GEOS Circle

Author

Eric W. Weisstein
November 2, 2005

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

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

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

Initialization

Additional Contributors

Frank Jackson
June 24, 2004 and November 2, 2005

Check Against Known Circles

t=RandomTriangle[];

r=CircleRadius[#[t]]&/@CentralCircles

CentralCircles[[Flatten[Position[r,Null]]]]

ToSidelengths[CircleRadius[circ]]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]

Position[r,Abs[#-0.4654139048155379`]<.01&]

IncidentCenters[KimberlingCenters,ToSidelengths[Center[circ]]]//Timing

Graphic

<<Graphics`Colors`
t=Triangle[{{0,0},{1,0},{.34,.5}}];
Show[Graphics[{
t,
{Red,
circ=GEOSCircle[t],
l=Line[Coordinates/@{X[650][t],X[20][t]}],
EqualSideTicksAboutMidpoint[l,1]
},
{Blue,EulerLine[t]},
{Green,OrthicAxis[t]},
{Orange,GergonneLine[t]},
{Yellow,SoddyLine[t]},
{Purple,
PerpSquare[{EulerLine[t],OrthicAxis[t]},.12,4],
PerpSquare[{SoddyLine[t],GergonneLine[t]},.12,3]
},
PointLabels[{
{X[20][t],"Z==(Subscript[X, 20])==(Subscript[L, E])⋂(Subscript[L, S])",Green},
{X[468][t],"(Subscript[X, 468])==(Subscript[L, E])⋂(Subscript[L, O])",Blue},
{X[650][t],"(Subscript[X, 650])==(Subscript[L, G])⋂(Subscript[L, O])",Brown},
{X[1323][t],"Fl==(Subscript[X, 1323])==(Subscript[L, G])⋂(Subscript[L, S])",Orange},
{Center[circ],"(Subscript[O, GEOS])",Red},
Sequence@@Transpose[{VertexPoints[t],{"A","B","C"}}]
},
LabelOffset->.2,PointSize[.015]]
}],AspectRatio->Automatic,PlotRange->{{-6,2},{-2.1,2.1}},ImageSize->500]

Angles

FullSimplify[Factor[Together//@ToSidelengths[Angle[GergonneLine,OrthicAxis]]/.Sqrt[s_]:>FullSimplify[Sqrt[Expand[s]],TriangleQ[]]]]/. (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])->4Area

FullSimplify[ToSidelengths[Angle[SoddyLine,EulerLine]]/.Sqrt[s_]:>FullSimplify[Sqrt[Expand[s]],TriangleQ[]]]/. (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])->4Area

Center

Center[GEOSCircle]

Symmetrize[ToSidelengths[Center[circ]]]

-Symmetrize[Together//@ToSidelengths[Midpoint[{X[650],DeLongchampsPoint}]]]

Factor[ToSidelengths[ ((a sa)/((a-b)(c-a)))+(((a^2)SA-SB SC)/(S^2))]]

Radius

CircleRadius[GEOSCircle]

(r=Factor//@Together//@ToSidelengths[CircleRadius[circ]])//Timing

(r2=FullSimplify[r^2,TriangleQ[]])//Timing

Sqrt[Factor[r2]]

Circle Function

CircleFunction[GEOSCircle]

Factor//@Together//@ToSidelengths[CircleFunction[circ]]

Simplify[%,TriangleQ[]]

ToAngles[-((a (a-b-c) ((a^2)-(b^2)-(c^2)))/(4 (a-b) b (a-c) c))]

IncidentCenters[KimberlingCenters,CyclicTrilinears[((a (a-b-c) Cos[A])/(2 (a-b) (a-c)))]]//Timing

Incident Points

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

Length[pts]

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

CenterFunction[X[#]]&/@{20,468,650,1323}

Concylic Check

circ=Circumcircle[Triangle[Trilinears[X[#]]&/@{20,468,650}]];

Together//@ToSidelengths[HalfAngleFormulas[ConcyclicQ[Quadrilateral[Trilinears[X[#]]&/@{20,468,650,1323}]]]]//Timing