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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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
50 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 ...
2
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2answers
93 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 ...
3
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1answer
57 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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23 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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0answers
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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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2answers
55 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
36 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 ...
4
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2answers
78 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
17 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
33 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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0answers
25 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
32 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 ...
3
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2answers
116 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
11 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 ...
18
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2answers
232 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
71 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 ...
33
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2answers
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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 ...
3
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2answers
76 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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0answers
17 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
37 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
31 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
57 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
41 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
50 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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2answers
70 views

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
22 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 ...
8
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2answers
149 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
53 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 ...
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0answers
28 views

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
46 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
41 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 ...
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1answer
30 views

Question about inequality in linear algebra

$V$ is inner product space. $u, v \in V$ are two orthogonal vectors. Prove that $\|v-u\| \geq \|v\|$. Because $\|v-u\|, \|v\| \geq 0$ it's enough to prove that $||v-u||^2 \geq \|v\|^2$. ...
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2answers
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1answer
18 views

Find Cartesian and Vector equations of the plane perpendicular to (1, 0, -2) and containing the point (1, -1, -3)

I'll work through my current progress until I reach the bit where I get stuck. We are given $$n = (1, 0, -2)$$ Thus Cartesian form will be $$x - 2z = r\cdot n$$ Now, $$r\cdot n = (1, -1, -3)\cdot ...
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1answer
35 views

Two perfect squares in a right triangle

Prove that there is no integer sided right triangle in which the lengths of two sides are simultaneously perfect squares
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2answers
103 views

Geometry: Find angle x in triangle

I have not been able to find a euclidean geometry solution to this, but any other solutions are also appreciated. Let ABC be a triangle with AB=CD and angles as marked in the diagram. Find the ...
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0answers
25 views

Orthogonal spaces, span-formula

I read in a book this formula: ($v_i$ are vectors of an euclidean vector space, each one $\neq$ 0) $(\cap v_i ^\bot )^\bot = \sum v_i^{\bot \bot}$, The intersection and the sum are build over a ...
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1answer
56 views

Geometry Homework [closed]

A quadrilateral $ABCD$ is inscribed in a circle of center $O$. Chords $AB$ and $CD$ intersect at $M$ such that $AM=6, MB=4, MD=3$ and $MC=8$. The tangents at $A$ and $B$ to the circle meet in ...
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1answer
44 views

A geometry homework question

Let $A,B,C$ and $D$ be $4$ points in the plane such that any combination of three or more of them are non-collinear. Let $[AB]$ and $[CD]$ intersect at $M$. Suppose that: $AM = 6$, $MB = 4$ and $MD = ...
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1answer
22 views

Travelling Sales Man problem

I am studying the Travelling Sales Man problem: Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and ...
0
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2answers
25 views

How to prove triangle inequality for euclidean norm on complex number?

We were asked to show that when: $\displaystyle \Vert Z\Vert = \left(\sum_{k=1}^{n} (x_k+iy_k)(x_k-iy_k)\right)^{1/2}$ that $\Vert Z+W\Vert \leq \Vert Z\Vert+\Vert W\Vert$ whenever $Z$ and $W$ are ...
4
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1answer
59 views

Is the following set (path) connected?

This is a homework question. $d,n\ge 2$. Let $L=\{(x_1,...,x_n)\in (\mathbb{R}^d)^n: x_i\in \mathbb{R}^d, x_i\ne x_j \forall i\ne j\}$. I tend to think it is not path connected because if you ...
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1answer
34 views

Position of a point on a line segent relative to the segent's length

I would like to ask for help with clarifying the following formula for calculation of relative position of a point on a line segment with respect to the line segment's length in two-dimensional ...
2
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2answers
67 views

How to draw an $405^\circ$ angle?

In a math test a question was to draw a $405^\circ$ angle. Is it formally correct to say draw an angle as I think that in geometry, an angle has just some formal definition. So what is the connection ...
3
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3answers
109 views

From Hartshorne's Geometry: Euclid and Beyond: contruct and inscribed equilateral triangle in a given circle

I haven't found a propert solution for this problem: (4.3) Given a circle, but not given its center, construct an inscribed equilater triangle in as few steps as possible. I managed to ...
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0answers
15 views

$\text{diam}(\Omega)$ is $\geq$ to at least one side of the minimal rectangular box containing $\Omega$?

For $\Omega\subset\mathbb{R}^n$ open and bounded, is it always the case that $\text{diam}(\Omega)$ is greater or equal to at least one side of the minimal rectangular box containing $\Omega$ ? Added : ...
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1answer
38 views

Euler charcteristic of the intersection of hyperplanes with a pointed cone

Consider a pointed closed convex polyhedral cone $C$ in $\mathbb R^n$. Let $S_1,..,S_n$ be hyperplanes in $\mathbb R^n$. Consider $S=C \cap \left(\bigcap\limits_{i=1}^n S_i\right)$. If $S$ is ...
3
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2answers
46 views

elementary geometry/algebra

I was looking at my geometry chapter summary on similar triangles, and I was a little confused with the result. I'm really tired right now and I am having difficulty leafing through the chapter to ...