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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0answers
62 views
Please help out, equidistant points
I have a test today in this geometry course, and this was on our review sheet if you could please help me out with this question it would be very helpful thank you
Distance in $\mathbb{R^3}$ from ...
10
votes
1answer
294 views
Proving collinear points
This problem is so hard that I cannot figure it out. I hope you guys can give me a small push on how to tackle this problem, as I have been thinking about this for, like a week. Here's the problem:
...
0
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1answer
68 views
Perimeter and area of a hexagon
The floor of a theater was built in the shape of a hexagon. The interior angles of the hexagon are the same and one side is 8 meters long. How long is the perimeter of the floor in feet?
What is the ...
0
votes
2answers
56 views
Vectors and euclidean spaces
I tried doing this problem and this is the problem
if $a, b,$ and $c$ are real numbers and $a+bx+cx^2 \geq 0$ for any real number $x$, explain why $b^2-4ac \leq 0$
this is how i am trying to do if ...
2
votes
1answer
107 views
How to formally prove that this proof is (not) correct?
In lemma 2 in this article's section 5 there is a proof below and at the end it states that equation $\|(1-\alpha)p_j+\alpha p_i-p_k\|=\|p_i-p_k\|$ has only one solution $\alpha=1$.
Examples can be ...
-1
votes
1answer
232 views
Using the three reflection theorem
There is this sample question from my book that I dont know how to go about.
please help out
Use the three reflections theorem to show that the only transformations of the Euclidean plane
are ...
4
votes
2answers
103 views
Congruent division of a shape in euclidean plane
Any triangle can be divided into 4 congruent shapes:
http://www.math.missouri.edu/~evanslc/Polymath/WebpageFigure2.png
An equilateral triangle can be divided into 3 congruent shapes.
Questions:
...
2
votes
2answers
93 views
Sets whose intersection with line segments have finite components
Certain subsets $A$ of $\mathbb{R}^n$ satisfy the property that given any line segment $L$, $A \cap L$ has a finite number of connected components. For instance, $A$ can be any finitary union of ...
7
votes
4answers
116 views
closest point to on $y=1/x$ to a given point
I feel like I'm missing something basic - given a point $(a,b)$ how do I find the closest point to it on the curve $y=1/x$? I tried the direct approach of pluggin in $y=1/x$ into the distance formula ...
3
votes
1answer
55 views
A particular ILP where the existence of a relaxed solution implies the existence of an integer solution
This question emerged from a discussion on my previous question Determining quickly whether this Integer Linear Program has any solution at all, which I would like to elaborate separately.
I am ...
2
votes
1answer
34 views
What do you call the 4-dimensional analog of a plane?
I keep wanting to call it a "space" but that conflicts with the 2-space, 3-space, 4-space, n-space nomenclature.
e.g., in 3-space, you can hold one variable constant to get a 2-space "slice" (a ...
4
votes
1answer
105 views
Determining quickly whether this Integer Linear Program has any solution at all
I've got an integer linear program of the form
$$
\begin{aligned}
\text{Minimize}&& c &= x_1 + \cdots + x_m \\
\text{subject to}&& A\mathbf{x} &\geq \mathbf{b} \\
\text{where}
...
0
votes
2answers
44 views
Affine geometry about parallelogram
If ABCD is a parallelogram and M,N,P,Q are points on it sides then MNPQ is a paralellogram iff the diagonals intersect at a common point (i.e the diagonals of MNPQ and ABCD intersect at the same ...
1
vote
3answers
159 views
Euclidean Isometry
This is part of my homework problems, it is not for assignment nothing to hand in first of all, i just dont get how to go about this proof. If i can see it it would be good.
Question is
Show that a ...
0
votes
1answer
24 views
Conditions for the existence of a conical combination of some given vectors such that it lies in a cone?
Let $v_1,v_2,\dots,v_n,u_1,u_2,\dots,u_r\in\mathbb{R}^n$. Can one find analytical conditions (not write the problem up as a convex optomisation problem and argue it can be solved this way) under which ...
