geometry assuming the parallel postulate of Euclid: in a plane, given a line and a point not on that line, there is exactly one line parallel to the given line through the given point.

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13
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366 views

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

I've thought about the following question for a month, but I'm facing difficulty. Question : Letting $S{^\prime}$ be the area of the inner pentagon made by the five diagonals of a convex pentagon ...
10
votes
0answers
187 views

Number of circles in configuration

Consider the $n^2$ lattice points $(i, j)$, where $1 \leq i, j \leq n$. Let the number of circles that pass through at least 3 points of this set be $C(n)$. What is a good way to count this? Is there ...
7
votes
0answers
39 views

Reference Request: Complete Proof of Braikenridge–Maclaurin Theorem

Where can I find a reference to a complete proof of the Braikenridge–Maclaurin theorem, which is stated as: If the three pairs of opposite sides of (an irregular) hexagon meet at three collinear ...
6
votes
0answers
612 views

Minimal number of moves to construct the challenges (circle packings and regular polygons) in Ancient Greek Geometry?

In the web game Ancient Greek Geometry, there are challenges to construct regular polygons and circle packings using ruler and compass constructions. The game measures the number of line and circles ...
6
votes
0answers
149 views

Biggest ball included in an intersection of balls

I would like to prove that for any family of balls $\{B(c_i,r_i)\}_i \subset \mathbb{R}^d$ such that $\{c_1, \dots, c_n\} \subset \bigcap_i B(c_i,r_i) $ and $\forall i, r_i \geq 1$, there exists a ...
5
votes
0answers
41 views

Generalizing convexity of sets

Let $X$ be a subset of some Euclidean space. We say that $X$ is convex if for any two points $p$, $q$ in $X$, the line segment joining $p$ and $q$ is also in $X$. But what if we loosen this definition ...
5
votes
0answers
59 views

Polar of a non centred ball.

Recall that the polar of a set $A\subset\mathbb{R}^n$ is the following set: $$A^{\circ} = \lbrace c \in \mathbb{R}^{n} |\; \langle c,x \rangle \leq 1 \quad \forall x \in A \rbrace$$ where $\langle ...
5
votes
0answers
84 views

Polygons with coincident area and perimeter centroids

Let $P$ be a simple, planar polygon. Define $c_a$ as the area centroid of $P$, i.e., the center of gravity of the closed shape $P$. Define $c_p$ as the perimeter centroid of $P$, the center of gravity ...
5
votes
0answers
445 views

Counting simple quadrilaterals in a rectangular lattice.

I've been trying to make an algorithm to find the number of all possible simple quadrilaterals in a N*M lattice. I already have a brute force solution but since this is a Project Euler problem I ...
5
votes
0answers
119 views

Characterization of Convex Polygons

John Lee's Axiomatic Geometry has an interesting characterization of convex and non-convex vertices for polygons. Let $P$ be a polygon. Consider a ray emanating from a vertex of $P$ which does not ...
5
votes
0answers
96 views

What property does this equation calculate?

It's pretty difficult to Google for the meaning of a formula. This is the equation, it has to do with ellipses and GIS coordinates. $$\nu =\frac{ a} {\sqrt{(1 - (e^2 \cdot \sin(\varphi))^2)}}$$ $a$ ...
4
votes
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73 views

The reverse pizza problem .

The pizza problem is a fairly well-known problem which sounds like this : You have a circular pizza and you need to cut it such that you and your friend would both receive half of the pizza . ...
4
votes
0answers
71 views

Modern Geometry Textbook

What's a good introduction (undergrad level) to modern axiomatic geometry? By that I mean Euclidean geometry, but using a more modern set of axioms such as Hilbert's or Tarski's.
4
votes
0answers
109 views

A surprising locus of points

I was playing with the setup for Desargues Theorem in Geogebra today and got a very odd-looking (to me) result. I imagine I could grind through this analytically and get an ugly-looking parametric ...
4
votes
0answers
211 views

Three properties of the Lebesgue measure on $\mathbb{R}^n$

I'm writing notes for my upcoming class in Game Theory and I realized some time ago that I only need three properties of the Lebesgue measure $\lambda$ on $\mathbb{R^n}$. It is a non-negative ...
4
votes
0answers
114 views

Unlit region in a Room lined with Mirror

Mathematician Ernst Straus wondered if a room lined with mirror can always be lit with a single match. He (or someone else) discovered that in the following room, light shone from A can't reach B: ...
4
votes
0answers
193 views

Which chapters of Euclid's elements would be helpful for drawing a grid?

