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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Equation of a line about which we are reflecting

Let $A$ be the matrix of a reflection about a line of the euclidean plane (w.r.t. the standard basis). How can I find the equation of the line?
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Reflections in Euclidean plane

Let $T: \mathbb{R}^2 \to \mathbb{R}^2$ be the counterclockwise rotation of $\frac{\pi}{2}$ and $S: \mathbb{R}^2 \to \mathbb{R}^2$ be the reflection w.r.t. the line $x+3y=0$. There exists a reflection ...
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On the same straight line there cannot be constructed two similar and unequal segments of circles on the same side.

So, according to this figure : http://aleph0.clarku.edu/~djoyce/java/elements/bookIII/propIII23.html We cannot have similar segments of circles and unequal ones be built on the same side of the same ...
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Points on a 2D plane spanned by a turtle graphics system

Suppose you have a turtle graphics system with a set "turning angle" $\delta$, in which the turtle can execute three commands: $F$: Go forward, by unit length, in the current direction. (The initial ...
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1answer
19 views

In a circle the angle at the center is double the angle at the circumference when the angles have the same circumference as base.

I have the following theorem : "In a circle the angle at the center is double the angle at the circumference when the angles have the same circumference as base." (Figure is in the link) ...
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How can I find the diagonal of a quadrilateral? [closed]

Given a quadrilateral $MNPQ$ for which $MN=26$, $NP=30$, $PQ=17$, $QM=25$ and $MP=28$ how do I find the length of $NQ$?
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Horn angles and Euclid's elements.

We have the following statement by Euclid : "I say further that the angle of the semicircle contained by the straight line BA and the circumference CHA is greater than any acute rectilinear angle, and ...
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1answer
34 views

Euclid's elements proposition 15 book 3

http://aleph0.clarku.edu/~djoyce/java/elements/bookIII/propIII15.html I have understood the proof in general. It is only a small detail which i'm not sure. Maybe it's because english isn't my first ...
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33 views

Euclid's elements proposition 13 book 3

"Then, since on the circumference of each of the circles ABDC and ACK two points A and C have been taken at random, the straight line joining the points falls within each circle, but it fell within ...
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1answer
73 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 ...
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1answer
41 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 ...
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60 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 ...
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41 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 ...
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1answer
29 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 $x$ ...
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58 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 ...
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2answers
27 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 ...
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1answer
65 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 ...
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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.
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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 $\left\{\phi_i, i=1,\dots,D-m\right\}$ ...
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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 ...
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2answers
35 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 ...
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42 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?
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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 ...
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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: ...
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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 $B\in\mathbb{R}^{n\times n}$ ...
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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 ...
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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 $\frac{x}{2}=\frac{y}{3}=\frac{z}{6}$
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22 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 ...
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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
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1answer
65 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.
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1answer
32 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 ...
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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 ...
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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 = ...
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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 ...
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1answer
33 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 ...
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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 $r \neq p,q$. ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
37 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 ...
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2answers
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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 ...
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51 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 ...
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3answers
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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 ...
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2answers
110 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 ...
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2answers
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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 ...
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71 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 ...
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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 ...
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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?