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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Discretization of Unit Vector in 3D

I cant think of a thing that I think is supposed to be easy... =/ Im glad if you could help me. Im working with a regular discretization of a 3d euclidean space. Cubic cells. Then, after a ...
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1answer
50 views

What is real $R$ so that every subset of Euclidean space with diameter one is inside a ball of radius $R$?

What is infimum of real numbers $R$ so that for every $n$ every $S \subseteq \mathbb{E}^n$, for which $d(S) = \sup\{|x-x'| \mid x,x' \in S \} = 1$, is inside some closed $n$-ball of radius $R$? In ...
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22 views

The necessary and sufficient condition for a regular n-gon to be constructible by ruler and compass.

I have a problem concerning the necessary and sufficient condition for a regular n-gon to be constructible by ruler and compass. $\bf My$ $\bf question:$ For a given positive integer $n$, how can we ...
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0answers
23 views

To a given straight line in a given rectilinear angle, to apply a parallelogram equal to a given triangle.

There's again one small detail on which I'm not sure. (Proposition 44 - book 1) http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI44.html Here's the quote : "Then HLKF is a parallelogram, HK ...
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Making a metric out of distance measure

I'm working with a pseudo-distance measure that is not a metric since it does not hold the triangle inequality. It is called Dynamic Time Warping. The problem is - I need to perform some projections, ...
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0answers
23 views

Triangles which are on the same base and in the same parallels equal one another.

I have a small question regarding proposition 37 of the elements of Euclid. http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI37.html The only problem I got with the proof is the fact that we ...
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4answers
122 views

What is the area of shaded region which is lies between outer and inner circle.

There is a outer circle with radius 2r and another inner circle with radius r whose center is the middle of big circle.As depicted in the following figure. Foo graph Image There is a sector of 120 ...
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Straight lines which join the ends of equal and parallel straight lines in the same directions are themselves equal and parallel

I have a small question regarding proposition 33 of the elements of Euclid. http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI33.html We want to prove that two lines joining equal parallels ...
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1answer
15 views

Action of the Euclidean group, generalizing linearity?

I have a vector $v \in \mathbb{R}^2$ and two elements $(A,a)$ and $(B,b)$ of the Euclidean group $E(2)$. If the relation $$[(A,a)(B,b)](v) = v$$ holds, can I say that $(A,a)(B,b)$ is the neutral ...
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A-Level/GCSE Geometry textbook? Geometry for STEP and MAT?

everyone. I have been looking for a book that covers the most elementary parts of Geometry, such as similar triangles, circles(arc, sector and others), Pythagorean theorem, Sine and Cosine Laws, so ...
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1answer
21 views

Be $m$ and $n$ two perpendicular lines, and …

Be $m$ and $n$ two perpendicular lines, and be distinct points $A$ and $B$ outside the lines and in the first quadrant. What is the shortest way to get from point $A$ to point $B$ by tapping the two ...
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0answers
19 views

On the solutions of a system of inequalities avoiding Helly's theorem

Let $a_1,b_1,\cdots,a_4,b_4\in\mathbb{R},r_1,\cdots,r_4\in(0,+\infty)$. Show that, if $\not\exists (x,y)\in\mathbb{R}^2$ such that $$ \begin{cases} (x-a_1)^2+(y-b_1)^2\le r_1\\ ...
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1answer
38 views

Performing projections with distance different to Euclidean

This is the first time I'm asking a question on math section of stackexchange, so please excuse me if this isn't the right place for such a question. I'm a programmer studying about an algorithm ...
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2answers
304 views

Diameters and Circles

I have a question (given by a teacher) that looks really easy but then when I thought about it, couldn't find a way to find the answer. It is a proof question relating to diameters: Prove that any ...
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2answers
91 views

Hexagon packing in a circle

Suppose I want to pack hexagons in a circle, as on the drawing below (red indicates "packed" hexagons). I am wondering what is known about this problem. Specifically, I am interested in an ...
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1answer
904 views

calculating volume of a horizontal cylindrical tank from depth

Has anyone found a good formula to convert a depth of fluid to a volume remaining in a tank for a cylindrical tank laying horizontal? (with or without half sphere end caps) Much Thanks!
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1answer
23 views

