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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68 views

Two prove two lines in a triangle are parallel

$D$, $E$, $F$ are the midpoints of sides $BC$, $CA$ and $AB$ respectively of a triangle $ABC$ right angled at $C$. If $EF$ and $DF$ (extended if necessary), meet the perpendicular from $C$ on $AB$ in ...
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1answer
34 views

Looking for an alternative proof of the angle difference expansion

I have thought about this for a while and have no progress. Does there exist a purely Euclidean Geometric proof of the Angle Difference expansion for Sine and Cosine, for Obtuse angles?
3
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3answers
38 views

Question about concyclic points on the coordinate axes

If the points where the lines $3x-2y-12=0$ and $x+ky+3=0$ intersect both the coordinate axes are concyclic,then the number of possible real values of k is (A)1 ...
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2answers
50 views

External bisectors of the angles of ABC triangle form a triangle $A_1B_1C_1$ and so on

If the external bisectors of the angles of the triangle ABC form a triangle $A_1B_1C_1$,if the external bisectors of the angles of the triangle $A_1B_1C_1$ form a triangle $A_2B_2C_2$,and so on,show ...
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0answers
107 views

Average Degree of a Random Geometric Graph

A set of $N$ points are distributed randomly on a unit square with uniform distribution. Two points $\mathbf{p}_i$ and $\mathbf{p}_j$ are said to be connected if $\|\mathbf{p}_i - \mathbf{p}_j\| \leq ...
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1answer
317 views

A problem on triangle and its perpendicular bisectors.

I'm trying to solve the following problem : "In △ABC, coordinates of $B$ are $(−3, 3)$. Equation of the perpendicular bisector of side $AB$ is $2x + y − 7 = 0$. Equation of the perpendicular ...
7
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1answer
110 views

A movement requires 1 dimension, a rotation requires 2 dimensions, a what requires three dimensions?

Movement A zero-dimensional object cannot move. A one-dimensional object can move in one dimension (the x-axis). A two-dimensional object can move in two dimensions (the x-axis and y-axis). A three ...
22
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3answers
1k views

A circle in the plane contains at most four lattice points?

Let $\cal C$ be a circle in ${\mathbb R}^2$ : $\cal C=\lbrace (x,y)\in{\mathbb R}^2 | (x-x_0)^2+(y-y_0)^2=r^2\rbrace$ for some constants $x_0,y_0,r$. What is the maximal number of points that can ...
2
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2answers
47 views

Prove line connecting intersection of tangents and opposite vertex bisects segment containing intersection of tangents and a vertex

Let $\triangle ABC$ be an isosceles triangle with $AB=BC$. Let $\Gamma$ be the circumcircle of $\triangle ABC$. Let the tangents at $A$ and $B$ intersect at $D$, and let $DC\cap\Gamma=E\neq C$. Prove ...
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4answers
88 views

Find the area of a triangle given the radius of its incircle and a tangential point

A friend gave me recently the following interesting problem and I would like to share a couple of solutions. Any additional contributions are welcome. A triangle $\vartriangle ABC$ is given and we ...
2
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1answer
45 views

One special case of Helly's theorem (for $\text{radius}=1$ circles)

There are $n$ points on the plane. Any $3$ of them can be covered with a radius $1$ circle. Prove that there is a radius $1$ circle that covers all the points. Came to this when tried to prove an easy ...
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1answer
367 views

Internal polygon formed by drawing diagonals in a regular polygon

In an n-sided (n>4) regular polygon, label the vertices {0, 1, ..., n-1}. For each vertex i, draw a pair of diagonals: from i to (i+2) mod n and from i to (i-2) ...
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4answers
77 views

How to prove that the line perpendicular to the radius is the tangent in the calculus sense?

Let $P=(p_1,p_2)$ be a point on an semicircle and $r$ be the line perpendicular to the radius $\overline{OP}$, like the picture below. Euclid showed (Book III, Proposition 16) that $r$ does not ...
54
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17answers
22k views

What is the most elegant proof of the Pythagorean theorem? [on hold]

The Pythagorean Theorem is one of the most popular to prove by mathematicians, and there are many proofs available (including one from James Garfield). What's the most elegant proof? My favorite ...
0
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1answer
41 views

equally spaced on circle question

Define $$\|\vec{x}\|:=\sqrt{\alpha^2+\beta^2},$$ where $\vec{x}:=(\alpha,\beta)\in \mathbb{R}^2.$ Set $$\mathbb{S}^1:=\{\vec{x}\in \mathbb{R}^2: \|\vec{x}\|=1\}\quad \quad and\quad \quad ...
15
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1answer
239 views

Hypercube and Hyperspheres

Let $n,k\in\mathbb{N}$. In this problem, the geometry of $\mathbb{R}^n$ is the usual Euclidean geometry. The lattice hypercube $ Q(n,k)$ is defined to be the set $ \{1,2,...,k\}^n ...
2
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2answers
31 views

Using the cross product to find the angle between two vectors in $\Bbb R^3$

Let $$u = \langle 1, −2, 3 \rangle \qquad \text{and} \qquad v = \langle −4, 5, 6 \rangle$$. Find the angle between $u$ and $v$, first by using the dot product and then using the cross product. ...
23
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2answers
385 views

When is a metric space Euclidean, without referring to $\mathbb R^n$?

