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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Calculating pairwise distance of two N-dimensional vectors given their length and angle

I am not a mathematician, so apologies in advance for any nomenclature blasphemy. Given the magnitudes of two vectors $b$ and $c$ and the angle between them $A$, I can calculate their distance in 2-D ...
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3answers
45 views

Find coordinates of the rotation center

My software is going to control a Laser. I know the Laser's current Position defined as $P_1$ with coordinates $(x,y)$ and the place where it will be after a clockwise rotation around a point $C$, ...
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2answers
19 views

Trigonometric inequality in an obtuse triangle

Let $ABC$ be an obtuse triangle with $A$ the obtuse angle. I conjecture that the following inequality is true $$\sin B + \sin C \le |\tan A|.$$ Show that it holds or give a counterexample.
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1answer
51 views

Why are values squared in distance formulas, such as the Pythagorean Theorem?

Why do you square the values in the Pythagorean Theorem or any distance formula wherein you're trying to find the distance between two points in two-dimensional, Euclidean space? for example, why ...
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3answers
1k views

Proof that every polygon with an inscribed circle is convex?

In many elementary (and not-so-elementary) Euclidean geometry texts, a (simple) polygon is said to be tangential  if it is convex and has an inscribed circle (i.e., a circle that intersects and ...
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2answers
48 views

When is the sign of inner products preserved?

I'm interested in the following question: Let $E$ be a real Euclidian space. What are the linear transformations $f$ of $E$ that preserve the sign of inner products? That is, for all vectors ...
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1answer
41 views

Proper Proof for Completeness of $\mathbb{R}$ with the Euclidean Metric

My code can't be uploaded because it doesn't work with the websites coding, but here is a pdf of my LaTex code. My question is, is this a proper proof? It feels as if I'm missing something important. ...
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2answers
69 views

Slope and euclidean geometry (grade 10)

Given Triangle $DEF$ with vertices $$D(-2,6),E(7,3), \text{ and } F (2,-3)$$ find: a) the equation of the altitude from vertex $E$ in standard form (include y=mx+b) b) Find where this altitude in a) ...
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2answers
696 views

Euclidean Geometry Intersection of Circles

Two circles intersect in the Cartesian Coordinate system at points $A$ and $B$. Point $A$ lies on the line $y=3$. Point $B$ lies on the line $y=12$. These two circles are also tangent to the x-axis at ...
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1answer
21 views

How can one prove that (AD) and (EB) are orthogonal?

ABC is a triangle, we make outside two squares CBGD and ACEH. I have to show that (AD) and (EB) are orthogonal. So I'm sure that there is many ways to solve this exercise. I did it using the scalar ...
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3answers
20 views

Greatest area using a string with the length of $l$

Suppose we have a string with length of $l$ what is the shape that has highest area? In other words,with a constant perimeter of $l$ what is the shape with the highest area? P.S:My own speculation ...
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1answer
12 views

If a quadratic form $f$ takes the minimum on a triangle in a vertex, what can I say about min of $f$ on edges of a subdivision?

Let $f(x)=x^2+y^2$ be the Euclidean square-norm and $A,B,C\in\mathbb{R}^2$ be vertices of a triangle $\Delta$ such that $f$ takes the maximum on $\Delta$ in $C$, the minimum in $A$ and takes the ...
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1answer
79 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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1answer
28 views

Geometric Construction Rhombus

Given two line segments, Construct a rhombus whose diagonals have lengths equal to the lengths of the two given segments. I can get to finding perpendicular bisectors of each line segment, but have ...
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30 views

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 ...
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0answers
49 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
57 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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22 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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29 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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1answer
24 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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51 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
61 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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23 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
49 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
997 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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1answer
45 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
513 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
72 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
18 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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34 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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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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2answers
105 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
357 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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35 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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1answer
96 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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19 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
39 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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3answers
329 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
978 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
38 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
128 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 ...
3
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2answers
172 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
121 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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37 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
61 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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99 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
70 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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10 views

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
33 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
30 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 ...