Isogonal Conjugate

Author

Eric W. Weisstein
February 15, 2003

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

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

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

Initialization

<<PlaneGeometry`

Figure

tri={{0,0},{1,0},{.2,.7}};
cev1={.2,.3,.1};
cev2=Table[1./cev1[[i]],{i,1,3}];
pts={TrilinearToCartesian[tri,cev1],"P"};
pts2={TrilinearToCartesian[tri,cev2],"Q"};

Show[Graphics[{
	Cevians[tri,cev1],
	{GrayLevel[.5],Cevians[tri,cev2]},
  {Dashing[{.02,.02}],AngleBisectors[tri]},
  {Thickness[.01],Triangle[tri]},
  {PointLabels[tri,pts,.03,{.02,.04}]},
  {GrayLevel[.5],PointLabels[tri,pts2,.03,{.02,.04}]}
}],AspectRatio->Automatic]

tri={{0,0},{1,0},{.15,.8}};
cev1={.2,.3,.1};
cev2=Table[1./cev1[[i]],{i,1,3}];
p=TrilinearToCartesian[tri,cev1];
q=TrilinearToCartesian[tri,cev2];
pt=Triangle[Join[CevianTriangle[tri,cev1][[1]][[{1,2}]],{p}]];
qt=Triangle[Join[CevianTriangle[tri,cev2][[1]][[{1,2}]],{q}]];
pts={p,"P",tri[[1]],"A",tri[[2]],"B",tri[[3]],"C",
	pt[[1,1]],"(Subscript[P, BC])",pt[[1,2]],"(Subscript[P, AC])"};
pts2={q,"Q",qt[[1,1]],"(Subscript[Q, BC])",qt[[1,2]],"(Subscript[Q, AC])"};

Show[Graphics[{
  {Thickness[.01],Triangle[tri]},
	{
				Line[{tri[[3]],p}],
				{Dashing[{.02,.02}],Cevians[tri,cev1]},
				pt,
				PointLabels[tri,pts,.03,{.02,.04}]
	},
  {GrayLevel[.5],
				Line[{tri[[3]],q}],
				{Dashing[{.02,.02}],Cevians[tri,cev2]},
				qt,
				PointLabels[tri,pts2,.03,{.02,.04}]
	}
}],AspectRatio->Automatic]

Points

CenterFunction/@{GergonnePoint,NagelPoint}

cf=CenterFunction/@Centers;

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

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