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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How to represent two coordinate system transformations as one

I'm working on a system of relative euclidean coordinate systems. I'd like to define every coordinate system relative to a global coordinate system, which I'll refer to as [0]. Then, for example, ...
10
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1answer
325 views

The number of the circles which are tangent to two circles and to a line

Suppose that we have two distinct circles and a line on a plane and that the distance between the centers of the circles is bigger than the sum of their radiuses. Also, suppose that the two circles ...
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0answers
12 views

Dandelin spheres and the asymptotes of a hyperbola

The other day, I was reading up on the synthetic geometry of conic sections a bit, and I wondered: is it possible to construct the hyperbola's asymptotes given just the intersecting plane and the ...
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0answers
15 views

What is the analog of the scalar triple product in two dimensions?

Is there a standard name and/or a notation for the analog of the scalar triple product in two dimensions? Namely, i am interested in the following operation: given two elements $\vec u$ and $\vec v$ ...
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12 views

Circumcentre of three points X, Y, Z, given distance from each to points A and B

I'm racking my brain trying to figure out where to start on this, and it's been too many years since working on these kinds of problems. I have six measurements which I'd like to use to calculate a ...
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6 views

What is the proper name of a point a long a smooth curve where the radius changes but not direction of curvature?

What you call a point a long a smooth curve where the radius changes? When it reverses curvature, it’s an “inflection point”. What if it doesn’t change direction, just radius? I seem to remember ...
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1answer
40 views

euclidean distance between one dimension points-how to? [on hold]

i am reading a research paper about round-robin scheduling algorithm that uses Euclidean distance to determine a time quantum based on similarity of burst times of all processes in the ready queue,but ...
3
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2answers
66 views

How long is the curve that a creature walks?

I have a problem in solving mathematical problem. Take a ball with radius 60 cm. A creature walk from the southpole to northpole by following the spiral curve that goes once around the ball every ...
3
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3answers
51 views

5 points on a plane with rational distances

Can you find 5 points on a plane whose Euclidean distances between them are all rational numbers and no 3 points out of them are co-linear? If the answer is yes, can we find a construction for ...
0
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1answer
12 views

Tranversal parallel lines theorem

I need to prove Tranversal parallel lines theorem that says: If two parallel lines are cut by a transversal, the corresponding angles are congruent, the alternate angles are congruent, and the ...
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0answers
52 views

Calculating Euclidean dissimilarity for a given cluster with itself

Suppose I have clusters $$A= \{(1,1)^T, (1,2)^T\} $$ $$B=\{(2,3)^T, (3,4)^T\} $$ $$C= \{(4,5)^T, (5,6)^T, (1,2)^T\} $$ I wish to use the Euclidean dissimilarity and Average linkage to calculate a ...
2
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1answer
24 views

A triangular inequality including squares of sides

Show that for any triangle ABC, the following inequality is true $$a^2 + b^2 + c^2 > \sqrt{3} \max\{|a^2-b^2|,|b^2-c^2|,|c^2-a^2|\}$$ where $a,b,c$ are the sides of the triangle
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13 views

Central symmetry

There's definition: Central symmetry $s_O:M \to M$ is bijection defined as $s_O(T)=T'$ if and only if $O$ is midpoint of $\overline{TT'}$. 1st: Prove that central symmetry is involution ($s_O \circ ...
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1answer
221 views

Proof of a certain lemma in geometry

In the following article: http://yufeizhao.com/olympiad/geolemmas.pdf in the proof of the fact about the diameter of the incircle on page 2, the author claims that the proof that $BD = CF$ follows ...
2
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1answer
22 views

How can I uniformly draw points from an ellipsoid?

Specifically, given a positive definite matrix $A \in \mathbb{R}^{n \times n}$, how can I efficiently generate points $x \in \mathbb{R}^n$ that satisfy $x^TAx \leq 1$? I know how to do this when the ...
1
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1answer
21 views

Pairs of isometries that jointly fix a set (revised)

Let $E$ be a Euclidean space, and let $X, Y\subseteq E$ be two disjoint subsets. Suppose that there exist isometries $i$ and $j$ such that $i(X)\cap j(Y)=\emptyset$ and $i(X)\cup j(Y)=X\cup Y$. Does ...
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0answers
17 views

Inequalities in a quadrilateral

In a quadrilateral ABCD, it is given that AB is parallel to CD and the diagonals AC and BD are perpendicular to each other. Show that ...
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1answer
23 views

Plane-geometry problem with circles and tangents

I have a problem that even my smartest colleagues were able to solve. This is to get the radius of the smallest circle in the drawing below. Using a computer program, I managed to get that lightning ...
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0answers
10 views

Convexity of circle in neutral geometry

I am trying to prove that a circle is convex in neutral geometry. i.e. If $A$ and $B$ are inside a circle $C$, than any point in $AB$ is also in $C$. But I have difficulty in proving it. The case ...
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2answers
27 views

Is this a Trapezium?

