Moses Circle

Author

Eric W. Weisstein
December 2, 2004

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.6}}];
colors={Red,Blue};
Show[Graphics[{
t,
{Green,NinePointCircle[t]},
{Red,MosesCircle[t]},
{Blue,LongestLine[Coordinates/@Through[{X[2028],X[2029]}[t]],1]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{
{X[39][t],"(Subscript[X, 39])",Red},
{X[115][t],"(Subscript[X, 115])",Green},
{X[1569][t],"(Subscript[X, 1569])",Blue},
{X[2028][t],"(Subscript[X, 2028])",Blue},
{X[2029][t],"(Subscript[X, 2029])",Blue}
}
],LabelOffset->.05]
}],
AspectRatio->Automatic,PlotRange->{{-.2,1.1},{-.1,.7}}
]

Derivation

Factor[Simplify[Together//@DistanceTrilinears[CircleEquation[X[39],Circumradius[]Tan[BrocardAngle[]]TrigExpand[Sin[2BrocardAngle[]]]]],TriangleQ[]]]

Cancel[CircleFunction[Symmetrize[%]]]

InputForm[%]

Circle Function

Factor[CircleFunction[MosesCircle]]

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

Through[{CenterName,CenterFunction}[X[599]]]

Circle Radius

FullSimplify[Together//@DistanceTrilinears[CircleRadius[MosesCircle]],TriangleQ[]]

FullSimplify[Circumradius[]Tan[BrocardAngle[]]TrigExpand[Sin[2BrocardAngle[]]]]

Factor[%]

Circle Center

Symmetrize[Factor[Together//@DistanceTrilinears[CircleCenter[MosesCircle][[1]]]]]

IncidentCenters[KimberlingCenters,CyclicTrilinears[-a ((b^2)+(c^2))]]//Timing

CenterName[X[39]]

Incident Centers

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

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

Check

Short/@(FullSimplify[DistanceTrilinears[TrigExpand[eqn/.Thread[{α,β,γ}->Trilinears[#]]]],
TriangleQ[]]&/@ctrs)//Timing

Tangent Circles

Simplify[Together//@ToSidelengths[TangentQ[MosesCircle,NinePointCircle]/.Cos[x_]:>TrigExpand[Cos[x]]],TriangleQ[]]