Anticomplementary Circle

Author

Eric W. Weisstein
January 16, 2005

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[{
t,
{Red,AnticomplementaryCircle[t]},
{Blue,AnticomplementaryTriangle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[AnticomplementaryTriangle[t]],{"A'","B'","C'"}}],
{
{Orthocenter[t],"H",Red}
}
],LabelOffset->.06]
}],AspectRatio->Automatic]

Radius

CircleRadius[AnticomplementaryCircle]//TraditionalForm

Center

IncidentCenters[{X[4]},Center[AnticomplementaryCircle]]

CenterName[X[4]]

Circle Function

CircleFunction[AnticomplementaryCircle]

IncidentCenters[{X[32]},CyclicTrilinears[%]]

CenterName[X[32]]

Incident Centers

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

First/@ctrs

Length[ctrs]

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

Derivation

circ1=Together//@Circumcircle@@TrilinearVertexMatrix[AnticomplementaryTriangle];

circ=circ1/.Circle[ctr_List,r_]:>Circle[-Symmetrize/@ToAngles[Symmetrize/@ctr],FullSimplify[r,TriangleQ[]]]

IncidentCenters[KimberlingCenters,{Sec[A],Sec[B],Sec[C]}]//Timing

CenterName[X[4]]

Circumradius[]

eqn=Symmetrize[Numerator[Together//@ToSidelengths[TrilinearEquation[circ]]/.Sqrt[x_]Sqrt[y_]:>Simplify[Sqrt[x y],TriangleQ[]]]]

CircleFunction[eqn]

IncidentCenters[KimberlingCenters,{((a^2)/(b c)),((b^2)/(a c)),((c^2)/(a b))}]//Timing

CenterName[X[32]]