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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Express the length of the as a function of x

I am having problems understanding how to extract this information into a formula. ...
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1answer
49 views

What are the coordinates of a point on a rigid body after a rotation in 3D Euclidean space, given the initial coordinates and a center of rotation

Main question Let ($x_p$, $y_p$, $z_p$) be the initial coordinates of a point $P$ on a rigid body in a right-handed 3D Euclidean space. Let ($x_r$, $y_r$, $z_r$) be the coordinates of a center of ...
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103 views

A Modern Alternative to Euclidean Geometry

First of all, I want to master Geometry, I have knowledge on high school geometry and I was thinking of learning Euclidean Geometry. I bought a copy of Euclid's Elements, it is very interesting, ...
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87 views

A question about 4 concyclic points

In a triangle $ABC$, let $I$ denote its incenter. Points $D, E, F$ are chosen on the segments $BC, CA, AB$, respectively, such that $BD + BF = AC$ and $CD + CE = AB$. The circumcircles of triangles ...
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60 views

Why are these lines tangent?

I was trying the problems at http://euclidthegame.org and for level 20, ending up using, but couldn't see the reason behind the following: We have a circle centred on B and a point A outside the ...
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1answer
33 views

Proof about isometries, symmetry and reversing orientation.

Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is an isometry of the reals. Prove that $f$ is a symmetry around a point if and only if $f$ reverses orientation of $\mathbb{R}$. The orientation of ...
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2k views

Euclid the Game Level 2: Construct a line that bisects the given angle.

This is the level I'm trying to complete: My last idea was this, but I still don't get a message that I completed the level. Why is this not correct ? Last time I did mathematics was 10 years ...
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1answer
37 views

Proof about symmetry in isometries.

Suppose $f: \Bbb R \rightarrow \Bbb R$ is an isometry of the reals. Prove that $f$ is a symmetry about a point if and only if $f$ has a unique fixed point. Part 1: The assumption is $f$ is a ...
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36 views

Proof about isometries

Suppose $f\colon\mathbb R\to\mathbb R$ is an isometry of the reals. Prove $f$ is a non-trivial translation iff $f$ has no fixed points. Assumption: $f$ is a non-trivial translation (trivial ...
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67 views

A surprising locus of points

I was playing with the setup for Desargues Theorem in Geogebra today and got a very odd-looking (to me) result. I imagine I could grind through this analytically and get an ugly-looking parametric ...
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23 views

Given a pair of conics, construct (synthetically) their shared tangent lines

There are various well-known ways to construct the common tangents to a pair of circles; this is an easy one. I also just learned that we can use Pascal's Theorem to construct a tangent to a conic at ...
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53 views

Is triangle congruence SAS an axiom?

I was wondering if there is a way to prove SAS in triangle congruence with Euclidean axioms. Thank you for your help!
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Euclidean geometry please help me

http://i62.tinypic.com/5osi8n.jpg please help me just wrote exam and wanted to know whether it was correct or not. I said (m+n)(m-n)=(m-n)^2 so I think it is not A right angled triangle what do you ...
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1answer
47 views

Find the equation of a plane that is perpendicular to another plane, parallel to a line and goes through a point

Find the equation of a plane which is perpendicular to the plane $$\pi_1\equiv x-3y-z+1=0,$$ parallel to a line $$l\equiv\frac{x - 2}{2} = \frac{y -3}{-3} = \frac{z}{1}$$ and goes through point $P ...
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81 views

Three Circles Meeting at One Point

We have three triples of points on the plane, that is, $X=\{x_1, x_2, x_3\}$, $Y=\{y_1, y_2, y_3 \}$, and $Z=\{z_1, z_2, z_3\}$, where $x_i, y_i, z_i$ are points on the plane. I was wondering if there ...
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32 views

Mapping between two unknown 3D coordinate systems from common motion

Coordinate systems A and B are rigidly linked in an unknown way. The platform then moves and the motion vectors [RA|TA] and [RB|TB] are calculated in each coordinate system. They are parallel but not ...
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Axis of rotation of composition of rotations (Artin's Algebra)

Say $R_1$, $R_2$ are rotations in $\mathbb{R}^3$ with axes and angles $(v_1,\theta_1), (v_2,\theta_2)$ respectively. Since $SO_3$ is a group, we have that $R_2 \circ R_1$ is a rotation with some axis ...
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251 views

Difference between Euclidean space and vector space?

