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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Did Euclid prove that Pi is Constant?

Pi is defined the ratio of the circumference of a circle to its diameter, but of course different circles have different circumferences and diameters, so in order for it to be well-defined we need to ...
2
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2answers
93 views

Largest bounded square

Suppose I have a triangular land-plot, but some part of it (the yellow part) is unusable. I want to build a square house on the usable (white) part. The house may be rotated (but must be square). What ...
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0answers
26 views

Where are $A,B,C$ in the regular $n$-gon such that $\min (|AB|+|BC|,|BC|+|CA|,|CA|+|AB|)$ gives the max?

Let $F_n$ be the regular $n$-gon of edge-length $1$. Let us consider taking three points $A, B, C$ in $F_n$. Suppose that you can take a point on the edge of $F_n$. Supposing that $|AB|$ represents ...
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1answer
55 views

Question about Right Angles

I am stumped on the following question: Prove that the measure of a right angle is 90. I so far have tried extending the lines making the angles but I can't get anything. I am not sure what kind of ...
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0answers
81 views

Points in the interior of an angle

I would like some help for the following question about proving that a point is in the interior of an angle (it involves betweenness of rays too): Prove that if ray BA - ray BC - ray BD and point P ...
2
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0answers
129 views

Infimum over area of certain convex polygons

Let us identify the Euclidean plane with the complex plane $\Bbb C$. For every $n\ge 1$, consider the following collection of finite sets in $\Bbb C^{n+1}$: $$S_n:=\{(z_0,z_1,\dots,z_n)\in\Bbb ...
3
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2answers
69 views

Unit length vectors that sum to zero

Let's say we have a collection of $n$ vectors in $\mathbb{R}^2$ where $n$ is odd. Suppose each vector has unit length and that the sum of the vectors is zero. Is it necessarily true that the vectors ...
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3answers
162 views

Show centers of squares formed by a parallelogram form a square.

A homework question I have been having some issue with - Given parallelogram $ABCD$, generate 4 squares from the sides of the $ABCD$. Given the 4 centers of the squares $W, X, Y, Z$ (formed by their ...
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1answer
47 views

Projection on a hyperplan and a hypercube intersection

I need to project an array y onto a hyperspace defined by (a.x) = c where a is an array in R^N However, x needs to belong in the hypercube {0 <= x_i <= 1, for all i from 1 to n} Therefore from ...
2
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1answer
70 views

cutting a cake without destroying the square toppings

There is a square cake. It contains N toppings - N disjoint axis-aligned squares. The toppings may have different sizes, and they do not necessarily cover the entire cake. I want to divide the cake ...
0
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1answer
123 views

Are side lengths enough to find the ratio of the diagonals of a quadrilateral?

Is it possible to find the ratio of two diagonals of a quadrilateral when the length of all sides are given??
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2answers
628 views

Number of Lines Passing Through a Given Point in the Plane

How can one prove that infinite number of lines pass through a given point in plane, using Euclid's axioms (or Hilbert's, if necessary)?
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2answers
65 views

About the inscribed sphere and the exspheres of a $n$-dimensional simplex

Let us consider $n$-dimensional simplex $K$ in $n$-dimensional Euclidean space. Let $r_0$ be the radius of the inscribed sphere of $K$, and let be $r_1, r_2, \cdots, r_{n+1}$ be each radius of the ...
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0answers
64 views

Pointing a not stabilized camera using imu data and matrix rotation in the euclidean space

this is my scenario: I have a support with a pan/tilt camera and an imu. This support can be moved by changing the pitch, and roll. From the same position of the support, but independent from the ...
4
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1answer
187 views

Online Math Open Contest 2 Problem 50

In tetrahedron $SABC$, the circumcircles of faces $SAB$, $SBC$, and $SCA$ each have radius $108$. The inscribed sphere of $SABC$, centered at $I$, has radius $35.$ Additionally, $SI = 125$. Let $R$ be ...
2
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1answer
50 views

About the 'minimum triangle' which includes a convex bounded closed set

Question : Is the following true? "Letting $K$ be a convex bounded closed set on a plane, then there exists a triangle $M$, which includes $K$, such that $|M|\le 2|K|$. Here, $|M|,|K|$ is the area of ...
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1answer
56 views

