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Questions tagged [geometric-construction]

Questions on constructing geometrical figures using a limited set of tools. The compass and straightedge are almost always allowed, while other tools like angle trisectors and marked rulers (neusis) may be allowed depending on context.

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Compass and straightedge construction of a square of an arbitrary line segment

If I have some arbitrary length line $AB$ and a unit length line $CD$, how can I construct a line whose length is equal to the square of the length of line $AB$ using a compass and a straightedge?
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5answers
445 views

On The Construction Of An Ellipse

You know how when you construct an ellipse, you take a rope, fix it to 2 points, and stretch that rope? When the rope is being stretched, let's call the part of the string attached to the first ...
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2answers
90 views

On The Specifics Deriving The Equation Of Ellipse

I am trying to learn how to derive the equation of an ellipse, from this website (https://people.richland.edu/james/lecture/m116/conics/elldef.html). I am struggling, however, to prove to myself why ...
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1answer
73 views

Creating a Straightedge, Compass and “New Tool” problem.

I've been trying to create a problem that involves similar concepts as straightedge and compass construction. The idea is to add a new tool that allows for a different type of graph to be drawn ...
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1answer
60 views

Constructions: A straight line segment of length pi units. [duplicate]

A line segment of length 22/7 units or 3.14 units can be drawn. But how can a line segment be drawn of exactly pi units?
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1answer
44 views

Largest circle contained in a region delimited by 4 circumferences

I'm working in a region delimited by 4 circumferences, concentric 2 to 2, with opposite centers (shaded area) which vary with respect to a parameter $\alpha$. $$r<(x-a(\alpha))^2+y^2<R$$ $$r&...
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2answers
117 views

How to find the exact value of $\cos(\frac{2\pi}{17})$ with WolframAlpha?

I tried to find the exact value of $\cos(\frac{2\pi}{17})$ with WolframAlpha but only obtained a decimal approximation. Is there any way to find this exact value with WolframAlpha?
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1answer
32 views

Geometric construction to divide a segment

Given a segment AB, I would like to construct using only straightedge and compass, a point C on the segment AB such that $\frac{AC}{CB}$ is equal to $\frac{\phi}{2}$, where $\phi$ is the golden ratio, ...
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5answers
238 views

Where is the hole in this argument asserting the constructibility of all regular polygons?

Some engineers have a so-called "general" method for constructing any (regular) polygon with the classical instruments only, given the length of its side (they may recognise that it appears to be ...
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2answers
97 views

Prove the regular 12-gon is constructible.

Prove, both geometrically and then algebraically, that the regular 12-gon is contructible. I'm pretty stuck on this one and trying to get my head around constructibility, so far I've seen that ...
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1answer
57 views

Constructing a triangle between an angle with an arbitrary centroid.

Let $\overrightarrow {OA}$ and $\overrightarrow {OB}$ be two ray with common end point $O$. Let $G$ be a point lying in the interior of the $\angle AOB$. Construct a $\triangle OCD$ such that the ...
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1answer
876 views

Prime Numbers and Architecture

Prime Numbers are widely used in technology and cryptography. They are sometimes also used in small scales such as building gears and evolution of life cycle of Cicada insect. Are they in any way used ...
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60 views

How/when do we use circle inversions to solve problems?

Given an angle AOB and a point M inside it, construct a segment PQ such that M is the midpoint of PQ P is on side OA Q is on side OB So i've been thinking about this problem and of course the best ...
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2answers
64 views

Constructing a right-angled triangle given the half-perimeter $s$ and an altitude $h_c$

I would like to receive some help about the next problem: Problem: Construct a right-angled triangle given the half-perimeter $s$ and an altitude $h_c$, where $\angle(BCA) = 90°$. What i did: 1) ...
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1answer
61 views

With edge and compass construction, given cubes of volumes $a^3,b^3$, can one construct a cube of volume $a^3+b^3$?

Suppose one has cubes A and B of volumes $r_A^3$ and $r_B^3$. Using only ruler and compass constructions, I need to determine whether it is possible to construct a cube of volume $r_A^3+r_B^3$. I ...
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1answer
85 views

How to derive coordinates of the vertices in the unit-distance embedding of the Golomb Graph?

