The euclidean-geometry tag has no wiki summary.

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### Is Euclid dead? [closed]

Apparently Euclid died about 2,300 years ago (actually 2,288 to be more precise), but the title of the question refers to the rallying cry of Dieudonné, "A bas Euclide! Mort aux triangles!" ...

**53**

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**9**answers

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### 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 ...

**36**

votes

**14**answers

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### 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 ...

**35**

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**1**answer

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### Probability that a stick randomly broken in five places can form a tetrahedron

The following problem was brought to my attention by a doctoral dissertation on Mathematics Education, but - as far as I know - the solution remains unknown.
I have already asked this question on ...

**28**

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**2**answers

648 views

### Term for “uncheckable constructions”

Is there a term for "uncheckable geometric constructions"?
Say, Angle Trisection and Doubling the Cube are checkable;
i.e., if the answer is given one can do finite Compass-and-straightedge ...

**25**

votes

**3**answers

977 views

### About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals

Question : Letting $S{^\prime}$ be the area of the inner pentagon made by the five diagonals of a convex pentagon whose area is $S$, then find the max of $\frac{S^\prime}{S}$.
...

**24**

votes

**1**answer

594 views

### The Eyeball Theorem generalized

I have not seen the 2D Eyeball Theorem—that tangents from the centers of two circles, each encompassing the other, intersect each circle in the same segment length—generalized to higher ...

**23**

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**3**answers

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### An elementary problem in Euclidean geometry [closed]

This problem was first put to me by Luke Pebody (who did not know the answer at the time) and after some work I am yet to find a proof or counterexample. I would be grateful of any insights.
Call a ...

**22**

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**3**answers

717 views

### Tetrahedron insphere iteration

I know that iterating the following incircle construction approaches an equilateral triangle in the limit:
Starting with any triangle $T$, one forms $T'$ by connecting ...

**20**

votes

**4**answers

902 views

### Pinball on the infinite plane

Imagine pinball on the infinite plane, with every lattice
point $\mathbb{Z}^2$ a point pin.
The ball has radius $r < \frac{1}{2}$.
It starts just touching the origin pin, and shoots off at angle ...

**18**

votes

**6**answers

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### Does every ellipse inside a tetrahedron inside a ball fit in a triangle inside the ball?

In three-dimensional euclidean space, consider the closed unit ball $B$. Let $T$ be a tetrahedron, and $E$ an ellipse, with $E \subset T \subset B$. Does there necessarily exist a triangle $T'$ with ...

**17**

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**4**answers

857 views

### Metrics for lines in $\mathbb{R}^3$?

I seek a metric $d(\cdot,\cdot)$ between pairs of (infinite) lines in $\mathbb{R}^3$.
Let $s$ be the minimum distance between a pair of lines $L_1$ and $L_2$.
Ideally, I would like these properties:
...

**17**

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**9**answers

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### 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 ...

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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 $v$ ...

**16**

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**8**answers

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### Determine if circle is covered by some set of other circles

Suppose you have a set of circles $\mathcal{C} = \{ C_1, \ldots, C_n \}$ each with a fixed radius $r$ but varying centre coordinates. Next, you are given a new circle $C_{n+1}$ with the same radius ...

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**6**answers

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### 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 ...

**15**

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**1**answer

444 views

### Is the perimeter of an ellipse with integer axes irrational?

Let $Q$ be an ellipse with integer-length axes $a$ and $b$:
$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \;.$$
The perimeter of $Q$ is given by the complete elliptic integral of the 2nd kind, $E(\;)$:
$4 ...

**14**

votes

**2**answers

945 views

### Why do all incidence theorems follow from Pappus' theorem?

In Hilbert and Cohn-Vossen's ``Geometry and the Imagination,"
they state in the last paragraph of Chapter 20 that "Any
theorems concerned solely with incidence relations in the
[Euclidean ...

**14**

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**3**answers

1k views

### Can Morley's theorem be generalized?

Morley's theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle.
In a talk some years ago, David Rusin made the ...

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**1**answer

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### 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.)
...

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**5**answers

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### Definition of area

I am looking for an attractive, but rigorous definition of area;
say in Euclidean plane. Probably there is no short definition. It is OK to make it even longer, but can it be built from useful parts ...

**13**

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**1**answer

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### Smallest regular simplex containing the unit cube in $R^n$

What is the length $e_n$ of the edge of the smallest $n$-dimensional regular simplex $S_n$ containing the $n$-dimensional unit cube $Q_n$?
In particular, is there $n$ such that ...

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**1**answer

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### 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 ...

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**6**answers

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### 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.
...

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**4**answers

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### The Ramanujan Problems.

I originally thought of asking this question at the Mathematics Stackexchange, but then I decided that I'd have a better chance of a good discussion here.
In the Wikipedia page on Ramanujan, there is ...

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**2**answers

641 views

### Right triangle with edge lengths equal to regular unit polygon edge lengths

This question came up naturally recently from a blog post of John Baez. There is an observation of Euclid that edges of a pentagon, hexagon, and decagon inscribed in a unit circle form the edges of a ...

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**1**answer

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### What is the limit of the “knight” distance on finer and finer chessboards?

Consider the "infinite chessboard" on the plane. Think of it as the lattice $X_1:=\mathbb{Z}^2$, and also finer chessboards $X_n$ corresponding to $\frac{1}{n}\cdot \mathbb{Z}^2$, $n\geq 1$. Given two ...

