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)

2
votes
1answer
19 views

Reference request- Darboux cubic of a triangle

Hi everyone on Math Stackexchange, I'm recently interested in the geometry of a triangle, and my studies now seems to require some knowledge on cubic curves related to a triangle, in particular the ...
1
vote
0answers
27 views

Center of mass of voronoi cells of 3d lattice

Let $v_1,v_2,v_3$ be linearly independent vectors in $\mathbb{R}^3$, and let $A$ be a matrix whose columns are $v_1,v_2,v_3$. i.e. $A = [v_1,v_2,v_3]$ Then, define a lattice $\Lambda$ as $\Lambda = ...
0
votes
0answers
31 views

Name of the set of points equidistant from a line

I was reading about geometrical shapes in $n$-dimensional Euclidean spaces and programming some objects that would share some of their properties in different dimensions, like $n$-spheres. I had ...
0
votes
1answer
27 views

Coordinate proof of a rectangle

So I took a state test today and I'm not sure if I messed up or if I will get partial credit for my work, but here goes. We had to prove a quadrilateral was a rectangle and I showed that all the ...
1
vote
2answers
35 views

Proving two planes are parallel (question about the equation)

If I have two planes: $$5x + y - z = 7$$ $$-25x -5y + 5z = 9$$ I can see that from the first plane I get the vector $\langle5,1,-1\rangle$ from the coefficients and then from the second plane I get ...
2
votes
1answer
55 views

Understanding Norms on Vector Spaces

Let $\|\cdot\|$ be a norm (not necessarily the standard norm) on $\mathbf R^2$ and $S$ be the set of all the vectors $v$ such that $\|v\|=1$. For any point $p\in S$, let $\ell_p$ denote the line ...
0
votes
4answers
37 views

Circles and right angles

The following is a standard fact about circles: THEOREM: Let $p$ and $q$ be two antipodal points on a circle in $\mathbb{R}^2$ and let $r$ be another point on the circle such that $r \neq p,q$. ...
1
vote
2answers
40 views

Bisector of two lines in the euclidean space $\mathbb{E}_3$

Let $$r: \begin{cases} x + z = 0 \\ y + z + 1 = 0\end{cases}$$ and $$s: \begin{cases} x - y - 1 = 0 \\ 2x - z -1 = 0\end{cases}$$ be two lines in the euclidean space $\mathbb{E}_3$. It is easily ...
1
vote
2answers
52 views

Relationship between the side lengths of a tetrahedron and an inscribed tetrahedron with vertices at the centroids

Suppose that $OABC$ is a regular tetrahedron with sides having centroids $\lbrace E,F,G,H\rbrace$ also forming a regular tetrahedron. What is the relationship between the side lengths of $OABC$ and ...
-1
votes
2answers
57 views

Bisecting line segments in a tetrahedron. [closed]

Suppose that $OABC$ is a regular tetrahedron with base $ABC$. Suppose further that $T$ is the mid-edge of $AC$, $Q$ is the mid-edge of $OB$, $P$ is the mid-edge of $OA$, and $U$ is the mid-edge of ...
2
votes
3answers
66 views

Given point in triangle, prove that it is the centroid

So the question goes like this: Given a triangle ABC, there is a point M within that triangle such that [AMB]=[AMC]=[BMC]. Prove that M is the centroid of the triangle. ([AMC] denotes the area of ...
10
votes
3answers
472 views

Which statements are equivalent to the parallel postulate?

I would like to have a long-ish list of statements that are equivalent to the parallel postulate. If a line segment intersects two straight lines forming two interior angles on the same side that ...
0
votes
2answers
115 views

Mathematics of photography

From mathematics perspective, cameras do convert the 3d shapes into 2d shapes in the photos. If we consider a 3D coordinate system X-Y-Z which the origins is the camera (or its lens or things like ...
1
vote
2answers
28 views

Need help understanding wiki's informal description of an affine space.

The following was taken from https://en.wikipedia.org/wiki/Affine_space: The following characterization may be easier to understand than the usual formal definition: an affine space is what is left ...
1
vote
0answers
74 views

Abc is a triangle

Abc is a triangle (drawing of the triangle with measurements up the side of each side) Make a full size drawing of triangle abc in the space below The line AB has been drawn for you. Leave in all ...
1
vote
1answer
33 views

Lines on a quadric surface

I want to show that: Let $Q$ be a quadric surface of rank 3 in $\mathbf{P}^3$(over $\mathbf{C}$), then any line in $Q$ must pass through the only singular point of $Q$. I know that up to a transform ...
6
votes
1answer
81 views

Can eight circles be constructed from three circles?