Normally, the Euclidean space is introduced as $\mathbb R^n$. However, I've now been thinking about how one might define the $n$-dimensional Euklidean space only from the properties of the metric. ...
0
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0answers
30 views

Plotting Distance Constrained Points on a Plane

Does anybody know of some algorithmic way to tell if it is possible to plot a set of distance constrained points on a cartesian plane. Or, better still, a method to determine the minimum number of ...
2
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4answers
802 views

Two circles inside a semi-circle

Two circles of radius 8 are placed inside a semi-circle of radius 25.The two circles are each tangent to the diameter and to the semi-circle.If the distance between the centers of the two circles is ...
4
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1answer
55 views

8 cubes ($2$x$2$x$2$) crossed by a straight line

There are 8 cubes forming a bigger cube whose dimension is $2$ x $2$ x $2$. Let a straight line (or a laser) try to pierce through as many small cubes as possible. At most how many small cubes can be ...
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2answers
33 views

Models of Incidence Geometry

First I'd like to thank you for reading this, there's a lot here. Secondly I am having a lot of trouble negating the first axiom which I think is hindering my ability to apply it to these types of ...
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1answer
21 views

Equivalence of euclidean and analytic geometry [closed]

I read about the axioms of euclidean geometry. How is analytic geometry rigorously defined? What are the axioms? And most important: How to prove that all the results proved in analytic geometry are ...
0
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2answers
248 views

Area of quadrilateral inside a triangle.

Let us consider an isoscles triangle ABC where $AB=AC=\sqrt{13}$,and length of $BC=4$.The altitude on $BC$ from $A$ meets $BC$ at $D$.Let $F$ be the midpoint of $AD$.We extend $BF$ such that it meets ...
2
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1answer
15 views

Incident vector for lines in a 2D-Euclidean Geometry over Finite field

Consider the 2-D $EG(2,2^2)$ geometry. Let $\alpha$ be a primitive element of $GF(2^{2\times 2})$. The incident vector for the line $\mathcal{L} = \{\alpha^7, \alpha^8, \alpha^{10}, \alpha^{14}\}$ is ...
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0answers
17 views

Replacing Euclid's fifth postulate by any of two equivalent ones.

It is well-known that Euclid's fifth postulate has (at least) two equivalent substitutes. However, I fail to see how equivalent they are. How does one prove that the following postulates are ...
2
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2answers
69 views

Triangle Geometry and Circles Problem

I have discovered something using Geogebra and I am positive it is true. I have tried to prove and my solution works but it is extremley convoluted. I'm hoping someone can provide a simple proof of ...
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votes
4answers
51 views

Find the distance between the point $(0,0,0)$ and the plane $2x+3y+z=1$ [closed]

Find the distance between the point $(0,0,0)$ and the plane $2x+3y+z=1$. So I know in order to find the distance I need two points. How do I find a point in the plane?
2
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1answer
62 views

Point on the Plane, a Triangle, and a Lower Bound of a Ratio Sum

Let $ABC$ be a triangle on the Euclidean plane. At which point $P$ on the plane does the ratio sum $\frac{PA}{BC}+\frac{PB}{CA}+\frac{PC}{AB}$ attain its minimum value? Prove also that, for any ...
4
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1answer
38 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 ...
7
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1answer
179 views

Why does the Pythagorean Theorem have its simple form only in Euclidean geometry?

Below are the right-angled forms of the Pythagorean Theorem in elliptic, Euclidean, and hyperbolic geometry, respectively. $$\cos\left(\frac{c}{R}\right) = ...
10
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2answers
11k views

What is the difference between Euclidean and Cartesian spaces?

For me those are references to the same thing. On Wikipedia there are references to both but I still don't see the difference. http://en.wikipedia.org/wiki/Cartesian_coordinate_system#Cartesian_space ...
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0answers
19 views

what is exactly the fine-scale geometry of Euclidean space?

