van Lamoen Circle

Author

Eric W. Weisstein
November 30, 2004

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

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

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

Plot

Off[TrilinearToCartesian::"orient"];

t=Triangle[{{0,0},{1,0},{.3,.5}}];
g=Coordinates[Centroid[t]];
tt=Map[Triangle[Append[#,g]]&,Map[Coordinates,Transpose[{BeginPoint/@Medians[t],#[VertexPoints[t]]}],{2}]&/@{RotateRight,RotateLeft},{2}];
cc=Flatten[Map[Circumcenter,tt,{2}]];
Show[Graphics[
{
t,
{Red,Medians[t]},
{Blue,Map[Circumcircle,tt,{2}]},
{Green,Circumcircle[Coordinates/@Take[cc,3]]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{{Point@TrilinearToCartesian[t,CyclicTrilinears[((10 (a^4)-13 (a^2) (b^2)+4 (b^4)-13 (a^2) (c^2)-10 (b^2) (c^2)+4 (c^4))/a)]]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]],"(Subscript[X, 1153])",Green}},
Append[#,Red]&/@Transpose[{VertexPoints[MedialTriangle[t]],{"(Subscript[M, A])","(Subscript[M, B])","(Subscript[M, C])"}}],
{{Centroid[t],"G",Red}},
Append[#,Blue]&/@Transpose[{cc,
{"(Subscript[O, CA])","(Subscript[O, AB])","(Subscript[O, BA])","(Subscript[O, BC])","(Subscript[O, CB])","(Subscript[O, AC])"}
}]
],
LabelOffset->.03
]
}],AspectRatio->Automatic,PlotRange->All]

Derivation

Three Circumcircle centers

Simplify[CircleEquation[{Trilinears[Centroid],TrilinearVertexMatrix[MedialTriangle][[3]],{0,1,0}}]]//Factor

cc=CircleCenter[3 (a^2) c (α^2)-3 a b c α β+2 (a^3) α γ-4 a (b^2) α γ+2 a (c^2) α γ-4 (a^2) b β γ+2 (b^3) β γ-b (c^2) β γ+2 (a^2) c (γ^2)+2 (b^2) c (γ^2)-(c^3) (γ^2)]

Simplify[CircleEquation[{Trilinears[Centroid],TrilinearVertexMatrix[MedialTriangle][[1]],{0,1,0}}]]//Factor

dd=CircleCenter[((a^3) (α^2)-2 a (b^2) (α^2)-2 a (c^2) (α^2)+(a^2) b α β-2 (b^3) α β+4 b (c^2) α β-2 (a^2) c α γ+4 (b^2) c α γ-2 (c^3) α γ+3 a b c β γ-3 a (c^2) (γ^2))]

Simplify[CircleEquation[{Trilinears[Centroid],TrilinearVertexMatrix[MedialTriangle][[3]],{1,0,0}}]]//Factor

ee=CircleCenter[(-3 a b c α β+3 (b^2) c (β^2)+2 (a^3) α γ-4 a (b^2) α γ-a (c^2) α γ-4 (a^2) b β γ+2 (b^3) β γ+2 b (c^2) β γ+2 (a^2) c (γ^2)+2 (b^2) c (γ^2)-(c^3) (γ^2))]

Circle Equation

Together//@DistanceTrilinears[CircleEquation[{cc,dd,ee}]]//Factor

eqn=(8 (a^8) (α^2)-36 (a^6) (b^2) (α^2)+48 (a^4) (b^4) (α^2)-16 (a^2) (b^6) (α^2)-36 (a^6) (c^2) (α^2)+69 (a^4) (b^2) (c^2) (α^2)+6 (a^2) (b^4) (c^2) (α^2)+48 (a^4) (c^4) (α^2)+6 (a^2) (b^2) (c^4) (α^2)-16 (a^2) (c^6) (α^2)-8 (a^7) b α β+12 (a^5) (b^3) α β+12 (a^3) (b^5) α β-8 a (b^7) α β+78 (a^5) b (c^2) α β-78 (a^3) (b^3) (c^2) α β+78 a (b^5) (c^2) α β-162 (a^3) b (c^4) α β-162 a (b^3) (c^4) α β+76 a b (c^6) α β-16 (a^6) (b^2) (β^2)+48 (a^4) (b^4) (β^2)-36 (a^2) (b^6) (β^2)+8 (b^8) (β^2)+6 (a^4) (b^2) (c^2) (β^2)+69 (a^2) (b^4) (c^2) (β^2)-36 (b^6) (c^2) (β^2)+6 (a^2) (b^2) (c^4) (β^2)+48 (b^4) (c^4) (β^2)-16 (b^2) (c^6) (β^2)-8 (a^7) c α γ+78 (a^5) (b^2) c α γ-162 (a^3) (b^4) c α γ+76 a (b^6) c α γ+12 (a^5) (c^3) α γ-78 (a^3) (b^2) (c^3) α γ-162 a (b^4) (c^3) α γ+12 (a^3) (c^5) α γ+78 a (b^2) (c^5) α γ-8 a (c^7) α γ+76 (a^6) b c β γ-162 (a^4) (b^3) c β γ+78 (a^2) (b^5) c β γ-8 (b^7) c β γ-162 (a^4) b (c^3) β γ-78 (a^2) (b^3) (c^3) β γ+12 (b^5) (c^3) β γ+78 (a^2) b (c^5) β γ+12 (b^3) (c^5) β γ-8 b (c^7) β γ-16 (a^6) (c^2) (γ^2)+6 (a^4) (b^2) (c^2) (γ^2)+6 (a^2) (b^4) (c^2) (γ^2)-16 (b^6) (c^2) (γ^2)+48 (a^4) (c^4) (γ^2)+69 (a^2) (b^2) (c^4) (γ^2)+48 (b^4) (c^4) (γ^2)-36 (a^2) (c^6) (γ^2)-36 (b^2) (c^6) (γ^2)+8 (c^8) (γ^2));

CircleFunction[eqn]((R^2)/(Circumradius[]^2))//Factor

Circle Function

CircleFunction[VanLamoenCircle]

InputForm[%]

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

Circle Center

DistanceTrilinears[CircleCenter[eqn]]//Factor

Simplify[- (a-b-c) (a+b-c) (a-b+c) (a+b+c) (10 (a^4)-13 (a^2) (b^2)+4 (b^4)-13 (a^2) (c^2)-10 (b^2) (c^2)+4 (c^4))/(2a Times@@CyclicTrilinears[a+b-c]Semiperimeter[]),TriangleQ[]]//Factor

DistanceTrilinears[CircleCenter[VanLamoenCircle][[1]]]//Factor

IncidentCenters[{X[1153]},CircleCenter[VanLamoenCircle]]//Timing

IncidentCenters[KimberlingCenters,CircleCenter[VanLamoenCircle]]//Timing

Circle Radius

CircleRadius[eqn]

Simplify[Together//@DistanceTrilinears[CircleRadius[VanLamoenCircle]]//Factor,TriangleQ[]]

FullSimplify[%10,TriangleQ[]]

-(√(Factor[(2 ((a^2)+(b^2))-(c^2)) (2 (a^2)-(b^2)+2 (c^2)) ((a^2)-2 ((b^2)+(c^2))) (2 (a^4)+2 (b^4)-5 (b^2) (c^2)+2 (c^4)-5 (a^2) ((b^2)+(c^2)))]))/(18 (a-b-c) (a+b-c) (a-b+c) (a+b+c))((R^2)/(Circumradius[]^2))//Factor

Incident Centers

IncidentCenters[KimberlingCenters,VanLamoenCircle]//Timing