Given three sufficiently spaced circles in a plane, is it possible, using a straight edge and compass, to construct the eight circles that are tangent to all three given circles?
1
vote
2answers
45 views

Discretization of Unit Vector in 3D

I cant think of a thing that I think is supposed to be easy... =/ Im glad if you could help me. Im working with a regular discretization of a 3d euclidean space. Cubic cells. Then, after a ...
1
vote
1answer
53 views

What is real $R$ so that every subset of Euclidean space with diameter one is inside a ball of radius $R$?

What is infimum of real numbers $R$ so that for every $n$ every $S \subseteq \mathbb{E}^n$, for which $d(S) = \sup\{|x-x'| \mid x,x' \in S \} = 1$, is inside some closed $n$-ball of radius $R$? In ...
3
votes
1answer
26 views

The necessary and sufficient condition for a regular n-gon to be constructible by ruler and compass.

I have a problem concerning the necessary and sufficient condition for a regular n-gon to be constructible by ruler and compass. $\bf My$ $\bf question:$ For a given positive integer $n$, how can we ...
0
votes
0answers
43 views

To a given straight line in a given rectilinear angle, to apply a parallelogram equal to a given triangle.

There's again one small detail on which I'm not sure. (Proposition 44 - book 1) http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI44.html Here's the quote : "Then HLKF is a parallelogram, HK ...
0
votes
0answers
34 views

Making a metric out of distance measure

I'm working with a pseudo-distance measure that is not a metric since it does not hold the triangle inequality. It is called Dynamic Time Warping. The problem is - I need to perform some projections, ...
0
votes
0answers
24 views

Triangles which are on the same base and in the same parallels equal one another.

I have a small question regarding proposition 37 of the elements of Euclid. http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI37.html The only problem I got with the proof is the fact that we ...
4
votes
4answers
189 views

What is the area of shaded region which is lies between outer and inner circle.

There is a outer circle with radius 2r and another inner circle with radius r whose center is the middle of big circle.As depicted in the following figure. Foo graph Image There is a sector of 120 ...
0
votes
0answers
18 views

Straight lines which join the ends of equal and parallel straight lines in the same directions are themselves equal and parallel

I have a small question regarding proposition 33 of the elements of Euclid. http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI33.html We want to prove that two lines joining equal parallels ...
0
votes
1answer
18 views

Action of the Euclidean group, generalizing linearity?

I have a vector $v \in \mathbb{R}^2$ and two elements $(A,a)$ and $(B,b)$ of the Euclidean group $E(2)$. If the relation $$[(A,a)(B,b)](v) = v$$ holds, can I say that $(A,a)(B,b)$ is the neutral ...
0
votes
2answers
63 views

A-Level/GCSE Geometry textbook? Geometry for STEP and MAT?

everyone. I have been looking for a book that covers the most elementary parts of Geometry, such as similar triangles, circles(arc, sector and others), Pythagorean theorem, Sine and Cosine Laws, so ...
0
votes
1answer
21 views

Be $m$ and $n$ two perpendicular lines, and …

Be $m$ and $n$ two perpendicular lines, and be distinct points $A$ and $B$ outside the lines and in the first quadrant. What is the shortest way to get from point $A$ to point $B$ by tapping the two ...
3
votes
0answers
23 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\\ ...
1
vote
1answer
42 views

Performing projections with distance different to Euclidean

This is the first time I'm asking a question on math section of stackexchange, so please excuse me if this isn't the right place for such a question. I'm a programmer studying about an algorithm ...
0
votes
2answers
346 views

Diameters and Circles

I have a question (given by a teacher) that looks really easy but then when I thought about it, couldn't find a way to find the answer. It is a proof question relating to diameters: Prove that any ...
6
votes
2answers
174 views

Hexagon packing in a circle

Suppose I want to pack hexagons in a circle, as on the drawing below (red indicates "packed" hexagons). I am wondering what is known about this problem. Specifically, I am interested in an ...
2
votes
1answer
29 views

Side-angle-side and side-angle-angle as proved by Euclid in the Elements (Proposition 26)

I have small question regarding this proposition : http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI26.html To prove that one side is equal to another, Euclid assumes that one side is bigger ...
2
votes
1answer
30 views

What is the fraction of volume of unit hypersphere centered at one of the vertices of hypercube to that of hypercube?

consider a hyper-cube of n-dimension having a length of "r" units across each dimension. If a unit n-dimensional sphere is present at one of the vertices of the hyper-cube. what fraction of volume of ...
0
votes
2answers
34 views

Given a triangle $ABC$, make it a point $D$ on the side $AB$.

