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)

1
vote
0answers
22 views

Distance from a point to the involute of a circle

I know that the involute of circle of radius $r$ centered at $(0,0)$ is given by the following parametric form: $$\begin{cases} x(\theta) = r \big(\cos(\theta) + \theta\ \sin(\theta) \big),\\ ...
0
votes
1answer
27 views

Find all nearest points

I have two sets: $$P = \{p_1, p_2, ..., p_n\}$$ $$Q = \{q_1, q_2, ..., q_m\}$$ For each $p_i$ point I need to find all nearest points in $Q$. I.e., $$p_i \rightarrow \{ q_{i_1}, q_{i_2}, ..., ...
1
vote
0answers
21 views

Given five points and a line find the points of the line that lie in the conic through the five points [on hold]

So I'm given 5 points in general position and a line, I already know the method using Pascal's theorem to find points in the conic but I dont know how to find specifically the ones that lie on a given ...
2
votes
2answers
34 views

what is the lowest point of a tilted elliptical plate?

I'd like to know the lowest point $z_\min$ of an ellipse with radius $r_x, r_y$ in (Euclidian) XY that's tilted in XYZ - first rotated around X axis by $\gamma$, then rotated around Y axis by ...
0
votes
1answer
25 views

In Neutral Geometry, prove that the opposite sides of a rectangle are congruent.

I'm having some trouble proving a theorem of Neutral Geometry. First, allow me to clearly state what we are allowed to assume in Neutral Geometry: Hilbert's incidence axioms Hilbert's order axioms ...
0
votes
1answer
42 views

On intervals chosen randomly within the unit circle.

Let $S = \{(x,y)\in R^2 : x^2 + y^2 = 1\}$ be the unit circle in $R^2$. Let $(X_1, Y_1), (X_2, Y_2)$ be independent, both having uniform distribution over $S$. Let $D$ denote the Euclidean distance ...
1
vote
1answer
21 views

Lines cutting regions

15 lines are drawn in a plane such that 4 of them are parallel. a. What is the maximum number of regions into which the plane is divided? b. How many of the regions are finite(bounded)? a) The ...
15
votes
1answer
71 views

If the Greeks had been four dimensional, would they have been able to derive the pi squared coefficient for the hypersphere volume without calculus?

I was reading about Archimedes' pre-calculus proof of the volume of the sphere and I realized that the trick he uses (volume of hemisphere + volume of cone = volume of cylinder) doesn't generalize to ...
0
votes
1answer
16 views

How to test for two segments intersecting, excluding their endpoints?

Given two segments, for example S1=(2,1)-(2,3) S2=(7,8)-(2,3) the intersection test would be false. However, if I use the cross-products test the result is true. I understand the two ...
1
vote
0answers
15 views

Calculating pairwise distance of two N-dimensional vectors given their length and angle

I am not a mathematician, so apologies in advance for any nomenclature blasphemy. Given the magnitudes of two vectors $b$ and $c$ and the angle between them $A$, I can calculate their distance in 2-D ...
1
vote
2answers
16 views

Trigonometric inequality in an obtuse triangle

Let $ABC$ be an obtuse triangle with $A$ the obtuse angle. I conjecture that the following inequality is true $$\sin B + \sin C \le |\tan A|.$$ Show that it holds or give a counterexample.
1
vote
3answers
37 views

Find coordinates of the rotation center

My software is going to control a Laser. I know the Laser's current Position defined as $P_1$ with coordinates $(x,y)$ and the place where it will be after a clockwise rotation around a point $C$, ...
0
votes
1answer
51 views

Why are values squared in distance formulas, such as the Pythagorean Theorem?

Why do you square the values in the Pythagorean Theorem or any distance formula wherein you're trying to find the distance between two points in two-dimensional, Euclidean space? for example, why ...
0
votes
2answers
55 views

Triangle Inequality Like Equation [closed]

If we are in $R^2$ and define $d(a,b)$ as the set of points between $a$ and $b$ we can create an equation like this: $$d(x,z) \subseteq d(x,y) \times d(y,z)$$ where the $\subseteq$ is the subset ...
2
votes
0answers
41 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 ...
2
votes
2answers
43 views

When is the sign of inner products preserved?

I'm interested in the following question: Let $E$ be a real Euclidian space. What are the linear transformations $f$ of $E$ that preserve the sign of inner products? That is, for all vectors ...
-3
votes
2answers
39 views

Slope and euclidean geometry (grade 10)

Given Triangle $DEF$ with vertices $$D(-2,6),E(7,3), \text{ and } F (2,-3)$$ find: a) the equation of the altitude from vertex $E$ in standard form (include y=mx+b) b) Find where this altitude in a) ...
13
votes
1answer
104 views

Quadrilateral $APBQ$.

Quadrilateral $APBQ$ is incsribed in a circle $\omega$ with $\angle P = \angle Q = 90^{\circ}$ and $AP = AQ < BP$. Let $X$ be a variable point on segment $\overline{PQ}$. Line $AX$ meets $\omega$ ...
1
vote
1answer
39 views

Proper Proof for Completeness of $\mathbb{R}$ with the Euclidean Metric

My code can't be uploaded because it doesn't work with the websites coding, but here is a pdf of my LaTex code. My question is, is this a proper proof? It feels as if I'm missing something important. ...
1
vote
1answer
16 views

How can one prove that (AD) and (EB) are orthogonal?

ABC is a triangle, we make outside two squares CBGD and ACEH. I have to show that (AD) and (EB) are orthogonal. So I'm sure that there is many ways to solve this exercise. I did it using the scalar ...
0
votes
3answers
20 views

Greatest area using a string with the length of $l$

Suppose we have a string with length of $l$ what is the shape that has highest area? In other words,with a constant perimeter of $l$ what is the shape with the highest area? P.S:My own speculation ...
0
votes
1answer
12 views

If a quadratic form $f$ takes the minimum on a triangle in a vertex, what can I say about min of $f$ on edges of a subdivision?

Let $f(x)=x^2+y^2$ be the Euclidean square-norm and $A,B,C\in\mathbb{R}^2$ be vertices of a triangle $\Delta$ such that $f$ takes the maximum on $\Delta$ in $C$, the minimum in $A$ and takes the ...
0
votes
1answer
16 views

Geometric Construction Rhombus

Given two line segments, Construct a rhombus whose diagonals have lengths equal to the lengths of the two given segments. I can get to finding perpendicular bisectors of each line segment, but have ...
7
votes
0answers
24 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 ...
3
votes
0answers
45 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 $RT$. ...
19
votes
1answer
480 views

Can I represent groups geometrically?

I have just taken up abstract algebra for my college and my professor was giving me an introduction to groups, but since I like geometric definitions or ways of looking at stuff, I kept thinking, "How ...
1
vote
0answers
22 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
28 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 ...
1
vote
1answer
22 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
4answers
53 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
19 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
38 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
69 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
27 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
34 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
35 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
21 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
17 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
56 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
37 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
64 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
109 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
32 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
30 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
29 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
45 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
22 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 ...