Tangential Mid-Arc Circle

Author

Eric W. Weisstein
January 16, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/TangentialMid-ArcCircle.nb.

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

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.7}}];
Show[Graphics[{
t,
{Blue,TangentialMidArcTriangle[t]},
{Red,TangentialMidArcCircle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[TangentialMidArcTriangle[t]],{"A'","B'","C'"}}],
{
{Center[TangentialMidArcCircle[t]],"X",Red}
}
],LabelOffset->.04
]
}],AspectRatio->Automatic]

Center

IncidentCenters[KimberlingCenters,Center[TangentialMidArcCircle]]//Timing

Center[TangentialMidArcCircle][[1]]

Circle Function

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

Radius

CircleRadius[TangentialMidArcCircle]//TraditionalForm

Incident Centers

IncidentCenters[KimberlingCenters,TangentialMidArcCircle]//Timing

Derivation

Circle[TangentialMidArcCircle]

r=Distance[Center[TangentialMidArcCircle],TrilinearVertexMatrix[TangentialMidArcTriangle][[1]]];

Factor[(Together//@ToSidelengths[HalfAngleFormulas[CircleRadius[TangentialMidArcCircle]^2]])//.Sqrt[x_]Sqrt[y_]:>FullSimplify[Sqrt[x y],TriangleQ[]]]

FullSimplify[%,TriangleQ[]]