Given a triangle $ABC$, make it a point $D$ on the side $AB$. Show that $\overline {CD}$ is smaller than the length of one of the sides $BC$ and $AC$. Ideas? The triangular inequality will not. I ...
1
vote
0answers
58 views

Proofs of the three-perpendiculars theorem

I have to prove this theorem in three different ways. I have already proved it geometricaly and using vectors, but I can not think of any other way. Theorem: If PQ is perpendicular to a plane XY and ...
0
votes
1answer
15 views

Subset convex of plane

A plan of the subset is $convex$ if the segment connecting any two of its points is fully contained therein. The simplest examples of $convex$ $sets$ are the plan itself and any half-plane. Show that ...
0
votes
2answers
44 views

Hard problem about law of the cosine

I have been trying without success to prove by contradiction the following problem: Given 5 segments $x_1\leq x_2\leq x_3\leq x_4\leq x_5$ each three of which are sides of a triangle. Prove that ...
2
votes
1answer
13 views

Solve a convex quadrilateral with four sides and equality of two adjacent angles analytically?

Given the length of four sides of a convex quadrilateral and knowing that two adjacent angles are equal, the quadrilateral is determined. I want to know whether there's a formula representing the ...
1
vote
2answers
42 views

Denial of the 5th postulate of Euclid

I am trying to recover the denial of the Playfair's axiom but it is logical a bit strange. "To a given line and a point not on it, there is only one line through this point parallel to it". This ...
22
votes
2answers
572 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 ...
6
votes
4answers
429 views

Some theorems in euclidean geometry have incomplete proofs

I have seen that, in euclidean geometry, proofs of some theorems use one instance of the 'geometric shape'(on which the theorem is based) to proof the theorem. Like, the proof of 'A straight line ...
13
votes
4answers
2k 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 ...
2
votes
1answer
55 views

An inequality about inner product in $\mathbb{R}^2$.

Let $a_i,b_i,r_i,s_i$ be positive integers for $i\in\{1,2\}$. $r_i$ and $s_i$ are non-zero for $i\in\{1,2\}$. Let $a=\left(\frac{1}{a_1},\frac{1}{a_2}\right), ...
4
votes
1answer
31 views

Very naive questions in elementary geometry

I was wondering whether the following questions are difficult to solve : Consider a triangle ABC (defined in euclidean geometry). Let M be inside the triangle ABC such that the triangles AMB, AMC ...
0
votes
1answer
23 views

How to show express $y $ in terms of angle $\theta$?

$ABC$ is a straight line with $AB = BC = 3$ units. $B$ is the centre of the circle with radius of $2$ units. $P$ is a point on the circle. $\widehat{B_1} = \theta$, $\widehat{A} = x$, $QC \perp AC$ ...
0
votes
0answers
44 views

Euclid's elements proposition 17. The sum of two angles in a triangle is less than 180 degres.

I have a very short question on this proposition : http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI17.html I understand the way the theorem was proved. Euclid proves that angles ABC and ACB ...
-1
votes
1answer
746 views

“Pythagoras Theorem” - Why is “theorem” or “theory” used rather than “law” in mathematics?

Why is Pythagoras Theorem a "theory" but not a "law"? I mean we use it many times in school and to build stairs etc. and it has been proven, however it is still called a theory. What are the ...
0
votes
1answer
42 views

On the definition of sphere in analytic geometry…

Last year, when I was teaching mathematics (analytic geometry) for one of my clever freands, I arrived to the definition of sphere. I said Fix $r>0$, An sphere is the set of all triples ...
0
votes
1answer
12 views

How do I get vectors orthogonal to the one generated by the spherical coordinate formula?

Given a formula: F : ℝ → ℝ → ℝ3 F(θ,φ) = (cos(φ)*sin(θ), sin(φ)*sin(θ), cos(θ)) what are the formulas: ...