Cosine Circle

Author

Eric W. Weisstein
November 22, 2004

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

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

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

Figure

t=Triangle[{{0,0},{1,0},{.15,.7}}];
v=Vertices[t];
k=SymmedianPoint[t];
aa=Antiparallels[t,k];
ap=Flatten[aa/. Line->Identity,1];
pts={k,K,
v[[1]],(Subscript[A, 1]),v[[2]],(Subscript[A, 2]),v[[3]],(Subscript[A, 3]),
ap[[1]],(Subscript[P, 1]),ap[[2]],(Subscript[Q, 2]),ap[[3]],(Subscript[P, 2]),
ap[[4]],(Subscript[Q, 3]),ap[[5]],(Subscript[P, 3]),ap[[6]],(Subscript[Q, 1])};

Show[Graphics[{
t,
PointLabels[tri,pts,.02,{.03,.04}],
aa,
CosineCircle[t] 
}],AspectRatio->Automatic]

Radius

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

Circumradius[]Tan[BrocardAngle[]]

Derivation

FullSimplify[R Sqrt[c^2+(1-c)^2Tan[ω]^2]/.c->0,0<ω≤Pi/6]

Circle Function

FullSimplify[Together//@DistanceTrilinears[CircleFunction[CircleEquation[Trilinears[SymmedianPoint],R Tan[ω]]]/.ω->BrocardAngle[]/.R->Circumradius[]],TriangleQ[]]

LawOfCosines[%]

IncidentCenters[KimberlingCenters,CyclicTrilinears[CircleFunction[CosineCircle]]]

Through[{CenterName,CenterFunction}[X[69]]]

Center

FullSimplify[Together//@DistanceTrilinears[CircleCenter[CosineCircle]]]/.

IncidentCenters[KimberlingCenters,CircleCenter[CosineCircle]]

CenterName[X[6]]

IncidentCenters

(ctrs=IncidentCenters[KimberlingCenters,CosineCircle])//Timing

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