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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Why do these angle values imply collinearity?

I have another question about this proof of Morley's theorem. Towards they end it says $\angle RR_{2}R_{1}=\gamma^{+}\implies R_{1}$ is on segment $AR_{2}$. I dont understand why this is true. Figure ...
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More trouble with Morley's theorem proof

A followup to my previous question. In this proof they say $\angle R_{1}RR_{2}=\beta+\alpha-\gamma$. I understand why this is true, but I dont understand how the proof made that logical jump? In ...
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62 views

Trouble with Morley's theorem proof.

I am reading through this proof of Morley's theorem, and I am confused about one part. Why does $\angle ARR_{2}=\beta+\frac{\pi}{3}$? Also, why does the diagram put $\beta+\frac{\pi}{3}$ on $\angle ...
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25 views

squared euclidian distance as a heuristic for A*-algorithm

I've read that for the A*-algorithm the squared euclidian distance is not a good heuristic, because it might lead to wrong shortest paths. I further found two counterexamples, but I don't understand ...
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2answers
52 views

Is it possible to employ the circle as a primitive concept to measure the area of the rectangle?

Historically, it seems that we have found more practical uses for the area of the rectangle than the area of the circle. The definition of the area of the rectangle was given and used as a primitive ...
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70 views

Cutting a square of area $A$ through It's mid point yields two polygons of area $A/2$ for arbitrary cuts?

A square of area $A$ is cut by a straight line at It's mid point : It consists now of two rectangles, with area $A/2$. I want to find a proof that I could rotate the line and the area of the two ...
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1answer
26 views

Seeking for construction s.t. every intersection contains at least 3 lines

In Euclidean geometry, is there some set of lines in s.t. there are at least 2 intersections, but every intersection contains at least 3 lines, and no lines in the set are parallel? I tried for a ...
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1answer
57 views

n-Ball Volume and surface with $n \rightarrow \infty$

I am thinking about something I just read: The volume of the n-ball is given by $V_n(r) = \frac{\pi^{n/2}}{\Gamma (\frac n 2 + 1)}r^n$ and its surface area is $S_n(r) = \frac{\pi^{n/2}}{\Gamma (\frac ...
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42 views

$3x+3y-1,4x^2+y-5,4x+2y$ are sides of an equilateral triangle

I am completely lost in this one $3x+3y-1,4x^2+y-5,4x+2y$ are sides of an equilateral triangle, its area is closest to the which integer?
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2answers
47 views

On congruent chords

Let $C_i=C(A_i,r_i)$ two secant circles intersecting each other at $R,S$, with $r_1\neq r_2$. Let $M$ be the median point of $A_1A_2$. Let $t\perp RM$ at $R$, intersecting the circles at $X,Y$. I'd ...
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40 views

Stuck at Extended Euclidean Algorithm to solve equation

I'm trying to solve the following function via the Extended Euclidean Algorithm, but I'm stuck at the last step where I need to sub in sub 2. ...
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76 views

A convex $n$-gon and the $n$-gon made by its $n$ medians

For a convex $n$-gon $P_1P_2\cdots P_n$, let $M_i$ be the mid-point of the line segment $P_iP_{i+1}\ (i=1,2,\cdots,n)$ where $P_{n+1}=P_1$. Also, let $Q_1Q_2\cdots Q_n$ be an inner $n$-gon made by $n$ ...
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1answer
49 views

Given three points, can one construct a hyperbolic curve using classical geometric construction method?

Using straight edge and compass method, can a hyperbolic curve be drawn through three given points?
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97 views

Dynamic Geometry Software for Straight-edge and Compass Constructions

Geogebra is a very good dynamic geometry software. It has so many default tools, e.g. parallel line, angle bisector, tangent to the circle, inscribed and circumscribed circles, etc. But I want the ...
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110 views

Cutting a solid torus by a plane parallel to its rotary axis

I've been interested in finding a cutting method such that the area of the cut surface is max when we cut a $3$ dimensional solid by a plane. Then, here is my question. Question : How about the ...
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3answers
51 views

Product of rotation and translation is a rotation

I have a homework question that I'm not sure how to answer. ...
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1answer
56 views

Connecting Coordinate Geometry and Plane Geometry

What is it that allows us to take theorems proven in Euclidean geometry (i.e. with Euclid's five postulates or Hilbert's Axioms) and then apply them outside of Euclidean geometry. For example in ...
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41 views

How does this proof of law of sines determine equal angles?

