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3
votes
0answers
315 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
6answers
682 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
1answer
435 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
1answer
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
1answer
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
1answer
554 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
3answers
776 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
11answers
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 ...
17
votes
6answers
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
1answer
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
1answer
584 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
9answers
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
1answer
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
1answer
490 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
6answers
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
1answer
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
1answer
587 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
9answers
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
4answers
836 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
7answers
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
2answers
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
1answer
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
1answer
944 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
14answers
39k 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 ...