Questions on the construction of geometrical figures using a limited set of "tools".

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Is there a direct proof that pi is not the root of an algebraic equation whose degree is a power of 2 [duplicate]

All known proofs that the circle cannot be squared are based on Lindemann's theorem that $\pi$ is not analgebraic number. But this seems to be a case of using an atomic bomb to kill a fly. What ...
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2answers
87 views

How to make π degree angle?

Can we make π degree angle? π is a decimal and angles are divided into minutes and seconds, but, I think (I'm not sure), we can still divide 1 degree into decimal parts (we can divide 1 degree into ...
2
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1answer
120 views

Triangle Construction being known its perimeter, height and angle

"Construct a triangle $ABC$ knowing his perimeter, the angle $\widehat{A}$ and the height relative to $BC$, i.e., $h_a$." It really looks to be an easy one, but I wasn't able to do it... :( Any hint? ...
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1answer
72 views

Tangent Circumference Construction

"Construct a circumference that is tangent to a given circumference and tangent to a line $r$ through a point $A$ of this line." I've done the line perpendicular to $r$ through $A$, cause we know ...
2
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1answer
62 views

Construct a triangle given an angle and two medians

Construct, with ruler and compass, a triangle $ABC$ knowing the angle $\widehat{A}$ and $m_a$ and $m_b$, where $m_a$ and $m_b$ are the medians relative to the vertices $A$ and $B$, respectively.
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0answers
123 views

History of the neusis construction of cube roots?

A simple neusis (marked ruler) construction of $\sqrt[3]{2}$ is given in many places, for example wikipedia. My question is: what is the history of this construction? As far as I can determine, all ...
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1answer
50 views

Visualizing construction of a certain function

Consider a map $f$ defined on $\mathbb{R}^3$ with the following properties:\ 1) $f$ fixes the poles $(0,0,\pm1)$.\ 2) $f$ is symmetric in the plane $\{x=0\}$ and the plane $\{y=0\}$.\ 3) $F$ is ...
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0answers
63 views

Smallest field containing $\mathbb{Q}$ and closed under square root

I'm following Isaacs' Algebra and I need to prove that the field $K$ of constructible numbers is the smallest subfield of $\mathbb{C}$ such that is closed under taking square root. I already know ...
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0answers
10 views

Maximization of an integral based on Vittali covering

Let $f \in L^1(Y)$ and $f\geq 0$. Where $Y = [-1,1]^N$, suppose $\{B(y_k,\epsilon_k)\}_k$ forms a Vitalli covering of $Y$, satisfying \begin{equation}\begin{aligned} &(i) B(y_k,\epsilon_k) ...
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1answer
35 views

Constructing a special point in quadrilateral

This problem appeared in my mind when I was working on another problem, I found it interesting but still don't know how to solve it yet. So i decided to post it here, I hope we can discuss and you guy ...
5
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1answer
152 views

History of the three “impossible” compass-and-straightedge problems

I'm preparing a presentation about constructible numbers and I wanted to know some of the history about it to motivate the topic. I wanted to know if the classical Greek problems (doubling the cube, ...
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1answer
34 views

Geometric Construction

"Given in position the points $A,B$ and $P$ and a segment $m$, draw through $P$ a line $r$ in such a way that $A$ and $B$ be in opposite sides of $r$ and that the sum of the distances from $A$ and ...
3
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1answer
97 views

Regular Polyspiral (Geometry)

Criteria: -Each side of every polygon has to be the same length. -Every successive polygon has to have one more side. -Each additional polygon has to start on the opposite right side (assuming the ...
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4answers
222 views

Pentagon Geometry

$ABCDG$ is a pentagon, such that $\overline{AB} \parallel \overline{GD}$ and $\overline{AB}=2\overline{GD}$. Also, $\overline{AG} \parallel \overline{BC}$ and $\overline{AG}=3\overline{BC}$. ...
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0answers
43 views

Is this element constructible from this elements?

Let the figure below. According to same notation of the figure verify if it's possible to construct the point $\displaystyle \zeta=e^{\frac{2\pi i}{13}}$ with straight-edge and compass from ...
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1answer
123 views

Dynamic Geometry Software for teach construction

I would like to know about a software that will help me show construction steps to the students with using a Compass/Straight Edge/Protractor/Divider. Here is example video below. ...
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1answer
75 views

Are there numbers that we can't get with a usual compass and ruler, but can get with 3D compass and ruler?

