Stammler Circle

Author

Eric W. Weisstein
January 21, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.15,.7}}];
Show[Graphics[{
t,
{Blue,circs=StammlerCircles[t,{1,2,3}]},
{Green,StammlerTriangle[t]},
{Red,StammlerCircle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"},{LabelOffset->-.1{1,1},LabelOffset->.1{1,-1},{}}}],
Append[#,Red]&/@Transpose[{VertexPoints[StammlerTriangle[t]],{"(Subscript[O, A])","(Subscript[O, B])","(Subscript[O, C])"}}],
{
{Center[StammlerCircle[t]],"O",Red}
}
]]
}],AspectRatio->Automatic,PlotRange->{{-1.5,2.5},{-1.9,2.2}}]

Center

Center[StammlerCircle]

IncidentCenters[Take[KimberlingCenters,10],Center[StammlerCircle]]//Timing

Radius

CircleRadius[StammlerCircle]//TraditionalForm

Circle Function

CircleFunction[StammlerCircle]//TraditionalForm

IncidentCenters[Take[KimberlingCenters,10],CyclicTrilinears[CircleFunction[StammlerCircle]]]//Timing

Incident Centers

(ctrs=IncidentCenters[KimberlingCenters,StammlerCircle])//Timing

First/@ctrs

Length[ctrs]

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

Derivation

eqn=TrilinearEquation@@(TrilinearVertexMatrix[StammlerTriangle]);

circ=Circumcircle[eqn];//Timing

IncidentCenters[Take[KimberlingCenters,3],Center[circ]]//Timing

CenterFunction[X[3]]

Factor[Factor[Together//@ToSidelengths[TrilinearEquation[Circle[Trilinears[X[3]],2Circumradius]]]]/.Sqrt[x_]Sqrt[y_]:>FullSimplify[Sqrt[x y],TriangleQ[]]]

eqn=(-3 (a^3) b c (α^2)+(a^4) c α β-8 (a^2) (b^2) c α β+(b^4) c α β-2 (a^2) (c^3) α β-2 (b^2) (c^3) α β+(c^5) α β-3 a (b^3) c (β^2)+(a^4) b α γ-2 (a^2) (b^3) α γ+(b^5) α γ-8 (a^2) b (c^2) α γ-2 (b^3) (c^2) α γ+b (c^4) α γ+(a^5) β γ-2 (a^3) (b^2) β γ+a (b^4) β γ-2 (a^3) (c^2) β γ-8 a (b^2) (c^2) β γ+a (c^4) β γ-3 a b (c^3) (γ^2));

CircleFunction[eqn]

InputForm[%]

IncidentCenters[KimberlingCenters,%20]//Timing

CenterName[X[6]]

Factor[TrigToExp[eqn/.Thread[{α,β,γ}->#]&/@TrilinearVertexMatrix[StammlerTriangle]/.f_Cos:>TrigExpand[f]]]