCircle,DrozFarnyCircle[2]]]],TriangleQ[]]//Timing

Graphic

t=Triangle[{{0,0},{1,0},{-.2,.7}}];
circs={OrthopticCircleOfTheSteinerInellipse,PolarCircle,DrozFarnyCircle[2],StevanovicCircle};
centers={"H","(Subscript[X, 351])","(Subscript[X, 3])","(Subscript[X, 650])"};
colors={Red,Green,Blue,Purple};
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->{{-1,1.1},{-1,1}}]