With recent interest in the Hadwiger-Nelson problem on the chromatic number of the plane, thanks to de Grey's theorem that $CNP\ge5$, I've been looking at unit-distance embeddings of various graphs ...
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0answers
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the Support function of reuleaux triangle

I am looking for the explicit support functions (and also reuleaux polygones ) but all I find is general properties , could you give a Book or a paper when can i found the expolicit support ...
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2answers
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For compass and straightedge problems, are you allowed to use the compass as a ruler?

For compass and straightedge problems, you could have a line between two points A and B, and want to make a line the same size between C and line DE. If you placed the two points of the compass ...
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2answers
217 views

Which roots of irreducible quartic polynomials are constructible by compass and straightedge?

A problem in Artin's Algebra, 2nd ed. (16.9.17) reads: Determine the real numbers $\alpha$ of degree 4 over $\mathbb{Q}$ that can be constructed with ruler and compass, in terms of the Galois ...
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1answer
156 views

A question regarding construction of a graph.

I was doing following construction. We know $C_4$ and $C_5$ are $2$-self-centered graphs. When we add a new vertex $x$ and $y$ to $C_4$ and $C_5$, resp, (shown in fig) the new graph contains exactly ...
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1answer
144 views

Ruler and Compasses without Ruler (Mohr-Mascheroni)

In 1797 Lorenzo Mascheroni published result that Every geometric construction that can be carried out by compasses and ruler may be done without ruler. This theorem named after Lorenzo Mascheroni ...
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1answer
138 views

Straightedge and compass construction: focus of a parabola given $A,B,V$

$A,B,V$ are three distinct, non-collinear points in the plane and we want to find, through straightedge and compass, the focus $F$ of a parabola $\wp$ with vertex at $V$ and going through $A,B$. ...
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0answers
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Construction an isosceles right triangle with transformation

We have a Point $A$ on the Plane $P$ and two circles $C1,C2$ on the same plane with radii $R1,R2$ on the same plane. ($R1$ and $R2$ are not necessarily equal). We want to construct an isosceles right ...
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Question about Constructible Points

A point $P$ is constructible from $\{ O , I \}$ [that is, from $(0,0)$ and $(1,0)$] iff $P$ is constructible from $\mathbb{Q} \times \mathbb{Q}$. I am confused on where to go with this. Should I be ...
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3answers
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Is the fourth root of 2 constructible? [duplicate]

Is $2^{1/4}$ (fourth root of two) constructible using only a straight edge and compass? How would you construct it? I understand that a number is constructible if it can be done in a finite number of ...
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1answer
465 views

How to cut fried eggs with mathematical elegance and perfection

Suppose you have fried $N$ eggs and your entire, perfectly circular pan is filled with egg white and $N$ perfectly circular, non-overlapping egg yolks of equal size. How would you cut the egg white ...
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3answers
92 views

How to construct a triangle given two sides and their bisector?

Suppose I have two triangle sides $AB$ and $AC$, and the length of the angle bisector of $A$. How can I construct (straightedge and compass) the triangle? (This question is from one of the earlier ...
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1answer
178 views

Doubling the cube with unit sticks

In the January 2000 issue of Erich Friedman's Problem of the Month, the problem of bracing distances – building a rigid unit-distance graph where two vertices are the required distance apart – was ...
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2answers
124 views

A tangent to a circle with a straight edge

Given a circle on an Euclidean plane and a point $A$ outside the circle, find a line through $A$, tangent to the circle. You're allowed to use a straight edge only.
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1answer
62 views

What is the “parameter on the squares of the ordinates” w.r.t. a parabola?

I'm going through some constructions and derivations in the English Translation of On Burning Mirrors - Diocles, Pg 44 but I'm unable to understand a particular statement w.r.t. to the construction ...
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1answer
127 views

Construct a square with vertices on a given point, line, and circle.

How to construct a square ABCD given point C, circle and a line so that point A lies on the line and point D lies on the circle?
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Geometric construction of golden angle

Is it possible to construct the golden angle using only a compass, ruler and pencil?
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0answers
68 views

How do I prove the construction of a Pentadecagon for a given side length?

I was recently tutoring geometry at a university. The students knew how to construct different polygons. Therefore I wanted to know how to prove the construction. One task was constructing a ...
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1answer
123 views

How to show that a 30-gon is constructible?

