Circumcenter

Author

Eric W. Weisstein
August 2, 2007

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

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

©2007 Wolfram Research, Inc. except for portions noted otherwise

Figure

t=Triangle[{{0,0},{1,0},{.2,.7}}];
Graphics[{
t,
{Red,Circumcircle[t]},
{
Blue,
Line[{Coordinates[o],#}]&/@Vertices[t]
},
{Red,PointLabels[{{"O",Coordinates[o=Circumcenter[t]],LabelOffset->.05}}]},
PointLabels[Transpose[{{"A","B","C"},Vertices[t],{LabelOffset->-.05,LabelOffset->{.05,-.05},LabelOffset->{-.02,.06}}}]]
}]

Distances

OCl

FullSimplify[ToSidelengths[Distance[Circumcenter,ClawsonPoint]^2],TriangleQ[]]

Factor[%]

ToAngles[%]

OF

FullSimplify[Together//@ToSidelengths[Distance[Circumcenter,FeuerbachPoint]^2/.Cos[x_]:>TrigExpand[Cos[x]]],TriangleQ[]]

Factor[%]

ToAngles[%]

OG

Factor[FullSimplify[ToSidelengths[Distance[Centroid,Circumcenter]],TriangleQ[]]^2]

ToAngles[%]

Factor[ToSidelengths[Circumradius^2-(1/9)(a^2+b^2+c^2)]]

OGe

Factor[Together//@ToSidelengths[Distance[Circumcenter,GergonnePoint]^2]]

ToAngles[%]

OH

Factor[FullSimplify[ToSidelengths[Distance[Orthocenter,Circumcenter]],TriangleQ]^2]

ToAngles[-Denominator[%]]

ToAngles[%%]

Factor[ToSidelengths[9Circumradius^2-a^2-b^2-c^2]]

OI

Factor[FullSimplify[ToSidelengths[Distance[Circumcenter,Incenter]],TriangleQ]^2]

ToAngles[%]

expr1=((( (1/(4Δ))(Sqrt[a b c ((a^3)+(b^3)+(c^3)-a (a b+a c-b c)-b (a b-a c+b c)-c (-a b+a c+b c))])))^2);

expr2=((a b c ((a^3)-(a^2) b-a (b^2)+(b^3)-(a^2) c+3 a b c-(b^2) c-a (c^2)-b (c^2)+(c^3)))/(16(Δ^2)));

FullSimplify[expr1/expr2]

FullSimplify[expr1-expr2]

OK

Factor//@FullSimplify[Factor[FullSimplify[ToSidelengths[Distance[Circumcenter,SymmedianPoint]],TriangleQ[]]^2],TriangleQ[]]

ToAngles[%]

OL

Factor[FullSimplify[ToSidelengths[Distance[Circumcenter,DeLongchampsPoint]^2],TriangleQ[]]]

ToAngles[%]

OM

Factor[Together//@ToSidelengths[Distance[Circumcenter,Mittenpunkt]^2]]

ToAngles[%]

ONa

Factor[Together//@ToSidelengths[Distance[Circumcenter,NagelPoint]^2]]

PowerExpand[Sqrt[ToAngles[%]]]

ONi

Factor[Simplify[ToSidelengths[Distance[Circumcenter,NinePointCenter]^2/.Cos[x_]:>TrigExpand[Cos[x]]],TriangleQ[]]]

ToAngles[%]

OSp

Factor[Together//@ToSidelengths[Distance[Circumcenter,SpiekerCenter]^2]]

ToAngles[%]

Circumcenter and Orthocenter are Isogonal Conjugates

tri=Triangle[{{0,0},{1,0},{.2,.7}}];

{IsogonalConjugate[tri,Circumcenter[tri]],Orthocenter[tri]}

{IsogonalConjugate[tri,Orthocenter[tri]],Circumcenter[tri]}

Diagram

tri=Triangle[{{0,0},{1,0},{.2,.7}}];
v=Vertices[tri];
cev1=Trilinears[Circumcenter];
cev2=1/cev1;
pts=Transpose[{VertexPoints[tri],{"A","B","C"}}];
redpt={{Circumcenter[tri],"O"}};
bluept={{Orthocenter[tri],"H"}};

Show[Graphics[{
Circumcircle[tri],
{
Blue,
Cevians[tri,cev2],
PerpSquare[#,.03,4]&/@Transpose[{Sides[tri],Altitudes[tri]}],
PointLabels[bluept,LabelOffset->.03]
},
{
Red,
Cevians[tri,cev1],
PerpSquare[#,.03,2]&/@Transpose[{Sides[tri],PerpendicularBisectors[tri]}],
{Dashing[{.02,.02}],PerpendicularBisectors[tri]},
PointLabels[redpt,LabelOffset->.03]
},
{
Green,
Thickness[.002],
EqualAngleDivisionTicks[tri,2,.1,.02,.08,VertexArc->True],
Thickness[.005],
AngleBisectors[tri]
},
{Thickness[.01],tri},
PointLabels[pts,LabelOffset->.03]
}
],AspectRatio->Automatic]