If we have unit segment, we can use a compass and ruler to make segments whose length represents many numbers (all rational, sqrt(2)), but there are "unreachable" ...
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4answers
216 views

Construction of a triangle

I need to construct a triangle with given information: $c = 6$, $h = 4$ and $\alpha - \beta = 30º$. I'll put approximate result for any clarification.
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0answers
170 views

Which power means are constructible?

The three classic Pythagorean means $A$, $G$, $H$ (arithmetic, geometric, and harmonic mean respectively) of positive real $a$ and $b$ have a cute geometric construction, as does the quadratic mean ...
3
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1answer
46 views

Computational compass-and-straightedge constructions

I recently came across Ancient Greek Geometry, a web toy by Nico Disseldorp, and it got me wondering: is there a way to exactly model the points generated by geometric constructions, starting from two ...
5
votes
1answer
90 views

Is the non-existence of a general quintic formula related to the impossibility of constructing the geometric median for five points?

In particular, in the Computation section of in the Wikipedia page for geometric median, there is this statement: ...but no such formula is known for the geometric median, and it has been shown ...
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2answers
49 views

Construction of a point with resprect to a triangle

For a given triangle $ABC$, how to construct a point $P$ such that $PA \colon PB \colon PC = 1 \colon 2 \colon4$?
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2answers
101 views

Drawing a triangle from medians

Is it possible to draw a triangle, if the length of its medians $(m_1, m_2, m_3)$ are given only? Someone asked me this question, but I can not see it. Is it really possible? UPDATE Apart from the ...
2
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1answer
68 views

How to embed this circle tangent to the other circles?

I want to construct a circle that would be tangent to the $3$ circles and would have its diameter lie somewhere on the segment $BI$. $EF$ includes the diameters of the $3$ given circles. $EB=BF$. ...
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17answers
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How to create circles and or sections of a circle when the centre is inaccessible

I am doing landscaping and some times I need to create circles or parts of circles that would put the centre of the circle in the neighbours' garden, or there are other obstructions that stop me from ...
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1answer
72 views

A construction of a triangle mapping with a homothety

Given an acute triangle $ABC$ draw a triangle $PQR$ such that $AB=2PQ,BC=2QR,CA=2RP$, and the lines $PQ,QR,RP$ pass through $A,B,C$ respectively. Note $A,B,C,P,Q,R$ are distinct. This is a problem ...
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0answers
47 views

Geometric question (middle line of a parallelogram)

Let $ABCD$ be a parallelogram and $I,J$ are two points such that $AI = ID$ and $BJ = JD$. Show that ($IJ$) is parallel to ($AB$).
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0answers
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Geometrical construction of 7th roots of unity given parabola x^2

Given parabola $x^2$ on plane, how can I construct 7th roots of unity? I was straying with my only idea, that sum of squares of real and imaginary part of roots of 1 equals 1 and belongs to ...
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0answers
35 views

Given a pair of conics, construct (synthetically) their shared tangent lines

There are various well-known ways to construct the common tangents to a pair of circles; this is an easy one. I also just learned that we can use Pascal's Theorem to construct a tangent to a conic at ...
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1answer
259 views

Geometric proof of dot product distributive property

I'm working my way through a text book for fun in order to keep my math brain fresh and came across this simple yet perplexing problem. "Demonstrate geometrically that the dot product is ...
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1answer
87 views

How to prove by induction the constructibility of a line segment of length $\sqrt{n}$?

How to prove the following statement by induction? If a line of unit length is given, then a line of length $\sqrt{n}$ can be constructed with straightedge and compass for each positive integer $n$. ...
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1answer
58 views

Prove : for every integer $n\ge1$, if the regular $n^2$-gon is constructible, then $n$ has no odd prime divisors.

