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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191 views

categorical description of the Minkowski sum of polytopes

Consider the category $\textbf{Poly}$ of polytopes, where the objects are convex hulls of finite subsets of $\mathbb{R}^d$ for arbitrary $d \in \mathbb{N}$ and where the morphisms are affine maps ...
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1answer
78 views

About a solid which satisfies $\sum_{i=1}^{n}x_i=0, |x_i|\le1\ (i=1,2,\cdots,n)$

For $n\ge 2\in\mathbb N$, let $S_n$ be the volume of a $(n-1)$ dimensional solid which satisfies $$\sum_{i=1}^{n}x_i=0, |x_i|\le1\ (i=1,2,\cdots,n).$$ Then, here is my question. Question : Can ...
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2answers
360 views

Problem in understanding a proof there are five Platonic solids.

Thanks to several comments by Gerry Myerson, it is now clear that I wasn't clear, up to a state where I seriously confused myself. In a renewed attempt: Recently, I've been thinking about Platonic ...
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7answers
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How to find the distance between two planes?

The following show you the whole question. Find the distance d bewteen two planes \begin{eqnarray} \\C1:x+y+2z=4 \space \space~~~ \text{and}~~~ \space \space C2:3x+3y+6z=18.\\ \end{eqnarray} ...
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1answer
76 views

Finding the “middle 2” of four lines

This may seem like an overly abstract problem, but it's the best generalization I could make of a specific problem I'm trying to tackle. This problem works in 2-dimensional Euclidean space. A ...
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0answers
119 views

Characterization of Convex Polygons

John Lee's Axiomatic Geometry has an interesting characterization of convex and non-convex vertices for polygons. Let $P$ be a polygon. Consider a ray emanating from a vertex of $P$ which does not ...
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0answers
151 views

Tiling an L-shape with “almost square”s

ABSTRACT: Define an "almost square" as a rectangles with aspect ratio in $[{1 \over 2},2]$. What is the minimal number of interior-disjoint almost-squares required to tile the following L-shape (where ...
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1answer
237 views

How to find the points of tangency of a parabola using Calculus?

How can someone find the points of tangency of a parabola in this situation? I need to find two points of tangency so that the triangle formed by the two tangent lines at those points and the x axis ...
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1answer
667 views

Is Euclid's Fourth Postulate Redundant?

Euclid's Elements start with five Postulates, including the fifth one, the famous Parallel Postulate. Less well, known however is the Postulate that forms the basis for motivation behind the fifth: ...
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1answer
165 views

Which is the correct definition of stationary point for real-valued functions in Euclidean space?

Given a multivariable real-valued function $f$ whose first partials all exist (but which aren't all continuous) at $p$, it is possible that $f$ is not (totally) differentiable at $p$. But since the ...
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1answer
113 views

Inequality in triangle

Let $ABC$ be a triangle and $M$ a point on side $BC$. Denote $\alpha=\angle BAM$, $\beta=\angle CAM$. Is the following inequality true? $$\sin \alpha \cdot (AM-AC)+\sin \beta \cdot (AM-AB) \leq 0.$$
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1answer
130 views

Finding the radius of a circle

Given a point A that outside a circle so that $AT$ is tangent to the circle in point $T$ And $AC$ is a secant to that circle in points $B,C$. From points $B,C$ we build heights to $AT$ ...
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1answer
52 views

Proving a triangle is isoceles

In the graphic we have an isosceles triangle, and the problem is Calculate $\text{m}\angle BCD$ I added the point $E$ at distance $x$ from $C$ because it causes $DE=x$, after playing with ...
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2answers
242 views

Which is the correct Hessian matrix (the standard matrix of a bilinear form)?

Please note the typo in the first entry: $\frac {\partial^2f} {\partial x_1\partial x_2}$ should instead be $\frac{\partial^2f} {\partial x_1\partial x_1}$. Also, this Hessian matrix need not be ...
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2answers
389 views

Decomposable Families of Shapes

There are two types of golden triangles in the world, as shown in the following picture: Here $\varphi = \dfrac{1+\sqrt{5}}{2}$ denotes the golden ratio. Each of these golden triangles can be ...
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1answer
43 views

Can we represent a symmetric curve by a parameter with symmetry?

Question : Can we represent the following curve $C$ by one parameter $t$ as $x=f(t),y=g(t),z=h(t)$ with symmetry? The curve $C$ in the $xyz$ space is defined as $$\begin{cases} x^2+y^2+z^2=1 ...
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1answer
151 views

Proving the diameter is two times the radius

I am stuck on the following question: Prove that each diameter is twice as long as each radius. I drew a circle, with center O and diameter AB. Is there a theorem that could help me say that ...
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0answers
88 views

Are there impossible boolean constructions?

I was reading about logic and I remember, for example: That with the binary $\mathtt{NAND}$ connector can be used to assemble all the other binary connectors - I already know that there are primitive ...
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2answers
433 views

Diameters and Circles

I have a question (given by a teacher) that looks really easy but then when I thought about it, couldn't find a way to find the answer. It is a proof question relating to diameters: Prove that any ...
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5answers
1k views

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 ...
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2answers
114 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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1answer
95 views

Question about Right Angles

I am stumped on the following question: Prove that the measure of a right angle is $90^\circ$. I so far have tried extending the lines making the angles but I can't get anything. I am not sure what ...
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0answers
147 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 ...
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2answers
208 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
368 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
108 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 ...
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1answer
86 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 ...
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1answer
355 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
3k 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
86 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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1answer
256 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 ...
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1answer
75 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
118 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
2k 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
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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
52 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
139 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 ...
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1answer
103 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
90 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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2answers
233 views

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

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
77 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
62 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
99 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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2answers
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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
615 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 ...
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1answer
29 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
172 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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1answer
359 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
193 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
219 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 ...