Questions on the construction of geometrical figures using a limited set of "tools".

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15
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2answers
3k 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), ...
13
votes
3answers
25k views

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 ...
66
votes
1answer
3k views

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 ...
14
votes
2answers
1k 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 ...
29
votes
10answers
10k 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 ...
2
votes
3answers
938 views

Determine the centre of a circle

Given a circle $O$ on a paper, we do not know the centre point. Can we draw the centre only using a ruler (by which we can only draw straight lines)? One fact I know: we can draw the tangent line at ...
7
votes
3answers
2k views

Equilateral triangle geometric problem

I have an Equilateral triangle with unknown side $a$. The next thing I do is to make a random point inside the triangle $P$. The distance $|AP|=3$ cm, $|BP|=4$ cm, $|CP|=5$ cm. It is the red ...
6
votes
1answer
148 views

Is the non-existence of a general quintic formula related to the impossibility of constructing the geometric median for five points?

In particular, in the Computation section of in the Wikipedia page for geometric median, there is this statement: ...but no such formula is known for the geometric median, and it has been shown ...
2
votes
3answers
317 views

How to construct the point of intersection of a line and a parabola whose focus and directrix are known?

I found this problem in Polya's "How to solve it". It goes as follows Using only a straight edge and a compass, construct the point(s) of intersection of a given line and a parabola whose focus ...
11
votes
3answers
2k 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 ...
10
votes
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 ...
3
votes
2answers
2k views

construct inverse point with respect to the circle by the use of the compass alone

If the given point P lies inside a circle C ,with center O,the circle of radius OP about P intersects C in two points. How to construct point P' inverse to point P with respect to the circle C by ...
5
votes
3answers
2k views

Can a circle's circumference be divided into arbitrary number of equal parts using straight edge and compass only?

Can a circle's circumference be divided into arbitrary number of equal parts using straight edge and compass only? In other words, are all the $\frac{2\pi}{k} , k \in \mathbb N^+$ angles constructible?...
3
votes
2answers
290 views

Drawing a triangle from medians

Is it possible to draw a triangle, if the length of its medians $(m_1, m_2, m_3)$ are given only? Someone asked me this question, but I can not see it. Is it really possible? UPDATE Apart from the ...
0
votes
1answer
75 views

Construct the triangle with given angle bisectors

given three lines $\ell_1,\ell_2, \ell_3 $ which intersect in one point $P$. How can one construct a triangle such that the given lines become its angle bisectors? So far I tried to find conditions ...
17
votes
1answer
9k 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 ...
3
votes
2answers
106 views

Construction using a straight edge only

Given a circle, its diameter and a point on the circle, find a procedure to construct a line perpendicular to the diameter using only a straight edge. The perpendicular must pass through the given ...
11
votes
1answer
134 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 ...
8
votes
0answers
686 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 ...
6
votes
3answers
5k views

Constructing a circle through a given point, tangent to a given line, and tangent to a given circle

While browsing around about problems similar to the problem of Apollonius, I have found references to constructions of all types of circles. For example, not only is it possible to construct a circle ...
5
votes
1answer
403 views

On Constructions by Marked Straightedge and Compass

Pierpont proved that a regular $n$-gon is constructible by (singly) marked straightedge and compass if and only if $n = k \, p_1 \cdots p_{s}$, where $k = 2^{a_1} 3^{a_2}$ for $a_i \geq 0$ and $p_i = ...
4
votes
2answers
191 views

Does there exist a tool to construct a perfect sine wave?

For example, a perfect circle can be constructed using a compass and a perfect ellipse can be constructed using two pins and a piece of string, because a circle can be defined as the locus of points ...
3
votes
3answers
324 views

How to construct a line with only a short ruler

Suppose I want to draw a line between the points A and B but I only have a ruler that covers only something between a fifth and a quarter of the distance between the two points. Also available a ...
8
votes
2answers
326 views

Dividing an angle into $n$ equal parts

My question is simply: for which values of $n$ is it possible to divide any given angle into $n$ equal parts using only a compass and a straight edge? I know that it is possible for $2$ and not ...
3
votes
1answer
37 views

On constructing a triangle given the circumradius, inradius, and altitude .

