Gibert Point

Author

Eric W. Weisstein
November 20, 2005

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

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

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

Nice Form

Factor[CenterFunction[X[1141]]]

Symmetrize[(1/a)(((a^2)-a b+(b^2)-(c^2)) ((a^2)+a b+(b^2)-(c^2)) ((a^2)-(b^2)-a c+(c^2)) ((a^2)-(b^2)+a c+(c^2)) ((a^4)-2 (a^2) (b^2)+(b^4)-(a^2) (c^2)-(b^2) (c^2)) ((a^4)-(a^2) (b^2)-2 (a^2) (c^2)-(b^2) (c^2)+(c^4)))]

Symmetrize[ToAngles[Factor[Trilinears[X[1141]]]]]

Cartesian[%[t]]

t=Triangle[{{0,0},{1,0},{0.3`,0.5`}}]

X[1141][t]

Simplify[CyclicTrilinears[x(c^2y^2-b^2z^2)/.Thread[{x,y,z}->{a α,b β,c γ}]]]/(a b c)

Cancel[CyclicTrilinears[x(c^2y^2-b^2z^2)/.Thread[{x,y,z}->{a α,b β,c γ}]]/(a b c)]

Collinearity

Simplify[ToSidelengths[CollinearQ[{KosnitaPoint,NinePointCenter,X[1141]}]/.c:(_Cos|_Sec):>TrigExpand[c]]/.Sqrt[s_]Sqrt[t_]:>Simplify[Sqrt[s t],TriangleQ[]]]

By Reflection

q=Symmetrize[Factor[ToSidelengths[InversePoint[KosnitaPoint,Circumcircle]/.c:(_Cos|_Sec):>TrigExpand[c]]/.Sqrt[s_]Sqrt[t_]:>Simplify[Sqrt[s t],TriangleQ[]]]]

Symmetrize[ToAngles[q]]

rt=Factor[ToSidelengths[Reflection[q,#]&/@Sides[ReferenceTriangle]]]

Together//@PerspectiveQ[Triangle[ReferenceTriangle],Triangle[rt]]

gp=Factor[Perspector[Triangle[rt],Triangle[ReferenceTriangle]]]

Symmetrize[ToAngles[Factor[ToAngles[Symmetrize[Factor/@Symmetrize[Factor[gp]]]]]]]

Symmetrize[ToAngles[Factor[Trilinears[X[1141]]]]]

KimberlingLookup[q]

Name[X[1157]]

t=Triangle[{{0,0},{1,0},{1/10,5/10}}];
Show[Graphics[{
{Yellow,Sides[t,2]},
t,
{Red,myrt=Triangle[Coordinates[Cartesian[#[t]]]&/@rt]},
{Green,l=Line[Coordinates/@{X[1157][t],#}]&/@VertexPoints[myrt],
MapIndexed[EqualSideTicksAboutMidpoint[#1,#2[[1]],.02]&,l]
},
{Purple,Circumcircle[t//N]},
{Blue,LongestLine[Append[#,Coordinates[X[1141][t]]]]&/@Transpose[First/@{myrt,t}]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
Append[#,Red]&/@Transpose[{VertexPoints[myrt],{"A'","B'","C'"}}],
{
{X[1141][t],"(Subscript[X, 1141])",Blue},
{X[1157][t],"(Subscript[X, 1157])",Green}
}
],LabelOffset->.05,PointSize[.025]]
}],AspectRatio->Automatic,PlotRange->{{-.5,1.2},{-.5,1.2}}]

Derivation

Cubic

Kn=CyclicSum[SA x((4SC^2-a^2b^2)z^2-(4SB^2-c^2a^2)y^2)/.Thread[{x,y,z}->{a α,b β,c γ}]]

