Isotomic Conjugate

Author

Eric W. Weisstein
February 15, 2003

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

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

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

Figure

IsotomicPoints[tri_List,ctr_List]:=Module[{cev={},d={},unitvec=UnitVec[tri],i},
  cev=Take[CevianTriangle[tri,ctr][[1]],3];
  d=Table[Dist[cev[[i]],tri[[Sub1[i]]]],{i,1,3}];
  Map[Point,Table[tri[[Add1[i]]]+d[[i]]unitvec[[i]],{i,1,3}]]
]
IsotomicLines[tri_List,ctr_List]:=Module[
  {
   isopt=IsotomicPoints[tri,ctr]
  },
  Table[Line[{tri[[i]],isopt[[i,1]]}],{i,1,3}]
]

tri={{0,0},{1,0},{.2,.7}};
ctr={.1,.3,.4};
Show[Graphics[{{GrayLevel[0.5],Cevians[tri,ctr]},
  IsotomicLines[tri,ctr],{PointSize[.02],IsotomicConjugatePoint[tri,ctr]},
  {GrayLevel[0.5],PointSize[.02],Point[TrilinearToCartesian[tri,ctr]]},
  {Thickness[.01],Triangle[tri]}}],AspectRatio->Automatic]

Points

CenterFunction/@{GergonnePoint,NagelPoint}

cf=CenterFunction/@Centers;

(conj=Sort[({#1,Centers[[Sequence@@#2]]}&@@@Select[Transpose[{Centers,Flatten[Position[cf,1/#/a^2,{1},1]]&/@cf}],#[[2]]≠{}&])])//ColumnForm

conj/.X[n_]:>CenterName[X[n]]//ColumnForm