Proof with congruence of angles

I came across a proof exercise from my proof work-book that I am stuck on. The questions says: Suppose we have angle PQR with P, Q, and R non-collinear, and ray QS distinct from ray QR such that ...
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2answers
257 views

Circles in Complex Planes

Points on the circle centre C and radius r are given by the equation $|Z-C|=r$ or $(Z-C)(\overline{Z}-\overline{C})=r^2$. Where $Z = x + iy$. When multiplied out, I understand that we have ...
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1answer
42 views

Properties of a Square

So I have that squares A and B are congruent and one vertex of B is at the center of A. The question is what is the ratio of the shaded area to the area of square A. My question is if two square are ...
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3answers
46 views

Prove that $l_1$ and $l_2$ are parallel if and only if $a_1b_2-a_2b_1=0$

For $b_1$ and $b_2$ non-zero, consider the lines $l_1=\{(x,y) \in \mathbb{R}^2 | a_1x + b_1y + c_1=0\}$ and $l_2=\{(x,y) \in \mathbb{R}^2 | a_2x + b_2y + c_1=0\}$. Assuming I only know Euclid's ...
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2answers
94 views

Type of isometry

What type of isometry of $\mathbb{R}^3$ is the one given by sending $(x,y,z)$ to $(y,z,x)$ for each $x,y,z\in\mathbb{R}$? How can I find this isometry? Does anyone have a hint? My solution is that ...
9
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1answer
89 views

Inner product space, points cannot be placed inside a ball of a given radius

I've found a very nice problem and I don't know how to go about solving it. Let $(E, || \cdot ||)$ be an inner product space, $x_1, ..., x_n \in E$. Prove that if for $i \neq j$ we have ...
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1answer
72 views

Repeating an operation infinitely makes any convex $n$-gon a regular $n$-gon?

For any convex $n$-gon $P_{0,1}P_{0,2}\cdots P_{0,n}$, let us consider the following operation : Operation : Let $k=0,1,\cdots$. Take $n$ points $P_{k+1,i}\ (i=1,2,\cdots,n)$ outside of $n$-gon ...
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1answer
77 views

How is the circle that fits beneath two adjacent circles related?

This is hard to search and probably easy to solve, but I keep finding articles about intersecting circles, and that is not what I'm after. I don't know what to tag this under, so if you know how to ...
2
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1answer
61 views

Some property of half-planes in Euclidean and non-Euclidean geometry

Consider the Cartesian product of the set of real numbers $\mathbb{R}^2$ with the standard Euclidean metrics. An open half-plane is any set of pairs of real numbers $x$ and $y$ such that $ax+b> y$, ...
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0answers
18 views

behaviour of an angle if the points are transformed

Let $\angle(OAB)$ denote the angle between the line segments $\overline{OA}$ and $\overline{OB}$, where $A$, $B$ and $O$ are points in $\mathbb R^2$. It would be interesting to learn about the ...
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0answers
50 views

Critique of my Solution / Is my solution correct

Question: "In triangle ABC, points E and D lie on AC and BC, respectively. Point F is inside the triangle such that $\angle CAF = \angle FAD$ and $\angle EBF$ and $\angle FBC$. Prove that $\angle AEB ...
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1answer
25 views

proving orthocenter circumcenter centroid using congruence

Prove by congruency that orthocenter circumcenter centroid are collinear. Please don't use similarity, trigonometry. I have no idea how to do it.
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1answer
58 views

projection onto vector spaces

How do you project a vector on to the euclidean ball? For example, if there is a vector $x ∈ R^n$ how does one project this onto the euclidean ball. What are the steps for projecting a vector onto a ...
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1answer
108 views

Prove using integration "circle is a polygon when number of sides-> infinity

Is there a proof of "if number of sidesof a regular polygon ->infinitythe regular polygon -> circle." using integration?
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1answer
226 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 ...
0
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1answer
27 views

What direction does a vector with more than two entries point at?

Say you are given theses two vectors: u = (1, -2, 4) v = (-2, 4, 8) Since there are three entries, how do you know if they point in the opposite/same/different direction?
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1answer
49 views

How is bisector of one side of a right angled triangle, drawn from right angled corner equal to the half of the bisected side?