Factor[ToSidelengths[Kn]]

eqn= ((a^8) b (α^2) β-3 (a^6) (b^3) (α^2) β+3 (a^4) (b^5) (α^2) β-(a^2) (b^7) (α^2) β-(a^6) b (c^2) (α^2) β+(a^4) (b^3) (c^2) (α^2) β-(a^4) b (c^4) (α^2) β+(a^2) b (c^6) (α^2) β+(a^7) (b^2) α (β^2)-3 (a^5) (b^4) α (β^2)+3 (a^3) (b^6) α (β^2)-a (b^8) α (β^2)-(a^3) (b^4) (c^2) α (β^2)+a (b^6) (c^2) α (β^2)+a (b^4) (c^4) α (β^2)-a (b^2) (c^6) α (β^2)-(a^8) c (α^2) γ+(a^6) (b^2) c (α^2) γ+(a^4) (b^4) c (α^2) γ-(a^2) (b^6) c (α^2) γ+3 (a^6) (c^3) (α^2) γ-(a^4) (b^2) (c^3) (α^2) γ-3 (a^4) (c^5) (α^2) γ+(a^2) (c^7) (α^2) γ+(a^6) (b^2) c (β^2) γ-(a^4) (b^4) c (β^2) γ-(a^2) (b^6) c (β^2) γ+(b^8) c (β^2) γ+(a^2) (b^4) (c^3) (β^2) γ-3 (b^6) (c^3) (β^2) γ+3 (b^4) (c^5) (β^2) γ-(b^2) (c^7) (β^2) γ-(a^7) (c^2) α (γ^2)+a (b^6) (c^2) α (γ^2)+3 (a^5) (c^4) α (γ^2)+(a^3) (b^2) (c^4) α (γ^2)-a (b^4) (c^4) α (γ^2)-3 (a^3) (c^6) α (γ^2)-a (b^2) (c^6) α (γ^2)+a (c^8) α (γ^2)-(a^6) b (c^2) β (γ^2)+(b^7) (c^2) β (γ^2)+(a^4) b (c^4) β (γ^2)-(a^2) (b^3) (c^4) β (γ^2)-3 (b^5) (c^4) β (γ^2)+(a^2) b (c^6) β (γ^2)+3 (b^3) (c^6) β (γ^2)-b (c^8) β (γ^2));

Centers incident on Cubic

IncidentCenters[Take[KimberlingCenters,All],ToSidelengths[Kn]]//Timing

Numerical

t=RandomTriangle[30]

t=Triangle[{{0.042372984038468844717676010108372268387491767932`30.,0.248963673865333311892947596039516193663324289571`30.},{0.615135460576648866224201557998582688809100534453`30.,0.393355744674972037939899147505685381229711713259`30.},{0.701987132801410537219242875455283007812679669725`30.,0.894743676525974646907662571465103192741737352844`30.}}];

soln=NSolve[ToSidelengths[{Numerator[Together[ToSidelengths[Kn]]]==0,
TrilinearEquation[Circumcircle]==0,
ToSidelengths[4Area^2-(a α+b β+c γ)^2]==0}]/.Thread[{a,b,c}->SideLengths[Evaluate[t]]],
{α,β,γ}];//Timing

X[1141][t]

Union[Cartesian[#[t]]&/@Select[{α,β,γ}/.soln,And@@(Im[#]==0&/@#)&],SameTest->Equal]

VertexPoints[t]

Analytic

(soln=Factor/@GroebnerBasis[{eqn,
TrilinearEquation[Circumcircle],
ToSidelengths[4Area^2-(a α+b β+c γ)^2]},
α,{β,γ},MonomialOrder->EliminationOrder])//Timing

Simplify[Solve[ToAngles[soln[[1,6]]]==0,α],TriangleQ[]]

res=Simplify[Factor//@(α/.Solve[soln[[1,7]]==0,α]),TriangleQ[]]

Cartesian[CyclicTrilinears[#][t]]&/@res

Cartesian[CyclicTrilinears[#/.Abs->Identity][t]]&/@res

Factor[Symmetrize[I res[[1]]/.Abs->Identity]]

Symmetrize[ToAngles[%]]

Graphic

t=Triangle[{{0,0},{1,0},{2/10.,1/2.}}];
Show[Graphics[{
t,
{Red,TrilinearCurve[t,eqn]},
{Blue,Circumcircle[t]},
PointLabels[Join[
Transpose[{VertexPoints[t],{"A","B","C"}}],
{X[#][t],Subscript["X",#],Red}&/@{4,5,13,14,30,79,80,265,621,622,1117},
{{X[1141][t],"(Subscript[X, 1141])",Orange}}
],LabelOffset->.08,PointSize[.025]]
}],AspectRatio->Automatic,PlotRange->{{-.7,1.2},{-.45,.7}}]