Holditch's theorem

In plane geometry, Holditch's theorem states that if a chord of fixed length is allowed to rotate inside a convex closed curve, then the locus of a point on the chord a distance p from one end and a distance q from the other is a closed curve whose enclosed area is less than that of the original curve by ${\displaystyle \pi pq}$. The theorem was published in 1858 by Rev. Hamnet Holditch. While not mentioned by Holditch, the proof of the theorem requires an assumption that the chord be short enough that the traced locus is a simple closed curve.