Brocard Angle

Author

Eric W. Weisstein
November 26, 2004

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

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

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

Identities

cot(ω)==cot(A)+cot(B)+cot(C)

Together//@ToSidelengths[Subtract@@{Cot[BrocardAngle],Cot[A]+Cot[B]+Cot[C]}]/.Sqrt[x_]Sqrt[y_]:>Simplify[Sqrt[x y],TriangleQ[]]/.Sqrt[x_]:>Simplify[Sqrt[x],TriangleQ[]]

cot(ω)==(((a^2)+(b^2)+(c^2))/(4 Δ))

ToSidelengths[Subtract@@{Cot[BrocardAngle],(((a^2)+(b^2)+(c^2))/(4Area))}]

cot(ω)==(((sin^2)(A)+(sin^2)(B)+(sin^2)(C))/(2 sin(A) sin(B) sin(C)))

Simplify[ToSidelengths[Subtract@@{Cot[BrocardAngle],((((Sin[A])^2)+((Sin[B])^2)+((Sin[C])^2))/(2Sin[A]Sin[B]Sin[C]))}],TriangleQ[]]

cot(ω)==((a sin(A)+b sin(B)+c sin(C))/(a cos(A)+b cos(B)+c cos(C)))

Simplify[ToSidelengths[Subtract@@{Cot[BrocardAngle],((a Sin[A]+b Sin[B]+c Sin[C])/(a Cos[A]+b Cos[B]+c Cos[C]))}],TriangleQ[]]

(csc^2)(ω)==(csc^2)(A)+(csc^2)(B)+(csc^2)(C)

Together//@ToSidelengths[Subtract@@{((Csc[BrocardAngle])^2),((Csc[A])^2)+((Csc[B])^2)+((Csc[C])^2)}]

sin(ω)==((2 Δ)/((a^2)(b^2)+(c^2) (a^2)+(b^2) (c^2)))

Simplify[Together//@ToSidelengths[Subtract@@{Sin[BrocardAngle],((2Area)/(Sqrt[a^2b^2+b^2c^2+c^2a^2]))}],TriangleQ[]]

(sin^2)(ω)==(((-a+b+c) (a-b+c) (a+b-c) (a+b+c))/(4 ((a^2) (b^2)+(c^2) (b^2)+(a^2) (c^2))))

Together//@ToSidelengths[Subtract@@{Sin[BrocardAngle]^2,(((-a+b+c)(a-b+c)(a+b-c)(a+b+c))/(4(a^2b^2+b^2c^2+c^2a^2)))}]

(sin^3)(ω)==sin(A-ω) sin(B-ω) sin(C-ω)

Simplify[(Together//@ToSidelengths[Subtract@@{Sin[BrocardAngle]^3,Sin[A-BrocardAngle]Sin[B-BrocardAngle]Sin[C-BrocardAngle]}/.Sin[x_]:>TrigExpand[Sin[x]]])/.Sqrt[x_]:>FullSimplify[Sqrt[x],TriangleQ[]],TriangleQ[]]

tan(ω)=((sin(A) sin(B) sin(C))/(cos(A) cos(B) cos(C)+1))

Factor[Together//@ToSidelengths[Subtract@@{Tan[BrocardAngle],((Sin[A]Sin[B]Sin[C])/(Cos[A]Cos[B]Cos[C]+1))}]/.Sqrt[x_]Sqrt[y_]:>Simplify[Sqrt[x y],TriangleQ[]]/.Sqrt[x_]:>Simplify[Sqrt[x],TriangleQ[]]]

Inequalities

<<NumericalMath`NMinimize`

Maximum Brocard Angle

ϵ=10^(-5);
NMinimize[{Cot[a1]+Cot[a2]+Cot[Pi-a1-a2],{a1+a2<Pi-ϵ,ϵ<a1<Pi-ϵ,ϵ<a2<Pi-ϵ}
},
{a1,a2},WorkingPrecision->20]

ArcCot[First[%]]

30. Degree

Minimum cot ω

Cotw[x_,y_]:=Plus@@Cot/@{x,y,Pi-x-y}

{l=Table[Cotw[#,y],{y,.001,Pi-#,.01}];
Min[l],3/2Tan[#/2]+1/2Cot[#/2]}&/@Range[.1,Pi,.1]

8 (((cot(A)+cot(B)+cot(C)))^3)≤a b c

InequalityInstance[a1+a2+a3==π&&a1>0&&a2>0&&a3>0&&8(Cot[a1]+Cot[a2]+Cot[a3])^3≤a1 a2 a3,{a1,a2,a3}]

CylindricalAlgebraicDecomposition[a1+a2+a3==π&&a1>0&&a2>0&&a3>0&&8(Cot[a1]+Cot[a2]+Cot[a3])^3≤a1 a2 a3,{a1,a2,a3}]