I was recently pondering about constructing triangles given different attributes of it. I am wondering whether we could construct a triangle given its Circumradius $R$ , Inradius $r$, and length ...
3
votes
1answer
101 views

Construction of a triangle with given angle bisectors [duplicate]

given three distinct lines $g,h,l$ meeting in one point $P$. I want to construct a triangle with vertices on $g,h,l$ such that those lines $g,h,l$ become its angle bisectors. In general, if we ...
3
votes
2answers
238 views

Apollonius circle

I'm given two points, $A$ and $B$, and two lengths, $b$ and $c$. I need to find the locus of point $C$ such that $BC:AC=b:c$. This link describes Apollonius circle of first type, but I can't seem to ...
2
votes
3answers
77 views

Straight Edge - Only Geometric Construction

Given a circle, its diameter and a given point on the diameter, find a procedure to construct a line perpendicular to the diameter using only a straight edge. The perpendicular must pass through the ...
1
vote
1answer
113 views

How to construct hyperbolically equidistant points on a line?

In Stillwells' "Sources of Hyperbolic Geometry " page 66 figure 3.3 shows an ((incomplete?) construction of hyperbolically equidistant points on a line. I tried to reconstruct the figure but did ...
0
votes
1answer
86 views

Triangle construction procedure

Two lines $L1,L_2$ pass through a common point $O. $ $L_2$ goes through points $P$ and $Q$. How to construct a circle through $P,Q$ to be tangent to $L_1?$ In a particular case, at the tangent ...
4
votes
2answers
2k views

Why can't we construct a square with area equal to a given equilateral triangle?

In modern geometry, given an equilateral triangle, one can't construct a square with the same area with the use of Hilbert tools. Why is this? The claim seems untrue to me, so there must be something ...
4
votes
1answer
84 views

Go from A to D in three equal steps

Given two parallel lines $r$ and $s$, line $p$, perpendicular to both, and points $A$ and $D$ on different sides of $p$ with respect to the parallel lines, how can I prove the existence of two points, ...
3
votes
0answers
5k views

Division of a line segment in the given ration internally

Before I state my problem description, would like to describe problem which was stated before my problem. So it is like this Given a line segment $AB$. You are required to divide it internally ...
3
votes
1answer
66 views

Predicting Spirals

I am currently in the process of analyzing a polyspiral, a spiral where each successive length drawn is increased at specified increment at the same angle. *Please note the angles selected are the ...
3
votes
2answers
203 views

Construction of a circle through a point and tangent to angle

given an angle $\angle (h,k)$, where $h,k$ are the legs of the angle. Let $P$ be some point in the interior of the angle. I want to construct a circle through P which is tangent to both legs $h,k$. ...
2
votes
0answers
52 views

Is there a direct proof that pi is not the root of an algebraic equation whose degree is a power of 2 [duplicate]

All known proofs that the circle cannot be squared are based on Lindemann's theorem that $\pi$ is not analgebraic number. But this seems to be a case of using an atomic bomb to kill a fly. What ...
2
votes
0answers
48 views

A circle can include all but one of n points, but which one can it be?

The answers to the question "Circle enclosing all but one of n points" demonstrate that, given $n$ points, it is possible to construct a circle such that all but one of the points is inside the circle ...
1
vote
1answer
50 views

Geometric Construction

"Given in position the points $A,B$ and $P$ and a segment $m$, draw through $P$ a line $r$ in such a way that $A$ and $B$ be in opposite sides of $r$ and that the sum of the distances from $A$ and $B$...
1
vote
1answer
176 views

To draw a straight line tangent to two given ellipses

How can I draw a a straight line that touches two ellipses? There are, like for two circles, 4 different solutions. I´m not interested in the analytical solution, but in the geometrical drawing, ...
0
votes
1answer
103 views

Constructing the reciprocal of a segment

How can one construct the reciprocal length of a line segment? For example, given any line segment a, how can $\frac{1}{a}$ be constructed? I was told that it can be solved by creating similar ...