I am drawing a $19 \times 19$ grid on my desk. For aesthetic purposes, I don't want to use a ruler. Rather, I want to use Euclidean theorems to 'prove' to myself that such and such line meets at a ...
3
votes
0answers
28 views

What does it take to have a precise definition of volume?

Many proofs in elementary geometry use an intuitive but imprecise definition of the area or the volume. For example, Euclid's first proof of the Pythagorean Theorem uses the fact that all triangles of ...
3
votes
0answers
32 views

Average distance between nearest neighbors for randomly placed points in a unit square?

The answers I found were generally about the distance between any two points in a square. I'm trying to find the average distance between nearest neighbors. Background on this is I'm processing 3D ...
3
votes
0answers
45 views

More problem on van Aubel configuration associated with parallelogram and cyclic quadrilateral

Problem 1: Let $ABCD$ be a parallelogram. Construct four squares on the sidelines $ABCD$. Let $NPMO$ be the Thebault’s square. Show that: 1-Centers of four circles $(NAP)$, $(PBM)$, $(MCO)$, $(OCN)$ ...
3
votes
0answers
63 views

A generalization of Zeeman-Gossard perspector theorem

I found a conjectures of generalization of the Zeeman Gossard theorem a year ago, but I no found a solution for the conjecture. I'm an electrical engineer, I am not a mathematician. I don't know how ...
3
votes
0answers
35 views

Ratio between subsegments of space diagonal of a cube

Let $ABCDEFGH$ be a cube where $ABCD$ is the ground face and $E$ is about $A$, $F$ above $B$, and so on. Consider the space diagonal $AG$, and call $P$ the orthogonal projection of $B$ on $AG$. The ...
3
votes
0answers
62 views

About pythagorean triples

In the circle of diameter $AB$ it is well known each point $C$ determines a right triangle $\Delta ABC$ and so it is with every point $D$ on the circle of diameter $AC$ determining a right triangle ...
3
votes
0answers
27 views

On the solutions of a system of inequalities avoiding Helly's theorem

Let $a_1,b_1,\cdots,a_4,b_4\in\mathbb{R},r_1,\cdots,r_4\in(0,+\infty)$. Show that, if $\not\exists (x,y)\in\mathbb{R}^2$ such that $$ \begin{cases} (x-a_1)^2+(y-b_1)^2\le r_1\\ ...
3
votes
0answers
67 views

Given $n$ points, can we always connect them such that every angle is at least $30°$?

Suppose $x_1, \ldots, x_n \in \mathbb{R}^2$ are given, all distinct. We can make a sequence of these points (for example $(x_2, x_1, x_3)$, if $n=3$). The question is, can we always make such a ...
3
votes
0answers
57 views

Area of an equilateral triangle

Prove that if triangle $\triangle RST$ is equilateral, then the area of $\triangle RST$ is $\sqrt{\frac34}$ times the square of the length of a side. My thoughts: Let $s$ be the length of $RT$. ...
3
votes
0answers
63 views

Pythagorean rectilinear polygons

Polygons all of whose edges meet at right angles are called rectilinear polygons. I am interested in rectilinear polygons with integer distance between each pair of vertices. Such rectilinear polygons ...
3
votes
0answers
60 views

A question concerning radians and arc length

I was asked by a colleague yesterday about how the formula for the arc length of a circle is derived. I wanted to give them a correct answer, so I said I'd get back to them once I'd thought about it ...
3
votes
0answers
87 views

How to determine the length of $x$ in this sketch?

Consider the following construction: Assume that you are given the lengths of the sides $a,b,c,d,e$ as well as the size of the angle $\phi$. The task is to determine the length of $x$ in terms of the ...
3
votes
0answers
262 views

Menelaus's Theorem clarification

If three points, one on each side of a triangle (extended if necessary) are collinear, then the product of the ratios of division of the sides by the points is -1 if internal ratios are considered ...
3
votes
0answers
243 views

Ellipse and circles

Playing with an ellipse I discovered the following properties and am now looking for a nice proof or references. Let's consider an ellipse with foci $A$ and $B$ and let $C$ be a point on it. Let $l$ ...
3
votes
0answers
89 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 ...
3
votes
0answers
93 views

Computing the proportion of vectors with the same sign

Let $s \propto (1, \ldots, 1)$ be a d-dimensional unit vector, so that $s_i = 1 / \sqrt{d}$. Let ${\cal V} = \{ u \in {\mathbb S}^{d-1}:u \cdot s = \cos \theta \}$ be a collection of unit vectors that ...
3
votes
0answers
261 views

Euclidean geometry and the Euclidean group

At Wikipedia's Erlangen program I read that "quite often, it appears there are two or more distinct geometries with isomorphic automorphism groups". Some examples are given. But what are examples ...
2
votes
0answers
26 views

How does a group of transformations lead to a geometry?