I often hear them used interchangeably ... they are very complicated to make any use of. Wikipedia words: Euclidean space: One way to think of the Euclidean plane is as a set of points ...
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23 views

What are geodesics in H$^2$?

Specifically, I am looking at a question that asks What axioms for a projective plane fail in the this space? Any two “points” are contained in a unique “line.” Any two “lines” contain a unique ...
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4answers
82 views

Euclidean Geometry in Physics

I've started tutoring my 13 year old niece in math. She learning geometry this year (it's a year-round school). Obviously, it'll just be basic Euclidean geometry -- though I might try to get to a ...
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1answer
49 views

Proving a strange vector inequality in the euclidean space

It seems to hold the following inequality in an euclidean reference frame $(x,y,z)$: $$\overrightarrow{U}\cdot\overrightarrow{U}\ge\sqrt{2}\left(\Omega_x+\Omega_y\right)$$ where: ...
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29 views

Find the equation of an $n$-Cone given the apex, a directional vector and the solid angle

Let $x$ be a point in $\mathbb R^n$ and $v$ a vector in $\mathbb R^n$. Find the equation of the cone with apex $x$ opening in the direction $v$ with a solid angle of $2\theta$. That is, on any ...
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1answer
143 views

Trying to use Cholesky decomposition of covariance matrix to sample error ellipsoid

I'm trying to construct an error ellipsoid from a covariance matrix (which exists for a 3D point) and then sample consistent xyz points in this region. In a previous question when I asked about this ...
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1answer
53 views

Parallelogram constructed through medians

Bdmo In $\Delta ABC$, Medians AD and CF intersect at P.Let Q be any point on AC.Construct QM and QN parallel to AD and CF respectively.Now the line joining M and N intersects CF and T and AD at ...
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1answer
99 views

What exactly does a Mobius Transformation do?

From what I understand, a Mobius transformation is of the form f(z) = $\frac{Az+D}{Cz+B}$ where A,B,C, and D may be real or complex What is f(z) doing to z exactly? And what are some of the ...
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34 views

How does a linear fractional function behave like a $2\times 2$ matrix?

So I did the math for this and got \begin{align*} A &= a_1a_2 + b_1c_2\\ B &= a_1b_2 + b_1d_2\\ C &= c_1a_2 + d_1c_2\\ D &= c_1b_2 + d_1d_2. \end{align*} My book does not talk ...
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1answer
60 views

Plane geometry.

I was reading proposition 14 of Euclid's elements and there is only one thing which I find weird: Why do we need postulate 4 to conclude that “the sum of the angles $\angle CBA$ and $\angle ABE$ ...
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1answer
175 views

What does it mean to call horizontal lines through O the “points at infinity” in real projective plane $RP^2$?

This is a picture from my book. I extended the line M to get a better idea of where $p_n$ is. It says the following: It is natural to call the horizontal lines through O the "points at infinity". ...
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1answer
55 views

The composition of two different glide reflections is a rotation

Denote by $G_{XY}$ a glide reflection which reflects around the $XY$-axis and then takes the point $X$ to $Y$. I would like to prove that the composition of two different glide reflections $G_{XY} ...
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25 views

n-gon Inequality Theorem Converse

In the plane, if we have an n-gon with side lengths $v_1$, $\ldots$,$v_n$, these lengths satisfy the "planar $n$-gon inequalities," ie. the length of each side is less than the sum of the lengths of ...
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applications of the identity $ab + \left(\frac{a+b}{2} - b\right)^2 = \left(\frac{a+b}{2}\right)^2$

I am reading euclid's elements I found the algebraic identity $ab + \left(\frac{a+b}{2} - b\right)^2 = \left(\frac{a+b}{2}\right)^2$ I ponder on usage of this identity for $2$ hours. but I can't ...
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57 views

Expected area of an inscribed triangle in a sphere

On the surface of a unit sphere, three points $A$, $B$ and $C$ are chosen in the following way: Points $A$ and $B$ are chosen randomly and independently on the whole surface After $A$ ...
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1answer
66 views

What is the length of GH?