I have already shown a 10-gon is constructible. The I am trying to use the fact that the angle cos(2$\pi$/10) is constructible and that cos(2$\pi$/10) = cos(3(2$\pi$/30)) = $4cos^3(2\pi/30)-3cos(2\pi/...
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1answer
252 views

Construction to show that block design exists

I am taking a mathematics course and we covered block designs. I have tried solving the following problem, but I can't find a final answer. "Give an explicit construction to show that a block design ...
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1answer
46 views

Find the image of the point $(1,3,4)$ on the plane $2x-y+z+3=0$.

Find the image of the point $(1,3,4)$ on the plane $2x-y+z+3=0$. Let the image be $(a,b,c)$. Equation of the line joining $(1,3,4)$ and $(a,b,c)$ is $(a-1,b-3,c-4)$ and it will be parallel to the ...
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2answers
354 views

Construct line segments of lengths $a/b$ and $\sqrt {a}$ [closed]

Given: 3 lines (longest one a, medium one b and unit 1) How can you construct line segments of lengths $a/b$, $\sqrt {a}$ and $a/n$ with $n$ a positive integer?
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2answers
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Construction of segment containing a point as the midpoint

Let $\angle A$ be given with two sides $l_1,l_2$, and a point $K$ in the interior of the angle. How could I construct two points $p_1,p_2$ on $l_1,l_2$ respectively, so that the midpoint of $p_1$ and $...
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2answers
46 views

Make a 60° angle on line $l$

We have got Line $l$ and point $P$ which is not on $l$. By using a compass and a non-graded ruler, draw a line from $P$ that makes a 60° angle with line $l$. Please help me!
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What is this “easy application of the Pythagorean theorem”?

I am reading Stillwell's Elements of Algebra, he gives this figure: $\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$ And then: What is this easy application of the Pythagorean theorem? I ...
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1answer
41 views

Ruler and compass construction field extension

I've got a problem to prove something, so I need your help ! :) Actually, we consider the complex plane and we have, for $z$, $z' \in \mathbb{C}$, and $r>0$ : $(zz') = \{ z+t(z'-z) \mid t \in \...
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1answer
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When are we supposed to use 'constructions' in mathematical geometery proofs?

In questions of Triangles or Circles where you are required to prove something, there are cases when the help of construction is required (for example, proving the converse of Pythagoras theorem by ...
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1answer
57 views

Geometrically construct sides on cube

How many degrees would I skew a circle to display it on each side (3) of a cube (die) perfectly centered from the viewpoint of the camera?
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1answer
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Proof of construction of inscribed regular hexagon

I honestly do not know why I'm so lost here, but I am. Really, I'm proving the construction of an equilateral triangle inscribed in a circle, but the inscribed regular hexagon is crucial to that proof ...
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Given the angles and diagonals of a quadrilateral, construct the quadrilateral using only a straightedge, a pencil, and a compass

I tried to do this by constructing 1 angle, and then constructing 2 circles with their radii equal to the diagonals with their centres on the vertex of the angle, but I couldn't get any further from ...
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1answer
119 views

Construct a triangle given a height, his base and the opposite angle

Let $\overline{AB}$ and $\overline{CD}$ be given segments and $\alpha$ a given angle. Construct the triangle $ABC$ with height $\overline{CD}$ corresponding to the side $\overline{AB}$, such that $\...
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1answer
61 views

Sequence of folds for finding intersection of two circles, given centers/radii

I know that any ratio that can be constructed by use of a straightedge and compass (and some which cannot) can be constructed by folding paper. I am not certain whether or not the same is true of ...
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1answer
46 views

Construction of a graph from cycle $C_6$ and $C_7$ with specific properties (of specific eccentricities)

This question is related to my previous problem asked: Construction of a graph with specific properties (of specific eccentricities) In the following given figures I tried to make a graph from $C_4$ ...
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2answers
525 views

Constructing a Regular Pentagon of a Desired Length

I was working on a problem that needed to construct a regular pentagon of a desired length. I couldn’t solve it so checked the solution. The solution in the book was as follows: Draw the line $AB$ of ...
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Optimal Compass and Straightedge Constructions

I was recently looking over some Islamic geometry patterns, and was struck by the complexity of the constructions needed to create seeming simple patterns. This got me wondering regarding optimal ...