# 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.

623 questions
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...