Gallatly Circle

Author

Eric W. Weisstein
November 29, 2004

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

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

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

Derivation

CenterName[X[39]]

Factor[Simplify[Together//@DistanceTrilinears[CircleEquation[X[39],Circumradius[]Sin[BrocardAngle[]]]],TriangleQ[]]]

eqn=Symmetrize[%]

Circle Function

Factor[CircleFunction[eqn]]

InputForm[%]

CircleFunction[GallatlyCircle]

IncidentCenters[KimberlingCenters,CyclicTrilinears[CircleFunction[GallatlyCircle]]]//Timing

Through[{CenterName,CenterFunction}[X[183]]]

Circle Radius

Square Root

FullSimplify[Together//@DistanceTrilinears[CircleRadius[GallatlyCircle]],a>0&&b>0&&c>0&&a+b>c&&b+c>a&&a+c>b]

Subtract@@@Table[With[{λ=√((a-b+c) (a+b+c) ((a^6) (c^2)-(b^6) (c^2)+(b^4) (c^4)-2 (a^5) c ((b^2)+2 (c^2))+(a^4) ((b^2)+(c^2)) ((b^2)+6 (c^2))+2 (a^3) c ((b^4)-3 (b^2) (c^2)-2 (c^4))+2 a (b^2) c ((b^4)+(b^2) (c^2)-(c^4))+(a^2) (-(b^6)+2 (b^4) (c^2)+7 (b^2) (c^4)+(c^6))))},{(a b c √((1/(2 (b^2) (c^2)+2 (a^2) ((b^2)+(c^2))))((a-b+c) (a+b+c) ((a^6) (c^2)+2 (a^3) (b^2) c ((b^2)+(c^2))+(a^4) ((b^2)+(c^2)) ((b^2)+6 (c^2))+(b^2) c (-(b^4) c+(b^2) (c^3)+λ)+(a^2) (-(b^6)+2 (b^4) (c^2)+7 (b^2) (c^4)+(c^6)+c λ)+a (2 (b^4) (c^3)+(b^2) λ+(c^2) λ)))))/((a^3) (b^2)-a (b^4)+(a^4) c+2 (a^2) (b^2) c-(b^4) c+3 (a^3) (c^2)+2 a (b^2) (c^2)+3 (a^2) (c^3)+(b^2) (c^3)+a (c^4)+λ),((a b c)/(2 (Sqrt[(a^2) (b^2)+(a^2) (c^2)+(b^2) (c^2)])))}/.Thread[{a,b,c}->SideLengths[Evaluate[Triangle[Table[Random[Real],{3},{2}]]]]]
],{20}]//Chop//Union

Square It

Subtract@@@Table[With[{λ=√((a-b+c) (a+b+c) ((a^6) (c^2)-(b^6) (c^2)+(b^4) (c^4)-2 (a^5) c ((b^2)+2 (c^2))+(a^4) ((b^2)+(c^2)) ((b^2)+6 (c^2))+2 (a^3) c ((b^4)-3 (b^2) (c^2)-2 (c^4))+2 a (b^2) c ((b^4)+(b^2) (c^2)-(c^4))+(a^2) (-(b^6)+2 (b^4) (c^2)+7 (b^2) (c^4)+(c^6))))},{(((1/(2 (b^2) (c^2)+2 (a^2) ((b^2)+(c^2))))((a-b+c) (a+b+c) ((a^6) (c^2)+2 (a^3) (b^2) c ((b^2)+(c^2))+(a^4) ((b^2)+(c^2)) ((b^2)+6 (c^2))+(b^2) c (-(b^4) c+(b^2) (c^3)+λ)+(a^2) (-(b^6)+2 (b^4) (c^2)+7 (b^2) (c^4)+(c^6)+c λ)+a (2 (b^4) (c^3)+(b^2) λ+(c^2) λ)))))/((a^3) (b^2)-a (b^4)+(a^4) c+2 (a^2) (b^2) c-(b^4) c+3 (a^3) (c^2)+2 a (b^2) (c^2)+3 (a^2) (c^3)+(b^2) (c^3)+a (c^4)+λ)^2,(1/(4 ((a^2) (b^2)+(a^2) (c^2)+(b^2) (c^2))))}/.Thread[{a,b,c}->SideLengths[Evaluate[Triangle[Table[Random[Real],{3},{2}]]]]]
],{20}]//Chop//Union

FullSimplify[With[{λ=√((a-b+c) (a+b+c) ((a^6) (c^2)-(b^6) (c^2)+(b^4) (c^4)-2 (a^5) c ((b^2)+2 (c^2))+(a^4) ((b^2)+(c^2)) ((b^2)+6 (c^2))+2 (a^3) c ((b^4)-3 (b^2) (c^2)-2 (c^4))+2 a (b^2) c ((b^4)+(b^2) (c^2)-(c^4))+(a^2) (-(b^6)+2 (b^4) (c^2)+7 (b^2) (c^4)+(c^6))))},
((1/(2 (b^2) (c^2)+2 (a^2) ((b^2)+(c^2))))((a-b+c) (a+b+c) ((a^6) (c^2)+2 (a^3) (b^2) c ((b^2)+(c^2))+(a^4) ((b^2)+(c^2)) ((b^2)+6 (c^2))+(b^2) c (-(b^4) c+(b^2) (c^3)+λ)+(a^2) (-(b^6)+2 (b^4) (c^2)+7 (b^2) (c^4)+(c^6)+c λ)+a (2 (b^4) (c^3)+(b^2) λ+(c^2) λ))))/((a^3) (b^2)-a (b^4)+(a^4) c+2 (a^2) (b^2) c-(b^4) c+3 (a^3) (c^2)+2 a (b^2) (c^2)+3 (a^2) (c^3)+(b^2) (c^3)+a (c^4)+λ)^2],a>0&&b>0&&c>0&&a+b>c&&b+c>a&&a+c>b]//Timing

