# 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.

617 questions
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### Relation between $[L \cap M : K \cap M]$ and $[L : K]$, and the Wantzel theorem

The well-known Gauß-Wantzel Theorem states that a real number $x$ can be constructed using straightedge and compass only if the minimal polynomial of $x$ (over the field $\mathbf Q$) has degree of ...
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### What's a minimal origami construction realizing a cube root?

The constructible numbers are those that can be achieved as lengths of line segments via compass and straightedge, starting with a segment of length $1$. The origami (constructible) numbers are those ...
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### Construction of a right-angled triangle

Let $ABC$ a triangle with right angle at $C$. Let $D\in AB$ such that $CD\perp AB$. Construct the triangle given $AC$ and $BD$. Using the relation $AC^2=AD(AD+BD)$ I tried to construct a segment of ...
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### Dividing Unit Square in to 7 equal areas

Is it possible to divide say a UNIT Square in to seven equal areas. Two trivial solutions are dividing using $6$ Horizontal(Vertical) lines as shown below: Is it possible to divide it using Straight ...
1answer
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### Can origami math solve polynomial equations of degree greater than 3?

I heard that straight edge + compass can solve up to quadratic equations. I've also heard the Origami/Paper-folding can solve cubic equations. But can it solve higher-degree polynomial equations (e.g. ...
2answers
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### What is the tower of fields a number must be in in order to be constructed by MARKED RULER and compass?

I know that for unmarked ruler and compass a number (distance) is constructible from $\mathbb Q$ iff it lies in a finite tower of field extensions $\mathbb Q=K_{0}...K_{n}$, where $[K_{i} :K_{i+1}] =2$...
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### About the 3d Surface

I am creating the Enneper Surface in Geogebra with the following equations. Is the surface or equation correct, and does the Enneper surface look like this? I'm confused because Wikipedia has a ...
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### Is $\cos(11^\circ)$ constructible? [duplicate]

I'm trying to prove whether $\cos(11^\circ)\over \sqrt{1+\sin(15^\circ)}$ is constructible. I suspect it is not, and would like to use the triple angle identity to use RRT and prove there is no ...
4answers
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### Find the length x such that the two distances in the triangle are the same

I have been working on the following problem Statement Assume you have a right angle triangle $\Delta ABC$ with cateti $a$, $b$ and hypotenuse $c = \sqrt{a^2 + b^2}$. Find or construct a point $D$ ...
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### Given three non-overlapping circles, find the triangle of minimum perimeter with one vertex on each circle

G. Polya "Mathematics and plausible reasoning" Chapter 9, problem 2: Three circles in a plane, exterior to each other, are given in position. Find the triangle with minimum perimeter that has one ...
1answer
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### Are there two non-congruent quadrilaterals with same sets of sides and angles?

Are there two non-congruent quadrilaterals with same sets of sides and angles? It is relatively easy to construct such pentagons. Trying to construct quadrilaterals allows for seemingly multiple ...
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### Given a compass + straightedge, construct convex quadrilateral tangent to circumcircle

Given a compass, a straightedge, and a convex quadrilateral $ABCD$, construct a point $X$ such that the circumcircle of $\triangle{AXB}$ is tangent to the circumcircle of $\triangle{CXD}$, and the ...
5answers
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### “Given three parallel straight lines. Construct a square three of whose vertices belong to these lines.” Are all three lines required?

Given three parallel straight lines. Construct a square three of whose vertices belong to these lines. What does "belongs" mean in the context of this question? Do the three lines have to be used ...
1answer
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### Constructing a graph with radius two.

From cycles $C_n$, $n\geq6$, I was trying to form a new graph by adding a single vertex to $C_n$ so that the added vertex has eccentricity two and rest have three. I tried for $C_6$ and $C_7$ as given....
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### Is classical Euclidean geometry Turing complete?

By "classical Euclidean geometry" I don't mean the study of $\mathbb R^2$ with the Euclidean metric, or a model of Hilbert's Grundlagen axioms, or the study of those properties of $\mathbb R^2$ that ...
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### What kind of geometric constructions require marking of a unit length?

Wondering if there is a rule to determine if a geometric construction or question requires marking a unit length. For example, constructing a square root length or product length (a*b) requires unit ...
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### Show that every point in the interior of one circle is the orthocentre of another triangle inscribed in another circle

Let $C_1$ and $C_2$ be two circles in the plane with radius R and 3R respectively. Show that every point in the interior of $C_2$ is the orthocentre of some triangle inscribed in $C_1$. I gave a ...
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### Question on given compass and ruler construction definition wrt. angle bisection

I am trying to understand chapter 4 in John M. Howie's book called "Fields and Galois Theory" (published by Springer). In chapter 4 regarding geometric constructions he gives the following definition: ...
3answers
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### what geomatric series formula is used?

trying to follow a solution related to geometric series, but not sure what formula is used here. Any pointer is appreciated.Image here, can't embed image yet. I do try to plug in the Sn formula, but ...
0answers
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### Find the line that is closest to 4 skew lines

If I have 4 skew lines in $\mathbb{R}^3$, how can I find the line $L_c$, that is closest to all of them? I know that with 3 skew lines, there is always a line that intersects all of them, in fact ...
2answers
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### Prove that the following equation has no constructible solutions.

Prove that the following equation has no constructible solution: $\ x^3 - 6x + 2\sqrt{\pi} = 0$ The way I am trying to approach is that: I want to transform the equation into some integer ...
0answers
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### Galois theory: Gauss-Wantzel theorem, proof explanation

I am using Ian Stewart Galois theory book and it says that for $A =$ primitive $p^2$ root of unity $A$ has min poly of $m(t)= 1+ t^p +.....+t^{p(p-1)}$ and so $p(p-1)$ is a power of two. why is ...
1answer
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### How does one verify a ruler-compass construction is valid?

I happened upon this paper by Ramanujan in which he gives an approximation for the side length of a square with area nearly equal to that of a given circle. I don't have much experience with ...
1answer
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### Inscribe an equilateral triangle inside a triangle

Given a triangle ΔABC, how to draw all possible inscribed equilateral triangles with given side whose vertices lie on different sides of ΔABC?
0answers
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### How do I determine if a complex number $w = f(z_1,\,z_2,\,z_3)$ is a known triangle center?

I've found a number of closely related functions, each of which takes three complex numbers $z_1,\,z_2$, and $z_3$ (which we can consider as the vertices of a triangle) as its arguments, and outputs a ...
1answer
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### How to construct a triangle with $BC=7.5$ cm. $\angle ABC$=$60$° and $AC-AB=1.5$ cm.

How to construct a triangle with $BC=7.5$ cm. $\angle ABC$=$60$° and $AC-AB=1.5$ cm. At first I constructed $BC$ then $\angle ABC$ ,but I don't know what to do next. Please help me.
1answer
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### Proof of Construction of regular pentagon by using compass and straightedge.

How to prove that the polygon constructed by the method mentioned in the following link is indeed a regular pentagon? Constructing a Regular Pentagon (Video on YouTube)
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### A conjecture about the sum of the areas of $3$ triangles built on the sides of any triangle (by means of centroid/orthocenter) [closed]

Given any triangle $\triangle ABC$, let us draw its orthocenter $D$. By means of this point, we can draw three circles with centers in $A,B,C$ and passing through $D$. These circles intersect in the ...