For every integer $n\ge1$, if the regular $n^2$-gon is constructible, then $n$ has no odd prime divisors. I know is has something to do with the fact the output Euler's Totient function on $n^2$ ...
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0answers
40 views

Dummit & Foote: Construction of the regular $17$-gon

In the last step of the exercise where we construct the $17$-gon, I have to draw a circle with a diameter whose endpoints are $(0, 1)$ and $(\eta_1', \eta_4')$ and show that it intersects the positive ...
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1answer
44 views

Is the given triangle unique?

I was reading Polya's How to Solve It when I came across the following problem. Construct a triangle with an angle, the length of altitude through that angle and the perimeter of the triangle given. I ...
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1answer
47 views

Ramanujan (LPS) graph construction

In the original LPS article they mention two constructions for Ramanujan graphs, one with order $q^3$ nodes and the other in order $q$. My question is regarding the second construction (which is ...
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2answers
128 views

How to construct the point of intersection of a line and a parabola whose focus and directrix are known?

I found this problem in Polya's "How to solve it". It goes as follows Using only a straight edge and a compass, construct the point(s) of intersection of a given line and a parabola whose focus ...
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1answer
88 views

Logic behind a Geometric Construction of regular heptadecagon

I'm reading a Chinese book "Methods of Mathematical Physics" by Wu Chongshi. During introduction of complex analysis, it explains a Geometric Construction of regular heptadecagon. Task: to achieve ...
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57 views

proof of construction of angle 360/17

One construction for a 17-sided polygon is shown below. ...
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0answers
28 views

To construct a right triangle given the hypotenuse and sum of two legs [duplicate]

NOTE: I want a hint only. A compass and a straightedge construction:Given a hypotenuse and the sum of lengths of the legs,we need to construct a right triangle. MY TRY: From any ray $BE$, ,let ...
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2answers
33 views

Some help needed with a geometry question

What is a formula for all integers n for which a regular polygon with n sides can be constructed using a ruler and compass construction?
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1answer
24 views

Constructing a line given two other lines, two angles, and a distance.

Two non parallel lines $l$ and $m$ are given. For given two angles $A$ and $B$ we have to construct a line $n$ such that it makes angles $A$ and $B$ with lines $l$ and $m$ respectively. Line $n$ ...
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2answers
39 views

Construction of pentagon with all sides and alternative two angles given.

All sides of a pentagon ABCDE are given. Angles A & C are given. Can such pentagon be created by straightedge and compass?
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1answer
15 views

Geometric Sequence with Normal Distribution Problem

Given: The running time (in seconds) of an algorithm on a data set is approximately normally distributed with mean 3 and variance 0.25. a. What is the probability that the running time of a run ...
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1answer
25 views

Constructing “strange” sets by removing elements: Is there any comparison?

Recently I had a thought about some sets with special properties that are constructed from a starting set $A$ and successively removing elements from it. I am not an expert in set theory but perhaps ...
2
votes
2answers
323 views

triangle construction given side, angle and median

I can't figure out the solution to this, it looks to me like it doesn't have any solution but I need some proof. problem: Construct a triangle ABC with given $a=6 cm$ $\alpha=75^\circ $ and ...
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2answers
52 views

Construction by compass and straightedge.

l,m,n are three concurrent line concurrent at point A. Given a point B on line l. Is it possible to construct point C on line n such that line m is a median of triangle ABC
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1answer
100 views

What are some alternative ways of describing n-dimensional surfaces using control points other than Bezier surfaces?

I'm interested in problems involving geometric constraints and curve subdivision. I noticed that most of these problems describe the curves/surfaces using the Bezier form. I wanted to know if there ...
10
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1answer
109 views

Using a compass and straightedge, what is the shortest way to divide a line segment into $n$ equal parts?

Sometimes I help my next door neighbor's daughter with her homework. Today she had to trisect a line segment using a compass and straightedge. Admittedly, I had to look this up on the internet, and ...
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1answer
142 views

To draw a straight line tangent to two given ellipses

How can I draw a a straight line that touches two ellipses? There are, like for two circles, 4 different solutions. I´m not interested in the analytical solution, but in the geometrical drawing, ...
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1answer
142 views

hyperbolic geometry (and circle ) construction problem

Was thinking about hyperbolic geometry, the Poincare Disk Model and Sweikarts constant and combined them all in a construction puzzle that I was unable to solve. My construction puzzle: Given: A ...