Equi-Brocard Center

Author

Eric W. Weisstein
November 26, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneGeometry/Equi-BrocardCenter.nb.

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

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

Figure

t=Triangle[{{0,0},{1,0},{.3,.5}}];

ts=TriangulationTriangles[t,X[368][t]]

BrocardAngle/@TriangulationTriangles[t,X[368][t]]

Angle/@Prepend@@@Transpose[{Most/@Vertices/@ts,Coordinates/@(BrocardPoint[1][#]&/@ts)}]

colors={Red,Blue,Green};
Show[Graphics[{
Transpose[{colors,EqualAngleTicks[#,1,.1,.01,1]&/@Prepend@@@Transpose[{Most/@Vertices/@ts,Coordinates/@(BrocardPoint[1][#]&/@ts)}]}],
Transpose[{colors,Text["ω",Flatten[#[[1,1,2]]&/@EqualAngleTicks[#,1,.1,.03,1, VertexArc->False]]]&/@Prepend@@@Transpose[{Most/@Vertices/@ts,Coordinates/@(BrocardPoint[1][#]&/@ts)}]}],
ts,
Transpose[{colors,BrocardAngles[1][#]&/@ts}],
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{{X[368][t],"(Subscript[X, 368])",Purple}},
Transpose[{BrocardPoint[1][#]&/@ts,{"(Subscript[Ω, A])","(Subscript[Ω, B])","(Subscript[Ω, C])"},colors}]
],
LabelOffset->.03]
}],
AspectRatio->Automatic,PlotRange->All]

Derivation

Equations

aa=FullSimplify[Distance[{1,0,0},{alpha,beta,gamma}],TriangleQ[]]

Simplify[LawOfCosines[(Sqrt[((b c (b (beta^2) c+beta (-(a^2)+(b^2)+(c^2)) gamma+b c (gamma^2)))/(((a alpha+b beta+c gamma))^2))])/.Thread[{alpha,beta,gamma}->{α,β,γ}]],TriangleQ[]]

bb=FullSimplify[Distance[{0,1,0},{alpha,beta,gamma}],TriangleQ[]]

cc=FullSimplify[Distance[{0,0,1},{alpha,beta,gamma}],TriangleQ[]]

cot(ω)==(((a^2)+(b^2)+(c^2))/(4 Δ))

eqns=FullSimplify[{((aa^2+cc^2+b^2)/(Area[{aa,b,cc}])),((aa^2+bb^2+c^2)/(Area[{aa,bb,c}])),((a^2+bb^2+cc^2)/(Area[{a,bb,cc}]))}^2,TriangleQ[]]

Factor/@eqns

gcd=PolynomialGCD@@%

eqns2=Cancel[eqns/gcd]

eqns3=-I PowerExpand[Sqrt[eqns2]]

eqns3={alpha gamma (2 (a^2) (alpha^2) b+(a^3) alpha beta+3 a alpha (b^2) beta+(a^2) b (beta^2)+(b^3) (beta^2)-a alpha beta (c^2)+b (beta^2) (c^2)+2 a alpha b c gamma-(a^2) beta c gamma+3 (b^2) beta c gamma+beta (c^3) gamma+2 b (c^2) (gamma^2)),alpha beta (2 (a^2) (alpha^2) c+2 a alpha b beta c+2 (b^2) (beta^2) c+(a^3) alpha gamma-a alpha (b^2) gamma-(a^2) b beta gamma+(b^3) beta gamma+3 a alpha (c^2) gamma+3 b beta (c^2) gamma+(a^2) c (gamma^2)+(b^2) c (gamma^2)+(c^3) (gamma^2)),beta gamma ((a^3) (alpha^2)+a (alpha^2) (b^2)+3 (a^2) alpha b beta+alpha (b^3) beta+2 a (b^2) (beta^2)+a (alpha^2) (c^2)-alpha b beta (c^2)+3 (a^2) alpha c gamma-alpha (b^2) c gamma+2 a b beta c gamma+alpha (c^3) gamma+2 a (c^2) (gamma^2))};

Select[{alpha,beta}/.Solve[{Equal@@eqns3,gamma==1}/.{a->2,b->41/10,c->5},{alpha,beta}],And@@(Im/@#=={0,0})&]//N

SideLengths[Triangle[2,41/10,5]]

p=TrilinearToCartesian[Triangle[2,41/10,5],{0.36493959122867764`,0.9589356276127483`,1}]

BrocardAngle/@(Triangle[Append[#,p]]&/@Partition[%32[[1]],2,1,1])

Solve

k==((2 Δ)/(a α+b β+c γ))

(soln=Solve[{Equal@@eqns3,2Area[]==(a alpha+b beta+c gamma)},{alpha,beta,gamma}])//Timing

...ran out of memory on byblis2...

(soln=Solve[{Equal@@eqns3,2Δ==(a alpha+b beta+c gamma)},{alpha,beta,gamma}])//Timing

Gröbner Basis

k==((2 Δ)/(a α+b β+c γ))

Factor/@GroebnerBasis[Append[Subtract@@@Partition[eqns3,2,1],4Area[]^2-(a alpha+b beta+c gamma)^2],{alpha},{beta,gamma},MonomialOrder->EliminationOrder]

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Factor/@CoefficientList[myalpha[[1]][x],x]

CyclicTrilinears[myalpha]/.Thread[{a,b,c}->SideLengths[Evaluate[Triangle[2,41/10,5]]]]//N

p=TrilinearToCartesian[Triangle[2,41/10,5],%]

BrocardAngle/@(Triangle[Append[#,p]]&/@Partition[Vertices[Triangle[2,41/10,5]],2,1,1])