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

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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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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
132 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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0answers
34 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
49 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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85 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
44 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 ...
2
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1answer
70 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
201 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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73 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
81 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
75 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
47 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
416 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 ...
4
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1answer
61 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
135 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
311 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: ...
2
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1answer
124 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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0answers
166 views

Spherical Construction Problem about using a ruler and a compass

I've known the following theorem: Theorem 1: On a plane, if we have both of a primitive ruler and a primitive compass, then we can do the same construction as we can do by using a macro-ruler or a ...
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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 ...
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1answer
81 views

How to draw or construct a brachistochrone

Since the brachistochrone is such a beautiful curve in our planet, I want to build one somewhere around 1.60 m high. I need a quick way to trace the curve on the material to be cut, e.g. a wide sheet ...
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0answers
303 views

Minimal number of moves to construct the challenges (circle packings and regular polygons) in Ancient Greek Geometry?

In the web game Ancient Greek Geometry, there are challenges to construct regular polygons and circle packings using ruler and compass constructions. The game measures the number of line and circles ...
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1answer
85 views

Occurrence of $e$ in intersecting circles.

Consider two identical circles that share a radius such that they intersect. The radii of the circles are $\pi\over 2$. If this new shape sits such that its major axis is horizontal and the shortest ...
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Can one trisect $\arccos(6/7)$?

Is this proof correct? Proof: Here $\theta = \arccos(6/7)$. Now to show we can't trisect $\theta$, we show that $\theta/3$ is not constructible by finding the irreducible polynomial in $\mathbb ...
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1answer
517 views

Equilateral and equiangular polygon

Can we have an equilateral polygon $n \geq 5$, which is not equiangular? Ot does every odd n-gon which is equilateral must be equiangular? Is a construction of an equilateral but not equiangular n-gon ...
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1answer
103 views

Descartes' theorem: Find the midpoint of the $4$th circle

I'm thinking about Descartes' theorem "Wikipedia". I understood how to find the radius with algebra. Now I'm trying to use ruler and compass to find the midpoint of the 4th circle. I thought about ...
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1answer
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How can I construct a square using a compass and straight edge in only 8 moves?

I'm playing this addictive little compass and straight edge game: http://www.sciencevsmagic.net/geo/ I've been able to beat most of the challenges, but I can't construct a square in 8 moves. To ...
2
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1answer
232 views

Geometric proof that if n is a non-perfect square, then √n is irrational.

I know there is a geometric proof of the irrationality of √2. I thought maybe this one could be generalized for √n when n is a non-perfect square, but I could not find something like that anywhere. ...
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3answers
601 views

Pair of compasses drawing a square (from children's fiction)

I have read a children's book where alien race of "square people" used a pair of compasses that drafted a perfect square when used. Now I wanted to explain to the child that it is not possible to have ...
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352 views

To construct an ellipse, being a projection of a great circle, given two points on it

I'm looking for a geometric construction which would allow me to draw an ellipse, which is supposed to be an orthographic projection of a great circle of a sphere, given two points on it. The ...
4
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2answers
126 views

Construction of triangle

I don't know how to prove or disprove the following problem: How to construct triangle if elements $a$, $b$, $\beta-\gamma$ are given? Is it constructible (if not, how to prove it)? Any help is ...
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2answers
77 views

Are the lengths from this recursive construction a geometric sequence?

In his 1999 review of Edward Tufte's Visual Explanations in the Notices of the AMS (third page), Bill Casselman gives a very pretty proof of the irrationality of the golden mean. More precisely, ...
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326 views

Importance of construction of polygons

Why did the ancient Greeks give so much importance to the construction of regular polygons with $n$-sides using only ruler and compass and tried to study for what $n$ was such a construction possible? ...
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139 views

Software to Explain Geometrical Construction

I would like to know about a software that will help me SHOW construction of geometrical figures. Maybe something as simple as constructing a triangle. BUT I need to see a compass and a Straight ...
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0answers
48 views

Geometric calculation: two kneading discs

I have two kneading discs of a screw overlapping with each other at 60 deg. I know the cross section area of one disc and I want to know what will be the overlap area if the other disc is rotated at ...
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2answers
583 views

