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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69 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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1answer
90 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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0answers
91 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
87 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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2answers
365 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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3answers
217 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
176 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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2answers
48 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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2answers
382 views

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 $...
6
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1answer
143 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
677 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
927 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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1answer
2k 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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0answers
44 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
58 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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4answers
3k 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 non-...
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2answers
1k 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
200 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 ...
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0answers
200 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 (i.e....
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1answer
80 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
405 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
17k views

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
78 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
122 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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161 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
274 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 ...
5
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1answer
813 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
183 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 ...
3
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1answer
121 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
139 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 $AT$ in ...
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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
271 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
439 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
196 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
92 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
531 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
2k 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 ...
2
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2answers
116 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
101 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
150 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 C^{n+...
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2answers
251 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
449 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
111 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
88 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
386 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
88 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 ...
4
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1answer
263 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
76 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 ...