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

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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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99 views

Minimum components required to construct a unique polygon

What is the minimum components required to construct a unique n sides polygon? If k is a such number for n sided polygon for any k given components polygon can be constructed by compass and ...
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1answer
20 views

construction of a line.

Two non parallel lines l & m are given. For given two angles A & B we have to construct a line n such that it makes angles A & B with lines l & m respectively. Line n intersects l ...
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2answers
35 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
24 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 ...
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33 views

geometric construction puzzle is my construction right?

I think I solved my own question of hyperbolic geometry (and circle ) construction problem but am not sure if it is correct, it looks that way but can somebody help me with prooving it? My ...
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2answers
165 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
48 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
89 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 ...
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1answer
102 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
101 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
124 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 ...
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63 views

Computing an area with geometric methods

Consider the construction below (it is self-evident, no tricks or misleading drawing) What is the area enclosed by the curve that goes through the points H,J,M,L ? I solved the problem using ...
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1answer
65 views

Geometric construction of 2 soap bubbles meeting

When 2 soap bubbles meet, you get a dividing wall between the soap bubbles. Plateau concluded that the three spheres (2 soap bubbles and the dividing wall) meet at angles of 120. Here's an ...
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1answer
87 views

What are four ways to quadrisect any triangle?

What are four ways to quadrisect any triangle with compass and straightedge? I have a few already: Draw a median and from the midpoint, draw two medians to the remaining sides. Draw a median and ...
2
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1answer
139 views

The Basel Problem and Theodorus' Spiral

I've been trying to find a classical solution to the famous Basel Problem solved by Euler. To those unfamiliar the problem is to find the sum infinite series made up of the reciprocals of square ...
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1answer
54 views

Why do people not use “partially directed” graphs?

Are there structures in use which are a mix of directed and undirected graphs? I.e. the effective edge-set consists of both directed and undirected vertex pairs. In the case that the graph is simple, ...
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1answer
374 views

Thinking outside of the box

You want to draw a circle with a 4 inch radius. A trivial task for you and your trusty compass. When you go to grab your compass which has not had much love for a while you find it is rusted shut; ...
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1answer
81 views

A problem of forming equal angles in plane geometry

C and D are two points on the same side of a straight line AB. Find a point X on AB such that angles CXA and DXB are equal.
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5answers
431 views

How to construct three mutually orthogonal circles in stereographic projection?

I'm new to spherical geometry and I enjoy doing ruler-and-compass constructions, so I'm trying to teach myself to do them in stereographic projection. I'm finding it challenging, to put it mildly. ...
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1answer
72 views

How to construct hyperbolically equidistant points on a line?

In Stillwells' "Sources of Hyperbolic Geometry " page 66 figure 3.3 shows an ((incomplete?) construction of hyperbolically equidistant points on a line. I tried to reconstruct the figure but did ...
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1answer
87 views

Constructing image under homography from known information

If $f$ is a homography of the real projective line, $f^2=id$ (is an involution), and $f$ has exactly two fixed points, how can I construct (geometrically) the image of an arbitrary point?
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1answer
330 views

Geometrical construction problem (2)

There are given: $[BC],[DE]$, the angles $\phi_1$, $\phi_2$ and a point $A$. Build a isosceles triangle $MAN$ (with$\angle{MAN}$ = $\pi/2$) with the apex of the right angle in $A$ such as ...
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1answer
32 views

Geometrical place of one point

Through one of the intersection points of two given circles, a line is build which intersects 2nd time the 2 circles in $A$ and $B$. Determine the geometrical place of the middle of $AB$. thanks for ...
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3answers
257 views

How to build a trapezoid

Build a trapezoid knowing its diagonals, the angle between them, and, also, the sum of $2$ adiacent sides. I appreciate your time and help!
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1answer
113 views

what is the most sided sturdy regular n-polygon that can be made with lego?

Was puzzeling with this question: What is the most sided regular n-polygon that can be made with lego? It has to be sturdy (the polygon should stay in shape when pushed around) made with the normal ...
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1answer
190 views

Could Euclid have bisected a line segment without his method of superposition?

In Book I Proposition 10 of the Elements, Euclid performs the bisection (i.e. finding a midpoint) of a line segment. In the course of doing so, he uses Book I Proposition 4, the Side-Angle-Side ...
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3answers
131 views

Construct triangle given inradius and circumradius

If we know the inradius $r$ of a triangle and the circumradius $R$ we can find out the distance between the incircle $I$ and the circumcircle $O$: $OI^2 = R^2-2Rr$. Therefore we can draw the incircle ...
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2answers
881 views

History of Compass/Straight Edge Construction

I'm interested in learning the origin of compass/straight-edge constructions. In particular, I am interested in the historical interplay between Euclid's axioms for plane geometry, and ...
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173 views

Prove that $45°$ angle can be trisected with straightedge and compass.

