Mathematics Stack Exchange
Mathematics Stack Exchange

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.

The most common use of "geometric construction" refers to the "compass and straightedge" constructions in classical Euclidean geometry. The notion has been extended also to (a) compass/straightedge constructions in non-Euclidean geometries and (b) allowing different sets of tools such as a marked straightedge (neusis) or origami.

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Direct proof that $\pi$ is not constructible

Is there a direct proof that $\pi$ is not constructible, that is, that squaring the circle cannot be done by rule and compass? Of course, $\pi$ is not constructible because it is transcendental and so is not a root of any polynomial with rational…
lhf
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Inscribing square in circle in just seven compass-and-straightedge steps

Problem Here is one of the challenges posed on Euclidea, a mobile app for Euclidean constructions: Given a $\circ O$ centered on point $O$ with a point $A$ on it, inscribe $\square{ABCD}$ within the circle — in just seven elementary steps. Euclidea…
PDE
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What is the next number on the constructibility sequence? And what is the asymptotic growth?

Let us systematically generate all constructible points in the plane. We begin with just two points, which specify the unit distance. With the straightedge, we may construct the line joining them. And with the compass, we may construct the two…
JDH
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How to create circles and or sections of a circle when the centre is inaccessible

I am doing landscaping and some times I need to create circles or parts of circles that would put the centre of the circle in the neighbours' garden, or there are other obstructions that stop me from just sticking a peg in the ground and using a…
Maths Fail
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What is the (mathematical) point of straightedge and compass constructions?

The ancient discipline of construction by straightedge and compass is both fascinating and entertaining. But what is its significance in a mathematical sense? It is still taught in high school geometry classes even today. What I'm getting at is…
I. J. Kennedy
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Compass-and-straightedge construction of the square root of a given line?

Given A straight line of arbitrary length The ability to construct a straight line in any direction from any starting point with the "unit length", or the length whose square root of its magnitude yields its own magnitude. Is there a way to…
Justin L.
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Representing the multiplication of two numbers on the real line

There is a simple way to graphically represent positive numbers $x$ and $y$ multiplied using only a ruler and a compass: Just draw the rectangle with height $y$ in top of it side $x$ (or vice versa), like this But is there a way to draw the number…
temo
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Equivalence of different ways of geometrical multiplication

There are at least five ways to multiply two natural numbers $a$ and $b$ given as integer points $A$ and $B$ on the number line by geometrical means. Two of them include counting, the others are purely geometric. I wonder (i) if there are other ways…
Hans-Peter Stricker
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A conjecture about the sum of the areas of three triangles built on the sides of any given triangle

Given any triangle $\triangle ABC$, and given one of its side, we can draw two lines perpendicular to that side passing through its two vertices. If we do this construction for each side, we obtain the points $D,E,F$ where two of these perpendicular…
user559615
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Elliptical version of Pythagoras’ Theorem?

Consider any right triangle $\triangle ABC$. We focus on one side, $AC$, and we take the midpoint $E$ of this side. Then, we draw the circle with center in $E$ and passing by $A,C$. If we take the perpendicular to $AC$ passing by $E$, we define a…
user559615
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How well-studied is origami field theory?

It's well known that angle trisection cannot be done with straightedge and compass alone, as Theorem 1. If $z \in \mathbb C$ is constructible with straightedge and compass from $\mathbb Q$, then $$\mathbb Q (z) : \mathbb Q = 2^n.$$ But the minimal…
YuiTo Cheng
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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 clarify a move is: Drawing a line Drawing a…
fredley
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Finding the circles passing through two points and touching a circle

Given two points and a circle, construct a/the circle through the two points and touching the given circle. I came across this problem in History of Numerical Analysis by H. Goldstein. I spent some time on this. I have a method of constructing it…
abel
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Minimum operations to find tangent to circle

I've been playing the game Euclidea 3, and I can't really wrap my mind around one of the minimal solutions: https://www.youtube.com/watch?v=zublg6ZevKo&feature=youtu.be&t=9 The object is to get a tangent line on right of the blue circle with only 3…
Gillespie
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Can a regular heptagon be constructed using a compass, straightedge, and angle trisector?

Euclid has a magical compass with which he can trisect any angle. Together with a regular compass and a straightedge, can he construct a regular heptagon?
Arthur from Belgium
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