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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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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40 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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31 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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1answer
51 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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97 views

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

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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16 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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1answer
28 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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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
42 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
17 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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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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380 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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1answer
17 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
24 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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0answers
14 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 ...
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3answers
48 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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1answer
35 views

How to calculate Cartesian coordinates for an element after rotation has been applied?

I have a square on a Cartesian coordinate system with origin (0,0) on top left (yellow arrow from the picture). The initial coordinate of the square from the ...
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32 views

Midpoint of side, center of gravity of triangle, … [on hold]

$\Delta ABC$ is any triangle having: $I$ the midpoint of $[BC]$ , $J$ the midpoint of $[AC]$ , $G$ the center of gravity. $\vec{BA} + \vec{CA}=$ $\vec{IA}$ $6 \cdot \vec{IG}$ $3 ...
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2answers
80 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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1answer
34 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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22 views

Can you always cover a circle in a finite number of steps with this “radar” algorithm?

Suppose you have a disc $C$ of radius $V$ with center $c$ and you randomly place a point $p$ in it. $p$ Behaves as follows: at every time-step, $p$ calculates its angle to $c$, and moves a distance of ...
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12 views

Geometry involving area of rhombus and interior isosceles triangles

Points E, F, G, and H lie inside a rhombus ABCD, such that the triangles AEB, BHC, CGD, and DFA are isosceles right triangles with hypotenuses AB, BC, CD, and DA.The sum of areas of ABCD and EFGH is ...
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54 views

Segments on a family of parallel lines

Let $\{l_i:i\in I\}$ be a family of parallel lines on the plane $\mathbb{R}^2$. Suppose for each $i\in I$ there is a closed segment $s_i\subset l_i$. Moreover, for each triple $i_1,i_2,i_3$ there ...
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1answer
32 views

How many degrees of freedom are in a flat metric and how does one count them?

I think that there are zero degrees of freedom in a flat metric, but I do not know how to count them. I know that any symmetric metric tensor has $n(n-1)/2$ degrees of freedom, where $n$ is the ...
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(Kiselev) Construction of an arc of a circle

Using only compass, construct a 1 degree arc on a circle, if a 19 degree arc of this circle is given. The first thing that is stumping me is if we can use a straight edge? In any case, I can think of ...
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2answers
64 views

Difficult geometry question involving Pythagoras theorem?

Hello mathematicians, I was given this question by my teacher and after spending a couple of hours looking over it have not been able to solve it. I understand it involves radians which I have ...
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0answers
16 views

Hahn Banach theorem and supporting hyperplane theorem

The question is out of Rudin Functional analysis Chapter 3 problem 1. Call a set $H \subset \mathbb{R}$ a hyperplane if there exists real numbers $a_1,\ldots, a_n, c$ (with $a_i \neq 0$ for at least ...
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2answers
27 views

Equilateral Triangle equality

Let ABC be an equilateral triangle, and P be an arbitrary point within the triangle. Perpendiculars PD, PE, PF are drawn to the three sides of the triangle. Show that, no matter where P is chosen, PD ...
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20 views

Intersection of two convex lattices polygons

A convex lattice polygon is a polygon whose vertices are points on the integer lattice. Let P and Q two convex lattice polygons with n ,(resp. m) vertices. Let R be the convex lattice polygon ...
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2answers
28 views

How to check if point $x \in \mathbb{R}^n$ is in a $n$-simplex?

Is there any universal solution how to check if a point $x \in \mathbb{R}^n$ is in $n$-simplex for any number of dimensions (any $n$)? Is it possible to use Barycentric coordinates for any $n$? I ...
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110 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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9 views

semidefinite matrices and Euclidean distance matrices

Is this statement true? If $X$ and $D$ are related as $D_{ij} = X_{ii} + X_{jj} - 2X_{ij}$ then $$ X\succeq 0 \iff D \text{ is a Euclidean distance matrix}. $$ Clearly $\Rightarrow$ is true; make a ...
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220 views

To whom do we owe this construction of angles and trigonometry?

I've come across what is, to me, the most precise, beautiful and thorough definition of what we know of as the angle between two vectors. I say this because most literature either skims over things ...
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1answer
66 views

Algebraic calculation steps.

