Holditch's theorem

Holditch's theorem states that when a fixed-length chord moves around a closed curve, a point dividing it into lengths a and b traces a curve whose enclosed area differs from the original by πab.

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Chord: A line segment joining two points on a curve. The theorem begins with a chord whose length stays fixed as its endpoints move.
Hugh Philip Holditch: A British mathematician and Anglican clergyman associated with the theorem bearing his name. Holditch published the result that gives this area difference for a moving chord.
Closed curve: A continuous curve whose endpoint coincides with its starting point. The chord’s endpoints travel around a closed boundary in the theorem’s setup.
Linkage mechanism: A system of rigid links connected by joints to produce constrained motion. A rigid link can model the fixed chord and the motion of its interior point.
Area: A measure of the two-dimensional extent enclosed by a plane figure. Holditch’s theorem concerns the difference between the areas enclosed by two traced curves.
The Lady's and Gentleman's Diary: A British annual miscellany that published mathematical problems, solutions, and essays. Holditch’s theorem appeared in this recreational-mathematics publication.
Curve parametrization: A function that assigns coordinates to points on a curve using a parameter. Parametrizing the boundary describes how the chord endpoints move.
Kinematics: The study of motion without regard to the forces that cause it. The theorem’s construction is a geometric description of constrained planar motion.
Rolling curve: A curve generated by a point attached to a curve that rolls along another curve without slipping. The moving chord is carried around the boundary, producing a related family of traced curves.
Nineteenth-century geometry: The study of geometric structures and transformations developed during the nineteenth century. The theorem belongs to an era when geometric properties of moving figures drew sustained attention.