Symmedial Circle

Author

Eric W. Weisstein
December 20, 2004

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

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

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

Graphic

Cartesian[Center[SymmedialCircle][t]]

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[
{
t,
{Blue,Symmedians[t]},
{Green,SymmedialTriangle[t]},
{Red,SymmedialCircle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[SymmedialTriangle[t]],{"(Subscript[K, A])","(Subscript[K, B])","(Subscript[K, C])"}}],
{
{SymmedianPoint[t],"K",{Blue,LabelOffset->{-.035,.05}}},
{Cartesian[Center[SymmedialCircle][t]],"X",Red}
}
],LabelOffset->{.05,-.02},PointSize[.025]]
}
],AspectRatio->Automatic,PlotRange->All]

Radius

CircleRadius[SymmedialCircle](R/(Circumradius[]))//TraditionalForm

Incident Centers

IncidentCenters[KimberlingCenters,SymmedialCircle]//Timing

CenterName[X[125]]

Derivation

eqn=Numerator[Symmetrize[Factor[ToSidelengths[TrilinearEquation@@TrilinearVertexMatrix[SymmedialTriangle]]]]]

facs=Factor[Coefficient[eqn,#]&/@{(α^2),(β^2),(γ^2),α β,α γ,β γ}]

Center

ctr=Symmetrize/@Center[eqn]

InputForm[%7[[1]]]

IncidentCenters[KimberlingCenters,ctr]//Timing

Circle Function

l=Factor[CircleFunction[eqn]]

InputForm[%[[1]]]

IncidentCenters[KimberlingCenters,l]//Timing

Radius

r=Together//@ExpandAll[Together//@CircleRadius[eqn]]

FullSimplify[%,TriangleQ[]]

Factor//@%

Circumradius[]√(((a^4)-3 (a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)-(c^4)) ((a^4)+(a^2) (b^2)-(b^4)+(a^2) (c^2)+3 (b^2) (c^2)-(c^4)) ((a^4)-(a^2) (b^2)-(b^4)-3 (a^2) (c^2)-(b^2) (c^2)+(c^4)))/(2 ((a^2)+(b^2)) ((a^2)+(c^2)) ((b^2)+(c^2)) )//InputForm