I am reading Vinberg's algebra text, and on page 144 he says "Of course, not every transformation group leads to a geometry which is interesting and also important for some applications. All such ...
2
votes
0answers
23 views

Equation curves

Introduce equation curves to the canonical form, finding an appropriate rectangular coordinate system. a) $5x^2+12xy-22x-12y-19=0$ b) $9x^2+24xy+16y^2-230x+110y-475=0$ Could somebody do one task. I ...
2
votes
0answers
97 views

What is the best approximate of points on a sphere?

I have a unit radius sphere with a set $S$ of $n$ points on it. How can I find a map $f:S\to \mathbb{R}^4$ which minimizes $$\sum_{x,y\in S} \bigg( d_{\text{geodesic}} (x,y)^{2} - ...
2
votes
0answers
182 views

Assumptions needed for proof of the Pythagorean Theorem from examples

There are lots of geometric dissection and reassembling proofs of the Pythagorean Theorem (PT). I want to know what are the necessary ideas to allow a proof using discrete instances. For example, we ...
2
votes
0answers
57 views

N-squares conjecture

When I studied the conjecture 1, I found a generalization as follows: Conjecture: Let $A_iB_iC_iD_i$ for $i=1,2,...,n$ be $n$ squares in a plane, with $A_i \rightarrow B_i \rightarrow C_i ...
2
votes
0answers
35 views

A generalization of the Parry circle theorem

I found a conjectures of generalization of the Parry circle theorem a more than year ago, but I no a found solution for the conjectures. I'm an electrical engineer, I am not a mathematician. I don't ...
2
votes
0answers
49 views

A generalization of the Lester circle theorem

I found a conjectures of generalization of the Lester circle theorem a more than year ago, but I no a found solution for the conjectures. I'm an electrical engineer, I am not a mathematician. I don't ...
2
votes
0answers
75 views

Proof of Square Cube Law

Can someone offer a general proof of the square cube law of surface area and volume growth? For any fixed three dimensional shape it's clear how to proceed, but considering a general shape, how can ...
2
votes
0answers
68 views

Collinearity problem (Newton-Gauss line)

I had some troubles with this problem : Let $ABCD$ be a convex quadrilateral. $M$ and $N$ are the midpoints of the diagonals $AC$ and $BD$. The sides $AB$ and $CD$ are extended until they ...
2
votes
0answers
28 views

Name of the segment connecting a point's coordinate axis projections?

Given any point $(x,y)$ in the real plane consider the corresponding line segment connecting $(x,0)$ with $(0,y)$. See diagram. Is there a name for this special segment? (I believe that in ...
2
votes
0answers
74 views

Alternative proof for the equality of two angles in an isosceles triangle.

From the answers of my previous question, I got an idea to prove equality of two angles in an isosceles triangle. In that question the equality of two angles in a right-angled-isosceles triangle was ...
2
votes
0answers
59 views

Triangle side-length problem

my problem is the following. A triangle ABC is given. P is a point on $\overline{AB}$. $k_1, k_2, k$ are the radii of the in-circles of APC, BPC, ABC. $s_1, s_2, s$ are radii of the ex-circles of ...
2
votes
0answers
25 views

Statue and a flag distances

Next to a flagpole is a statue that measures 9m high. The upper end of the flagpole with the bottom of the statue form an angle of 53.13 degrees to the floor, and the upper end of the flagpole to the ...
2
votes
0answers
62 views

Why is unit circle not sufficient to bound the partial sums?

I want to find vectors $\textbf{v}_1, \dots,\textbf{v}_n$ in $\mathbb{R}^2$ with that $\sum_{i=1}^n\textbf{v}_i=\textbf{0}$ and $\Vert \textbf{v}_i\Vert\leq 1$ for all $i=1,\dots,n$, such that for ...
2
votes
0answers
90 views

Huzita Axiom 6 - Computing the Origami Trisection of an Angle

The Galois theory proof of the improssiblity of angle trisection rests on studying the triple angle formula $\cos 3\theta = 4 \cos^3 \theta - 3 \cos \theta$. Ruler and compass numbers can only be ...
2
votes
0answers
49 views

Intesection point of feet of altitudes

If triangle has vertexes at $(x_1,y_1),(x_2,y_2),(x_3,y_3)$, is the intersection points of feet of altitudes $$x_h = \frac{x_1x_2(y_2-y_1) + x_2x_3(y_3-y_2) + x_3x_1(y_1-y_3) + y_1^2(y_3-y_2) + ...