Equilateral triangle geometric problem

I have an Equilateral triangle with unknown side $a$. The next thing I do is to make a random point inside the triangle $P$. The distance $|AP|=3$ cm, $|BP|=4$ cm, $|CP|=5$ cm. It is the red ...
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2answers
129 views

Apollonius circle

I'm given two points, $A$ and $B$, and two lengths, $b$ and $c$. I need to find the locus of point $C$ such that $BC:AC=b:c$. This link describes Apollonius circle of first type, but I can't seem to ...
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3answers
385 views

Constructing irrational numbers

Which of the following numbers is constructible? 1) $3.14141414\ldots$ 2) $\sqrt{3}$ 3) $5^\frac{1}{4}$ 4) $2^\frac{1}{6}$ Also, Given a segment of length $\pi$, is it possible to construct, ...
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2answers
221 views

Doubling the cube neusis

Can anyone explain me in simple maths the neusis construction at http://en.wikipedia.org/wiki/Doubling_the_cube#Using_a_marked_ruler? Why and how does it produce the $\root 3 \of 2$?
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2answers
236 views

Is it fair to say that Kepler's equation involves squaring the circle?

I'm trying to understand the degree to which I'm at sea, anchored, or on firm ground, and how to firm up my understanding as needed. I think Kepler's equation reduces to a task (call it $t$, which ...
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1answer
130 views

How to draw right triangle whose hypotenuse is side of square and whose side is tangent to a circle?

Given a circle whose center (P) is at center of a square, and using pencil, compass, and straight-edge, how can I create a right triangle whose hypotenuse is side of square and whose side is tangent ...
3
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1answer
272 views

Finding a circle that touch two other circles and a line

Given two circles $(x1, y1, r1), (x2, y2, r2)$ and a line passing through two points $A(xa, ya)$ and $B(xb, yb)$. How to find a circle $(x3, y3, r3)$ that is tangent to line and two given circles? I ...
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5answers
737 views

Sangaku. How to draw those three circles with only a ruler and a compass?

I found in a book of Sangakus the following problem. Let $R_b$, $R_g$ and $R_r$ the radiuses of the blue, green and red circles $C_b$, $C_g$ and $C_r$. Prove that ...
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1answer
290 views

Construct two chords of equal length through points A and B (two arbitrary points INSIDE a circle) that are perpendicular to each other.

Its a construction problem I am having trouble with. I realize I need to use rotations and/or other isometries but I am really stuck. Any help would be really appreciated! Thanks!
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1answer
530 views

Draw a parallelogram given its sides and the angle between diagonals

I'm having trouble with this one: Draw a parallelogram knowing the lengths of its sides and the angle between the diagonals. Bonus points if the answer uses a translation, because that's where this ...
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1answer
43 views

proving that this family of angles, cannot be trisected

Given a field $ \Bbb Q \subset K \subset \Bbb C$. One can prove that $\beta \in \Bbb C$ is constructible over $K$ iff the galois group of the minimal polynomial over $K$, $m_{\beta}(x)\in K[x]$ is a ...
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1answer
145 views

Proving there is a rectangle inside an hexagone.

How can I prove this : Let a regular hexagon ABCDEF and M, N, R and S the respective midpoints of AB, CD, DE and FA: i) Prove that the MNRS is a rectangle. ii) compare the area of MNRS and the area ...
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2answers
470 views

Construct a triangle given one side, its height and inradius

I've been scratching my head with this problem: "Draw a triangle given one of its sides, the height of that side and the inradius." Now, I can calculate the area and obtain the semiperimeter. From ...
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2answers
521 views

Resource for learning straightedge and compass constructions

Does anyone know a good resource for learning about straightedge and compass constructions besides "The Elements?" I tutor geometry to middle-schoolers and high-schoolers and thought that including ...
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1answer
63 views

Construction of a field

Given the polynomial $$f(x)= x^4-16x^2+4$$ which has $a=\sqrt 3+\sqrt 5$ as one of its roots in $\Bbb C$, can you use $f(x)$ to construct a field $E$ of the form $Q[x]/I $ for some appropriate ideal ...