Cyclologic Center

Author

Eric W. Weisstein
August 12, 2026

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

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

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

Illustration

The three circles through AB'C', BC'A', and CA'B' meet at the cyclologic center P.

Module[{a, ap, b, bp, c, circleColor, circleData, circles, cp, labelBackground, 
   neutral, p, triangleColor2}, {a, b, c, ap, bp, cp} = {{-1.4606741573, -0.3370786517}, 
     {1.5068493151, -0.6849315068}, {0.1015228426, 1.421319797}, {1.2346793487, 0.9002851161}, 
     {0.4197123905, -0.1334839109}, {1.2265381482, -0.8415280637}}; p = {0, 0}; circleColor = Blue; 
   triangleColor2 = Red; neutral = LightDarkSwitched[Black, White]; 
   labelBackground = LightDarkSwitched[White, Black]; circleData[{u_, v_, w_}] := 
    Module[{center}, center = LinearSolve[2*{v - u, w - u}, {v . v - u . u, w . w - u . u}]; 
      {center, Norm[center - u]}]; circles = circleData /@ {{a, bp, cp}, {b, cp, ap}, {c, ap, bp}}; 
   Graphics[{Directive[circleColor, AbsoluteThickness[2], Opacity[0.65]], Circle @@@ circles, 
     Directive[neutral, AbsoluteThickness[2], Opacity[1]], Line[Append[{a, b, c}, a]], 
     Directive[triangleColor2, AbsoluteThickness[2], Opacity[1]], Line[Append[{ap, bp, cp}, ap]], 
     Directive[circleColor, PointSize[0.022]], Point[p], Directive[neutral, PointSize[0.014]], 
     Point[{a, b, c}], Directive[triangleColor2, PointSize[0.014]], Point[{ap, bp, cp}], 
     Text[Style["P", 17, circleColor, Background -> labelBackground, FontFamily -> "Times", 
       FontSlant -> "Italic"], p + {-0.12, 0.14}], 
     Text[Style["A", 15, neutral, Background -> labelBackground, FontFamily -> "Times", 
       FontSlant -> "Italic"], a + {-0.12, 0.1}], 
     Text[Style["B", 15, neutral, Background -> labelBackground, FontFamily -> "Times", 
       FontSlant -> "Italic"], b + {0.1, 0.05}], 
     Text[Style["C", 15, neutral, Background -> labelBackground, FontFamily -> "Times", 
       FontSlant -> "Italic"], c + {0, 0.13}], Text[Style["A'", 15, triangleColor2, 
       Background -> labelBackground, FontFamily -> "Times", FontSlant -> "Italic"], 
      ap + {0.12, 0.1}], Text[Style["B'", 15, triangleColor2, Background -> labelBackground, 
       FontFamily -> "Times", FontSlant -> "Italic"], bp + {-0.02, 0.14}], 
     Text[Style["C'", 15, triangleColor2, Background -> labelBackground, FontFamily -> "Times", 
       FontSlant -> "Italic"], cp + {0.1, -0.08}]}, PlotRange -> {{-2, 2}, {-1.5, 2}}]]