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

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2
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0answers
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Is there any construction method that yields all algebraic numbers?

Compass and Straightedge = rationals and square roots Origami = rationals, square roots and cube roots (I think) How far can we get if we use other tools, like rulers, protractors, pieces of string, ...
-5
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3answers
36 views

Extend length of the vector [on hold]

I have a vector $AB$ defined by the points $A$ and $B$ in $x,y$ coordinates. I need to find the coordinates of the point $C$ on the the line defined by $A$ and $B$ for any value of $L$ length I set, ...
0
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2answers
46 views

Circle with center point and tangential to lines

I have defined Points all points (3 blue, and one green). All points have the same distance to A point. Yellow lines are bisectors. I have equations of AB and ...
4
votes
1answer
96 views

Geometry construction problem

Given two circles $S_1$ and $S_2$, a line $l_1$, and a length $a$ that is less than the sum of the diameters of the circles, construct a line $l$, parallel to $l_1$, so that the sum of the chords that ...
1
vote
2answers
30 views

Construct a midpoint of two parallel lines with only straightedge

Say I have a large plank of wood that I'm trying to cut in half the long way, but I only have a straightedge (no compass). How can I mark the midpoint between the long edges? (As a note, this isn't a ...
1
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2answers
44 views

Geometric Construction Problem

Given three non-collinear points A, B, and C, construct three circles that are pairwise tangent at these points. Are there any cases where such circles do not exist? I am not sure how to start the ...
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0answers
37 views

How to calculate the edge sizes of a Goldberg polyhedron?

I want to build a paper model of a Goldberg polyhedron, a Icosahedral G(2,2). But i cant find the formula to calculate the 2 sizes needed for the edges. For a G(4,1) the sizes are here : ...
8
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4answers
378 views

Construct quadrangle with given angles and perpendicular diagonals

The following came up when I worked on the answer for a different question (though it was ultimately not used in this form): Proposition. Given positive angles $\alpha,\beta,\gamma,\delta$ with ...
0
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2answers
22 views

Find a point C on an infinite line AB which, when connecting two other points M and N, would form congruent angles

On an infinite line $AB$, find a point $C$ such that the rays $CM$ and $CN$ connecting $C$ with two given points $M$ and $N$ situated on the same side of $AB$ would form congruent angles with the ...
0
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1answer
19 views

construction of the rectangle with the highest area

I have 2 times a square with side length 2, 2 times a square with side length 3, 1 times a square with side length 4 and 1 times a square with side length 5. I have to create the rectangle with the ...
0
votes
1answer
23 views

Find circle for two points, one with given angle.

I have point A and B. I also have a vector v. How can I mathematically find a circle whose tangent at point C has the same angle as v where point C is the same as B and the circle also contains point ...
2
votes
1answer
35 views

More general constructible numbers?

I've recently learned about the field of constructible numbers (those which can be constructed with compass and straight-edge). A theorem in this subject states that a number $z$ (real or complex) is ...
3
votes
1answer
43 views

Where to place a bridge over the highway?

I've got a problem to solve. I had 2 different ideas which didn't actually work for all cases. On the both sides of highway there are 2 houses K and L (as in the attached picture). Line, that passes ...
0
votes
2answers
32 views

Trisection of an angle with straightedge and a compass

Suppose there exists an angle Z such that cos Z = -11\16 Prove or disprove that such an angle can be trisected with a straightedge and a compass. Well, we know that an angle is constructible iff its ...
2
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0answers
41 views

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 ...
1
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2answers
71 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
votes
1answer
52 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? ...
0
votes
1answer
47 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
votes
1answer
37 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.
2
votes
0answers
62 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 ...
1
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1answer
49 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 ...
2
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0answers
45 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
4 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) ...
0
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1answer
30 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
115 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
29 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
votes
1answer
57 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 ...
6
votes
4answers
206 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 $\overline{AG}=3\overline{BC}$. ...
2
votes
0answers
41 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 ...
1
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1answer
50 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. ...
6
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1answer
66 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
201 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
156 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
votes
1answer
41 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
79 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 ...
2
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2answers
47 views

Construction of a point with resprect to a triangle

For a given triangle $ABC$, how to construct a point $P$ such that $PA \colon PB \colon PC = 1 \colon 2 \colon4$?
3
votes
2answers
86 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
votes
1answer
51 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. $EB=BF$. ...
40
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17answers
2k views

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 ...
1
vote
1answer
53 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
38 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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votes
0answers
50 views

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 ...
1
vote
0answers
25 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 ...
0
votes
1answer
150 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
59 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
56 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 $n^2$ ...
0
votes
0answers
35 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 ...
1
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1answer
41 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 ...
0
votes
1answer
30 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 ...
2
votes
2answers
75 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 ...