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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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), ...
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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 ...
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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 ...
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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 ...
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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 ...
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Geometry Construction Problems

Recently I've been trying my hand at a few geometrical construction problems using just a straight edge and a compass. So far I have constructed the following: an equilateral triangle a square a ...
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Pair of compasses drawing a square (from children's fiction)

I have read a children's book where alien race of "square people" used a pair of compasses that drafted a perfect square when used. Now I wanted to explain to the child that it is not possible to have ...
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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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Using a compass and straightedge, what is the shortest way to divide a line segment into $n$ equal parts?

Sometimes I help my next door neighbor's daughter with her homework. Today she had to trisect a line segment using a compass and straightedge. Admittedly, I had to look this up on the internet, and ...
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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 divide a pizza into $n$ parts?

Let's say you have invited $(n-1)$ people for dinner. You decide that the main course consists of one pizza for each guest, so you order $n$ pizzas. Unfortunately, the pizza guy on the scooter trips ...
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Minimal number of moves to construct the challenges (circle packings and regular polygons) in Ancient Greek Geometry?

In the web game Ancient Greek Geometry, there are challenges to construct regular polygons and circle packings using ruler and compass constructions. The game measures the number of line and circles ...
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Evaluating $\sum_{k=0}^\infty \left(\frac{1}{5k+1} - \frac{1}{5k+2} - \frac{1}{5k+3} + \frac{1}{5k+4} \right)$

I saw this problem somewhere recently and I was having some difficulty getting started on it. The problem is twofold. The first is to evaluate: \sum_{k=0}^\infty \left(\frac{1}{5k+1} - \frac{1}{5k+...
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Find the maximum area possible of equilateral triangle that inside the given square

How can I find the maximum area possible of equilateral triangle that inside a square whose sides have length a. And how does that triangle look like? Can we construct it (with compass and ...
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How do you know if a line is straight?

How do you know a line is straight? How can you check in a practical way if something is straight - without assuming that you have a ruler? How do you detect that something is not straight? If you ...
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How to construct a square equal to a given triangle.

I have a triangle $ABC$ and I want to construct a square of the same area as that of the triangle using ruler and compass. Consider the following image. I first locate the mid-points of $AB$ and $BC$...
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How to construct three mutually orthogonal circles in stereographic projection?

I'm new to spherical geometry and I enjoy doing ruler-and-compass constructions, so I'm trying to teach myself to do them in stereographic projection. I'm finding it challenging, to put it mildly. ...
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 $\... 1answer 206 views Given three non-overlapping circles, find the triangle of minimum perimeter with one vertex on each circle G. Polya "Mathematics and plausible reasoning" Chapter 9, problem 2: Three circles in a plane, exterior to each other, are given in position. Find the triangle with minimum perimeter that has one ... 4answers 1k views Why is adding the same as extending a length? I've come to realize that the more I study some math subjects the more I question some results or ideas that seemed trivial or obvious to me. My question is about the real numbers and their geometric ... 1answer 13k views Constructing the incenter of a triangle in only six steps Lately I have become hooked on the game Euclidea. One of the problems gives a triangle and asks you to construct the incenter, or as it is put, "the intersection of angle bisectors." It is stated ... 8answers 2k views Divide a line segment in the ratio$\sqrt{2}:\sqrt{3}.$"Divide a line segment in the ratio$\sqrt{2}:\sqrt{3}.$" I have got this problem in a book, but I have no idea how to solve it. Any help will be appreciated. 4answers 969 views Proof without words of a simple conjecture about any triangle Given the midpoint (or centroid)$D$of any triangle$\triangle ABC$, we build three squares on the three segments connecting$D$with the three vertices. Then, we consider the centers$K,L,M$of the ... 2answers 6k views History of Compass/Straight Edge Construction I'm interested in learning the origin of compass/straight-edge constructions. In particular, I am interested in the historical interplay between Euclid's axioms for plane geometry, and compass/... 2answers 2k views Importance of construction of polygons Why did the ancient Greeks give so much importance to the construction of regular polygons with$n$-sides using only ruler and compass and tried to study for what$n$was such a construction possible? ... 2answers 200 views A conjecture related to any triangle Given one side$AC$of any triangle$\triangle ABC$, we can draw the couple of circles with center in$A$and passing through$C$and with center in$C$and passing through$A$, obtaining two points$...
Given any triangle $\triangle ABC$, we build the hyperbole with foci in $A$ and $B$ and passing through $C$. Similarly, we can build other two hyperboles, one with foci in $A$ and $C$ and passing ...