Anticomplement

Author

Eric W. Weisstein
October 25, 2005

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

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

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

Line Derivation

Solve[(l α+m β+n γ/.{α->1,β->1})==0,γ]

p1=AnticomplementPoint[{1,1,((-l-m)/n)}]

Solve[(l α+m β+n γ/.{α->0,β->1})==0,γ]

p2=AnticomplementPoint[{0,1,-(m/n)}]

((a b c n)/2) Factor[LineEquation[{p1,p2}]]

Factor/@(Coefficient[%,#]&/@{α,β,γ})

InputForm[%]

Anticomplement of a Line is a Line

Anticomplement[Line[{l,m,n}]]

Line[OrthicAxis]

Together//@ToSidelengths[Anticomplement[Line[OrthicAxis]]]

Together//@ToSidelengths[Anticomplement[Line[AntiorthicAxis]]]

Together//@ToSidelengths[Anticomplement[OrthicAxis]]

Lines

Print[Length[circs=Lines]];
t=Triangle[Table[Random[Real,{0,1},20],{3},{2}]];
kc=KimberlingCenters;
rules=Join[Thread[{a,b,c}->SideLengths[Evaluate[t]]],Thread[{A,B,C}->Angles[Evaluate[t]]]];

Length[k=Rest[#]/First[#]&/@((Trilinears/@KimberlingCenters)/.rules)]//Timing

circs=Lines

search::zero="Bad zero encountered `1`";
l={};
Do[
Print[l0={
i,
circs[[i]],
rc=Anticomplement[circs[[i]]][[1]]/.rules;
(
pt=Switch[rc,
_?(#=={0,0,0}&),{0,0,0},
Or@@Thread[#==0]&,Message[search:zero,#];rc,
_,Rest[rc]/First[rc]
];
If[pt=={0,0,0},{0},
If[#=={},{},kc[[#]]]&[Flatten[Position[k,_?(#==pt&)]]]
]
)//Timing
}];
AppendTo[l,l0],
{i,Length[circs]}
]

TableForm[{
#[[2]],
If[#[[-1,-1]]=={},"*",#[[-1,-1,1]]],
If[#[[-1,-1]]=={},"*",LineName[#[[-1,-1,1]]]]
}&/@l,TableDepth->2]

Circles

Circumcircle-Nine-Point Circle

Factor[TrilinearEquation[Circumcircle]/.Thread[{α,β,γ}->Anticomplement[{α,β,γ}]]]

Factor[ToSidelengths[TrilinearEquation[NinePointCircle]]]

de Longchamps Circle-Polar circle

Factor[TrilinearEquation[DeLongchampsCircle]/.Thread[{α,β,γ}->Anticomplement[{α,β,γ}]]]

Factor[ToSidelengths[TrilinearEquation[PolarCircle]]]

New centers

New centers

Anticomplement of the Brocard Axis

Anticomplement[BrocardAxis][[1,1]]

InputForm[%]

Anticomplement of the Fermat Axis

Symmetrize[Factor[Anticomplement[FermatAxis][[1,1]]]]

InputForm[%]

Anticomplement of the Lemoine Axix

Symmetrize[Factor[Anticomplement[LemoineAxis][[1,1]]]]

InputForm[%]

Anticomplement of the Soddy Line

Symmetrize[Factor[Anticomplement[SoddyLine][[1,1]]]]

InputForm[%]

Point Derivation

eqn1=Factor[Distance[{app,bpp,gpp},Trilinears[Centroid]]^2-4Distance[Trilinears[Centroid],{ap,bp,gp}]^2]

eqn2=Numerator[Together[Det[{{ap,bp,gp},{app,bpp,gpp},Trilinears[Centroid]}]]]

Factor//@Solve[{
Last[eqn1]==0,
eqn2==0,
gpp==1
},{app,bpp}]

Anticomplement[tri_]:=CyclicTrilinears[tri.{-a,b,c}/a]

CyclicTrilinears[{α,β,γ}.{-a,b,c}/a]

Examples

k=KimberlingCenters;

Do[
Print[{i,IncidentCenters[k,AnticomplementPoint[Trilinears[X[i]]],MaxIterations->1]//Timing}],
{i,50}]

Do[
Print[{i,If[MemberQ[k,X[i]],IncidentCenters[k,ComplementPoint[Trilinears[X[i]]],MaxIterations->1]//Timing,InputForm[CenterFunction[X[i]]]]}],
{i,982,Length[k]}]

Do[
Print[{i,If[MemberQ[k,X[i]],IncidentCenters[k,ComplementPoint[Trilinears[X[i]]],MaxIterations->1]//Timing,InputForm[CenterFunction[X[i]]]]}],
{i,1142,Length[k]}]

Print[{#,IncidentCenters[KimberlingCenters,AnticomplementPoint[Trilinears[X[#]]],MaxIterations->1]//Timing}]&/@{618,619,623,624,629,630,1125}

(Subscript[X, 13]) ((Subscript[X, 616]))

Trilinears[X[13]][[1]]//InputForm

Anticomplement[Trilinears[X[13]]][[1]]//InputForm

CenterFunction[X[616]]

StringReplace[Trilinears[X[616]][[1]],
Thread[{"S<sub>A</sub>","S<sub>B</sub>","S<sub>C</sub>"}->(" "<>#<>" "&/@(ToString/@InputForm/@CyclicTrilinears[(b^2+c^2-a^2)/2]))
]]//ToExpression//InputForm

LawOfCosines[%]//InputForm

t=Triangle[Table[Random[Real],{3},{2}]];

ExactTrilinears[t,#]&/@CyclicTrilinears/@{((-a Csc[A+(π/3)]+b Csc[B+(π/3)]+c Csc[C+(π/3)])/a),(1/a)(-2 b c ((a^2)+(1/4) (Sqrt[3]) (Sqrt[(a+b-c) (a-b+c) (-a+b+c) (a+b+c)])) Cos[A]+(a^2) b c Cos[B] Cos[C])}/.Thread[{a,b,c}->SideLengths[Evaluate[t]]]/.Thread[{A,B,C}->Angles[t]]