2
votes
1answer
61 views
How to choose $x$ evenly distributed points from within an n-ball
I would like to know how to choose $x$ evenly distributed points from within an n-ball. I think a formal way of defining this is that we want to choose $x$ points from within the n-ball such that we ...
10
votes
3answers
411 views
What is a point?
In geometry, what is a point?
I have seen Euclid's definition and definitions in some text books. Nowhere have I found a complete notion. And then I made a definition out from everything that I know ...
-4
votes
1answer
173 views
Geometric Definitions: What is a straight line? What is a circle?
What is a straight line? I need a geometric definition of it. The equation of a straight line is known to me.I am saying about a straight line of 2D plane.
What is a circle? I need a geometric ...
2
votes
3answers
323 views
Explain why vertical angles must be congruent
I know why vertical angles are congruent but I dont know why they must be congruent
1
vote
5answers
54 views
Negation of “If $ l$ is line and P a point not on $l $, then every line through P intersects ”$l$"
To negate "If $ l$ is line and P a point not on $l $, then every line through P intersects "$l$":
I came up with "$l$ is a line and $P$ is a point that is not on $l$, and no line through $P$ ...
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votes
1answer
54 views
A continuous area-preserving mapping is an isometry?
Suppose that $f\colon\mathbb R^2\to\mathbb R^2$ is a continuous map which preserves area in the Euclidean sense. Can we say that $f$ is an isometry?
Note. We donot assume that $f$ is differentiable.
2
votes
1answer
55 views
The structure of realization spaces of polyhedral graphs
Given a polyhedral graph with $v$ vertices, $e$ edges and $f$ faces, each possible realization of the graph as a geometric (convex) polyhedron corresponds to a point in $\mathbb{R}^{3v}$, ...
2
votes
1answer
67 views
Solve for Angles in a Matrix
So, I have a linear equation, and I need to solve for some variables in said equation. However, since I don't know much about matrices, I don't know how to solve for the variables. The equation in ...
3
votes
1answer
39 views
What are the use cases related to cluster analysis of different distance metrics?
I'm trying to use different distance metrics like Euclidean, Manhattan, cosine, chebyshev among other distance metrics in my k-means algorithm to calculate distances between the data points and the ...
0
votes
0answers
34 views
A triangle construction problem to find triangular orbits in an elliptic billiard
In an elliptic billiard, there is a triangular orbit that starts from any point on the ellipse.
1) How to find such orbits with straightedge and compass?
2) Is it possible to construct a triangle ...
4
votes
0answers
100 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 ...
0
votes
1answer
107 views
How to Calculate the Direction of a Vector
Let's say I have points $a$, $b$, and $c$. We also have $\vec{ab}$ and $\vec{ac}$. Finally, we know neither vector's direction. (That is, the vector's angle on each axis as if the vector were ...
-4
votes
2answers
117 views
Just next to a certain real number, there is a real number. [closed]
{ReekMaths will clarify the question later.Please do nothing with the present question.}A straight line in 2 dimensional geometry is composed of infinitely many points. According to this, we assume ...
0
votes
1answer
69 views
Hilbert norm and Euclidean distance
For real matrix $X$ where $d_{i,j}^2(X)$ indicates the euclidean distance squared between the rows $i,j$ of $X$, if $d_{i,j}^2(X)=||f(X_i.)-f(X_j.)||_H$ then what would the function $f(.)$ be? Is ...
4
votes
3answers
112 views
Special properties of $\mathbb{R}^3$
Are there any special (nontrivial) properties of $\mathbb{R}^3$ that distinguish it from any other $\mathbb{R}^n$? If there are, what are some of the important ones?
1
vote
1answer
54 views
Euclidean K-Center Problem
My google searches have brought me to rather long papers explaining the Euclidean K-Center problem.
Can someone please provide a high-level explanation?
Thanks.
3
votes
1answer
226 views
Common tangent to two circles with Ruler and Compass
Given two circles (centers are given) -- one is not contained within the other, the two do not intersect -- how to construct a line that is tangent to both of them? There are four such lines.
6
votes
0answers
113 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 ...
3
votes
1answer
28 views
Geometric description of the algebraic degree of the circumradius over the rationals
Let $A,B,C$ be three non-collinear points in a plane with rational coordinates. Let $r$ be the circumradius of the triangle $ABC$. A simple argument (see here) shows that $r$ has degree $2^p$ over ...
