Mixtilinear Circle

Author

Eric W. Weisstein
December 21, 2004

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

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

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

Graphic

t=Triangle[{{0,0},{1,0},{.2,.5}}];
Show[Graphics[{
t,
Circumcircle[t],
{Blue,MixtilinearIncircles[t]},
{Green,MixtilinearTriangle[t]},
{Red,MixtilinearCircle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{Center/@MixtilinearIncircles[t],{"(Subscript[O, A])","(Subscript[O, B])","(Subscript[O, C])"}}],
{
{Circumcenter[t],"O"}
}
],LabelOffset->.03]
}],AspectRatio->Automatic,PlotRange->All]

Incident Centers

IncidentCenters[KimberlingCenters,MixtilinearCircle]//Timing

Center

Center[MixtilinearCircle]

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

Circle Function

CircleFunction[MixtilinearCircle]

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

Derivation

eqn=Factor[Together//@ToSidelengths[TrilinearEquation@@(Center/@Circles[MixtilinearIncircles])]]

ctr=Symmetrize/@Factor[Center[eqn]]

InputForm[ctr[[1]]]

Together//@CircleRadius[eqn]

Map[Factor,Refine[%,TriangleQ[]],{3}]

InputForm[%]

Factor[CircleFunction[(-12 (a^4) (b^2) c (α^2)+20 (a^3) (b^3) c (α^2)-4 (a^2) (b^4) c (α^2)-4 a (b^5) c (α^2)-12 (a^4) b (c^2) (α^2)-8 (a^3) (b^2) (c^2) (α^2)-12 (a^2) (b^3) (c^2) (α^2)+20 (a^3) b (c^3) (α^2)-12 (a^2) (b^2) (c^3) (α^2)+8 a (b^3) (c^3) (α^2)-4 (a^2) b (c^4) (α^2)-4 a b (c^5) (α^2)-3 (a^6) c α β-2 (a^5) b c α β+19 (a^4) (b^2) c α β-28 (a^3) (b^3) c α β+19 (a^2) (b^4) c α β-2 a (b^5) c α β-3 (b^6) c α β+2 (a^5) (c^2) α β-6 (a^4) b (c^2) α β-12 (a^3) (b^2) (c^2) α β-12 (a^2) (b^3) (c^2) α β-6 a (b^4) (c^2) α β+2 (b^5) (c^2) α β+7 (a^4) (c^3) α β-4 (a^3) b (c^3) α β+58 (a^2) (b^2) (c^3) α β-4 a (b^3) (c^3) α β+7 (b^4) (c^3) α β-4 (a^3) (c^4) α β+4 (a^2) b (c^4) α β+4 a (b^2) (c^4) α β-4 (b^3) (c^4) α β-5 (a^2) (c^5) α β+6 a b (c^5) α β-5 (b^2) (c^5) α β+2 a (c^6) α β+2 b (c^6) α β+(c^7) α β-4 (a^5) b c (β^2)-4 (a^4) (b^2) c (β^2)+20 (a^3) (b^3) c (β^2)-12 (a^2) (b^4) c (β^2)-12 (a^3) (b^2) (c^2) (β^2)-8 (a^2) (b^3) (c^2) (β^2)-12 a (b^4) (c^2) (β^2)+8 (a^3) b (c^3) (β^2)-12 (a^2) (b^2) (c^3) (β^2)+20 a (b^3) (c^3) (β^2)-4 a (b^2) (c^4) (β^2)-4 a b (c^5) (β^2)-3 (a^6) b α γ+2 (a^5) (b^2) α γ+7 (a^4) (b^3) α γ-4 (a^3) (b^4) α γ-5 (a^2) (b^5) α γ+2 a (b^6) α γ+(b^7) α γ-2 (a^5) b c α γ-6 (a^4) (b^2) c α γ-4 (a^3) (b^3) c α γ+4 (a^2) (b^4) c α γ+6 a (b^5) c α γ+2 (b^6) c α γ+19 (a^4) b (c^2) α γ-12 (a^3) (b^2) (c^2) α γ+58 (a^2) (b^3) (c^2) α γ+4 a (b^4) (c^2) α γ-5 (b^5) (c^2) α γ-28 (a^3) b (c^3) α γ-12 (a^2) (b^2) (c^3) α γ-4 a (b^3) (c^3) α γ-4 (b^4) (c^3) α γ+19 (a^2) b (c^4) α γ-6 a (b^2) (c^4) α γ+7 (b^3) (c^4) α γ-2 a b (c^5) α γ+2 (b^2) (c^5) α γ-3 b (c^6) α γ+(a^7) β γ+2 (a^6) b β γ-5 (a^5) (b^2) β γ-4 (a^4) (b^3) β γ+7 (a^3) (b^4) β γ+2 (a^2) (b^5) β γ-3 a (b^6) β γ+2 (a^6) c β γ+6 (a^5) b c β γ+4 (a^4) (b^2) c β γ-4 (a^3) (b^3) c β γ-6 (a^2) (b^4) c β γ-2 a (b^5) c β γ-5 (a^5) (c^2) β γ+4 (a^4) b (c^2) β γ+58 (a^3) (b^2) (c^2) β γ-12 (a^2) (b^3) (c^2) β γ+19 a (b^4) (c^2) β γ-4 (a^4) (c^3) β γ-4 (a^3) b (c^3) β γ-12 (a^2) (b^2) (c^3) β γ-28 a (b^3) (c^3) β γ+7 (a^3) (c^4) β γ-6 (a^2) b (c^4) β γ+19 a (b^2) (c^4) β γ+2 (a^2) (c^5) β γ-2 a b (c^5) β γ-3 a (c^6) β γ-4 (a^5) b c (γ^2)+8 (a^3) (b^3) c (γ^2)-4 a (b^5) c (γ^2)-4 (a^4) b (c^2) (γ^2)-12 (a^3) (b^2) (c^2) (γ^2)-12 (a^2) (b^3) (c^2) (γ^2)-4 a (b^4) (c^2) (γ^2)+20 (a^3) b (c^3) (γ^2)-8 (a^2) (b^2) (c^3) (γ^2)+20 a (b^3) (c^3) (γ^2)-12 (a^2) b (c^4) (γ^2)-12 a (b^2) (c^4) (γ^2))]]

InputForm[%[[1]]]