# 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.

354
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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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39 views

### 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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**1**answer

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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 [1] 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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votes

**1**answer

67 views

### 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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votes

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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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### The abc-conjecture as an inequality for inner-products?

The abc-conjecture is:
For every $\epsilon > 0$ there exists $K_{\epsilon}$ such that for all natural numbers $a \neq b$ we have:
$$ \frac{a+b}{\gcd(a,b)}\,\ <\,\ K_{\epsilon}\cdot \text{rad}\...

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### Logical completeness of Hilbert system of axioms

This is really a question about references. The entry in Russian Wikipedia about Hilbert's axioms states, in particular, that completeness of Hilbert's system was proven by Tarski in 1951. The ...

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### Euclid Book 1 Proposition 4 [closed]

In Euclid's The Elements, Book 1, Proposition 4, he makes the assumption that one can create an angle between two lines and then construct the same angle from two different lines. I do not see ...

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367 views

### addition theorems for hypersine

I learned from Wolfram MathWorld about hypersine, as being a dimensional analog trig function for hypersolid angles. There it is being defined by
The hypersine ($n$-dimensional sine function) is a ...

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### Does any real projective plane incidence theorem follow from axioms?

Is it known whether any projective geometry statement that holds true in the real projective plane (equivalently, can be deduced from Hilbert axioms) follows from the standard projective axiomatics?
...

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### The $r$-dimensional volume of the Minkowski sum of $n$ ($n\geq r$) line sets

Let $n$ line sets be $\mathcal{S}_i=\{a\mathbf{h}_i:0 \le a \le 1\}$, for $1 \le i \le n$, where $\{\mathbf{h}_1,\cdots,\mathbf{h}_n\}$ is a vector group of rank $r$ in the $r$-dimensional Euclidean ...

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### For regular tetrahedron $ABCD$ with center $O$, and $\overrightarrow{NO}=-3\overrightarrow{MO}$, is $NA+NB+NC+ND\geq MA+MB+MC+MD$?

Let $ABCD$ be a regular tetrahedron with center $O.$ Consider two points $M,N,$ such that $\overrightarrow{NO}=-3\overrightarrow{MO}.$ Prove or disprove that
$$NA+NB+NC+ND\geq MA+MB+MC+MD$$
I ...

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votes

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397 views

### Distance between point inside a triangle and its vertices [closed]

How to determine the distance between an arbitrary point inside a triangle and its vertices if side lengths are given. Is there any correlation between these distances or their sum and the lengths of ...

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### Strange formula for area of a convex polygon

Consider a convex $n-$gon in $\mathbb{R}^2$ with sides contained in the lines $y=k_ix+b_i, 1\leq i\leq n.$ Then its area equals to
$$
S=\frac{1}{2}\sum_{i=1}^{n} \frac{(b_{i+1}-b_i)^2}{k_{i+1}-k_i}.
$$...

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### Total length of a set with the same projections as a square

Take some convex polygon $P$. I'm mostly asking about the unit square, but would also appreciate thoughts on general polygons. We want to take a family of line segments inside $P$ that have the same ...

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### Collections of points maximally spaced with respect to one another

The icosahedron and dodecahedron are well known to share symmetry groups. This partially accounts for the fact that one can form a type of compound of the two where each of the vertices in the ...

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### Covering the disk with a family of infinite total measure - the convex sequel

Let $(U_n)_n$ be an arbitrary sequence of open convex subsets of the unit disk $D(0,1)\subseteq \mathbb{R}^2$ s.t. $\sum_{n=0}^\infty \lambda(U_n)=\infty$ (where $\lambda$ is the Lebesgue measure). ...

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### Covering the disk with a family of infinite total measure

Let $(U_n)_n$ be an arbitrary sequence of open subsets of the unit disk $D(0,1)\subseteq \mathbb{R}^2$ s.t. $\sum_{n=0}^\infty \lambda(U_n)=\infty$ (where $\lambda$ is the Lebesgue measure). Does ...

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### Open problems from antiquity solved with analytic geometry

E. T. Bell wrote in Men of Mathematics:
Though the idea behind it all is childishly simple, yet the method of analytic geometry is so powerful that very ordinary boys of seventeen can use it to prove
...

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768 views

### Euler's rotation theorem revisited - Elementary geometric proofs

This is a very elementary topic but I thought it might be worth giving it a try here, I would be very interested in any comments - I originally posted it to Maths SE.
Euler's Rotation Theorem, proved ...

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281 views

### Can set-like objects obeying ZFC be constructed in Euclidean geometry?

Is it possible to base set theory on Euclidean geometry, by carefully defining various notions from set theory in terms of geometric objects, so that the ZFC axioms can be shown to hold for them? ...

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### On the area-perimeter ratio of a convex limited set

(Previously asked on MSE)
Let $C\subset \mathbb{R}^2$ be a convex limited set. We define the average radius of $C$ as
$$a_C=\frac{\int_{v\in C}d(v,C)dxdy}{A(C)}$$
Where $d(v,C)$ is the distance ...

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votes

**1**answer

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### Frégier and Frégier's Theorem

A curious and interesting gem is Frégier's theorem, quoted here from David Wells:
Choose any point $P$ on a conic, and make it the vertex of a right
angle which rotates about $P$. Then the ...