3
votes
1answer
73 views
Minimizing the distance to a finite set of points in the plane
Let $A_1,A_2, \ldots ,A_n$ be a finite set of $n$ distinct points in ${\mathbb Q}^2$. For any $\varepsilon >0$, let $B(A_i,\varepsilon)$ denote the closed ball with center $A_i$ and radius ...
1
vote
2answers
109 views
How to get coordinates of point knowing distance from x,y and angle?
I have such a problem :
I am given :
x,y
$\|a\|$
$\alpha$
$\vec{v}$ and $\|v\|$
I need to get the coordinates of point X1Y2.
1
vote
1answer
46 views
graph theory theorem
The maximum number of points in a plane such that the distance of any of these points from a given point in the plane is less than the distance of it from any other point is five.
1
vote
1answer
91 views
Separation Theorem in Euclidean Space.
I want to show the following:
Let $A,B \subseteq \mathbb{R}^n$ disjoint, nonempty, closed and convex sets. Then there exists a $h \in \mathbb{R}^n$, such that $A$ and $B$ gets separated in the ...
0
votes
0answers
44 views
Constructing steiner tree
Given 4 nodes with edge values as stated below, is it possible to build a minimum spanning tree using Steiner tree?
...
15
votes
1answer
246 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, ...
7
votes
5answers
446 views
Sangaku. How to draw those three circles with only a ruler and a compass?
I found in a book of Sangakus the following problem.
Let $R_b$, $R_g$ and $R_r$ the radiuses of the blue, green and red circles $C_b$, $C_g$ and $C_r$.
Prove that
...
1
vote
1answer
46 views
Does the following condition characterise convexity of a set?
Conjecture: A set $X \subseteq \mathbb{R}^n$ is convex if and only if the following holds.
For any $x \in X$ and any vector $v \in \mathbb{R}^n$ such that $x+v \notin X$, it holds that for any scalar ...
1
vote
2answers
243 views
Is the area of intersection of convex polygons always convex?
I am interested specifically in the intersection of triangles but I think this is true of all convex polygons am I correct? Also is the largest possible inscribed triangle of a convex polygon always ...
1
vote
1answer
183 views
How to find the intersection of the area of multiple triangles
I have a couple of questions regarding finding the intersection of triangles. I have a system of 16 projectors that all have slightly different color gamuts. The color gamuts are represented by a ...
4
votes
2answers
107 views
Image of a half space in $\mathbb{R}^n\setminus \{\vec{0}\}$ under the inversion $x \mapsto \frac{x}{|x|^2}$
Let $n \ge 2$, and consider the inversion $\Phi\colon \mathbb{R}^n \setminus \{\vec{0}\} \to \mathbb{R}^n \setminus \{\vec{0}\}$ given by $x \mapsto \frac{x}{|x|^2}$. For $a > 0$, define the half ...
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votes
3answers
214 views
Proof of test of collinearity and coplanarity
Statement : If there are 3 points with position vectors a, b and c. Then the points are collinear if and only if there exist scalars x,y,z, not all zero,such that x a + y b +z c = 0 where x+y+z =0.
...
0
votes
1answer
44 views
Prove that the set of all indirect isometries (reflections and glide reflections) is not a group.
I kind of have an idea that "having the Identity in the set" part of the definition of a group should be used to disprove it but I am having trouble formalising the proof. Any help would be really ...
2
votes
1answer
89 views
What kind of isometry is a composition of a glide reflection with itself? Justify
Is there a simple algebraic proof?
Thanks!
0
votes
1answer
73 views
Draw a square around a point.
I have a point on the graph at position X,Y , and I have to draw a square around that point of side X m. I have described my problem in the image.
I have taken an example square of 3 m. The ...
0
votes
1answer
145 views
Construct two chords of equal length through points A and B (two arbitrary points INSIDE a circle) that are perpendicular to each other.
Its a construction problem I am having trouble with. I realize I need to use rotations and/or other isometries but I am really stuck. Any help would be really appreciated!
Thanks!
