Hexyl Circle

Author

Eric W. Weisstein
January 21, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[{
t,
{Blue,HexylTriangle[t]},
{Red,HexylCircle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[HexylTriangle[t]],{"A'","B'","C'"}}],
{
{Incenter[t],"I",Red}
}
],LabelOffset->.05]
}],AspectRatio->Automatic]
Null

Center

Center[HexylCircle]//TraditionalForm

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

Radius

CircleRadius[HexylCircle]//TraditionalForm

Circle Function

CircleFunction[HexylCircle]//TraditionalForm

IncidentCenters[KimberlingCenters,CyclicTrilinears[CircleFunction[HexylCircle]]]//Timing

Incident Centers

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

Derivation

v=2a b c Factor[ToSidelengths[TrilinearVertexMatrix[HexylTriangle]]]

Factor[Together//@TrilinearEquation@@v]

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

circ=Circumcircle[eqn];//Timing

Symmetrize[-Center[circ][[1]]]

IncidentCenters[Take[KimberlingCenters,10],{1,1,1}]

CenterName[X[1]]

r2=FullSimplify[Factor[CircleRadius[circ]^2],TriangleQ[]]

Simplify[Sqrt[r2 ((Circumradius^2)/(Circumradius^2//ToSidelengths))],Circumradius>0]

ToAngles[CircleFunction[eqn][[1]]]

TeXForm[-(((a^4)-2 (a^3) b+2 a (b^3)-(b^4)-2 (a^3) c-2 a (b^2) c-2 a b (c^2)+2 (b^2) (c^2)+2 a (c^3)-(c^4))/(4 (Area^2)))]

IncidentCenters[KimberlingCenters,CircleFunction[eqn]]//Timing