I once read that in hyperbolic geometry, two hyperbolas can be parallel. In a trapezium, you have four sides and a pair of parallel lines, therefore is it possible to have a trapezium with two ...
4
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1answer
28 views

Repeated projection of points onto lines

Consider a point $P$ on the Euclidean plane, and lines $l_1,l_2,\ldots,l_n$. Project $P$ onto $l_1$. Then project the resulting point onto $l_2$. Then project the resulting point onto $l_3$, and so ...
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39 views

how to find angle between two added up vectors in cartesian space

I would like to find the angle between two vectors (theta) -> v1 From i to i+1 v1=(xi1-xi , yi1-y1) and v2 from i+1 to i+2 v2=(xi2-xi1, yi2-yi1), which are shown as in the figure (but v1 and v2 can be ...
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1answer
33 views

Nice geometry with areas

Consider triangle $ABC$. Let $H$ ortocentre and $O$ circumcentre, then prove that area of one of triangles $AOH$,$BOH$,$COH$ is equal to the sum of areas of other two.
2
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1answer
69 views

Pairs of isometries that jointly fix a set.

Let $E$ be a Euclidean space, and let $X, Y\subseteq E$ be two disjoint subsets. Suppose that there exist isometries $i$ and $j$ such that $i(X)\cap j(Y)=\emptyset$ and $i(X)\cup j(Y)=X\cup Y$. Does ...
1
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1answer
30 views

Can these phenomena occur within Non-Euclidean geometries?

I've enrolled in an undergraduate seminar on the subject of non-euclidean geometry. I wanted to ground myself a little before-hand, because popular media has lead me to believe that non-euclidean ...
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1answer
39 views

By proposition 3.21 is ether acute, right or obtuse

Let A, B, C be points on a line with A ∗ B ∗ C. Let D be a point not on that line such that angle ABD is obtuse. Show that angle CBD is acute. Do NOT use the Measurement Theorem. (Hint: By ...
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1answer
25 views

Do collinear lines or overlapping collinear line segments intersect?

I am writing a function to find the intersection of a pair of lines and another function to find the intersection of a pair of line segments. The parallel case and the single point intersection case ...
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2answers
24 views

On subgroups of isometries and their respective fixed-points

I am working on the following problemset: Let $G < \DeclareMathOperator{\Iso}{Iso}\Iso(E)$ be a finite subgroup of the isometries in the euclidean plane. Denote: $$G = \{g_1, \ldots , g_n \} ...
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2answers
59 views

A geometry problem involving geometric mean

$ABCD$ is a quadrilateral inscribed in a circle of center $O$. Let $BD$ bisect $OC$ perpendicularly. $P$ is a point on the diagonal $AC$ such that $PC=OC$. $BP$ cuts $AD$ at $E$ and the circle ...
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0answers
31 views

Distance on a 3-sphere

The arc-length $l$ between two points on on a 2-sphere of radius $R$ is given by $l=R\theta $ where $\theta$ is the subtended angle. I can rewrite this in terms of the euclidean distance $d$ between ...
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31 views

“Polysticks” in 3d

Consider a finite set of three-dimensional Euclidean vectors with integer components. How many three-dimensional closed loops can I construct with them? How many of them are elementary, i.e., cannot ...
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22 views

On power of the point

Let $P$ be a point outside the circle $O$ and $A$ a point on $O$ where $PA$ is tangent to $O$. Let $B, C$ be points on the $O$ such that $P$, $B$, and $C$ are collinear. Then $PA^2=PB\cdot PC$. Is ...
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1answer
26 views

Neutral Geometry and Proclus' Axiom

Find a counter-example to Proclus' Axiom in the Klein disk. What can you conclude about Euclid's Parallel Postulate in the Klein disk?
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3answers
42 views

Has triangle an angle?

