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 . The theorem was published in 1858 by Rev. Hamnet Holditch.[1][2] 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.[3]
Observations
The theorem is included as one of
Clifford Pickover's 250 milestones in the history of mathematics.[1]
Some peculiarities of the theorem include that the area formula is independent of both the shape and the size of the original curve, and that the area formula is the same as for that of the area of an Caius College, Cambridge
.
Proof
Assume the outer curve is a circle of area 1. It is trivial to compute the area of the inner curve (also a circle) is . The general (2-D) case follows.
Extensions
Bromanacute angle is traversed. Nevertheless, the generalization shows that if the chord is shorter than any of the triangle's altitudes, and is short enough that the traced locus is a simple curve, Holditch's formula for the in-between area is still correct (and remains so if the triangle is replaced by any convex polygon with a short enough chord). However, other cases result in different formulas.
References
- ^ ISBN 978-1-4027-5796-9
- ^ Holditch, Rev. Hamnet (1858), "Geometrical theorem", The Quarterly Journal of Pure and Applied Mathematics, 2: 38
- ^ MR 0618595
Further reading
- B. Williamson, FRS, An elementary treatise on the integral calculus : containing applications to plane curves and surfaces, with numerous examples (Longmans, Green, London, 1875; 2nd 1877; 3rd 1880; 4th 1884; 5th 1888; 6th 1891; 7th 1896; 8th 1906; 1912, 1916, 1918, 1926); Ist 1875, pp. 192–193, with citation of Holditch's Prize Question set in The Lady's and Gentleman's Diary for 1857 (appearing in late 1856), with extension by Woolhouse in the issue for 1858; 5th 1888; 8th 1906 pp. 206–211
- J. Edwards, A Treatise on the Integral Calculus with Applications, Examples and Problems, Vol. 1 (Macmillan, London, 1921), Chap. XV, esp. Sections 478, 481–491, 496 (see also Chap. XIX for instantaneous centers, roulettes and glisettes); expounds and references extensions due to Woolhouse, Elliott, Leudesdorf, Kempe, drawing on the earlier book of Williamson.
- Kılıç, Erol; Keleş, Sadık (1994), "On Holditch's theorem and polar inertia momentum", Communications Faculty of Sciences University of Ankara, Series A1: Mathematics and Statistics, 43 (1–2): 41–47 (1996), MR 1404786
- Cooker, Mark J. (July 1998), "An extension of Holditch's theorem on the area within a closed curve", S2CID 123443685
- Cooker, Mark J. (March 1999), "On sweeping out an area", S2CID 125103358
- MR 3024916
External links
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