Thabit ibn Qurra's Generalization of the Pythagorean Theorem

Author

Eric W. Weisstein
August 30, 2026

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

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

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

Figure

Thabit ibn Qurra's construction on an arbitrary triangle.

Module[{a = {2.2, 3.1}, b = {0, 0}, c = {6, 0}, alpha, bp, cp}, alpha = VectorAngle[b - a, c - a]; bp = {x, 0} /. FindRoot[VectorAngle[a - {x, 0}, b - {x, 0}] == alpha, {x, 1.1}]; cp = {x, 0} /. FindRoot[VectorAngle[a - {x, 0}, c - {x, 0}] == alpha, {x, 4.4}]; Graphics[{{Directive[LightDarkSwitched[Darker[StandardGray, 0.55], Lighter[StandardGray, 0.55]], AbsoluteThickness[2]], Line[{b, a, c, b}], Line[{a, bp}], Line[{a, cp}]}, {Directive[RGBColor[0.72, 0.24, 0.18], PointSize[0.018]], Point[{a, b, bp, cp, c}]}, MapThread[Text[Style[#1, FontFamily -> "Times", FontSlant -> Italic, 15], #2] & , {{"A", "B", "B'", "C'", "C"}, {a + {0, 0.25}, b + {-0.2, -0.25}, cp + {0, -0.25}, bp + {0, -0.25}, c + {0.2, -0.25}}}]}, Axes -> False]]