Orthologic Triangles

Author

Eric W. Weisstein
August 29, 2026

This notebook downloaded from https://mathworld.wolfram.com/notebooks/PlaneGeometry/OrthologicTriangles.nb.

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

©2026 Wolfram Research, Inc. except for portions noted otherwise

Illustration

The dashed blue perpendiculars concur at P1, while the dotted orange perpendiculars in the reciprocal direction concur at P2.

Module[{a,ap,b,bp,c,cp,firstCenter,firstLines,labelBackground,lineIntersection,neutral,normalLines,secondCenter,secondLines,sideLines},a={-2.1,-0.8};b={2.0,-0.9};c={0.45,2.0};firstCenter={-0.25,0.15};lineIntersection[{p1_,d1_},{p2_,d2_}]:=p1+d1 LinearSolve[Transpose[{d1,-d2}],p2-p1][[1]];sideLines={{{-0.2,-1.25},Reverse[firstCenter-a] {-1,1}},{{1.15,0.4},Reverse[firstCenter-b] {-1,1}},{{-1.05,0.75},Reverse[firstCenter-c] {-1,1}}};ap=lineIntersection[sideLines[[2]],sideLines[[3]]];bp=lineIntersection[sideLines[[3]],sideLines[[1]]];cp=lineIntersection[sideLines[[1]],sideLines[[2]]];normalLines={{ap,Reverse[b-c] {-1,1}},{bp,Reverse[c-a] {-1,1}},{cp,Reverse[a-b] {-1,1}}};secondCenter=lineIntersection[normalLines[[1]],normalLines[[2]]];firstLines=Line/@{{a,firstCenter},{b,firstCenter},{c,firstCenter}};secondLines=Line/@{{ap,secondCenter},{bp,secondCenter},{cp,secondCenter}};neutral=LightDarkSwitched[Black,White];labelBackground=LightDarkSwitched[White,Black];Graphics[{Directive[Blue,AbsoluteThickness[2]],Line[Append[{a,b,c},a]],Directive[Orange,AbsoluteThickness[2]],Line[Append[{ap,bp,cp},ap]],Directive[Blue,Dashed,AbsoluteThickness[1.5]],firstLines,Directive[Orange,Dotted,AbsoluteThickness[1.5]],secondLines,Directive[PointSize[0.018],neutral],Point[{a,b,c,ap,bp,cp}],Directive[PointSize[0.025],Blue],Point[firstCenter],Directive[PointSize[0.025],Orange],Point[secondCenter],MapThread[Text[Style[#1,16,neutral,Background->labelBackground,FontFamily->"Times"],#2]&,{{(Subscript[Style["A",FontSlant->"Italic"], 1]),(Subscript[Style["B",FontSlant->"Italic"], 1]),(Subscript[Style["C",FontSlant->"Italic"], 1]),(Subscript[Style["A",FontSlant->"Italic"], 2]),(Subscript[Style["B",FontSlant->"Italic"], 2]),(Subscript[Style["C",FontSlant->"Italic"], 2]),(Subscript[Style["P",FontSlant->"Italic"], 1]),(Subscript[Style["P",FontSlant->"Italic"], 2])},{a+{-0.18,-0.18},b+{0.18,-0.18},c+{0,0.2},ap+{-0.2,0.1},bp+{0.2,0.1},cp+{0.2,-0.15},firstCenter+{-0.18,0.18},secondCenter+{0.18,0.18}}}]}]]