What is the length of GH? Can you help me? I personally believe that this can't be solved?
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69 views

Calculate x if CD is a diameter of the circle

I cannot solve question 8c. Any advice on how to solve it? I was thinking of somehow making CD equal to CP.
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1answer
88 views

Generating Pythagorean Triples from Others via Dissections

Roger Alperin's paper Modular Tree of Pythagoras shows it is possible to generate Pythagorean triples from others. If $a,b,c$ are the sides of a right triangle $a^2 + b^2 = c^2$ then we can derive ...
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1answer
67 views

Show how to map the semicircle $x^2 +y^2 = 1$, $y > 0$, onto $(x−1)^2+y^2 = 4$, $y > 0$, by a combination of $z \to z+l$ and $z \to kz$.

I need some help with this one! One can begin to understand the geometric significance of linear fractional transformations of the half plane by studying the simplest ones, $z \to z+l$ and $z \to kz$ ...
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Regular polygons constructed inside regular polygons

Let $P$ be a regular $n$-gon, and erect on each edge toward the inside a regular $k$-gon, with edge lengths matching. See the example below for $n=12$ and $k=3,\ldots,11$.       Two ...
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1answer
12 views

Is there relations between earth mover's distance and vector norms?

Say I have two vectors $a$ and $b$. Can I estimate $\mbox{EMD}(a,b)$ via some combination of things like $\|a-b\|_p$ and such?
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Euclidean geometry and irrational numbers.

I was wondering, given a square that is $1 \times 1$, how can we know that the diagonal is an irrational length geometrically??? We could use the Pythagorean Theorem to see that the diagonal of a ...
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1answer
21 views

What do you call a projection of a hyperplane into a finite hypercube that keeps paraxial lines straight?

Similar to the Poincaré disc for hyperbolic space, is there a bijection from $\mathbb{R}^n$ into, say, $[-1,1]^n$, while any paraxial orthotope in $\mathbb{R}^n$ remains a paraxial orthotope after the ...
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1answer
39 views

Find Circle from Tangent Line and Two Points

I have two points a,b and a line L. I want an equation to find point c that is the center of the circle that touches a, b and L Thanks.
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2answers
112 views

How to prove this Gram determinant

Let $E$ be an Euclidian oriented vector space of dimension $3$ and $x,y,u,w \in E$. How do we prove (without coodinates) $$ \det \begin{pmatrix} \langle x,u \rangle & \langle x,w \rangle \\ ...
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40 views

Find the angles defining an hyperspherical cap

For the hyperspherical cap of dimension $n+1$ find all the angle $\phi_1, \phi_2, \ldots, \phi_n$ which defines the cap? I mean, I know a cap is usually define by its height $h$ and its base $a$. ...
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2answers
94 views

Why are polyhedra related to the prime numbers 2, 3 and 5, but not to the prime number 7?

Just take a quick glance at all the numbers in these Wikipedia pages on polyhedra: http://en.wikipedia.org/wiki/Platonic_solid http://en.wikipedia.org/wiki/Archimedean_solid ...
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3answers
213 views

Smallest square containing a given triangle

Given a triangle $T$, how can I calculate the smallest square that contains $T$? Using GeoGebra, I implemented a heuristic that seems to work well in practice. The problem is, I have no proof that it ...
5
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2answers
238 views

What is the smallest possible angle of this polygon?

A convex polygon contains a square with side-length 1 and is contained in a parallel square with side-length 2 (which is its smallest containing square). What is the smallest possible angle of the ...
2
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2answers
176 views

Area of Intersection of Circle and Square

Given a point $(x,y)\in [0,1]^2$ and $r > 0$, I would like to derive a general formula for the area of the intersection of the circle of radius $r$ centered at $(x,y)$ and the unit square. What is ...
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71 views

How to distribute a set number of points evenly in a rectangle?

So my motivation here is actually because I want to evenly distribute plants in a green house and ideally I would like to maximize the distance between the plants and the walls. It seems like there is ...
2
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1answer
62 views

How to prove $C=2πr$?

Everybody know this formula,but why the relation between $C$ and $r$ is linear relation? Not $C=2πr^{0.99}$ or $C=2πr^{1.01}$how to prove it,what axiom is it based on?
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1answer
29 views

Number of Lines formed by joining points formed by the intersection of lines in a plane

There are $\mathbf{n}$ lines in a plane no two of which are parallel. They intersect at $^{n}C_2$ distinct points in space. How many new lines are constructed by joining these points. The essential ...