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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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 ...
64
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2answers
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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 ...
41
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18answers
13k views

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 ...
34
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10answers
14k views

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 ...
25
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3answers
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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 ...
24
votes
4answers
8k views

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), ...
24
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3answers
691 views

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 ...
22
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2answers
1k views

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 ...
22
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2answers
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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 ...
21
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1answer
13k views

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 ...
21
votes
1answer
338 views

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 $$...
18
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2answers
4k views

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?
18
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2answers
290 views

Which power means are constructible?

The three classic Pythagorean means $A$, $G$, $H$ (arithmetic, geometric, and harmonic mean respectively) of positive real $a$ and $b$ have a cute geometric construction, as does the quadratic mean $Q$...
17
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2answers
5k views

Trisect unknown angle using pencil, straight edge & compass; Prove validity of technique

This question was posed by my high school geometry teacher, for extra credit: Is it possible, using only a pencil, a straightedge (not a ruler) and a compass to trisect an angle of unknown value? ...
16
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2answers
6k views

Is there a way to draw a 1 degree angle using only ruler and compass?

There are ways to draw $180^\circ, 90^\circ, 45^\circ, 30^\circ, 60^\circ, \dots$ angles. But is there a way to draw a $1^\circ$ angle? In other words how to divide a circle into $360$ equal ...
15
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2answers
6k views

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 ...
14
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3answers
14k views

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 ...
14
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1answer
2k views

Construction of a regular pentagon

In Robert Dixon's Mathographics, a regular pentagon is constructed with straightedge and compass only. It is the pentagon $ABCDE$ pictured below. I am having trouble seeing why the central angles ...
14
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4answers
3k views

Finding tangents to a circle with a straightedge

There is a geometric construction that I heard years ago and I still haven't figured out why it works despite several attempts. Playing with pen, paper and GeoGebra makes me confident that it does ...
13
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6answers
2k views

Geometric proof of existence of irrational numbers.

It is easy, using only straightedge and compass, to construct irrational lengths, is there a way to prove, using only straightedge and compass, that there are constructible lengths which are ...
13
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5answers
2k views

Sangaku. How to draw those three circles with only a ruler and a compass?

I found in a book of Sangakus the following problem. Let $R_b$, $R_g$ and $R_r$ the radiuses of the blue, green and red circles $C_b$, $C_g$ and $C_r$. Prove that $$\frac{1}{\sqrt{R_r}}=\frac{1}{\...
12
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6answers
2k views

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 ...
12
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3answers
1k views

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 ...
12
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2answers
931 views

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 ...
12
votes
1answer
241 views

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 ...
12
votes
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 ...
12
votes
1answer
839 views

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 ...
12
votes
3answers
459 views

same distance from a point to 2 non-parallel lines

There are 2 nonparallel lines $a,b$ and point $E$ which doesn't belong to any of them and lies anywhere between them. EDIT: Task is to find two couples of points F, G and H, I $\in$ y such that $|EF|=|...
12
votes
1answer
283 views

Construct point on a circle such that the reflection in that point is horiztonal

Let $P$ be a point in the plane outside the unit circle. There is a unique point $Q$ on the circle such that a light ray from $P$ is reflected in the circle at $Q$ and emerges parallel to the $x$-...
11
votes
2answers
756 views

Thinking outside of the box

You want to draw a circle with a 4 inch radius. A trivial task for you and your trusty compass. When you go to grab your compass which has not had much love for a while you find it is rusted shut; ...
11
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3answers
3k views

Construction of a right triangle

It's a high school level question which we can't seem to solve. Here it is: Given $2$ lines, one of the length of the hypotenuse and the other with the length of the sum of the $2$ legs, construct ...
11
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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 ...
11
votes
1answer
908 views

What is the reflection across a parabola?

Reflection across a line is well familiar, reflection across a circle is the inversion, the point at a distance $d$ from the center is reflected into a point on the same ray through the center, but at ...
11
votes
2answers
236 views

Geometric notion of addition for the real projective line

The real projective line $\mathbb{RP}^1 = \mathbb{R} \cup {\infty}$ is usually identified with (or defined as) the set of lines passing through the origin in $\mathbb{R}^2$. Thus, the number $m\in \...
11
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0answers
982 views

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 ...
10
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8answers
283 views

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+...
10
votes
6answers
7k views

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 ...
10
votes
3answers
2k views

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 ...
10
votes
1answer
889 views

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 $BC$...
10
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5answers
1k views

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. ...
10
votes
4answers
652 views

Construct quadrangle with given angles and perpendicular diagonals

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 $\...
10
votes
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 ...
9
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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 ...
9
votes
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 ...
9
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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.
9
votes
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 ...
9
votes
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/...
9
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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? ...
9
votes
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 $...
9
votes
2answers
174 views

A conjecture about the intersections of three hyperboles related to any triangle

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 ...