Factor[Simplify[a b c Factor[Sqrt[(1/(4 (b^2) (c^2)+4 (a^2) ((b^2)+(c^2))))]],a>0&&b>0&&c>0&&a+b>c&&b+c>a&&a+c>b]]

From Definition

Factor[FullSimplify[Circumradius[]Sin[BrocardAngle[]],TriangleQ[]]]

Circle Center

IncidentCenters[KimberlingCenters,CircleCenter[GallatlyCircle]]//Timing

Incident Centers

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

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

Check

Short/@(FullSimplify[DistanceTrilinears[TrigExpand[eqn/.Thread[{α,β,γ}->Trilinears[#]]]],
TriangleQ[]]&/@ctrs)//Timing

Figure

t=Triangle[{{0,0},{1,0},{.3,.5}}];
Show[Graphics[{
t,
{Red,GallatlyCircle[t]},
{Blue,LongestLine[Coordinates[X[#][t]]&/@{2026,2027},1]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{{X[39][t],"(Subscript[X, 39])",{Red,LabelOffset->-.03}}},
Append[#,Blue]&/@{
{X[2026][t],"(Subscript[X, 2026])"},
{X[2027][t],"(Subscript[X, 2027])"}
}
],PointSize[.025],LabelOffset->.05]
}],AspectRatio->Automatic,PlotRange->{{-.1,1.1},{-.1,.6}}]

CenterName[X[#]]&/@{2026,2027}

CenterFunction[X[#]]&/@{2026,2027}

Derivation

Trilinears       f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = e sin(A + ω) + cos(A + ω)
Barycentrics  (sin A)f(A,B,C) : (sin B)f(B,C,A) : (sin C)f(C,A,B)
Of the two points of intersection of the Brocard axis, X(3)X(6) with the Gallatly circle, X(2026) is the one nearer to X(3).
X(2026) lies on these lines: 3,6   2009,2039   2010,2040
X(2026) = reflection of X(2027) in X(39)

(Sqrt[1-4 Sin[BrocardAngle[]^2]])

eqn1=Factor[Numerator[Together[CircleEquation[GallatlyCircle]]]]

FullSimplify[DistanceTrilinears[TrigExpand[LineEquation[BrocardAxis]]],TriangleQ[]]

eqn2=(a c (-(a^2)+(c^2)) β+(b^3) (c α-a γ)+b (-(c^3) α+(a^3) γ));

Flatten[Solve[eqn1==0&&eqn2==0&&2Area[]==(a α+b β+c γ),{α},{β,γ}]]

pts=Last/@%29;

FullSimplify[pts,TriangleQ[]]

(%34 /. Sqrt[x_]:>Sqrt[Factor[x]])

{(a ((Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) ((b^2)+(c^2)) ((a^4)+(b^4)-(b^2) (c^2)+(c^4)-(a^2) ((b^2)+(c^2)))+(Sqrt[((a^2) (b^2)+(a^2) (c^2)+(b^2) (c^2)) ((a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4))]) Abs[(b^4)+(c^4)-(a^2) ((b^2)+(c^2))]))/(4 Factor[((a^4)+(b^4)-(b^2) (c^2)+(c^4)-(a^2) ((b^2)+(c^2))) ((b^2) (c^2)+(a^2) ((b^2)+(c^2)))]),(a ((Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) ((b^2)+(c^2)) ((a^4)+(b^4)-(b^2) (c^2)+(c^4)-(a^2) ((b^2)+(c^2)))-(Sqrt[((a^2) (b^2)+(a^2) (c^2)+(b^2) (c^2)) ((a^4)-(a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)+(c^4))]) Abs[(b^4)+(c^4)-(a^2) ((b^2)+(c^2))]))/(4 Factor[((a^4)+(b^4)-(b^2) (c^2)+(c^4)-(a^2) ((b^2)+(c^2))) ((b^2) (c^2)+(a^2) ((b^2)+(c^2)))])}

p1=Point[TrilinearToCartesian[t,CyclicTrilinears[(a ((Sqrt[-(a-b-c) (a+b-c) (a-b+c) (a+b+c)]) ((b^2)+(c^2)) ((a^4)+(b^4)-(b^2) (c^2)+(c^4)-(a^2) ((b^2)+(c^2)))+(Sqrt[((a^4)+(b^4)-(b^2) (c^2)+(c^4)-(a^2) ((b^2)+(c^2))) ((b^2) (c^2)+(a^2) ((b^2)+(c^2)))]) Abs[(b^4)+(c^4)-(a^2) ((b^2)+(c^2))]))/(4 ((a^4)+(b^4)-(b^2) (c^2)+(c^4)-(a^2) ((b^2)+(c^2))) ((b^2) (c^2)+(a^2) ((b^2)+(c^2))))]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]]]

p1=Point[{0.41939624536930736`,0.22097145147809227`}]