Chord-Chord Angle

Author

Eric W. Weisstein
August 12, 2026

This notebook downloaded from https://mathworld.wolfram.com/notebooks/PlaneGeometry/Chord-ChordAngle.nb.

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

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

Illustration

The angle APC equals half the sum of the measures of the intercepted arcs AC and BD.

Module[{a, angleColor, b, c, chordColor1, chordColor2, cross, d, label, neutral, p}, 
  a = {Cos[2.62], Sin[2.62]}; b = {Cos[6.02], Sin[6.02]}; c = {Cos[0.55], Sin[0.55]}; 
   d = {Cos[4.05], Sin[4.05]}; cross[u_, v_] := u[[1]]*v[[2]] - u[[2]]*v[[1]]; 
   p = a + (cross[c - a, d - c]/cross[b - a, d - c])*(b - a); 
   neutral = LightDarkSwitched[Black, White]; chordColor1 = Blue; chordColor2 = Red; 
   angleColor = Blue; label[s_, q_, offset_] := 
    Text[Style[s, 16, neutral, Background -> LightDarkSwitched[White, Black], 
      FontFamily -> "Times", FontSlant -> "Italic"], q + offset]; 
   Graphics[{Directive[neutral, AbsoluteThickness[2]], Circle[], 
     Directive[chordColor1, AbsoluteThickness[2.5]], Line[{a, b}], 
     Directive[chordColor2, AbsoluteThickness[2.5]], Line[{c, d}], 
     Directive[angleColor, Opacity[0.2]], Disk[p, 0.25, {ArcTan @@ (a - p), ArcTan @@ (c - p)}], 
     Directive[angleColor, Opacity[1], AbsoluteThickness[2.5]], 
     Circle[p, 0.25, {ArcTan @@ (a - p), ArcTan @@ (c - p)}], Directive[neutral, PointSize[0.015]], 
     Point[{a, b, c, d, p}], label["A", a, {-0.09, 0.08}], label["B", b, {0.09, -0.06}], 
     label["C", c, {0.09, 0.07}], label["D", d, {-0.08, -0.08}], label["P", p, {0.08, -0.11}], 
     Text[Style["θ", 18, angleColor, FontFamily -> "Times"], p + {-0.12, 0.27}]}]]