共焦椭圆和双曲线形成的四边形的对角线长度相等。

hbghlyj

由两个共焦椭圆和两个共焦双曲线形成的曲边四边形的对角线长度相等。

资料

Ivory's theorem
Ivory’s Theorem revisited
Recalling Ivory's Theorem
Dictionary of Mathematical Eponymy: Ivory's Theorem
Ivory's theorem in hyperbolic spaces - Institute of Geometry
ivory's theorem - in the minkowski plane
Incircular nets and confocal conics
Fun Problems in Geometry and Beyond16. Izmestiev I., Tabachnikov S., Ivory's theorem revisited, J. Integrable Syst. 2 (2017), xyx006, 36 pages, arXiv:1610.01384.
IVORY. In phr. Ivory's theorem, in Math., on the correspondences produced by confocal ellipsoids (sometimes called ivorys), a basic proposition in the theory of attraction. [Formulated in 1809 by James Ivory (1765–1842) of Dundee, Professor of Mathematics at the Royal Military College.]
Video-Ivory's and Arnold's theorems on the sphere and in the hyperbolic space
Mathematical Aspects of Classical and Celestial Mechanics 2007 · ‎Mathematics However, the question whether the generalized Sundman theorem is valid for the ... in the exterior are ellipsoids confocal to the layer (Ivory's theorem).
A Treatise on Analytical Statics - Page 111 Edward John Routh · 1922 Ivory's theorem . To find the attraction of a solid . homogeneous ellipsoid at an external point P whose coordinates are E , n ' , s .
The Messenger of Mathematics 1903 · ‎Mathematics ON A GENERALISATION OF IVORY'S THEOREM . By A. L. Dixon , Merton College , Oxford . IN Cayley's theory of distance ( sixth memoir on quantics , Coll .
The Universe of Quadrics - Page 306 Ivory's Theorem This famous theorem can be directly extracted from Theorem 7.2.1. Below, we combine it with Lemma 7.2.6. Theorem 7.2.2 (Ivory's Theorem in ...
the values of simplicity and generality in chasles's geometrical Ivory's theorem (my figure). In his 1838 memoir on the attraction of ellipsoids, Chasles was able to use this theorem to derive a much shorter
A GENERALIZATION OF IVORY'S THEOREM. FROM THE STANDPOINT OF CONFORMAL MAPPING. by HIROSHI HARUKI · 1973
A Generalization of Ivory’s Theorem
The Universe of Conics: From the ancient Greeks to 21st century developments 477页
Remarks on Joachimsthal Integral and Poritsky Property 8. Izmestiev, I., Tabachnikov, S.: Ivory's theorem revisited.
Between Rigidity and Flexibility 3.2 Flipping bipartite frameworks
Given: Net of confocal conics in the Euclidean plane $E^2$: Ivory’s Theorem: $X_1X'_2 = X'_1X_2$
H. Stachel, J. Wallner, Ivory’s theorem in hyperbolic spaces, Sibirsk. Mat. Zh..pdf (415.91 KB, Downloads: 29)

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hbghlyj

设 $ E(a) $ 为焦点为 $ F_{1}=(c,0),\;F_{2}=(-c,0) $ 的椭圆，其方程为
$${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{a^{2}-c^{2}}}=1\ ,\quad a>c>0\ $$
和 $ H(u) $ 为共焦双曲线，其方程为
$$ {\frac {x^{2}}{u^{2}}}+{\frac {y^{2}}{u^{2}-c^{2}}}=1\ ,\quad c>u\ . $$
计算 $ E(a) $ 和 $ H(u) $ 的交点，得到四个点：
$$ \left(\pm {\frac {au}{c}},\;\pm {\frac {\sqrt {(a^{2}-c^{2})(c^{2}-u^{2})}}{c}}\right) $$

hbghlyj

为了简化计算，不失一般性地设 $ c=1 $（任何其他共焦网格都可以通过均匀缩放得到），并在椭圆和双曲线的四个交点中选择正象限的点（其他符号组合在类似计算后得到相同结果）。

设 $ E(a_{1}),E(a_{2}) $ 为两个共焦椭圆，$ H(u_{1}),H(u_{2}) $ 为两个具有相同焦点的共焦双曲线。网格矩形的四个点的对角线为
\begin{aligned}P_{11}&=\left(a_{1}u_{1},\;{\sqrt {(a_{1}^{2}-1)(1-u_{1}^{2})}}\right),&P_{22}&=\left(a_{2}u_{2},\;{\sqrt {(a_{2}^{2}-1)(1-u_{2}^{2})}}\right),\\[5mu]P_{12}&=\left(a_{1}u_{2},\;{\sqrt {(a_{1}^{2}-1)(1-u_{2}^{2})}}\right),&P_{21}&=\left(a_{2}u_{1},\;{\sqrt {(a_{2}^{2}-1)(1-u_{1}^{2})}}\right)\end{aligned}为：
\begin{aligned}|P_{11}P_{22}|^{2}&=(a_{2}u_{2}-a_{1}u_{1})^{2}+\left({\sqrt {(a_{2}^{2}-1)(1-u_{2}^{2})}}-{\sqrt {(a_{1}^{2}-1)(1-u_{1}^{2})}}\right)^{2}\\[5mu]&=a_{1}^{2}+a_{2}^{2}+u_{1}^{2}+u_{2}^{2}-2\left(1+a_{1}a_{2}u_{1}u_{2}+{\sqrt {(a_{1}^{2}-1)(a_{2}^{2}-1)(1-u_{1}^{2})(1-u_{2}^{2})}}\right)\end{aligned}最后的表达式在交换 $ u_{1}\leftrightarrow u_{2} $ 时不变。正是这种交换导致了 $ |P_{1\color {red}2}P_{2\color {red}1}|^{2} $。因此 $ |P_{11}P_{22}|=|P_{12}P_{21}| $

Ivory 甚至证明了他的定理的三维版本（见 Blaschke，第 111 页）：
对于由共焦二次曲面形成的三维矩形长方体，连接相对点的对角线长度相等。