I read axiomatic geometry and found the following definitions: Points $A$ and $B$ and all those points that lie between those points is a line segment. If $AB$ and $AC$ are two rays that does not ...
0
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1answer
22 views

Perimeter Of A Simple Triangle

Here in $ \triangle ABC$ $ AC=4 , DE= EF =1, \angle ABC=90^{\circ} $. The perimeter of the triangle $ \triangle ABC$ can be written as $ \sqrt {m } + n $ where $m$ and $n$ are non-negative ...
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1answer
26 views

Similarity. Finding distance.

Consider the figure. It is supposed to be a tennis court. A ball is served at $F$. It's trajectory is a straight line. The ball touches the ground at $A$. Find the distance $\mathbf x'$ from $A$ to ...
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1answer
30 views

Distance of a point from a line

Let $w\in\mathbb{R}^{2}$ be a vector and denote $W=sp\{w\}$. Then $W^{\perp}$ is also a one dimensional space i.e is a line, denote this line as $l_{w}$. Given a line we can shift it from the origin ...
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1answer
34 views

Meaning of “circumference”

I am French and I have to solve a math problem written in English. The wording is the following : " In triangle ABC, the angle bisector of angle A intersects line BC at D and the circumference of ...
4
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1answer
26 views

Minimum dimension to hold $N$ points with given distances?

Suppose you're given $N$ points along with an $N\times N$ matrix $D$ with entries $d_{ij}$ giving the distances between the points (assume that the $d_{ij}$ satisfy the usual requirements of a ...
2
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2answers
40 views

Square inside triangle

Is there a triangle ABC having following properties: ABC is not right angled $d(A,B)\ne d(A,C)$ If one draws the largest possible square such that one of the side of square is a subset of $AB$ and ...
2
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1answer
33 views

Barycentric Coordinates of the circumcenter of an arbitrary triangle

Given points $A(1, 0, 0), B(0, 1, 0), C(0, 0, 1)$ in barycentric coordinates, and points $P(x_P, y_P, z_P), Q(x_Q, y_Q, z_Q), R(x_R, y_R, z_R)$, what would the barycentric coordinates of the ...
2
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0answers
32 views

A condition for both cyclic and tangential quadrilaterals

I'm looking for a nice condition that characterizes quadrilaterals that are both cyclic and tangential (i.e. there exists a circle that touches each side). I know that both concepts have some nice ...
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0answers
32 views

questions about the taxicab geometry

Given: distance function given by the sum of the absolute values: ̃$d(AB) = |a1 − b1| + |a2 − b2|$, (1) What does the circle with center $O = (0, 0)$ and radius $1$ look like in taxi-cab geometry? ...
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1answer
68 views

Finding circle with two points on it and a tangent from one of the points

Two points P1(x1,y1) and P2(x2,y2) are known. In addition, a line slope passing through P1 is known. The aim is to construct a circle (or circular arc) that it passes through both P1 and P2 and it is ...
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30 views

Verifying certain congruence axioms in taxicab geometry

Given: I need some help I've shown what I have so far d(A, B) = |a1 − b1| + |a2 − b2| where A = (a1, a2) and B = (b1, b2). Some people call this ...
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2answers
115 views

Finding an angle between side and a segment from specified point inside an equilateral triangle

Here is the question: $\overset{\Delta}{ABC}$ is an equilateral triangle. D is a point inside triangle. $m(\widehat{BAD})=12^\circ$ $m(\widehat{DBA})=6^\circ$ $m(\widehat{ACD})=x=?$ I managed to ...
4
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0answers
90 views

Why exactly is Bourbaki difficult?

I keep hearing people say that Bourbaki is difficult for most undergraduates but I still don't understand why. Surely if it starts from definitions/axioms then practically anyone should be able to ...
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2answers
23 views

How to compute volume of a circle defined by L1 distance?

In n dimension space, given a central $x=(x_1,x_2......x_n)$ and radius r, a circle C is defined as all point $y=(y_1,y_2,.....y_n)$ satisfy $ \sum_{i=1}^n\left\lvert y_i-x_i\right\rvert <= r$ ...
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1answer
27 views

what is the area of the polygon with given constraints?

What is the area of the polygon formed by all points $(x, y)$ in the plane satisfying the inequality $ ||x| – 2 | + | |y| – 2 | ≤ 4 $ ?
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1answer
48 views

What *can* Euclid prove?

It is well-known that Euclid's axioms for geometry are not up to modern standards of rigor: in particular, there are a lot of times when he used "obvious" facts about the geometric objects which were ...