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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Questions about elipse

Given the center of an elipse and three of its points, is this elipse completely determined? What is the easiest way to show that five points of an elipse are enough to determine the elipse?
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1answer
48 views

How can we draw a line?

We want to join by a line two distinct points $A$ and $B$. We have only a ruler of length $l>0$ and a pen. If $AB>l$ how can we do this?
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1answer
21 views

Specific triangle, symmetral of angle proof

In triangle $ABC$, angle $\gamma = 120$. Prove that $|\overline{CC'}|=\tfrac{ab}{a+b}$, where $\overline{CC'}$ is symmetral of angle $\gamma$ inside triangle. Look at image. I can't use areas, ...
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1answer
20 views

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, ...
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1answer
337 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
14 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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13 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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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? [closed]

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 ...
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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 ...
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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 ...
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1answer
13 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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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 ...
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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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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 ...
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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 ...
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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
18 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
12 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 ...
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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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44 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.
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1answer
70 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 ...
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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
40 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
26 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
61 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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0answers
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 ...
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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 ...
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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 ...
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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
34 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 ...
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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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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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0answers
31 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
116 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
24 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$ ...