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Holditch's theorem - mklabgr - 07-07-2026
Holditch's theorem
Summary
Holditch’s theorem is a remarkable result in plane geometry that reveals an unexpected relationship between motion and area. Imagine a fixed-length chord sliding continuously inside a smooth, convex closed curve while always keeping both of its endpoints on the boundary. As the chord moves, a chosen point that divides it into two fixed segments traces out a second closed curve. Holditch’s theorem states that the area enclosed by this new curve is always smaller than the area of the original curve by exactly πpq, where p and q are the distances from the chosen point to the two ends of the chord.
What makes this theorem especially surprising is that this difference depends only on the position of the point along the chord—not on the size or shape of the surrounding curve—provided the traced path remains a simple closed curve.
First published by Hamnet Holditch in 1858, the theorem has fascinated mathematicians for generations because of its elegant blend of geometry, kinematics, and invariance. Its area formula matches that of an ellipse with semi-axes p and q, hinting at deeper geometric connections, and it has inspired numerous extensions that apply the underlying idea to polygons, more general curves, and even problems in mechanics and motion.
Today, Holditch’s theorem is celebrated as one of the most elegant and surprising results in classical geometry, demonstrating how a simple geometric construction can uncover a universal mathematical law with lasting significance.
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