The euclidean-geometry tag has no usage guidance.

**18**

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**3**answers

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### Angle of a regular simplex

I find the following question embarrassing, but I have not been able to either resolve it, or to find a reference.
What is the vertex angle of a regular $n$-simplex?
Background: For a vertex $v$ ...

**26**

votes

**14**answers

6k views

### Open problems in Euclidean geometry?

Which are some (research level) open problems in Euclidean geometry ?
(Edit: I ask just out of curiosity, to understand how -and if- nowadays this is not a "dead" field yet)
I should clarify a ...

**5**

votes

**2**answers

546 views

### Can we alter the axioms of Euclidean space to have $\mathbb{Q}^3$ as a unique model?

I posted this question at math.stackexchange.com but didn't get an answer.
Motivation
Physicists are in search for a model of discrete space(-time) for a long time. So I wondered why not start with ...

**3**

votes

**0**answers

319 views

### Characterizations of Euclidean space

I posted this question at math.stackexchange.com but didn't get an answer. Is it a dumb question, eventually?
There are three ways of characterizing the abstract Euclidean space $E^n$ that are quite ...

**10**

votes

**6**answers

683 views

### Decomposing the plane into intervals

I posted this on Stack Exchange and got a lot of interest, but no answer.
A recent Missouri State problem stated that it is easy to decompose the plane into half-open intervals and asked us to do so ...

**6**

votes

**1**answer

437 views

### Using mirrors to make a non-convex polygon visible from a fixed interior point

Take a point $A$ inside a non-convex polygon $P$. Is it always possible to place a finite set of mirrors given by straight segments (not necessarily along the boundary of $P$, any position inside $P$ ...

**3**

votes

**1**answer

487 views

### Is always possible to slice a pizza in a fair way

Given a pizza, represented by the unit disk $D_1(0,0)=\{(x,y)\in\mathbb{R}^2\mid \|(x,y)\|\leqslant 1\}$, and given $N$ slices of $r$-pepperoni, represented by disks ...

**7**

votes

**1**answer

1k views

### Point cloud that maximizes the minimum pairwise distance in Euclidean space

I am interested finding the collection of points in the Euclidean space that has the maximal minimal pairwise distance subject to an average norm constraint, that is, how to maximize
$min_{i \neq j} ...

**5**

votes

**1**answer

557 views

### Elementary problem about triangles inside a convex polygon

Let P be a convex polygon with area A(P), and to each side of P, attach the largest area triangle possible that lies entirely within P. Must the sum S(P) of the areas of these triangles always satisfy ...

**9**

votes

**3**answers

788 views

### Neusis constructions

Is there some simple description of which complex numbers are "constructible" with straightedge and compass and neusis?
See http://en.wikipedia.org/wiki/Constructible_number and ...

**11**

votes

**11**answers

6k views

### Theorems in Euclidean geometry with attractive proofs using more advanced methods

The butterfly theorem is notoriously tricky to prove using only "high-school geometry" but it can be proved elegantly once you think in terms of projective geometry, as explained in Ruelle's book The ...

**18**

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**6**answers

3k views

### Euclid with Birkhoff

I'm looking for an short and elementary book which does Euclidean geomety with Birkhoff's axioms.
It would be best if it would also include some topics in projective (and/or) hyperbolic geometry.
...

**5**

votes

**1**answer

3k views

### Maximum number of mutually equidistant points in an n-dimensional Euclidean space is (n+1). Proof? [closed]

How to prove that the maximum number of mutually equidistant points in an n-dimensional Euclidean space is (n+1)?

**8**

votes

**1**answer

586 views

### Sticks and thread

In this recent question Math puzzles for dinner we had a nice time as we were asked to provide new maths puzzles for dinners. I suggested the following:
Given three equal sticks, and some
...

**53**

votes

**9**answers

6k views

### Geometric proof of the Vandermonde determinant?

The Vandermonde matrix is the $n\times n$ matrix whose $(i,j)$-th component is $x_j^{i-1}$, where the $x_j$ are indeterminates. It is well known that the determinant of this matrix is $$\prod_{1\leq ...

**13**

votes

**1**answer

1k views

### Origami Constructions: Intersecting two Circles

It is well known that every construction that can be performed with compass and straightedge alone can also be performed using origami, see:
R. Geretschlager. Euclidean Constructions and the Geometry ...

**2**

votes

**1**answer

491 views

### Reference: Countable Models of (Non-)Euclidean Geometry

Has there been a survey written on the model theory of first-order (non-)Euclidean geometry in the spirit of Hilbert and Tarski? I'm especially interested in two aspects of the model theory:
...

**16**

votes

**6**answers

1k views

### Tetrahedra with prescribed face angles

I am looking for an analogue for the following 2 dimensional fact:
Given 3 angles $\alpha,\beta,\gamma\in (0;\pi)$ there is always a triangle with these prescribed angles. It is ...

**2**

votes

**1**answer

1k views

### Finding a minimum bounding sphere for a frustum

I have a frustum (truncated pyramid defined by six planes) and I need to compute a bounding sphere for this frustum that's as small as possible.
I can choose the centre of the sphere to be right in ...

**5**

votes

**1**answer

588 views

### Malfatti Circles - Limiting point

"Three circles packed inside a triangle such that each is tangent to the other two and to two sides of the triangle are known as Malfatti circles" (for a brief historical account on this topic, see ...

**7**

votes

**9**answers

2k views

### Comprehensive reference for synthetic euclidean geometry

Euclidean geometry is a special case of the theory of Hilbert spaces; but in order to convince small children of basic facts, e.g. that the line segments from each of the vertices of a triangle to the ...

**6**

votes

**4**answers

845 views

### How to compute the average distance till intersection within a triangle in R^2?

Lots of simple questions because I am a noob.
You are given 3 points in R^2; A, B, C forming a triangle with area > 0. You pick an arbitrary point inside ABC and an arbitrary direction. After some ...

**4**

votes

**7**answers

2k views

### Side-Angle-Side Congruence and the Parallel Postulate

Is there a link between the side-angle-side congruence of triangles and the parallel postulate? Specifically, does it follow from Euclid's first four axioms alone? In fact, does it even follow from ...

**5**

votes

**2**answers

427 views

### Historical question re: ellipses obtained by certain geometrical constructions

I am a faculty member in the Forensic Science Program at PennState (UP). I am trying to obtain information of a historical nature concerning two closely related topics. I seek historical references ...

**14**

votes

**1**answer

3k views

### spiral of Theodorus

A long time ago when I was in college I read about making a spiral out of right triangles with sides 1 and $\sqrt{N}$. (A google search seems to indicate that this is called the Spiral of Theodorus.)
...

**2**

votes

**1**answer

945 views

### How to find the Fermat Point using the construction of the tangent to ellipse?

Be done the triangle ABC, it is known the method to finding the point Q that minimises the sum QA+QB+QC among all points Q in the plane (The Fermat point).
I want a hint for solving this problem using ...

**37**

votes

**14**answers

40k views

### If you break a stick at two points chosen uniformly, the probability the three resulting sticks form a triangle is 1/4. Is there a nice proof of this?

There is a standard problem in elementary probability that goes as follows. Consider a stick of length 1. Pick two points uniformly at random on the stick, and break the stick at those points. What ...