I have learnt math for a while, but I really do not have impressions on the concept of fine-scale geometry of Euclidean space. I hope this is not a trivial question as it sounds like. Any comments ...
1
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1answer
100 views

Find A such that $A^2 \neq I$ but $A^4 = I$ [duplicate]

Find a $3 \times 3$ matrix A such that $A^2 \neq I$ but $A^4 = I$, where $I$ is the $3 \times 3$ identity matrix. Is there a simpler way to solve this problem rather than bashing it out by ...
1
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1answer
33 views

Distance between two 3D lines

What is the distance between the 3D lines $x = \begin{pmatrix} 1 \\ 2 \\ -4 \end{pmatrix} + \begin{pmatrix} -1 \\ 3 \\ -1 \end{pmatrix} t$ and $y = \begin{pmatrix} 0 \\ 3 \\ 5 \end{pmatrix} + ...
10
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1answer
74 views

A generalisation of Napoleon's theorem. Is this result original?

I've found a generalisation of Napoleon's theorem to general polygons. Take any regular $n$-gon inscribed in a circle and stretch it (in any direction) so that the circle becomes an ellipse and the ...
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5answers
1k views

Why the surface of the sphere is not a Euclidean space?

In wiki, I see a description in manifold: "Although a manifold resembles Euclidean space near each point, globally it may not. For example, the surface of the sphere is not a Euclidean space, but in ...
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1answer
43 views

Point inside a triangle

let P be any point inside a triangle ABC.If $AP,BP,CP$ meet the sides $BC,CA,AB$ at points $D,E,F$ respectively,then prove that $PD+PE+PF< max(a,b,c)$ where $a,b,c$ are the lengths of the sides ...
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2answers
404 views

Bounding box enclosing circles, that complies with ratio constraints

Given a circle centered at $A$, with radius $R_a$ and another radius $R_b$, I need to find a center for circle $B$ such that both circles are tangential, and the bounding box including both circles ...
0
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0answers
24 views

Characteristic polynomials for matrix A, involving the Identity matrix

Let us say we have a square matrix A, where A's characteristic polynomial is defined as $P_A(t) = \det (t I - A)$ (In this problem, I represents the identity matrix which has the same dimensions as ...
3
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2answers
89 views

Seven points in the plane such that, among any three, two are a distance $1$ apart

Is there a set of seven points in the plane such that, among any three of these points, there are two, $P, R$, which are distance $1$ apart?
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5answers
1k views

Fit a equilateral triangle on three arbitrary parallel lines with an edge and compass

How can you fit a equilateral triangle on three arbitrary parallel lines with an edge and compass?
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0answers
21 views

Is a composition of two $n-1$-dimensional symmetries a composition of $n-2$-dimensional symmetries?

Let $X$ be a finite dimensional real Euclidean space and $S,T$ be symmetries with respect to $n-1$-dimensional subspaces of $X$. Is it possible to write $ST$ as a composition of symmetries with ...
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4answers
38 views

Geometry question about lines

If I have two points in euclidean space or the Cartesian plane whichever and both points lie on the same side of a straight line. Both above or both below- how can I show that the segment connecting ...
0
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0answers
27 views

Is my proof of 'inscribed angle theorem' different from the usual one?

The usual proof of inscribed circle theorem uses the fact that the sum of interior angles is equal to exterior angle. I've formulated a little different approach, although the underlying concept is ...
12
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1answer
82 views

Uniqueness of a configuration of $7$ points in $\Bbb R^2$ such that, given any $3$, $2$ of them are $1$ unit apart

This question from earlier today asks (paraphrasing here): Is there a configuration of $7$ points in the Euclidean plane such that, given any $3$ of the $7$ points, at least $2$ of them are $1$ ...
3
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4answers
1k views

What precisely is the difference between Euclidean Geometry, and non-Euclidean Geometry?

I was wondering, what it is precisely which defines the difference between Euclidean and non-Euclidean Geometry, in a few words/equations/diagrams? Would I be correct in understanding that ...
2
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3answers
60 views

How to find the sum of distances so that it is minimal?

Question: $A$ and $B$ are two points on the same side of a line $l$. Denote the orthogonal projections of $A$ and $B$ onto $l$ by $A^\prime$ and $B^\prime$. Suppose that the following distance are ...
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1answer
18 views

Generalization of the Saccheri-Legendre Theorem Proof

So I'm working on generalizing the Saccheri-Legendre Theorem to convex $n$-gons. $\underline{\text{Saccheri-Legendre Theorem:}}$ The sum of the angles of a triangle is at most $180^\circ$. A ...
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0answers
18 views

When to use which condition number? (which norm)?

The condition number is used to determine how sensitive b is to changes in A in the equation ...