Half-Moses Circle

Author

Eric W. Weisstein
December 2, 2004

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

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Half-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,
{Blue,MosesCircle[t]},
{Red,HalfMosesCircle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{
{X[39][t],"(Subscript[X, 39])",Red},
{X[2023][t],"(Subscript[X, 2023])",Red}
}
],LabelOffset->.03]
}],
AspectRatio->Automatic,PlotRange->All
]

Derivation

CenterName[X[39]]

Circle@@Through[{CircleCenter,CircleRadius}[MosesCircle]]

Simplify[Together//@ToSidelengths[CircleEquation[Circle[CircleCenter[MosesCircle],CircleRadius[MosesCircle]/2]]],TriangleQ[]]

Factor[%]

Factor[CircleFunction[(3 (a^7) b c (α^2)-2 (a^5) (b^3) c (α^2)-5 (a^3) (b^5) c (α^2)-2 (a^5) b (c^3) (α^2)-18 (a^3) (b^3) (c^3) (α^2)-8 a (b^5) (c^3) (α^2)-5 (a^3) b (c^5) (α^2)-8 a (b^3) (c^5) (α^2)-2 (a^6) (b^2) c α β+12 (a^4) (b^4) c α β-2 (a^2) (b^6) c α β-8 (a^6) (c^3) α β+12 (a^4) (b^2) (c^3) α β+12 (a^2) (b^4) (c^3) α β-8 (b^6) (c^3) α β+8 (a^4) (c^5) α β+22 (a^2) (b^2) (c^5) α β+8 (b^4) (c^5) α β-5 (a^5) (b^3) c (β^2)-2 (a^3) (b^5) c (β^2)+3 a (b^7) c (β^2)-8 (a^5) b (c^3) (β^2)-18 (a^3) (b^3) (c^3) (β^2)-2 a (b^5) (c^3) (β^2)-8 (a^3) b (c^5) (β^2)-5 a (b^3) (c^5) (β^2)-8 (a^6) (b^3) α γ+8 (a^4) (b^5) α γ-2 (a^6) b (c^2) α γ+12 (a^4) (b^3) (c^2) α γ+22 (a^2) (b^5) (c^2) α γ+12 (a^4) b (c^4) α γ+12 (a^2) (b^3) (c^4) α γ+8 (b^5) (c^4) α γ-2 (a^2) b (c^6) α γ-8 (b^3) (c^6) α γ+8 (a^5) (b^4) β γ-8 (a^3) (b^6) β γ+22 (a^5) (b^2) (c^2) β γ+12 (a^3) (b^4) (c^2) β γ-2 a (b^6) (c^2) β γ+8 (a^5) (c^4) β γ+12 (a^3) (b^2) (c^4) β γ+12 a (b^4) (c^4) β γ-8 (a^3) (c^6) β γ-2 a (b^2) (c^6) β γ-8 (a^5) (b^3) c (γ^2)-8 (a^3) (b^5) c (γ^2)-5 (a^5) b (c^3) (γ^2)-18 (a^3) (b^3) (c^3) (γ^2)-5 a (b^5) (c^3) (γ^2)-2 (a^3) b (c^5) (γ^2)-2 a (b^3) (c^5) (γ^2)+3 a b (c^7) (γ^2))]]

CenterFunction[BrocardMidpoint]

CircleRadius[MosesCircle]/2

Circle Function

CircleFunction[HalfMosesCircle]

Symmetrize[CircleFunction[HalfMosesCircle]]

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

CenterFunction[X[1]]

Circle Center

Symmetrize[ToSidelengths[CircleCenter[CircleEquation[HalfMosesCircle]][[1]]]]

CircleCenter[HalfMosesCircle]

Circle Radius

FullSimplify[Together//@ToSidelengths[CircleRadius[CircleEquation[HalfMosesCircle]]],TriangleQ[]]//Timing

CircleRadius[HalfMosesCircle]

Incident Centers

IncidentCenters[KimberlingCenters,HalfMosesCircle]//Timing

CenterName[X[2023]]