Side-angle-side and side-angle-angle as proved by Euclid in the Elements (Proposition 26)

I have small question regarding this proposition : http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI26.html To prove that one side is equal to another, Euclid assumes that one side is bigger ...
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1answer
21 views

What is the fraction of volume of unit hypersphere centered at one of the vertices of hypercube to that of hypercube?

consider a hyper-cube of n-dimension having a length of "r" units across each dimension. If a unit n-dimensional sphere is present at one of the vertices of the hyper-cube. what fraction of volume of ...
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2answers
32 views

Given a triangle $ABC$, make it a point $D$ on the side $AB$.

Given a triangle $ABC$, make it a point $D$ on the side $AB$. Show that $\overline {CD}$ is smaller than the length of one of the sides $BC$ and $AC$. Ideas? The triangular inequality will not. I ...
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1answer
330 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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Proofs of the three-perpendiculars theorem

I have to prove this theorem in three different ways. I have already proved it geometricaly and using vectors, but I can not think of any other way. Theorem: If PQ is perpendicular to a plane XY and ...
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1answer
12 views

Subset convex of plane

A plan of the subset is $convex$ if the segment connecting any two of its points is fully contained therein. The simplest examples of $convex$ $sets$ are the plan itself and any half-plane. Show that ...
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2answers
41 views

Hard problem about law of the cosine

I have been trying without success to prove by contradiction the following problem: Given 5 segments $x_1\leq x_2\leq x_3\leq x_4\leq x_5$ each three of which are sides of a triangle. Prove that ...
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2answers
359 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 ...
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1answer
12 views

Solve a convex quadrilateral with four sides and equality of two adjacent angles analytically?

Given the length of four sides of a convex quadrilateral and knowing that two adjacent angles are equal, the quadrilateral is determined. I want to know whether there's a formula representing the ...
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2answers
37 views

Denial of the 5th postulate of Euclid

I am trying to recover the denial of the Playfair's axiom but it is logical a bit strange. "To a given line and a point not on it, there is only one line through this point parallel to it". This ...
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2answers
541 views

Can I represent groups geometrically?

I have just taken up abstract algebra for my college and my professor was giving me an introduction to groups, but since I like geometric definitions or ways of looking at stuff, I kept thinking, "How ...
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4answers
413 views

Some theorems in euclidean geometry have incomplete proofs

I have seen that, in euclidean geometry, proofs of some theorems use one instance of the 'geometric shape'(on which the theorem is based) to proof the theorem. Like, the proof of 'A straight line ...
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2answers
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The shortest distance between any two distinct points is the line segment joining them.How can I see why this is true?

On a euclidean plane, the shortest distance between any two distinct points is the line segment joining them. How can I see why this is true?
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What is a point?

In geometry, what is a point? I have seen Euclid's definition and definitions in some text books. Nowhere have I found a complete notion. And then I made a definition out from everything that I know ...
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1answer
36 views

In any triangle the angle opposite the greater side is greater.

I have a small problem with the following : http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI18.html I did understand the proof, but the proposition claims that the angle opposite the ...
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1answer
53 views

An inequality about inner product in $\mathbb{R}^2$.

Let $a_i,b_i,r_i,s_i$ be positive integers for $i\in\{1,2\}$. $r_i$ and $s_i$ are non-zero for $i\in\{1,2\}$. Let $a=\left(\frac{1}{a_1},\frac{1}{a_2}\right), ...
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1answer
29 views

Very naive questions in elementary geometry

I was wondering whether the following questions are difficult to solve : Consider a triangle ABC (defined in euclidean geometry). Let M be inside the triangle ABC such that the triangles AMB, AMC ...
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1answer
21 views

How to show express $y $ in terms of angle $\theta$?

$ABC$ is a straight line with $AB = BC = 3$ units. $B$ is the centre of the circle with radius of $2$ units. $P$ is a point on the circle. $\widehat{B_1} = \theta$, $\widehat{A} = x$, $QC \perp AC$ ...
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0answers
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Euclid's elements proposition 17. The sum of two angles in a triangle is less than 180 degres.

