0
votes
0answers
26 views

Generalized Pitot Theorem

This is the generalized Pitot theorem: Let $a, b, c, d$ be four directed lines all touching one directed circle (directed objects are said to touch each other if they touch each other as ...
2
votes
3answers
395 views

Book with lots of geometry theorems

I want to study geometry and was looking for some book that has lots of theorems and covers almost all Euclidean geometry that is needed for High School and Maths Olympiads. Thanks.
3
votes
1answer
77 views

euclidean geometry books…

I consider myself poor in plane euclidean geometry. so I need a good geometry book which contains very good theory, and a collection a large number of solved problems, and the end of each part.This ...
1
vote
0answers
30 views

Seeking collection of geometry problems

I'm looking for a collection of geometry problems that does not surpass high-school to first-year university geometry.. Problem that could be found sometimes in SAT/GMAT test. They should all at ...
12
votes
2answers
273 views

Decomposable Families of Shapes

There are two types of golden triangles in the world, as shown in the following picture: Here $\varphi = \dfrac{1+\sqrt{5}}{2}$ denotes the golden ratio. Each of these golden triangles can be ...
1
vote
0answers
152 views

References for problems related to uniformly distributed points/ arcs on circle

I am looking for references to problems related to computing the statistical properties of uniformly randomly chosen points or arcs on the unit circle, e.g.: On a unit circle, $n$ points are ...
1
vote
0answers
62 views

What is known about $n$ independent random variables that yield a “converse” to uniform sample of a coordinate from a surface of an $n$-sphere?

It's well-known that to sample a coordinate $(Y_1,\ldots,Y_n)$ from a surface of an $n$-dimensional unit-radius sphere, it suffices to generate $n$ independent random variables $X_1,\ldots,X_n$ from ...
6
votes
3answers
193 views

Which books can I study (learn) better the topics about convergence in $\Bbb R^{n}$ and euclidean space?

I have question. Which books can I study (learn) better the topics about convergence in $\Bbb R^{n}$ and euclidean space ? Please can you give me an advice some book names? Thank you!
0
votes
1answer
87 views

Euclidean geometry applied to Ptolemy geocentric model

I'm looking for a good reference on how Ptolemy used the Euclidean geometry to calculate the planets positions.
21
votes
1answer
870 views

Does there exist a copy of Euclid's Elements with modern notation and no figures?

I am working through Euclid's Elements for fun, but I find the propositions difficult to understand without referencing the provided figures. Unfortunately, the figures usually give away the proofs, ...
2
votes
0answers
93 views

Five squares in a box.

Let $A,B,C,D$ and $E$ be five non-overlapping squares inside a unit square, of side lengths $a,b,c,d,e$, respectively. Prove that $$a+b+c+d+e \le 2.$$ (This problem is a continuation of my previous ...
3
votes
1answer
121 views

Is there a geometric proof to answers about the 3 classical problems?

I know that there is a solution to this topic using algebra (for example, this post). But I would like to know if there is a geometric proof to show this impossibility. Thanks.
0
votes
1answer
266 views

Proof of Archimedes Lemma about the Center of Mass

I am looking for a proof of the following statement, known as Archimedes' Lemma: If an object is divided into two smaller objects, the center of mass of the compound object lies on the line segment ...
4
votes
2answers
163 views

Higher-dimensional Extension of Triangle Geometry?

I am currently exploring generalizations of triangle geometry to higher dimensions. I know that "important questions" of Euclidean geometry have been already addressed and is considered obsolete; most ...
7
votes
1answer
165 views

A scalar product in the space of oriented volumes?

Let $L\colon \mathbb{R}^n \to \mathbb{R}^N$ be an injective linear map. By the Cauchy-Binet formula, $\det(L^TL)$ equals the sum of the squares of all minors of $L$ of order $n$: this looks just like ...
9
votes
5answers
2k views

Geometry Book Recommendation?

Can someone recommend a good basic book on Geometry? Let me be more specific on what I am looking for. I'd like a book that starts with Euclid's definitions and postulates and goes on from there to ...
0
votes
3answers
330 views

Looking for a book dealing with Euclidean Geometry

Good Day I am looking for a Book that contains detailed proofs of theorems in the field of Euclidean Geometry that are used to solve exams and school work of high school students. theorems like: ...
7
votes
1answer
855 views

Construction of a regular pentagon

In Robert Dixon's Mathographics, a regular pentagon is constructed with straightedge and compass only. It is the pentagon $ABCDE$ pictured below. I am having trouble seeing why the central angles ...
2
votes
1answer
598 views

Similar Triangle Theorem in the Incommensurable Case

The following is a geometry theorem whose proof is examinable in the Irish 'High School' Exam. Let $\Delta ABC$ be a triangle. If a line $L$ is parallel to $BC$ and cuts $[AB]$ in the ratio $s:t$, ...
15
votes
5answers
829 views

What is the modern axiomatization of (Euclidean) plane geometry?

I have heard anecdotally that Euclid's Elements was an unsatisfactory development of geometry, because it was not rigorous, and that this spurred other people (including Hilbert) to create their own ...
2
votes
2answers
351 views

High-dimensionality and intuition

Over the years I've come across (usually as a tangential remark in a lecture) examples of how our intuitions (derived as they are from the experience of living in 3-dimensional space) will lead us ...
6
votes
1answer
193 views

Triangle from lengths of angle bisectors

According to http://www.cut-the-knot.org/triangle/TriangleFromBisectors.shtml it is impossible to construct a triangle from the lengths of its angle bisectors. Is there a more comprehensive account of ...
2
votes
1answer
89 views

Arrangements of congruent rectangles

I have stumbled on an interesting problem. How many congruent rectangles on a plane can be arranged in such a way that they all touch each other but never overlap? I pondered this problem for a few ...
2
votes
0answers
84 views

Solving some geometry problems using involutions

Some geometry problems ( like this and this ) have short solutions if we use involutions. What references are there for solving geometry problems using involutions? I am particularly interested in ...
13
votes
7answers
7k views

Book recommendation on plane Euclidean geometry

I consider myself relatively good at math, though I don't know it at a high level (yet). One of my problems is that I'm not very comfortable with geometry, unlike algebra, or to restate, I'm much more ...
8
votes
1answer
285 views

Chebyshev center = center of mass?

I would like to know for which convex polyhedra $P$ in $\mathbb{R}^3$, is the center of the largest sphere enclosed in $P$ (a.k.a. the Chebyshev center, or the incenter) the same as the center of ...
-2
votes
1answer
351 views

Greek Geometry Algebra [closed]

Tell me everything you know about Euklid, Pythagoras etcetera. I'm about to begin reading about greek algebra used in calculating geometric figures. After reading I'm going to write a 15-20 pages ...
1
vote
2answers
637 views

Distinct Hamiltonian cycles of the icosahedron and dodecahedron

I am seeking a listing of the distinct Hamiltonian cycles following the edges of the icosahedron and the dodecahedron. By distinct I mean they are not congruent by some symmetry of the icosahedron or ...