Holditch定理

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在凸闭合曲线内有固定长度的弦$AB$，弦上的一点$P$到$A$的距离为$c$，到$B$的距离为$c'$.
则$P$的轨迹是一条闭合曲线，它封闭的面积等于原始曲线的面积减 $π cc'$.

Wikipedia
GEOMETRICAL THEOREM. By Rev. Hamnet Holditch. The Quarterly Journal of Pure and Applied Mathematics, 2: 38
If a chord of a closed curve, of constant length $c+c^{\prime}$, be divided into two parts of lengths $c, c^{\prime}$ respectively, the difference between the areas of the closed curve, and of the locus of the dividing point, will be $\pi c c^{\prime}$.

Solution

Let $A B$ be the chord in any position; $P$ the dividing point, so that $A P=c, B P=c^{\prime}$; let $Q$ be the point in which the chord intersects its consecutive position; let $[A]$ be the area of the given curve, $[P],[Q]$, those of the loci of $P, Q$, respectively; $A Q=r, B Q=c+c^{\prime}-r$.
Then$$\tag1[A]-[Q]=\frac{1}{2} \int_0^{2 \pi} r^2 d \theta$$but also$$[A]-[Q]=\frac{1}{2} \int_0^{2 \pi} (c+c'-r)^2 d \theta$$therefore$$\frac{1}{2} \int_0^{2 \pi} r^2 d \theta=\frac{1}{2} \int_0^{2 \pi} (c+c'-r)^2 d \theta$$or$$\left(c+c^{\prime}\right) \int_0^{2 \pi} r d \theta=\frac{1}{2} \int_0^{2 \pi}\left(c+c^{\prime}\right)^2 d \theta$$therefore$$\int_0^{2\pi} r d \theta=\pi\left(c+c^{\prime}\right) \tag2$$
Also $[P]-[Q]=\frac{1}{2} \int_0^{2 \pi}(c-r)^2 d \theta$,
therefore, by (1), $\begin{aligned}[t]
[A]-[P]&=\frac{1}{2} \int_0^{2 \pi}\left(2 c r-c^2\right) d \theta\\&=c \int_0^{2 \pi} r d \theta-\pi c^2\\&=\pi c\left(c+c^{\prime}\right)-\pi c^2, \mathrm{by}(2)\\&=\pi c c^{\prime}\end{aligned}$

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An Elementary Treatise on the Integral Calculus, containing Applications to Plane Curves and Surfaces, with numerous Examples. By B. Williamson, F.R.S.(London: Longmans, 1880.)
扫描版 Google Books 链接 (可下载)
书评: Of a third edition we need only remark that it is a carefully revised issue of the second, and point out the few important additions that have been made. In the discussion of Frullani's theorem(§119), a simple shape of the formula:, due to Mr. E. B. Elliott, is given, and reference made to other articles on multiple definite integrals by the same gentleman (and by Mr. Leudesdorf) in the Educational Times (1875) and in the Proceedings of the London Mathematical Society, 1876-7. A new article (119a) gives a proof of a simple character, by Zolotareff, of the remainder in Lagrange's series. § 147 contains a remarkable extension of Holditch's theorem, due to Mr. Elliott (Mess. of Math. February, 1878), and §147a gives the "singularly elegant" theorem discussed by Mr. Kempe (Mess. of Math. July, 1878), to which reference is made in Prof. Minchin's letter in Nature (December 23, 188o), in which he proves these theorems from other considerations. Various insertions of a minor character increase the volume by more than twenty pages. A good feature of the present edition is an index at the end of the work.

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