# Questions tagged [euclidean-geometry]

Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these.

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### Is this elementary formula for the parabolic segment new?

Recently (May 2020) a formula for the area of the parabolic segment (i.e. the region enclosed by a parabola and a line), in terms of the coefficients of the Cartesian equations, has been published by ...
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### Interpret certain expressions in terms of classical quadratic surfaces

I have two primary constraints, a linear one, \begin{equation} \label{C1} C_1=Q_1>0\land Q_2>0\land Q_3>0\land Q_1+3 Q_2+2 Q_3<1, \end{equation} and a quadratic one (incorporating $C_1$), \...
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### Comparing Euclidean norm of two normal vectors

Let $X_i$ ($i = 1,2$) be two random vectors in $\mathbb R^n$, with normal distribution with scalar covariance matrix $\sigma_i^2$ and center $\mu_i$ (in my case, $n = 2$). Is there a way to estimate ...
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### The lion and the zebras

The lion plays a deadly game against a group of $N$ zebras that takes place in the steppe (= an infinite plane). The lion starts in the origin with coordinates $(0,0)$, while the $N$ zebras may ...
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### What is a geometric construction corresponding to elliptic curve addition for Poncelet's Porism?

Background At least since Griffiths and Harris  we know that the geometric construction "draw the next tangent" appearing in Poncelet's Porism corresponds to addition of a constant in the elliptic ...
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### Distance between two polyhedra that takes incidence structure into account

Suppose that we have two polyhedra $P_1$ and $P_2$ in $\mathbb{R}^3$. I would like to define such a metric $\rho(P_1, P_2)$ that depends on several factors, but currently I don't know how to do it ...
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### Explicit construction of Kakeya sets using Perron tree

I have found many excellent notes online that illustrate how to construct a Kakeya needle set (with measure $<\varepsilon$.) Yet none of them gives full argument about the construction of a Kakeya ...
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### Two Questions on Tetrahedra and Platonic Solids [closed]

As was known to the ancients, two congruent regular tetrahedra can be inscribed in a cube and likewise 5 congruent regular tetrahedra can be inscribed in a regular dodecahedron. Is the converse to ...
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### On the crookedness of curves (Milnor's paper)

I am reading the paper (Ann Math. 1950) "on the total curvature of knots" by J. Milnor. I was trying to understand this part of the paper (here is a free access link to the paper): Let $P$ be a ...
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### Existence of $1$-Lipschitz map between triangles

Crosspost from math.SE Consider two (Euclidean) triangles $T$ and $T'$. Let's say that $T$ majorizes $T'$ if there exists a 1-Lipschitz map that sends vertices to vertices and sides to sides (for ...
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### The minimal volume of the intersection of two $\mathscr{l}_1$-ball in high dimension

We define $$B_1(r,c) = \{x\in\mathbb{R}^d : \Vert{}x-c\Vert_1 \le r\}$$ Now for arbitrary constant $r \ge s > 0$, given constant $\epsilon \in (r-s,r+s)$, considering $v \in \mathbb{R}^d$ such ...
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### Proving coincidence in Euclidean geometry by using finitely many constellations

Two polynomials $f(x)$ and $g(x)$ of degree $n$ are equal if they are equal for $n+1$ different $x$. Is anything like this true for Euclidean geometry? Say, I have three arbitrary points in the ...
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### Yau's problem: Construct a triangle given a side, an angle, and an angle bisector

In Shing-Tung Yau's autobiography The Shape of a Life, he mentions a problem that he came up with as a teenager. Suppose you know the length of one side of a triangle, one angle, and the length of ...