Orthocentroidal Circle

Author

Eric W. Weisstein
May 26, 2004

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

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

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

Code

OrthocentroidalCircle[t_Triangle]:=Module[{
g=Centroid[t],
h=Orthocenter[t]
},
Circle[Coordinates[Midpoint[{g,h}]],Distance[g,h]/2]
]

Definition

CircleFunction[OrthocentroidalCircle]

CircleEquation[OrthocentroidalCircle]

Figure

t=Triangle[{{0,0},{1,0},{.1,.7}}];
Show[Graphics[{
t,
{
Purple,
Line[Coordinates/@{Centroid[t],Orthocenter[t]}],
OrthocentroidalCircle[t]
},
PointLabels[Join[
Transpose[{Point/@Vertices[t],{"(Subscript[A, 1])","(Subscript[A, 2])","(Subscript[A, 3])"}}],
{
{Centroid[t],"G",Red},
{Orthocenter[t],"H",Blue}
}
],{PointSize[.025],LabelOffset->.04}
]
}],
PlotRange->All,AspectRatio->Automatic]

Radius

FullSimplify[Distance@@(DistanceTrilinears/@{Centroid,Orthocenter}),a>0&&b>0&&c>0]/2

(1/3) (Sqrt[((((((a^2)-(b^2)))^2) ((a^2)+(b^2))-((a^4)-3 (a^2) (b^2)+(b^4)) (c^2)-((a^2)+(b^2)) (c^4)+(c^6))/(-(a-b-c) (a+b-c) (a-b+c) (a+b+c)))])

r=(1/(12Δ)) (Sqrt[((((a^2)-(b^2)))^2) ((a^2)+(b^2))-((a^4)-3 (a^2) (b^2)+(b^4)) (c^2)-((a^2)+(b^2)) (c^4)+(c^6)]);

d/.{b->a,c->a}

Expand[((((a^2)-(b^2)))^2) ((a^2)+(b^2))-((a^4)-3 (a^2) (b^2)+(b^4)) (c^2)-((a^2)+(b^2)) (c^4)+(c^6)]

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

Circle Function

LawOfCosines[FullSimplify[Together//@CircleFunction[DistanceTrilinears[CircleEquation[FullSimplify[Midpoint[{Centroid,Orthocenter}]],(1/3) (Sqrt[((((((a^2)-(b^2)))^2) ((a^2)+(b^2))-((a^4)-3 (a^2) (b^2)+(b^4)) (c^2)-((a^2)+(b^2)) (c^4)+(c^6))/(-(a-b-c) (a+b-c) (a-b+c) (a+b+c)))])]]],TriangleQ[]]]

CyclicTrilinears[CircleFunction[OrthocentroidalCircle]]

IncidentCenters[KimberlingCenters,CyclicTrilinears[CircleFunction[OrthocentroidalCircle]]]

CenterName/@%

Center

FullSimplify[CircleCenter[OrthocentroidalCircle]]

FullSimplify[DistanceTrilinears[%]]

IncidentCenters[KimberlingCenters,CircleCenter[OrthocentroidalCircle]]

CenterFunction[X[381]]

Incident Points

(ctrs=IncidentCenters[KimberlingCenters,CircleEquation[OrthocentroidalCircle]])//Timing

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

Circle Contains Incenter

ctr=FullSimplify[Midpoint[{Orthocenter,Centroid}]]

ctr={((4 a Sec[A]+b Sec[B]+c Sec[C])/a),((a Sec[A]+4 b Sec[B]+c Sec[C])/b),((a Sec[A]+b Sec[B]+4 c Sec[C])/c)};

Simplify[DistanceTrilinears[ctr]]

d1=Together//@DistanceTrilinears[Distance[ctr,Trilinears[Incenter]]]

d2=Together//@DistanceTrilinears[CircleRadius[OrthocentroidalCircle]]

Numerical check

Table[d1/d2/.Thread[{a,b,c}->SideLengths[Triangle[Table[Random[],{3},{2}]]]],{1000}]//Max//Timing

Simplify[ExpandAll[d2]]

d22=FullSimplify[%,TriangleQ[]]

r2=FullSimplify[d1/d22,TriangleQ[]]^2

Table[r2/.Thread[{a,b,c}->SideLengths[Triangle[Table[Random[],{3},{2}]]]],{1000}]//Max//Timing

r2

Orthogonality

Check

Print[TimeConstrained[Together//@DistanceTrilinears[First[OrthogonalQ[#,OrthocentroidalCircle]]],60]//Timing]&/@{StevanovicCircle,LesterCircle}

Timing[expr=Together//@DistanceTrilinears[First[OrthogonalQ[LesterCircle,OrthocentroidalCircle]]/.Cos[x_]:>TrigExpand[Cos[x]]]]

Graphic

t=Triangle[{{0,0},{1,0},{1.4,.7}}];
circs={StevanovicCircle,LesterCircle,OrthocentroidalCircle};
centers={"(Subscript[X, 650])","(Subscript[X, 1116])","(Subscript[X, 381])"};
colors={Red,Green,Blue};
Show[Graphics[{
Transpose[{colors,Through[circs[t]]}],
PointLabels[Join[
Transpose[{CircleCenter[#[t]]&/@circs,centers,colors}]
],LabelOffset->.1
]
}],AspectRatio->Automatic,PlotRange->All]