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
1answer
71 views

Why is the Euclidian norm used to measure complex numbers?

Why is the Euclidian norm used to measure complex numbers? The complex numbers are numbers (or more precisely, pairs of numbers), and I can't see why are they essentially connected to the ...
1
vote
1answer
38 views

How many sets of four points in an MxN grid have one point contained by three other points?

Given a 3x3 grid: 1 2 3 8 9 4 7 6 5 We find 126 distinct sets of 4 points $$\binom{9}{4}$$ There are 8 cases such that when the points are connected with a line in clockwise direction, one point ...
3
votes
1answer
58 views

Euclidean triangle is determined by Angle, Median and radius of exterior circle

consider two triangles $\Delta(A,B,C), \Delta(A',B',C')$ in euclidean plane. I want to prove that these triangles are congruent if they are equal in the following data: they have the same angle at ...
0
votes
1answer
37 views

Online tool for making Geometric Constructions.

There was a website where it tasked you making different geometric shapes using only a compass and straightedge. I've looked for it and I can't find it or even discussion about it. What I do remember ...
1
vote
1answer
28 views

Lengths on the unit octahedron

Consider the face of the unit octahedron, defined by: $$O^2 = \{(x,y,z): |x|+|y|+|z|=1\}$$ Every point on the octahedron has between 0 and 3 positive coordinates. E.g, in $(0.2,-0.3,0.5)$, the $x$ ...
0
votes
2answers
57 views

Why is this point set a circle?

consider a circle in Euclidean plane $E$ and any point $A$ in the interior of the circle. Now consider all secants $s_A$ to the circle through the point $A$. The claim is now that the set of midpoints ...
2
votes
2answers
24 views

Proof of folding to trisect a right angle

If first you fold a normal (letter or A4) piece of paper in half: and then you fold one corner to meet the halfway line: Then you've trisected the right angle at bottom left - but how does one ...
1
vote
1answer
60 views

Find my coordinates from distance with unknown coordinates

I am trying to work out if there is a way to calculate some coordinates relative to each other simply by knowing $3$ or more distances from some unknown points. I do not have a distance matrix, I ...
1
vote
2answers
52 views

Show that $\triangle ABK \cong \triangle ABL$. [closed]

Show that $\triangle ABK \cong \triangle ABL$ where $D$ is the circumcenter,$K$ is the orthocenter and L lies on the circle.
0
votes
0answers
19 views

Solid angle in $D$ dimensions

Consider $d\Omega_D$, the element of solid angle in dimension $D$. Suppose an integrand depends only on $m-1$ of the angles (so that it doesn't depend on the set $\left\{\phi_i, i=1,\dots,D-m\right\}$ ...
3
votes
0answers
48 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 ...
0
votes
2answers
32 views

How do I calculate the angles between a point on a sphere and each unit vector in $\Bbb R ^3$?

Given the Cartesian coordinates of any point $p$ on the surface of a sphere in $\Bbb R ^3$, how do I calculate the angles between each axis $(x, y, z)$ and the vector $n$ defined by origin $o$ and ...
0
votes
2answers
40 views

any good sourcebook for plane geometry problems?

I wanted to find some good resource books for euclidean plane geometry. would anybody name some titles?
5
votes
2answers
82 views

What's interesting in latus rectum?

I'm a maths teacher in Italian secondary school and I've been spending some time trying to construct "meaningful" problems about conic sections. I particularly like problems which focus on practical ...
1
vote
0answers
22 views

Faster Alternative than Calculating Euclidian Distance to determine which Coordinate has Max Distance from a fixed coordinate (eg (0,0))

I am developing a program that needs me to determine which coordinate in a 2-d figure has maximum distance from a fixed coordinate. Let me demonstrate: 3 points: (1,3), (2,2), (5,0) ; Fixed point: ...
1
vote
0answers
8 views

Construct a matrix satisfying a linear restriction with bounded singular values

Suppose for some matrix $A\in\mathbb{R}^{n\times n}$, and some vectors $x,y\in\mathbb{R}^n$, $y=Ax$. Under what conditions (ideally on $A$) can one construct a matrix $B\in\mathbb{R}^{n\times n}$ ...
1
vote
1answer
23 views

Straightedge-Only for Perpendicularity

Given a triangle ABC and a midpoint M (of the line AB), is it possible to check whether the line CM is perpendicular to AB with a straightedge only? By this, I mean that points can be added ...
2
votes
1answer
25 views

Finding distance from point to line which is perpendicular to another line

Find the distance of the point $(1,1,1)$ from $x+y+z=1$ measured perpendicular to the line $\frac{x}{2}=\frac{y}{3}=\frac{z}{6}$
0
votes
0answers
19 views

Is there a term for two distance metrics that give the same ordering?

In order to calculate and sort things by distance from each other in a computer program I need to find an easy and fast distance metric to calculate. I can probably find one myself. However, first I ...
3
votes
1answer
43 views

Surface Area of unit n-sphere covered by rotating a unit vector around a fixed unit vector such that angle between the two vectors is always fixed.

Consider an n-dimensional unit sphere and unit vector from the origin with its tip lying on the surface of sphere. Consider another vector which makes some angle say $\epsilon$ with unit vector. From ...
4
votes
1answer
58 views

Moving circular disk between two parallel sinusoidal curves

Find the largest radius of the circle that can be "rolled" between the curves $y = sin(x)$ and $y = sin(x)+1$. After two weeks of research, I finally give up.
0
votes
1answer
28 views

Proof that an angle across line is equal to 180 degrees

When given a straight line, how do you prove that an angle across it is equal to 180 degrees, or two right angles? It feels like something that should be an axiom, but it isn't one of the 5 ...
2
votes
1answer
18 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
26 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
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 ...
0
votes
1answer
27 views

Angle between edge and lateral face of a regular pyramid

MABCD is a regular pyramid (ABCD is square and the lateral edges are equal). The angle between the base ABCD and the plane through BD, which is perpendicular to MC, is $\phi$. Then what is the sine of ...
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 $r \neq p,q$. ...
2
votes
1answer
54 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 ...
1
vote
2answers
34 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
34 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 ...
4
votes
2answers
91 views

Describing convex hulls in purely metrical terms

Let $X$ denote a Euclidean space; take $X = \mathbb{R}^n$ for concreteness. Now consider $x,y \in X$. Then the line segment joining $x$ and $y,$ hereafter denoted $[x,y]$, can be described in ...
8
votes
1answer
121 views

Instruct geometer moths so you can learn about their true geometry.

I had a space-ship wreck in an unknown world of some kind of moths. I could observe geometer moths working. Everything looked strange. The moths claimed that they were using only straight edges and ...
4
votes
1answer
34 views

Inequality involving lengths and triangles

I was quite sure this would have been asked before but I couldn't find it, so here goes: If $\displaystyle BC<AC<AB \hspace{5pt} (\alpha<\beta<\gamma)$, show $\displaystyle ...
-1
votes
2answers
51 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 ...
1
vote
2answers
47 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 ...
2
votes
3answers
58 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 ...
0
votes
2answers
98 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
27 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
68 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
32 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?
3
votes
1answer
24 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
35 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 ...
1
vote
2answers
44 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
52 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 ...
0
votes
0answers
29 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 ...
0
votes
1answer
23 views

Center and angle of complex function

Does a complex function of type $f(z)=az+b $ always have a center and angle (of rotation) or only when $b=0$ since $b\neq0$ represents a translation?
0
votes
0answers
16 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 ...