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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Reference Request: Complete Proof of Braikenridge–Maclaurin Theorem

Where can I find a reference to a complete proof of the Braikenridge–Maclaurin theorem, which is stated as: If the three pairs of opposite sides of (an irregular) hexagon meet at three collinear ...
12
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1answer
248 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 ...
3
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0answers
34 views

Area of an equilateral triangle

Prove that if triangle $\triangle RST$ is equilateral, then the area of $\triangle RST$ is $\sqrt{\frac34}$ times the square of the length of a side. My thoughts: Let $s$ be the length of $RT$. ...
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0answers
49 views

Why is unit circle not sufficient to bound the partial sums?

I want to find vectors $\textbf{v}_1, \dots,\textbf{v}_n$ in $\mathbb{R}^2$ with that $\sum_{i=1}^n\textbf{v}_i=\textbf{0}$ and $\Vert \textbf{v}_i\Vert\leq 1$ for all $i=1,\dots,n$, such that for ...
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0answers
20 views

Properties of $\Omega_{\epsilon}$

Let $\Omega$ be a bounded connected domain in $\mathbb R^n$ with compact smooth boundary. So $\partial \Omega$ can be viewed as a smooth submanifold of dimension n-1. Let $$\Omega_{\epsilon} : = \{ ...
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25 views

Euclidean geometry with ruler and compass

I was wondering, is there any book out there that is of the style of Euclid's Elements ? One which you have to use a compass and ruler for certain propositions like building a triangle, etc. Or would ...
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0answers
9 views

Finding closest rectangle to another using concept of closest edge [on hold]

I know coordinates and size of rectangles. My goal is to find 'the closest' rectangle to one special rectangle using the concept of closest edge and also to find distance between them??
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1answer
18 views

Prove that $GEBD$ is a square (see diagram).

$ABB_1A_1,BB_2C_1C,ACC_2A_2$ are squares. The problem itself is to prove that the area of $ABC$ and the area of $BB_1B_2,CC_1C_2,AA_1A_2$ are equal. If I could only prove that GEDB is a square it ...
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21 views

Euclidean Geometry [on hold]

What is the correct interpretation of the first flaw of euclidean geometry(the proof that shows every triangle is isosceles.)I mean why does this happen and why is this wrong.
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1answer
30 views

Relative distance to rotated object

An object (with center $O_{2}$) has been rotated by an angle $\alpha$. There are two images of the object taken by a camera (centered at $O_{1}$), and two points $x_{1}$ and $x_{2}$ that are actually ...
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24 views

On the centroid of a triangle

There's three different ways to see a triangle in the Euclidean plane: as three non-collinear points, say $A$, $B$, $C$; as the line segments connecting the three points, that we can parametrize as a ...
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4answers
45 views

Find area of rhombus

Given the following rhombus, where points E and F divide the sides CD and BC respectively, AF = 13 and EF = 10 I think the length of the diagonal BD is two times EF = 20, but i got stuck from there. ...
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17 views

Homomorphisms between countable spaces and Euclidean spaces?

Is there some place to start reading about homomorphisms between countable (discrete) spaces and Euclidean spaces or $l_2$? I know it is a rather general question, but I am not sure what I am looking ...
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1answer
33 views

Distribution of an angle between a random and fixed unit-length $n$-vectors

Suppose I have a random unit-length $n$-element vector $\mathbf{x}$ that is uniformly distributed on an $n$-dimensional sphere, and let vector $\mathbf{a}$ be some other unit-length $n$-element vector ...
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1answer
896 views

Parallel postulate from Playfair's axiom

Parallel postulate: If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, ...
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2answers
326 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 ...
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0answers
26 views

How can one show that the vector $AK=\frac{1}{3}AI$?

$ABC$ is a triangle then $I$ is the medium of $[CB]$ and $J$ the medium of $[AI]$ and $K$ the intersection of $(BJ)$ and $(AI)$. Then how can one show that $AK=\frac{1}{3}AC$ Do we have to add ...
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1answer
31 views

Every tetrahedron has a vertex whose adjacent edges can form a triangle

Prove that in every tetrahedron there exists a vertex $v$ such that the three edges incident at $v$ have lengths that can form a triangle. It can be proved using a tedious casework: assume by ...
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5answers
395 views

How to construct a line with a given equal distance from 3 Points in 3 Dimensions?

Important: I'm now convinced that 4 points are needes in order to reduce the solutions to a finite number. (Which is necessary because I need ALL solutions) In a computer science context I need to ...
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2answers
69 views

volume of the solid

Using geometry, calculate the volume of the solid under $z = \sqrt{49- x^2- y^2}$ and over the circular disk $x^2+ y^2\leq49$. I am really confused for finding the limits of integration. Any help?
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1answer
15 views

Are the connected components of the level sets of a $\mathcal{C}^1$ function path-connected?

I have a $\mathcal{C}^1$ (or even just $\mathcal{C}^0$) function $f:\mathbb{R}^n\rightarrow\mathbb{R}$, and have been trying to figure out when the connected components of its levels sets are also ...
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0answers
32 views

Generalizing convexity of sets

Let $X$ be a subset of some Euclidean space. We say that $X$ is convex if for any two points $p$, $q$ in $X$, the line segment joining $p$ and $q$ is also in $X$. But what if we loosen this definition ...
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0answers
15 views

How can one show that (DI) , (JB) and (AC) concurrents on G?

