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.

learn more… | top users | synonyms (1)

0
votes
0answers
14 views

Properties of $\Omega_{\epsilon}$

Let $\Omega$ be a bounded connected domain in $\mathbb R^n$ with compact smooth boundary. So $\partial \Omega$ can be viewed as a smooth submanifold of dimension n-1. Let $$\Omega_{\epsilon} : = \{ ...
1
vote
0answers
23 views

Euclidean geometry with ruler and compass

I was wondering, is there any book out there that is of the style of Euclid's Elements ? One which you have to use a compass and ruler for certain propositions like building a triangle, etc. Or would ...
0
votes
0answers
9 views

Finding closest rectangle to another using concept of closest edge [on hold]

I know coordinates and size of rectangles. My goal is to find 'the closest' rectangle to one special rectangle using the concept of closest edge and also to find distance between them??
1
vote
1answer
17 views

Prove that $GEBD$ is a square (see diagram).

$ABB_1A_1,BB_2C_1C,ACC_2A_2$ are squares. The problem itself is to prove that the area of $ABC$ and the area of $BB_1B_2,CC_1C_2,AA_1A_2$ are equal. If I could only prove that GEDB is a square it ...
-3
votes
0answers
20 views

Euclidean Geometry [on hold]

What is the correct interpretation of the first flaw of euclidean geometry(the proof that shows every triangle is isosceles.)I mean why does this happen and why is this wrong.
3
votes
4answers
44 views

Find area of rhombus

Given the following rhombus, where points E and F divide the sides CD and BC respectively, AF = 13 and EF = 10 I think the length of the diagonal BD is two times EF = 20, but i got stuck from there. ...
0
votes
0answers
14 views

Homomorphisms between countable spaces and Euclidean spaces?

Is there some place to start reading about homomorphisms between countable (discrete) spaces and Euclidean spaces or $l_2$? I know it is a rather general question, but I am not sure what I am looking ...
0
votes
0answers
23 views

On the centroid of a triangle

There's three different ways to see a triangle in the Euclidean plane: as three non-collinear points, say $A$, $B$, $C$; as the line segments connecting the three points, that we can parametrize as a ...
1
vote
1answer
30 views

Relative distance to rotated object

An object (with center $O_{2}$) has been rotated by an angle $\alpha$. There are two images of the object taken by a camera (centered at $O_{1}$), and two points $x_{1}$ and $x_{2}$ that are actually ...
-1
votes
0answers
26 views

How can one show that the vector $AK=\frac{1}{3}AI$?

$ABC$ is a triangle then $I$ is the medium of $[CB]$ and $J$ the medium of $[AI]$ and $K$ the intersection of $(BJ)$ and $(AI)$. Then how can one show that $AK=\frac{1}{3}AC$ Do we have to add ...
1
vote
1answer
15 views

Are the connected components of the level sets of a $\mathcal{C}^1$ function path-connected?

I have a $\mathcal{C}^1$ (or even just $\mathcal{C}^0$) function $f:\mathbb{R}^n\rightarrow\mathbb{R}$, and have been trying to figure out when the connected components of its levels sets are also ...
3
votes
0answers
31 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 ...
0
votes
0answers
15 views

How can one show that (DI) , (JB) and (AC) concurrents on G?

ABCD is a square , We add outside it two equilateral triangles ADJ and ABI How can I show that (DI) and (BJ) and (AC) occur in the same point ? Here can we demontrate that saying that IGB and JGD are ...
1
vote
1answer
31 views

Distribution of an angle between a random and fixed unit-length $n$-vectors

Suppose I have a random unit-length $n$-element vector $\mathbf{x}$ that is uniformly distributed on an $n$-dimensional sphere, and let vector $\mathbf{a}$ be some other unit-length $n$-element vector ...
0
votes
0answers
15 views

Packing spheres into a rectangular prism

So, this was a problem in the new standardized high school tests California has started using(CAASP). These new tests are completely done on the computer, and feature what they call Computer Adaptive ...
0
votes
0answers
16 views

Intersection of 3 positively sloped planes

Suppose I have three planes, each of which is 'positively sloped' in the sense that the first plane intersects the x-axis at a positive value, and the y and z-axes at a negative value. Similarly, the ...
2
votes
0answers
44 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 ...
0
votes
1answer
33 views

A quadrilateral with one pair of opposite right angles. Is this a rectangle?