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votes

**5**answers

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### Trigonometry / Euclidean Geometry for natural numbers?

Let $d(a,b) = 1 - \frac{2\gcd(a,b)^3}{ab(a+b)}$ be a metric on natural numbers without $0$.
The metric space $X = \{x_0,x_1,\cdots,x_n\},n>2$ is isometric embeddable in $\mathbb{R}^n$ if and only ...

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### A Erdős–Mordell Like inequality

Ono's inequality is true for acute triangle but false with general triangles. The inequality as follows is false with general triangls but I think it true with acute triangle (follows answer by Fedor ...

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### Mordell's theorem on rational quadrilaterals

Mordell proved that for any epsilon and any quadrilateral in the Euclidean plane there is an epsilon-close quadrilateral whose sides and diagonal are rational. Does this break down for five points in ...

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### Trilinear polarity from AG perspective

Consider a triangle $ABC$ in the projective plane $\mathbb{P}^2.$ For a point $p \in \mathbb{P}^2$ one can define its trilinear polar line $t(p)$ (see here). This defines a birational map to the dual ...

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### Which tetrahedra are scissor congruent to a cube?

Question: Which Euclidean tetrahedra are scissor congruent to cubes?
Consider a Euclidean tetrahedron $T$ in $\mathbb{R}^3$ with edge lengths $l_1,\ldots, l_6$ and dihedral angles $\alpha_1,\ldots, \...

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### Which polygons can be turned inside out by a smooth deformation?

Take a non-degenerate polygon with side lengths $\{a_1,\dots,a_n\}$ in a convex configuration. What is the condition on the $a_i$'s so that the polygon can be turned inside out by a continuous motion ...

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### «Euclidean» local systems

The moduli space of G-local systems on a surface is a fundamental object in mathematics. The cases $G=SU(2)$ and $G=SL_2(\mathbb{R})$ are of particular interest. Consider the group $E$ of isometries ...

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### Problem Understanding Euclid Book 10 Proposition 1 [closed]

this is embarrassing, but I am having trouble reading through Proposition 1 of Book 10 of Euclid's elements. I'm struggling with Euclid's terminology and don't have a clear picture of what divisions ...

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### Automatically solving olympiad geometry problems

Warning: I am only an amateur in the foundations of mathematics.
My understanding of this Wikipedia page about Tarski's axiomatization of plane geometry (and especially the discussion about ...

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### Trajectory definition using constrained projections on unknown surface

In $3D$ space where the $Z$ axis is up-down, I have the following:
A static camera $A$ at $(x_a, y_a, z_a)$;
A laser pointer $B$ at $(x_a, y_a, z_a + b)$ which can yaw or pitch by $1^\circ$ at a ...

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### A curious relation between angles and lengths of edges of a tetrahedron

Consider a Euclidean tetrahedron with lengths of edges
$$
l_{12}, l_{13}, l_{14}, l_{23}, l_{24}, l_{34}
$$
and dihedral angles
$$
\alpha_{12}, \alpha_{13}, \alpha_{14},
\alpha_{23}, \alpha_{24}, \...

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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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140 views

### 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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votes

**1**answer

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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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356 views

### 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 ...

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329 views

### Number of Reflections in a Circle between Two Points

For my research I am interested in the transmission characteristics between a transmitter (Tx) and a receiver (Rx) situated in a circular room. In particular, it is important for me to know the number ...

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### Reverse Loomis-Whitney Inequality for functions

I was wondering if the reverse Loomis-Whitney inequality holds for general functions:
Let $n\geq 2$. Let $(X_i,\mu_i)$, $1\leq i\leq n$ be measure spaces. Write $x=(x_1,\dots,x_n)$ and for each $1\...

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### Reference request: Oldest (non-analytic) geometry books with (unsolved) exercises?

Per the title, what are some of the oldest (non-analytic) geometry books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there.

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### What curve of positive curvature minimizes distance from the origin, given length and total curvature?

Let $\textit{F}$ be the family of $C^1$ curves in $\mathbb{R}^2$ of fixed length $\bar{l}$ and fixed tangent's turning angle $\bar{k}$.
What are the curves of positive curvature in $\textit{F}$ ...

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### History of the Taxonomy of Quadrilaterals

Question:
how did the classification of quadrilaterals come into being? Was there a single major contributor who coined terms like "rectangle", "square", "trapez/ium/oid", "kite", "deltoid", ...

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votes

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### Procedure to determine the equality of numbers in rationals plus square root

Consider the set $\mathbb{Q}^\sqrt{}$ of real numbers that can be constructed by applying finitely many of the five operations $+$, $-$, $\cdot$, $/$ and $\sqrt{}$ to a positive rational number. ...

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votes

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### Correlation between the first and a random position of an ergodic bit sequence

Edit: Since the geometric approach did not work, I try now another approach: phrasing the problem as a quadratic programme.
Probabilistic version.
Let $x=(x_1,x_2, \ldots) $ be an ergodic random ...

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### Symmetry group and irreducible representation

Let $S$ be a bounded geometric shape in the Euclidean space $E=\mathbb{R^n}$. Assume that the origin of $E$ is a fixed point of every element of the symmetry group $G(S)$ of $S$, and assume that $G(S) ...