In a right angled triangle ABC with right angle at B and D being the mid-point of side AC, is it possible to prove BD=AD=CD without using co-ordinate geometry or circle theorems etc? (Just by using ...
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41 views

The Graphic Representation of Physical Quantities (Vectors only)

We usually use the figure such as the one below for the graphic representation of physical quantities such as forces: What should we call this figure? Should we call it a ray? But a ray is defined ...
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1answer
140 views

Prove the product of two distinct, opposite rotations is a translation

My homework question is what is the product of rotations through opposite angles α,−α about two distinct points. The answer is clearly a translation, but I'm not sure how to prove it. My idea on how ...
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2answers
91 views

Product of two opposite, distinct rotations

My homework question is what is the product of rotations through opposite angles $\alpha, -\alpha$ about two distinct points. The answer is clearly a translation, but I'm not sure how to prove it. My ...
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1answer
90 views

Distance from point to vertices of convex hull

let $P = \{p_1, \ldots, p_k\}$ be any $k$ points on the unit $n$-sphere in $\mathbb{R}^{n+1}$ and $p_0 = 0$ the origin. Furthermore, let $CH(P\cup \{p_0\})$ denote the (possibly degenerate) convex ...
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4answers
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Prove that the sequence $(a_n)$ defined by $a_0 = 1$, $a_{n+1} = 1 + \frac 1{a_n}$ is convergent in $\mathbb{R}$

I will post the exercise below: Prove that the sequence $(a_n)$ defined by $a_0 = 1$, $a_{n+1} = 1 + \frac 1{a_n}$ for $n \in \mathbb N$ is convergent in $\mathbb R$ with the Euclidean metric, ...
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2answers
85 views

What square does not contain the middle?

Consider the square $S = [-1,1]\times[-1,1]$. Suppose we put a smaller square inside it, which is rotated with an angle $\alpha$ relative to the large square. What is the largest such square that does ...
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1answer
150 views

Line-preserving transformations

Is there a name for the class of transformations on the Euclidean plane (or projective plane) that preserves lines? They are not all affine transformations; consider a perspective projection $p$ in ...
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0answers
24 views

About the relation between two regular icosahedrons and a regular dodecahedron

Let $C$ be the regular icosahedron, each of whose vertex exists at the centroid of the each surface of the regular dodecahedron $B$, each of whose vertex exists at the centroid of the each surface of ...
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0answers
30 views

geometry problem when calculating the angle.

I want to calculate the alpha of the triangle, but I could not find out more equations. Would you please try to help me? http://postimg.org/image/6mkn7v9zx/ Thank you.
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2answers
139 views

Ratio of triangle A and B if the length of the sides are A:25,25, 30 and b:25,25, 40

If $A$ triangle's side length are $25,25$ and $30$ and $B$ triangle's length are $25,25$ and $40$ what is the ratio between the two areas of the triangles? #math
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About the diagonals whose length is an integer multiple of the edge length of a regular polygon

Are the followings true? 1. In every diagonal of every regular polygon, some diagonals of regular hexagon, whose lengths are twice as long as its edge length, are the only diagonals such that the ...
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1answer
62 views

About the area of integer-edge-length triangles

Let $a,b,c$ be three edge lengths of a triangle whose area is $S$. Then, here is my question. Question : Supposing that $a,b,c$ are natural numbers, then does there exists $(a,b,c)$ such that ...
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3answers
997 views

Equation of Earth's Orbit around Sun (ellipse)

The preihelion is the smallest distance from a planet to the sun, and aphelion is the greatest distance. The sun is one of the two foci. For the Earth, the perihelion is 147.1 million km and the ...
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0answers
47 views

Why are histograms being of non euclidean space [closed]

I have this confusion about non euclidean space. I am confused why histograms live in a non euclidean space. I was reading this paper
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20 views

Confusion related to definition of non euclidean space [duplicate]

What are non euclidean spaces. I am confused why histograms belong to non euclidean space
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1answer
235 views

Finding the angle between 2 points on a circle

forgive me if this isn't the right place to ask this question but I am trying to figure out the value of theta along a line tangent to a circle from a starting position on the circle to an ending one ...
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1answer
152 views

how to derive relation between solid angle and surface area and the radius of sphere using definite integral?

how to derive relation between solid angle and surface area and the radius of sphere ? I know $s=r^2\Omega$ but how they got it using integral ?