I can prove it's not a rectangle by drawing some lines, but is there a name for this kind of figure? Thanks.
2
votes
1answer
60 views

Area of triangle formed by angle bisector, altitude and median

Question:- Given a triangle ABC with side length a, b and c. Calculate the area of a triangle in terms of a, b and c formed by angle bisector from vertex A, altitude from vertex B and median from ...
2
votes
0answers
102 views

Is Euclidean geometry really a “dead” subject? If so, why? [closed]

It seems that Euclidean geometry is a "dead" subject nowadays. In the time of the Greeks, mathematicians and geometers were one and the same. Today, very few professional mathematicians study ...
0
votes
1answer
31 views

Every tetrahedron has a vertex whose adjacent edges can form a triangle

Prove that in every tetrahedron there exists a vertex $v$ such that the three edges incident at $v$ have lengths that can form a triangle. It can be proved using a tedious casework: assume by ...
0
votes
1answer
28 views

Non-Euclidean geometries

Does non-Euclidean geometry can be always immersed in Euclidean of dimension D+1? This is probably very basic question, but I'm just trying to understand why do you need to consider sometimes very ...
0
votes
0answers
28 views

An orthogonal transformation with determinant 1 rotate $\mathbb{R^3}$ around an axis.

I'm studying differential geometry and need confirmation on the solution of the following problem. A rotation is an orthogonal transformation $C$ such that det $C=+1$. Prove that $C$ does, in fact, ...
1
vote
1answer
44 views

How to find the focus of a parabola

To find the center of a circle, it's enough to choose three points on the circle and find the circumcenter of a triangle with those three points. When a parabola is drawn and the formula is not ...
0
votes
2answers
19 views

Vector proof to show the line connecting two points on a triangle is parallel to a side.

If $X$ and $Y$ are points on sides $AB$ and $AC$ of a triangle $ABC$ and $\dfrac{AX}{AB}=\dfrac{AY}{AC}$, then $XY\parallel BC$. I'm supposed to prove this using vectors, but we haven't done too much ...
0
votes
0answers
10 views

Prove that the intersections of the ray $f(x)\rightarrow x$ with the $n$-disk form a continuous function, with $f$ continuous

This appeared in a proof of the Brouwer fixed point theorem, in Introduction to Algebraic Topology, by Rotman, but it was left as an exercise. I could only prove this in 2 and 3 dimensions, not in ...
8
votes
5answers
394 views

How to construct a line with a given equal distance from 3 Points in 3 Dimensions?

Important: I'm now convinced that 4 points are needes in order to reduce the solutions to a finite number. (Which is necessary because I need ALL solutions) In a computer science context I need to ...
0
votes
1answer
19 views

Reduce distance between two points by %

I have two points, say A = (2, 6) and B = (5, 3). I want to move point B up to 70% closer to point A. I calculate Euclidean ...
1
vote
1answer
27 views

Proof for diagonals of a rectangle

If a rectangle is a figure with four sides and four rectangular angles, I would like to prove that the diagonals are congruent and both meet in the midpoints. However, I don't know where to start this ...
0
votes
0answers
16 views

Bound for the distance of projections onto the unit sphere

Given $x \in \mathbb{R}^n$, $x \neq 0$, let $x' = x/|x|$ (where $|\cdot|$ is the euclidean length) be its projection onto the unit sphere. I would like to prove that $$ |x' - y'| \leq 2 ...
1
vote
3answers
49 views

Proving algebraic equations with circle theorems

I got as far as stating that OBP=90˚ (as angle between tangent and radius is always 90˚), and thus CBO=90˚- 2x. CBO=OCB as they are bases in a isosceles. COB=180-90-2x-90-2x. But after this, i am ...
1
vote
1answer
35 views

How to calculate Cartesian coordinates for an element after rotation has been applied?

I have a square on a Cartesian coordinate system with origin (0,0) on top left (yellow arrow from the picture). The initial coordinate of the square from the ...
-4
votes
0answers
32 views

Midpoint of side, center of gravity of triangle, … [closed]

$\Delta ABC$ is any triangle having: $I$ the midpoint of $[BC]$ , $J$ the midpoint of $[AC]$ , $G$ the center of gravity. $\vec{BA} + \vec{CA}=$ $\vec{IA}$ $6 \cdot \vec{IG}$ $3 ...
2
votes
2answers
84 views

Area of rhombus and interior isosceles triangles

Points $E$, $F$, $G$, and $H$ lie inside a rhombus $ABCD$, such that the triangles $\triangle AEB$, $\triangle BHC$, $\triangle CGD$, and $\triangle DFA$ are isosceles right triangles with hypotenuses ...
3
votes
1answer
36 views

Isosceles trapezoid with inscribed circle

The area an isosceles trapezoid is equal to $S$, and the height is equal to the half of one of the non-parallel sides. If a circle can be inscribed in the trapezoid, find, with the proof, the radius ...
1
vote
0answers
22 views

Can you always cover a circle in a finite number of steps with this “radar” algorithm?