I have a very short question on this proposition : http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI17.html I understand the way the theorem was proved. Euclid proves that angles ABC and ACB ...
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621 views

“Pythagoras Theorem” - Why is “theorem” or “theory” used rather than “law” in mathematics?

Why is Pythagoras Theorem a "theory" but not a "law"? I mean we use it many times in school and to build stairs etc. and it has been proven, however it is still called a theory. What are the ...
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1answer
33 views

On the definition of sphere in analytic geometry…

Last year, when I was teaching mathematics (analytic geometry) for one of my clever freands, I arrived to the definition of sphere. I said Fix $r>0$, An sphere is the set of all triples ...
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1answer
11 views

How do I get vectors orthogonal to the one generated by the spherical coordinate formula?

Given a formula: F : ℝ → ℝ → ℝ3 F(θ,φ) = (cos(φ)*sin(θ), sin(φ)*sin(θ), cos(θ)) what are the formulas: ...
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6answers
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Is it possible to solve any Euclidean geometry problem using a computer?

By "problem", I mean a high-school type geometry problem. If no, is there other set of axioms that allows that? If yes, are there any software that does that? I did a search, but was not able to ...
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1answer
25 views

Geometric proof for $|| u ||^2 + || v ||^2 = \frac{1}{2}||u-v||^2 + 2||\frac{u+v}{2}||^2$

Is there an geometric proof for the following identity? $|| u ||^2 + || v ||^2 = \frac{1}{2}||u-v||^2 + 2||\frac{u+v}{2}||^2$. The norm here is normal Euclidean norm, and $u,v$ are vectors.
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0answers
25 views

Shortest path to find a highway

I remember this as a classic problem, but all Google results are video-game-related, so I guess I should ask it here: An adventurer got lost in the desert, but he knew that there was a highway ...
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1answer
45 views

Construct a regular pentagon in only 11 steps using ruler and compass. [closed]

One step is to draw a stright line or a circle (greek classical understsnding of step)
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212 views

Competition style problem circa 1992

We're given a triangle $ABC$. Going clockwise, let $B_1$ and $B_2$ be distinct points on the segment $AC$ ($B_1$ is between $A$ and $B_2$), let $A_1$ and $A_2$ be distinct points on the segment $CB$ ...
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Two lines intersect forming four angles [closed]

Two lines intersect forming four angles. If one of them is right, show that others are too straight. I am clueless how to start. Ideas?
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Diagonal of triangular bipyramid with 3 edges next to a point length 1 and orthogonal, and the lengths of three known.

I am working on a lighting system for a voxel game. It requires recursive euclidean distance calculation for successively further blocks, and the distance of each block from the light source needs to ...
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3answers
401 views

Euclidean geometry exercise

I would like some help to solve this: Consider a triangle $\triangle ABC$ with $\angle A$ a right angle and $BC=20$. Divide $BC$ into four congruent segments, that is, take the points ...
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3answers
386 views

Which statements are equivalent to the parallel postulate?

I would like to have a long-ish list of statements that are equivalent to the parallel postulate. If a line segment intersects two straight lines forming two interior angles on the same side that ...
2
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1answer
630 views

Intersection of Two Circles

I have two circles as: $C_1: (x-x_1)^2+(y-y_1)^2=r_1^2$ and $C_2: (x-x_2)^2+(y-y_2)^2 =r_2^2$ and these circles have non-empty intersection. In other words $\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}\leq ...
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1answer
468 views

Find locus of points relating to an ellipse

I would like to find the equation of the following locus. For a big circle C centered at (0,0), the locus of points that the sum of distances to Y-axis and to C is 1, say in the first quadrant, is ...
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3answers
26 views

Find two points on two lines in the plane where the line between the two points go through a third point and are equidistant from that point

I have the following situation (see pic below). I have two lines $B$, $C$, in the plane, the intersection point $a$, and a point $p$. I need to find the points $b$ and $c$ along $B$ and $C$ such that ...