34
votes
Which theorems have Pythagoras' Theorem as a special case?
The Law of cosines is the first that comes to my mind:
$$c^{2}=a^{2}+b^{2}-2ab\cos \gamma$$
(source: Wikipedia)
If $\gamma$ is a right angle, its cosine is 0 and all that remains is Pythagoras' ...
27
votes
Which theorems have Pythagoras' Theorem as a special case?
Parseval identities in the theory of Fourier series and integrals.
21
votes
Which theorems have Pythagoras' Theorem as a special case?
The parallelogram law says that if $\mathcal{V}$ is an inner product space and $\mathbf{v},\mathbf{w} \in \mathcal{V}$, then
$$
2\|\mathbf{v}\|^2 + 2\|\mathbf{w}\|^2 = \|\mathbf{v} + \mathbf{w}\|^2 + \...
21
votes
Which theorems have Pythagoras' Theorem as a special case?
The Pythagorean theorem is a limit of the general formula for spherical or hyperbolic space: $\cos(a\sqrt{\kappa})\cos(b\sqrt{\kappa})=\cos(c\sqrt{\kappa})$, where $\kappa$ is the curvature.
18
votes
Which theorems have Pythagoras' Theorem as a special case?
So far no one has mentioned the original generalization!
Early in Euclid's Elements, the Pythagorean theorem is stated by comparing square areas:
Book I, Proposition 47: In right-angled triangles the ...
16
votes
Which theorems have Pythagoras' Theorem as a special case?
This is probably the simplest: $(a+b)^2=a^2+b^2+2ab$, if you take $a,b$ elements of some inner product vector space and $(\cdot)^2$ means inner product with itself.
15
votes
Accepted
Does a compact contractible metric space have a point that is fixed by all isometries?
There are finite groups that act smoothly on a disk without a global fixed point. You can arrange the metric to be isometric, e.g. via the Mostow-Palais embedding theorem, which equivariantly and ...
- 26.7k
11
votes
Which theorems have Pythagoras' Theorem as a special case?
One of the most attractive generalisations is de Gua's theorem: If a tetrahedron $ABCD$ is rectangular at $A$, then $$ |BCD|^2=|ABC|^2+|ACD|^2+|ABD|^2$$ where the absolute value signs denote area.
...
10
votes
Which theorems have Pythagoras' Theorem as a special case?
Pythagoras' theorem is a special case of the three point identity for Bregman distances: Let $h$ be convex and lower semi-continuous on a Banach space - further assume differentiability of $h$ for ...
10
votes
Which theorems have Pythagoras' Theorem as a special case?
$$1=\cos^2 x+\sin^2x$$ (which can be proven without using Pythagoras) holds for arbitrary $x$ in $\mathbb C$ and yields Pythagoras for real $x$.
9
votes
Which theorems have Pythagoras' Theorem as a special case?
The discrete form of the parallel axes theorem for the second moment of area for $\,n\,$ points $\,A_k\,$ with centroid $\,G\,$ and an arbitrary point $\,P\,$ is $\,\sum_{k=1}^n PA_k^2 = n \cdot PG^2 +...
8
votes
Which theorems have Pythagoras' Theorem as a special case?
The Binet-Cauchy formula says that if $A$ and $B$ are a $n\times m$ and $B$ a $m\times n$ real matrices, respectively, and for $s\subseteq\{1,\ldots ,m\}$ with $|s|=n$ we denote by $A_s$ the $n\times ...
8
votes
Which theorems have Pythagoras' Theorem as a special case?
I like to think about Pythagoras theorem as a corollary/special case of the following theorem:
Theorem: Let $X$ be a finite dimensional real Banach space such that the group of linear isometries (that ...
5
votes
Accepted
Sphere in Urysohn space
This is not true, no.
There is a proof in Section 4.4 of this old paper of mine ; the key fact is that if $B$ is an open unit ball in the Urysohn space $\mathbb U$, then $\mathbb U$ is isometric to $\...
- 1,702
5
votes
Which theorems have Pythagoras' Theorem as a special case?
In a less “higher” maths fashion :
This Numberphile video somewhat says that Pythagoras theorem is a special case of Ptolemy’s theorem which is a more general view of properties of a cyclic ...
5
votes
Accepted
Can every smooth space curve be realized as an origami curved crease?
Note: I'm revising my answer to make the argument/construction more transparent. In the previous version, I stated an existence result about flat surfaces, but didn't indicate a proof (because, at ...
