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

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Construction of segment intercepting the sides of angle [on hold]

Given an angle and a point P in its interior, draw a line through P such that P is the midpoint of the segment intercepted by the sides of angle.
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43 views

Why $\pi$ is not Constructible with Circumference Length

If we use a compass to draw a circle with a diameter of length 1, then the circumference is $\pi$. From the definition given here (http://en.wikipedia.org/wiki/Constructible_number), it seems to me ...
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1answer
32 views

Existence of the Square in “Squaring the Circle” Problem

I understand that a square with area $\pi$ cannot be constructed using straightedge and compass. But such a square surely exists (and can be constructed through other means), right? If I'm right, I'm ...
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1answer
36 views

Construct a circle cutting two other circles at right angles

I have the following problem: On a line $l$ on this line are the centers of two circles $C_1$ and $C_2$ . Circles $C_1$ and $C_2$ do not intersect and are not tangent to eachother. (but one could be ...
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1answer
33 views

Construction based on circumcenter and incenter

Construct a triangle given the exact location of its circumcenter and its incenter, and the position of its angle bisector (including its direction), but not its length. I tried to consider the ...
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Is the set of numbers constructible with just a compass dense in $R^2$?

Suppose I have initially two points $0$ and $1$ in $R^2$. Given a set of $n$ points $P$ and $m$ circles $C$, suppose I am allowed to add any circle that has center $x$ and radius $r=|x-y|$ for $x,y ...
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2answers
61 views

Straightedge-only construction of segment of length $\sqrt{7}$, given regular hexagon with unit sides

Let's consider a regular hexagon with unit side length. Draw a line segment of length $\sqrt{7}$ using nothing except a straightedge (that is, an unmarked ruler). The position of the segment may be ...
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17 views

Does this construction method produce all possible convex pentagons (up to similarity)?

I read a question somewhere here about convex pentagons. I began to wonder if there was a way to list all possible convex pentagons and came up with the following method: 1) Draw a base line AB of ...
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44 views

Thread constructions in the Poincaré's disc

I just came across the following image (source) and realised something that should have been obvious to me a while ago: it should be possible to construct the envelopes of curves in the Poincaré disc ...
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3answers
131 views

How to construct a line with only a short ruler

Suppose I want to draw a line between the points A and B but I only have a ruler that covers only something between a fifth and a quarter of the distance between the two points. Also available a ...
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0answers
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Huzita Axiom 6 - Computing the Origami Trisection of an Angle

The Galois theory proof of the improssiblity of angle trisection rests on studying the triple angle formula $\cos 3\theta = 4 \cos^3 \theta - 3 \cos \theta$. Ruler and compass numbers can only be ...
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1answer
61 views

Construct a circle with straight edge and compass with some given conditions.

A line is given and two points in one half plane of the line are given. Construct a circle passing through these two points such that the given line is tangent to this circle. I have no idea how to ...
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1answer
47 views

which equations may be solved with compass and a marked ruler?

Ancient Greeks were not able to trisect a general angle with compass and straightedge: now we know that it is impossible, since we would need to solve a cubic equation while only linear and quadratic ...
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43 views

Where does the problem statement say sides are *equal*?

In the book How To Solve It, Part I, chapter/section 19, Pólya's hypothetical teacher poses the following problem to prove to the hypothetical student: Two angles are in different planes but each ...
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1answer
31 views

Given a $\triangle ABC$ construct another triangle with sides measuring the inverses of the altitudes of $\triangle ABC$

Given a $\triangle ABC$, we want to construct the $\triangle XYZ$ whose sides are the inverses of the altitudes of $\triangle ABC$ . If we denote the altitudes by $h_a,h_b,h_c$ then the sides of ...
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1answer
62 views

Construction of a triangle given some special points ($O,H,I$)

I'm a newbie in this site. I tried to search if this question was already answered but I'm not sure on how to do it. The problem is: given three distincts points $O,H,I$ namely the circumcenter, the ...
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0answers
29 views

how do the conic sections add to the possibilities of geometric construction?

If we are limited by what we can construct with compass and straight edge, then what becomes possible by expanding our toolkit to include all conic sections? The tools would be based on already ...
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1answer
27 views

smallest set of curves for constructing any real number and angle

If we are limited by what we can construct with compass and straight edge, then what are the fundamental curves required for constructing any real number? In other words, what is the smallest ...
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1answer
148 views

Geometry: How to find cube root, fourth root, fifth root… and so on?

As we know that square root of a number $n$ can be found by using a compass and a straight edge, given the line of length $n$. What I want to know is how to find cube root, fourth root, fifth root or ...
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1answer
81 views

Topology proof question?

How to prove that $X_1 := \{(x, y, z) ∈ \mathbb{R}^3 : x^2 + y^2 = 1\} / (S^1 × \{0\} )$ is homeomorphic to the union $X_2$ of two tangent spheres minus two points? What I know: Let $C$ be ...
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3answers
90 views

Construction of an ellipse

Is it possible to construct an ellipse with a line, compasses and a pencil? If yes, how and why is the construction correct?
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1answer
38 views

How do to derive the following SIMPLE geometric relationship between two points on a plane

Can someone show why: $$x' = L_1 \cos(a_1) + L_2\cos(a_1+a_2)$$ $$y' = L_1 \sin(a_1) + L_2\sin(a_1+a_2)$$ where $L_1$ and $L_2$ are the length of the red lines
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1answer
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Constructing an example of a infinite set of triangles on the rational line whose union does not contain the interior of a rectangle.

As part of my Topology course, I saw a proof the following proposition (as a consequence of Baire's category theorem): Proposition: Define $\triangledown_{t,h}$ as the interior of an equilateral ...
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0answers
62 views

Algebraic number of degree four that cannot be constructed with ruler and compass

The real number $b:=\frac{\sqrt{2a}+\sqrt{4\sqrt{a^2-3}-2a}}{2}$, where \begin{equation*} ...
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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, ...
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2answers
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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 ...
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2answers
182 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 ...
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2answers
61 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 ...
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2answers
60 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
50 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 : ...
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4answers
428 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 ...
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2answers
31 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 ...
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1answer
21 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 ...
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1answer
41 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 ...
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1answer
58 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 ...
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1answer
49 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 ...
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2answers
45 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 ...
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0answers
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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
86 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 ...
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1answer
96 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
68 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 ...
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1answer
51 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
104 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
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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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9 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
34 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 ...
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1answer
141 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
33 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
90 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 ...