External Similitude Center

Author

Eric W. Weisstein
January 20, 2005

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

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

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

Table

Length[pairs=Subsets[CentralCircles,{2}]]

t=Triangle[Table[Random[Real,{0,1},30],{3},{2}]];
kc=KimberlingCenters;
rules=Join[Thread[{a,b,c}->SideLengths[Evaluate[t]]],Thread[{A,B,C}->Angles[t]]];

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

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

Table

DisplayForm[Grid[{#[[1]],
If[#[[-1,-1]]=!={},CenterName[#[[-1,-1,1]]],"*"],
If[#[[-1,-1]]=!={},#[[-1,-1,1]],"*"]
}&/@Sort[{Sort[CircleName/@#[[2]]],#[[3]]}&/@Select[l,#[[-1,-1]]≠{}&]],RowLines->True,ColumnAlignments->Left
(*,RowBackgrounds->{{None,Yellow}}*)
]]

Test

t=Triangle[Table[Random[Real,{0,1},30],{3},{2}]];

X[1694][t]

CenterName[X[1694]]

ExternalSimilitudeCenter[ApolloniusCircle[t],BrocardCircle[t]]

Cartesian[ExternalSimilitudeCenter[ApolloniusCircle,BrocardCircle][t]]

CenterName[X[1693]]

CenterName[X[1693]]

InternalSimilitudeCenter[ApolloniusCircle[t],BrocardCircle[t]]

Cartesian[InternalSimilitudeCenter[ApolloniusCircle,BrocardCircle][t]]