- 97.6k
5
votes
Accepted
Fixed points on spherical buildings
Since spherical buildings are CAT(1), we get a fixed point if $$\mathop{\rm rad}S<\tfrac \pi 2.$$
- 37.4k
5
votes
Accepted
Symmetries of contractable subsets of $\Bbb R^n$
I posted a refined/generalized question Does a compact contractible metric space have a point that is fixed by all isometries? and received an answer that contained all essential ingredients to ...
- 9,471
4
votes
Convex sphere in R^3
The answer is positive, without the assumption on curvature (of course we do have curvature $\ge 0$, for any convex). Let $o$ be an interior point of the body $B$ whose boundary $\partial B$ is your ...
- 13.6k
4
votes
Which theorems have Pythagoras' Theorem as a special case?
There’s an “n-dimensional Pythagorean theorem” https://billcookmath.com/papers/2012-06_nD_pythag.pdf, saying that the square of the $k$-dimensional area of a $k$-dimensional parallelogram $P$ in $n$-...
4
votes
Can you perturb an inscribed polytope so all its edges grow?
OK, let me address the case of a simplex.
In fact, it follows from the `dual Kneser--Poulsen'conjecture, as stated, e.g., in this nice paper. A good thing is that the simplex has only $n+1$ vertices, ...
- 18.7k
3
votes
Accepted
Concyclic point made from Six arbitrary points
This is just a variant of Miquel’s pentagram theorem. Just apply a circle inversion in a circle centered at $P$, and you will obtain the same configuration as the pentagram theorem. I’m not sure if ...
- 46
3
votes
$O(n^{2-\epsilon})$ bound on choosing $n$ points on the hypersphere to maximize $\pm 1$ weighted sum of their $\binom{n}{2}$ inner products
This is correct.
Consider a random symmetric matrix $A=(a_{ij})$, $a_{ij}=a_{ji}=\epsilon_{ij}/2$, $a_{ii}=0$.
Let $u_1,\ldots,u_n$ be your $n$ points on the sphere $\mathcal{S}^{d-1}$. Denote $u_i=(...
- 86.9k
3
votes
$O(n^{2-\epsilon})$ bound on choosing $n$ points on the hypersphere to maximize $\pm 1$ weighted sum of their $\binom{n}{2}$ inner products
Let $\lambda$ be the largest eigenvalue of the symmetric matrix $M_{ij}$ with diagonal entries $0$ and off-diagonal entries $\epsilon_{ij}$.
Then
$$\sum_{1\leq i<j\leq n} \epsilon_{ij} v_i v_j^T =\...
- 115k
3
votes
Accepted
Functions $\mathbb{R}^2\to\mathbb{R}^2$ that preserve lines
Let $f : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a function that maps lines to lines, and suppose that there are three non-collinear points in the image of $f$.
Lemma 1: If $f[\ell_1]$ and $f[\ell_2]...
- 11.6k
3
votes
Open covering with bounded diameters
The answer is given by the Lebesgue's Covering Theorem (numbered as 1.8.20 in Engleking's book "Theory of dimensions: finite and infinite"): If $\mathcal F$ is a finite closed cover of the $...
- 32.9k
3
votes
What must a set of $n$ points in 2D space fulfill so that it is possible to connect them through tangent circles
The comment by mlk is right on the mark. The idea of this solution is basically what he wrote, but there a lot of messy details coming from the question of which direction the arcs are curving. Let ...
- 138k
2
votes
Projecting a given point onto a random $2$-dimensional plane in more than $3$ dimensions
Given $v_1$ and $v_2$ in $\mathbb{R}^d$ linearly independent, their Gram matrix $G$ is defined to be
$$ G = V^T V, $$
where $V = (v_1 v_2)$ is the $d \times 2$ matrix having $v_1$ as first column and $...
- 3,594
2
votes
What must a set of $n$ points in 2D space fulfill so that it is possible to connect them through tangent circles
(1) This paper's algorithm doesn't close smoothly at the joints, but it may
serve as a link into the circular-arc literature:
Duncan, Christian A., David Eppstein, Michael T. Goodrich, Stephen G. ...
- 144k
2
votes
Accepted
Closed almost geodesics in a Riemannian manifold
Any curve $\gamma:[a,b]\to M$ parametrized by arc length is an $\varepsilon$-geodesic for any $\varepsilon>0$.
The inequality $(1-\varepsilon)dist_M(\gamma(x),\gamma(y))\leq length[\gamma(x),\gamma(...
- 3,212
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