Suppose you have a disc $C$ of radius $V$ with center $c$ and you randomly place a point $p$ in it. $p$ Behaves as follows: at every time-step, $p$ calculates its angle to $c$, and moves a distance of ...
0
votes
0answers
12 views

Geometry involving area of rhombus and interior isosceles triangles

Points E, F, G, and H lie inside a rhombus ABCD, such that the triangles AEB, BHC, CGD, and DFA are isosceles right triangles with hypotenuses AB, BC, CD, and DA.The sum of areas of ABCD and EFGH is ...
4
votes
2answers
54 views

Segments on a family of parallel lines

Let $\{l_i:i\in I\}$ be a family of parallel lines on the plane $\mathbb{R}^2$. Suppose for each $i\in I$ there is a closed segment $s_i\subset l_i$. Moreover, for each triple $i_1,i_2,i_3$ there ...
1
vote
1answer
34 views

How many degrees of freedom are in a flat metric and how does one count them?

I think that there are zero degrees of freedom in a flat metric, but I do not know how to count them. I know that any symmetric metric tensor has $n(n-1)/2$ degrees of freedom, where $n$ is the ...
-1
votes
0answers
17 views

(Kiselev) Construction of an arc of a circle

Using only compass, construct a 1 degree arc on a circle, if a 19 degree arc of this circle is given. The first thing that is stumping me is if we can use a straight edge? In any case, I can think of ...
4
votes
2answers
67 views

Difficult geometry question involving Pythagoras theorem?

Hello mathematicians, I was given this question by my teacher and after spending a couple of hours looking over it have not been able to solve it. I understand it involves radians which I have ...
1
vote
0answers
16 views

Hahn Banach theorem and supporting hyperplane theorem

The question is out of Rudin Functional analysis Chapter 3 problem 1. Call a set $H \subset \mathbb{R}$ a hyperplane if there exists real numbers $a_1,\ldots, a_n, c$ (with $a_i \neq 0$ for at least ...
0
votes
2answers
30 views

Equilateral Triangle equality

Let ABC be an equilateral triangle, and P be an arbitrary point within the triangle. Perpendiculars PD, PE, PF are drawn to the three sides of the triangle. Show that, no matter where P is chosen, PD ...
0
votes
0answers
20 views

Intersection of two convex lattices polygons

A convex lattice polygon is a polygon whose vertices are points on the integer lattice. Let P and Q two convex lattice polygons with n ,(resp. m) vertices. Let R be the convex lattice polygon ...
0
votes
2answers
28 views

How to check if point $x \in \mathbb{R}^n$ is in a $n$-simplex?

Is there any universal solution how to check if a point $x \in \mathbb{R}^n$ is in $n$-simplex for any number of dimensions (any $n$)? Is it possible to use Barycentric coordinates for any $n$? I ...
3
votes
2answers
112 views

Proper axiomatization of Euclidean Geometry that Euclid would approve of

It is relatively common knowledge that Euclid's axiomatization is not sufficient to prove all the things that Euclid wants to, and that there are other axiomatizations out there that strengthen ...
0
votes
0answers
10 views

semidefinite matrices and Euclidean distance matrices

Is this statement true? If $X$ and $D$ are related as $D_{ij} = X_{ii} + X_{jj} - 2X_{ij}$ then $$ X\succeq 0 \iff D \text{ is a Euclidean distance matrix}. $$ Clearly $\Rightarrow$ is true; make a ...
17
votes
2answers
223 views

To whom do we owe this construction of angles and trigonometry?

I've come across what is, to me, the most precise, beautiful and thorough definition of what we know of as the angle between two vectors. I say this because most literature either skims over things ...
0
votes
1answer
70 views

Algebraic calculation steps.

Can somebody explain how the coefficients $a_{11}, a_{12}, a_{22}$ are derived after rotating the ellipse below ?? $\widetilde{s_{11}} = \frac{\sum_{j=1}^n(x_{jk} - \bar{x_k})}{n}$ Thank you in ...
31
votes
2answers
2k views

Why can't three unit regular triangles cover a unit square?

A square with edge length $1$ has area $1$. An equilateral triangle with edge length $1$ has area $\sqrt{3}/4 \approx 0.433$. So three such triangles have area $\approx 1.3$, but it requires four ...