I was reading over this proof of the law of sines and they say that $\angle CAB = \angle DOB$ because of "basic geometry". I do not get it though, how can you say that the angles are equal?
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Find the flaw in the attempted proof of the parallel postulate given by J. D. Gergonne

The attempted proof: Given $P$ not on line $l$, line $PQ$ perpendicular to $l$ at $Q$, line $m$ perpendicular to $PQ$ at $P$ and point $A \neq P$ on $m$. Then, let $PB$ be the last ray between rays ...
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1answer
82 views

Fair division of an octagon

A land-plot belongs to two partners. Its form is a regular octagon with area 1. They want to divide it such that one gets area $p$ and one gets area $1-p$, where $p \in (0,1)$ is a given constant. ...
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1answer
64 views

Problems with axioms and their potential uses in real life.

Okay, so I am looking for more examples that revolve around this topic. My topic is: "Is the assumption of basic truth needed in order to create a system of mathematics that is reliable?" Here, when ...
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2answers
196 views

Is it true that all of the euclidean geometry problem in the IMO(international mathematical olympiad) could all be solve by the analytical geometry?

Is it true that all of the euclidean geometry problem in the IMO(international mathematical olympiad) or even generalize to say that all the plane geometry problem and 3d-geometry could be solve by ...
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386 views

Is there a dissection proof of the Pythagorean Theorem for tetrahedra?

Of the many nice proofs of the Pythagorean theorem, one large class is the "dissection" proofs, where the sum of the areas of the squares on the two legs is shown to be the same as the area of the ...
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28 views

Seeking collection of geometry problems

I'm looking for a collection of geometry problems that does not surpass high-school to first-year university geometry.. Problem that could be found sometimes in SAT/GMAT test. They should all at ...
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1answer
49 views

How to enclose a ball more than once with a surface homeomorphic to $S^2$? In 3D.

In 3 dimensional space, how to enclose a monopole (either point-like or ball-like, both types exist.) more than once with a surface homeomorphic to $S^2$?
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3answers
166 views

What precisely is the difference between Euclidean Geometry, and non-Euclidean Geometry?

I was wondering, what it is precisely which defines the difference between Euclidean and non-Euclidean Geometry, in a few words/equations/diagrams? Would I be correct in understanding that ...
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2answers
63 views

What is the meaning of “integral point”?

While reading this paper (http://cowles.econ.yale.edu/P/cd/d04b/d0473.pdf) I encountered the concept of "integral point", used first in definition 5.1, on page 34. Does anybody know more details about ...
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0answers
76 views

Megiddo's algorithm for lines of least weighted sum distance from a set of points

I came across the following problem: Given a set of n points (coordinate in 2d plane) within a rectangular space, find out a line ($ax+by=c$), from which the sum of the perpendicular distances of all ...
2
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0answers
142 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
69 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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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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69 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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26 views

Median Partition

A triangle is divided into six smaller triangles by its medians. Prove that the circumcenters of these six triangles lie on a circle. (Floor van Loemen, Monthly April 2002) - (Geometry Unbound, p56). ...
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33 views

Proof of congruent angles from Hilbert's axioms

I'm looking for a proof, using Hilbert's plane axioms (compiled, for instance, here), of the congruence of the four blue angles. where the lines $CE$ and $HF$ are parallel. This is a well known ...
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26 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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73 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
97 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
207 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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95 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
87 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
97 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
45 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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1answer
113 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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244 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
39 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
54 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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71 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
93 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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4answers
576 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
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 ...