Circumcenter is the Orthocenter of its Pedal Triangle

tri=Triangle[{{0,0},{1,0},{.3,.6}}];
cc=Circumcenter[tri];
pedal=PedalTriangle[tri,Circumcenter];
bluepts={{Circumcenter[tri],"\!\(O\_\(\(A\_1\) \(A\_2\) A\_3\)\)"}};
redpts=Transpose[{VertexPoints[pedal],{"(Subscript[O, 1])","(Subscript[O, 2])","(Subscript[O, 3])"}}];
greenpts={{Orthocenter[pedal],"\!\(H\_\(\(O\_1\) \(O\_2\) O\_3\)\)"}};
pts=Transpose[{VertexPoints[tri],{"(Subscript[A, 1])","(Subscript[A, 2])","(Subscript[A, 3])"}}];

Show[Graphics[{
Circumcircle[tri],
{Thickness[.01],tri},
{Blue,PointLabels[bluepts,LabelOffset->.03]},
{Green,Altitudes[pedal],PointLabels[greenpts,LabelOffset->.03]},
{Blue,Dashing[{.02,.02}],PerpendicularBisectors[tri]},
{Red,pedal,PointLabels[redpts,LabelOffset->.03]},
PointLabels[pts,LabelOffset->.03]
},AspectRatio->Automatic,PlotRange->All]
]

Of Arbitrary Triangle

General Triangle

Circumcenter[Triangle[{{p1,q1,r1},{p2,q2,r2},{p3,q3,r3}}]]//Simplify

FullSimplify[%]

Named Triangles

t=RandomTriangle[20];

rules=Join[Thread[{a,b,c}->SideLengths[Evaluate[t]]],Thread[{A,B,C}->Angles[Evaluate[t]]]];
kc=KimberlingCenters;

Length[k=Rest[#]/First[#]&/@((Trilinears/@KimberlingCenters)/.TriangleRules/.rules)]//Timing

trips=Triangles;

search::zero="Bad zero encountered `1`";
l={};
Do[
Print[l0={
i,
trips[[i]],
rc=TimeConstrained[Circumcenter[trips[[i]]]/.TriangleRules/.rules,60];
(
pt=Switch[rc,
_?(#=={0,0,0}&),{0,0,0},
Or@@Thread[#==0]&,Message[search:zero,#];rc,
_,Rest[rc]/First[rc]
];
If[pt=={0,0,0},{0},
If[#=={},{},kc[[#]]]&[Flatten[Position[k,_?(#==pt&)]]]
]
)//Timing
}];
AppendTo[l,l0],
{i,Length[trips]}
]

Timing[l={#,KimberlingLookup[Circumcenter[#]]}&/@Triangles]

Not in Kimberling

First/@Cases[l,_?(#[[-1]]==={}&)]

Table

{Name[#[[2]]],#[[-1,-1,1]],Name[#[[-1,-1,1]]]}&/@DeleteCases[Sort[l,(OrderedQ[{Name[#1[[2]]],Name[#2[[2]]]}]&)],_?(#[[-1,-1]]=={}&)]//TableForm

Structures Passing through It

pt=Circumcenter;

Circles

First/@Select[{#,IncidentCenters[{pt},#]}&/@Circles,Last[#]=!={}&]

Circle Centers

First/@Select[{#,IncidentCenters[{pt},CircleCenter[#]]}&/@Circles,Last[#]=!={}&]

Conics

First/@Select[{#,IncidentCenters[{pt},#]}&/@Conics,Last[#]=!={}&]

Cubics

First/@Select[{#,IncidentCenters[{pt},#]}&/@TriangleCubics,Last[#]=!={}&]

Lines

First/@Select[{#,IncidentCenters[{pt},#]}&/@Lines,Last[#]=!={}&]

Acute Triangle

FullSimplify[Plus@@ToSidelengths[Distance[Trilinears[Circumcenter],#]&/@TrilinearVertexMatrix[MedialTriangle]],TriangleQ[]]

Factor[FullSimplify[Plus@@ToSidelengths[Distance[Trilinears[Circumcenter],#]&/@TrilinearVertexMatrix[MedialTriangle]],AcuteQ[]&&TriangleQ[]]]

Factor[Simplify[ToSidelengths[Circumradius+Inradius],TriangleQ[]]]