IsogonalConjugate::usage="It will be shown that the isogonal conjugate of the incenter is itself, that the isogonal conjugate of the circumcenter is the orthocenter and that the isogonal conjugate of the barycenter is the symmedian point of the triangle. Reference: Personal.";IsogonalConjugate[tri={{x1_,y1_},{x2_,y2_},{x3_,y3_}}]:=Module{},DEP=Det[{{x1,y1,1},{x2,y2,1},{x3,y3,1}}];IfDEP==0,Print[PasteButton["The points do not form a triangle"]],η=TriangleCenter[tri,"Incenter"];xi=η[[1]];yi=η[[2]];ζ=TriangleCenter[tri,"Circumcenter"];xcc=ζ[[1]];ycc=ζ[[2]];ξ=TriangleCenter[tri,"Centroid"];xb=ξ[[1]];yb=ξ[[2]];i=[tri,{xi,yi}];xii=i[[1]];yii=i[[2]];Print[PasteButton["The incenter and its isogonal conjugate (incenter)."]];Print["I(",N[xi],",",N[yi],")"," ","I(",N[xi],",",N[yi],")"];Print[" "];j=[tri,{xcc,ycc}];xci=j[[1]];yci=j[[2]];Print[PasteButton["The circumcenter and its isogonal conjugate (orthocenter)."]];Print["C(",N[xcc],",",N[ycc],")"," ","O(",N[xci],",",N[yci],")"];Print[" "];k=[tri,{xb,yb}];xbi=k[[1]];ybi=k[[2]];Print[PasteButton["The barycenter and its isogonal conjugate (symmedian point)."]];Print["B(",N[xb],",",N[yb],")"," ","S(",N[xbi],",",N[ybi],")"];Print[" "];SwatchLegend[{Red,Blue},{"Center","Isogonal conjugate"}]Show[Graphics[{EdgeForm[Black],White,Polygon[tri],Red,Disk[{xi,yi},.1],Blue,Disk[i,.1],Brown,Line/@Tuples[{tri,{{xi,yi},i}}]},AxesTrue,AxesOrigin->{0,0},ImageSize->700,PlotLabelStyle[Framed["The incenter and its isogonal conjugate (incenter)"],16,Blue,BackgroundYellow]]]Show[Graphics[{EdgeForm[Black],White,Polygon[tri],Red,Disk[{xcc,ycc},.1],Blue,Disk[j,.1],Brown,Line/@Tuples[{tri,{{xcc,ycc},j}}]},AxesTrue,AxesOrigin->{0,0},ImageSize700,PlotLabelStyle[Framed["The circumcenter and its isogonal conjugate (orthocenter)"],16,Blue,BackgroundYellow]]]Show[Graphics[{EdgeForm[Black],White,Polygon[tri],Red,Disk[{xb,yb},.1],Blue,Disk[k,.1],Brown,Line/@Tuples[{tri,{{xb,yb},k}}]},AxesTrue,AxesOrigin->{0,0},ImageSize700,PlotLabelStyle[Framed["The barycenter and its isogonal conjugate (symmedian point)"],16,Blue,BackgroundYellow]]]
In the command below, type the new coordinates of the points. Evaluate the notebook.
In[]:=
IsogonalConjugate[tri={{-2,5},{7,12},{9,-4}}]
I(4.06947,4.92508) I(4.06947,4.92508)
C(6.17722,3.77215) O(1.64557,5.4557)
B(4.66667,4.33333) S(3.48649,5.41216)
Out[]=