I want to prove that $45°$ angle can be trisected for this i have to show that $\sin 15°$ or $\cos 15°$ is constructible. How can i show that $\sin 15° \in \mathbb{Q}(\sqrt{2},\sqrt{3})$?
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1answer
124 views

Is it possible to approximate all angles with certain pythagorean triples?

With sticks $a,b$ and $c$ of length $3,4$ and $5$, you able to draw a right (tri)angle. But are also able to construct an angle $\cos\alpha=\frac35, \alpha=\arccos(\frac35)=$$53.13010...^°$. Is it ...
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0answers
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Exercise $7.7$ page $82$ Ian Stewart, Galois Theory

Prove that an angle $\theta$ can be trisected by ruler and compasses iff the polynomial $4t^{3}-3t-\cos\theta$ is reducible over $\mathbb{Q}\left(\cos\theta\right)$
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2answers
140 views

Two points on sides AB and AC of a triangle

How to determine using only the straightedge and compass the points P and Q on the sides AB and AC of a given triangle ABC such that the triangle APQ and the quadrilateral BPQC have the same surface ...
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40 views

Constructablilty of regular polygons on a sphere

There is a very clear theory of what polygons can be constructed in the plane. One of my professors said that he believed the same ones could be constructed on a sphere through stereographic ...
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1answer
63 views

Constructing a circle through 2 points

We have a triangle ABC with a circumscribed circle. Somewhere between BC we place a point D. There is a circle which goes through D and whose tangent at AB is A. This circle also intersects the ...
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95 views

Solving construction problems?

I recently encountered 'construction problems' in geometry. These were quite new to me and I didn't know the requirements they expected and prerequisites to solve them. I'll explain with an example. ...
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1answer
49 views

How to construct longitudinal from transversal waves and vice versa?

The above construction of a longitudinal wave out of a transversal wave has been encountered somewhere in an old physics textbook. There are several drawbacks with this construction. The maximum ...
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1answer
78 views

Pentagon construction

What is the simplest way to construct a regular pentagon using Euclid's Elements? Using the compass and straight edge is easy to get one side but how should the second side begin?
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1answer
354 views

Ruler and compass construction of the unit-distance petersen graph embedding

The Petersen graph is a unit distance graph, and this embedding is shown below, where each edge of the graph is one unit in length. Is there a ruler and compass construction for this embedding? If ...
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78 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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1answer
102 views

construct x =ab by using compass alone, if a and b are given segments.

I found the problem in the book "What is mathematics?". The following is a description of Mohr's constructions.(Macheroni problem) 9) Find $x = ab$, if $a$ and $b$ are given segments. I ...
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1answer
85 views

Perpendicular at a defined distance from point on line intersects another line in coordinates?

It approximatelly look likes the following picture The figure may be rotated at any angle. I know the coordinates of points A, B, C, D and the length of BF. ABD and CBD are equal (AD = CD and AB = ...
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0answers
50 views

Linejoin for fat lines?

I draw a figure with 2 fat lines. I need to draw a join between these lines correctly. Long red lines are in a middle of each fat line. What I know: coordinates of white points. the angle between ...
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3answers
432 views

Straight line problem : Find the number of points which lies between the figure : $(0,0) , (0,21); (20,0)$

Problem: Find the number of points which lies inside the triangle : $(0,0) , (0,21); (20,0) $ Approach : Let us take point $A = (0,0)$, $B = (0,21)$, and $C = (20,0)$. Since the figure ...
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1answer
64 views

Proof of necessary condition for constructibility of a number

I'm reading a proof of the necessary condition for a real number to be constructible, and it seems to leave out a few details that I can't really fill in. This is what I understand so far. We have to ...
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0answers
182 views

Application of Compass-and-straightedge construction

Nowadays with computers, Compass-and-straightedge construction doesn't look useful from my point of view. Probably I am just too narrow-minded, So I'm just curious, could anyone tell me any ...
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2answers
408 views

Why only compass and straightedge?

I've read and watched some lectures on euclidean geometry - not so advanced but I've seen the focus on constructions. Two instruments are used, compass and straightedge, I had the following doubts: ...
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1answer
146 views

Drawing Euclid?

I decided to study Euclid for fun. I have Oliver Bryne's edition. I also want, as much as possible, to construct the figures myself, to get a deeper understanding. How did people traditionally do ...
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2answers
2k views

Finding a line perpendicular to a line and passing through the intersection of other two lines

So here is the question as in my text book Find the equation of the line through the intersection of $2x - 3y + 4 = 0 $ and $3x + 4y - 5 = 0$ and perpendicular to $6x - 7y + c = 0$ so I ...