Can somebody explain how the coefficients $a_{11}, a_{12}, a_{22}$ are derived after rotating the ellipse below ?? $\widetilde{s_{11}} = \frac{\sum_{j=1}^n(x_{jk} - \bar{x_k})}{n}$ Thank you in ...
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2answers
2k views

Why can't three unit regular triangles cover a unit square?

A square with edge length $1$ has area $1$. An equilateral triangle with edge length $1$ has area $\sqrt{3}/4 \approx 0.433$. So three such triangles have area $\approx 1.3$, but it requires four ...
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2answers
74 views

Show that the triangles are congruent

Let $\triangle ABC$ be a acute-angled triangle so that $AB>AC$ and $\angle BAC = 60^\circ$. Let $O$ be the circumcenter and $H$ the orthocenter. Let $OH$ intersect $AB$ and $AC$ in $P$ and $Q$ ...
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14 views

Approximation of a trisecting an angle

I learned a proof that it is impossible to trisect an angle. Is there some research that if we have been given an angle, a ruler and a compass and we are allowed to draw $m$ circles and $n$ lines/line ...
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2answers
34 views

Number of integral coordinates in a given region.

The number of points, having both coordinates as integers, that lie in the interior of the triangle with vertices $(0,0) ,(0, 41$) and $(41,0)$ , is: (1) 901 (2) 861 (3) 820 (4) 780. I tried ...
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1answer
25 views

Parallel sides in regular polygons

So I've noticed a couple of things about regular polygons with an even number of sides but I'm having a hard time proving them, these are all very obvious, and I think perhaps induction is the best ...
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2answers
28 views

The geometry of unit vectors that have specific angle with a given vector

It is easy to see that for $S^2$ this space is nothing but a circle that is the intersection of a cone with aperture $2\alpha$ (where $\alpha$ is the predifined specific angle), and $S^2$. My question ...
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2answers
43 views

Intersecting lines from the vertices of a square with an arbitrary interior point.

This question is pretty hard to word without a picture, so I have attached one. I am wondering if there is a general way to find the area of the green or blue areas given the ordered pair of some ...
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0answers
32 views

Aspect Ratio of Cylinder, Pyramid and Dome

The aspect ratio can easily be defined for rectangular geometries ($AR = height/width$). Is there a definition for aspect ratio of a dome, cylinder, and pyramid (Here standard pyramid and dome were ...
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1answer
38 views

finding discrète coordinate of Intersection of two convex polygon?

I seek for cartésien coordinate of vertex's of the intersection area between two polygons ? We have two convex polygon's P & Q such that : all vertex of P (resp. Q) are in 2D cartésien plane. I ...
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I need a help for this problem [closed]

Given an isosceles triangle. Find the locus of the points inside the triangle such that the distance from that point to the base equals to the geometric mean of the distances to the sides. Any ideas ...
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1answer
15 views

Rotating points with horizontal and vertical tilt

From Wikipedia, I've seen that if I have a rotation to do in three dimensions, it must be around an axis in order to do so. However, I have a rotation along the z-axis along with the xy-plane (aka a ...
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2answers
122 views

Eritrea's Theorem

According to this newspaper, an Eritrean high school student named Saied Mohammed Ali has discovered a new geometric theorem. Another source seems to say that it's the following: Say you have a ...
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1answer
40 views

Prove all right angles are congruent?

Prove all right angles are congruent. I only have to prove one side to this argument, so I just need to the the other argument. So basically, if two angles are right, then they must be congruent is ...
4
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0answers
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Polar of a non centred ball.

Recall that the polar of a set $A\subset\mathbb{R}^n$ is the following set: $$A^{\circ} = \lbrace c \in \mathbb{R}^{n} |\; \langle c,x \rangle \leq 1 \quad \forall x \in A \rbrace$$ where $\langle ...
3
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1answer
41 views

Triangle geometry: $BC^2+AC^2=n\cdot AB^2$.

I am looking for information regarding which triangles $ABC$ satisfy $BC^2+AC^2=n\cdot AB^2$ for $n=1,2,3,...$. I'm sure that work has already been done in this area since it is a fairly simple ...
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1answer
40 views

Why is $A:X\to Y$ linear between two normed spaces is continuous iff bounded?

Why is it that every linear operator $f: \mathbb R^n \to \mathbb R^m$ is bounded and therefore continuous, but why is it that $A:X \to Y$ between two normed spaces is continuous iff bounded? That ...