ABCD is a square , We add outside it two equilateral triangles ADJ and ABI How can I show that (DI) and (BJ) and (AC) occur in the same point ? Here can we demontrate that saying that IGB and JGD are ...
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0answers
16 views

Packing spheres into a rectangular prism

So, this was a problem in the new standardized high school tests California has started using(CAASP). These new tests are completely done on the computer, and feature what they call Computer Adaptive ...
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2answers
99 views

3 circles and 3 squares all inscirbed into a right angled triangle problem

This is quite a tricky question for me, but this is how far I got: My drawing may not be precise, but I do know the points of tangency. I am a little stuck now, and I would appreciate it if someone ...
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2answers
325 views

About the hyperplane conjecture.

I have recently heard about the hyperplane conjecture and I would like to understand better the problematic behind this conjecture. The hyperplane conjecture: There exists a universal constant ...
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4answers
399 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
20 views

Vector proof to show the line connecting two points on a triangle is parallel to a side.

If $X$ and $Y$ are points on sides $AB$ and $AC$ of a triangle $ABC$ and $\dfrac{AX}{AB}=\dfrac{AY}{AC}$, then $XY\parallel BC$. I'm supposed to prove this using vectors, but we haven't done too much ...
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2answers
4k views

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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4answers
2k views

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
60 views

Area of triangle formed by angle bisector, altitude and median

Question:- Given a triangle ABC with side length a, b and c. Calculate the area of a triangle in terms of a, b and c formed by angle bisector from vertex A, altitude from vertex B and median from ...
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16 views

Intersection of 3 positively sloped planes

Suppose I have three planes, each of which is 'positively sloped' in the sense that the first plane intersects the x-axis at a positive value, and the y and z-axes at a negative value. Similarly, the ...
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1answer
33 views

A quadrilateral with one pair of opposite right angles. Is this a rectangle?

I can prove it's not a rectangle by drawing some lines, but is there a name for this kind of figure? Thanks.
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0answers
106 views

Is Euclidean geometry really a “dead” subject? If so, why? [closed]

It seems that Euclidean geometry is a "dead" subject nowadays. In the time of the Greeks, mathematicians and geometers were one and the same. Today, very few professional mathematicians study ...
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3answers
312 views

Geometry problem on circles from a competition

Triangle $\triangle ABC$ is an equilateral triangle whose side is $16$. A circle meets the sides of the triangle at $6$ points: it intersects $AC$ at $G$ and $F$ and $|AG|=2$, $|GF|=13$, $|FC|=1$. ...
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2answers
818 views

Prove using integration that $polygon → circle\space \text{as}\space number\space of\space sides → infinity$

Say we have a regular polygon $s$, with number of sides $n$: Is there a way to prove that as $n → ∞,\space $then $s → circle$ using integration?
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1answer
29 views

Non-Euclidean geometries

Does non-Euclidean geometry can be always immersed in Euclidean of dimension D+1? This is probably very basic question, but I'm just trying to understand why do you need to consider sometimes very ...
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2answers
107 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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2answers
163 views

Triangle geometry - synthetic proof

I'm looking for a nice synthetic proof of the following fact. Consider a non-isosceles triangle, pick a vertex. Assume that the median and the altitude passing through this vertex are isogonal ...
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2answers
112 views

Proper axiomatization of Euclidean Geometry that Euclid would approve of

It is relatively common knowledge that Euclid's axiomatization is not sufficient to prove all the things that Euclid wants to, and that there are other axiomatizations out there that strengthen ...
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0answers
28 views

An orthogonal transformation with determinant 1 rotate $\mathbb{R^3}$ around an axis.

I'm studying differential geometry and need confirmation on the solution of the following problem. A rotation is an orthogonal transformation $C$ such that det $C=+1$. Prove that $C$ does, in fact, ...
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1answer
44 views

How to find the focus of a parabola

To find the center of a circle, it's enough to choose three points on the circle and find the circumcenter of a triangle with those three points. When a parabola is drawn and the formula is not ...
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2answers
84 views

Area of rhombus and interior isosceles triangles

Points $E$, $F$, $G$, and $H$ lie inside a rhombus $ABCD$, such that the triangles $\triangle AEB$, $\triangle BHC$, $\triangle CGD$, and $\triangle DFA$ are isosceles right triangles with hypotenuses ...
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3answers
50 views

Proving algebraic equations with circle theorems

I got as far as stating that OBP=90˚ (as angle between tangent and radius is always 90˚), and thus CBO=90˚- 2x. CBO=OCB as they are bases in a isosceles. COB=180-90-2x-90-2x. But after this, i am ...
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Prove that the intersections of the ray $f(x)\rightarrow x$ with the $n$-disk form a continuous function, with $f$ continuous

This appeared in a proof of the Brouwer fixed point theorem, in Introduction to Algebraic Topology, by Rotman, but it was left as an exercise. I could only prove this in 2 and 3 dimensions, not in ...
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1answer
894 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
27 views

Proof for diagonals of a rectangle

If a rectangle is a figure with four sides and four rectangular angles, I would like to prove that the diagonals are congruent and both meet in the midpoints. However, I don't know where to start this ...
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1answer
19 views

Reduce distance between two points by %

I have two points, say A = (2, 6) and B = (5, 3). I want to move point B up to 70% closer to point A. I calculate Euclidean ...
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1answer
36 views

Isosceles trapezoid with inscribed circle

The area an isosceles trapezoid is equal to $S$, and the height is equal to the half of one of the non-parallel sides. If a circle can be inscribed in the trapezoid, find, with the proof, the radius ...
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0answers
16 views

Bound for the distance of projections onto the unit sphere

Given $x \in \mathbb{R}^n$, $x \neq 0$, let $x' = x/|x|$ (where $|\cdot|$ is the euclidean length) be its projection onto the unit sphere. I would like to prove that $$ |x' - y'| \leq 2 ...