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**1**answer

517 views

### Ratio of circumscribed/inscribed $(n{-}1)$-gons

As a discrete analog of the MO question,
"Löwner-John Ellipsoid: incribed and circumscribed,"
I've been wondering what might be the maximum ratio
of this quantity?
Let $P$ be a convex polygon of $n$ ...

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**0**answers

421 views

### Does every connected set that is not a line segment cross some dyadic square?

A dyadic square is a subset of $R^2$ of the form $x + 2^{-n} [0,1]^2$ with $x \in 2^{-m} Z^2$, for integers $m,n \geq 0$. We say that a set $A$ crosses a square $S$ if there exists a connected subset ...

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### 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 ...

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**3**answers

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### Efficient visibility blockers in Polya's orchard problem

Polya's orchard problem asks for which radius $\rho$ of trees at each lattice point within a distance $R$ of the origin block all lines of sight to the exterior of the orchard.
...

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**1**answer

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### Heronian triangle with two sides that are prime

Can any prime number form a Heronian triangle with a second prime as another side? I cannot find a second prime to form a Heronian triangle with either 23 or 167. I have checked up to the 10^7th prime ...

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**2**answers

538 views

### Maximum thickness of three linked Euclidean solid tori

Consider three circles of radius 1 in $\mathbb{R}^3$, linked with each other in the same arrangement as three fibers of the Hopf fibration. Now thicken the circles up into non-overlapping standard ...

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**2**answers

463 views

### Helix translates as geodesics

I believe one can fill $\mathbb{R}^3$ with
horizontal translates of the helix
$(\cos t, \sin t, t) \;,\; t \in \mathbb{R}$,
so that every point of $\mathbb{R}^3$
lies in exactly one helix.
I am ...

**10**

votes

**6**answers

670 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 ...

**10**

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**3**answers

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### (1-Lipschitz) + (length-preserving) = isometry

I am looking for an elementary way to prove the following theorem.
Theorem. Let $\alpha$ and $\beta$ be two simple convex closed curves in $\mathbb R^2$.
Assume
$$\mathop{\rm length} ...

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**3**answers

372 views

### The intersection of $n$ cylinders in $3$-dimensional space

A standard question in vector calculus is to calculate the volume of the shape carved out by the intersection of $2$ or $3$ perpendicular cylinders of radius $1$ in three dimensional space. Such ...

**10**

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**1**answer

514 views

### Polygons uniquely inducing arrangements

A beautiful, relatively recent result is that,
Every simple arrangement $\cal{A}$ of $n$ lines in the plane is induced by a simple $n$-gon $P$.
In a simple arrangement, every pair of lines ...

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**0**answers

347 views

### What is the field generated by an Archimedean Spiral?

Let $F \subset \mathbb R$ be the set of lengths of line segments that one can construct, starting from the points $(0,0)$ and $(1,0)$, using a straightedge, compass, and an Archimedean spiral - the ...

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votes

**4**answers

567 views

### Applications of n-dimensional crystallographic groups

I would like to know what are the applications of the theory of $n$-dimensional crystallographic groups (aka space groups)
1) in mathematics
2) outside of mathematics,
besides the applications to ...

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votes

**4**answers

379 views

### ten concurrent lines

this relates to a question asked at MSE. i was able to find an answer using complex numbers.
here is the question: there are five points on a circle. take any three points, through the centroid of ...

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**3**answers

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### 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 ...

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**1**answer

322 views

### Can Tarski decide constructibility in elementary geometry?

Can the decision routine for Tarski's Elementary geometry be extended to decide when an existence claim in that theory can be instantiated by a compass and straightedge construction?
The answer does ...

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**1**answer

262 views

### Which values can attain the minimum solid angle in a simplex

Given a simplex $S$ with a vertex $v$ by the solid angle at this vertex I mean the value $\hbox{vol}(B \cap S)/\hbox{vol}(B)$ where $B$ is a small enough ball centered at $v$ (for example, in the ...

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**1**answer

193 views

### Generalized Hlawka inequality

Let $E$ be a vector space over the real (the complex case is interesting too). We consider functions $f:E\rightarrow\mathbb R$ which satisfy the homogeneity property
$$f(\lambda x)=|\lambda|\,f(x).$$
...

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**0**answers

264 views

### How can we join two points with a small ruler? [closed]

We want to join by a line two distinct points $A$ and $B$. We have only a ruler of length $l>0$ and a pen. If $AB>l$ how can we do this? Imagine a method that works when $AB$ is really huge and ...

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**0**answers

797 views

### 3-piece dissection of square to equilateral triangle?

At a workshop it was suggested that it likely remains an open problem
whether or not there is a 3- or 2 -piece
dissection
of a square to an equilateral triangle.
Can anyone confirm that this is ...

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**3**answers

683 views

### On maximal regular polyhedra inscribed in a regular polyhedron

Let T, C, O, D, or I be regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron, respectively. Suppose that the outer polyhedron have edge-length 1.
For example, it's easy to prove that ...

**8**

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**1**answer

658 views

### How large can you draw an island on a map?

A cartographer friend asked me this question: could you classify (shapes of) islands by how much space they occupy on a map (comparatively to how much space is occupied by water) if you draw them as ...

**8**

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**3**answers

495 views

### In which geometries do triangles have an Euler line?

In Euclidean geometry, the centroid, orthocenter and circumcenter of a triangle lie on a line.
In which other geometries does this hold?