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

learn more… | top users | synonyms (1)

5
votes
0answers
44 views

What is reflection across 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 ...
1
vote
1answer
25 views

Which points in the interior of a parallelogram are as far as possible from the corners?

Question 1: Given a parallelogram $P=ABCD$, how does one construct/determine the points $X \in P$ which are as far as possible from the corners? That is, the points $X$ for which $$ ...
26
votes
10answers
8k 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 ...
0
votes
2answers
61 views

Construct two circles tangent to each other and to a line, and a circle tangent to all three

I saw a question that was nearly the same as this, but I couldn't understand the answers. Assume that everything that seems to be tangent should be tangent, and that everything that appears to be a ...
4
votes
0answers
63 views

Three planes in general position, one point in each, construct sections

I have three planes in general position, and in each plane an arbitrary point is selected : this gives us three points $R,S,T$. Is it possible to construct the intersection lines of the $(RST)$ plane ...
0
votes
1answer
19 views

Midpoint of a line segment with a marked straight edge

Given a line segment $AB$ and a marked straight edge. How can I construct the midpoint of the line segment with the marked straight edge only (i.e., in particular without a compass)? I have no idea, ...
0
votes
1answer
17 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 ...
7
votes
2answers
57 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 ...
2
votes
2answers
143 views

construct x =ab by using compass alone, if a and b are given segments.

I found the problem in the book "What is mathematics?". The following is a description of Mohr's constructions.(Macheroni problem) 9) Find $x = ab$, if $a$ and $b$ are given segments. I ...
4
votes
3answers
74 views

Construct the great circle (geodesic) in spherical or Riemanian geometry

Given: a circle $C$ with centre $M$ two points $P_1$ and $P_2$ inside circle $C$, so that $M$ is not on the line $P_1P_2$. Cunstruct an other circle $O$ so that: $P_1$ and $P_2$ are on ...
6
votes
1answer
139 views

Circle construction

I am stuck on this construction: "Show how to construct a circle to pass through two given points and to cut a given circle so that the common chord is of given length". Any clues?
40
votes
17answers
3k 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 ...
1
vote
0answers
45 views

Why $\pi$ is not Constructible with Circumference Length

If we use a compass to draw a circle with a diameter of length 1, then the circumference is $\pi$. From the definition given here (http://en.wikipedia.org/wiki/Constructible_number), it seems to me ...
0
votes
1answer
37 views

Existence of the Square in “Squaring the Circle” Problem

I understand that a square with area $\pi$ cannot be constructed using straightedge and compass. But such a square surely exists (and can be constructed through other means), right? If I'm right, I'm ...
0
votes
1answer
24 views

Constructing a line given two other lines, two angles, and a distance.

Two non parallel lines $l$ and $m$ are given. For given two angles $A$ and $B$ we have to construct a line $n$ such that it makes angles $A$ and $B$ with lines $l$ and $m$ respectively. Line $n$ ...
0
votes
1answer
40 views

Construct a circle cutting two other circles at right angles

I have the following problem: On a line $l$ on this line are the centers of two circles $C_1$ and $C_2$ . Circles $C_1$ and $C_2$ do not intersect and are not tangent to eachother. (but one could be ...
3
votes
3answers
362 views

Doubling the cube neusis

Can anyone explain me in simple maths the neusis construction at http://en.wikipedia.org/wiki/Doubling_the_cube#Using_a_marked_ruler? Why and how does it produce the $\root 3 \of 2$?
6
votes
3answers
915 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 ...
0
votes
1answer
35 views

Construction based on circumcenter and incenter

Construct a triangle given the exact location of its circumcenter and its incenter, and the position of its angle bisector (including its direction), but not its length. I tried to consider the ...
1
vote
0answers
32 views

Is the set of numbers constructible with just a compass dense in $R^2$?

Suppose I have initially two points $0$ and $1$ in $R^2$. Given a set of $n$ points $P$ and $m$ circles $C$, suppose I am allowed to add any circle that has center $x$ and radius $r=|x-y|$ for $x,y ...
1
vote
2answers
64 views

Straightedge-only construction of segment of length $\sqrt{7}$, given regular hexagon with unit sides

Let's consider a regular hexagon with unit side length. Draw a line segment of length $\sqrt{7}$ using nothing except a straightedge (that is, an unmarked ruler). The position of the segment may be ...
0
votes
0answers
20 views

Does this construction method produce all possible convex pentagons (up to similarity)?

I read a question somewhere here about convex pentagons. I began to wonder if there was a way to list all possible convex pentagons and came up with the following method: 1) Draw a base line AB of ...
3
votes
3answers
140 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 ...
1
vote
1answer
62 views

Construct a circle with straight edge and compass with some given conditions.

A line is given and two points in one half plane of the line are given. Construct a circle passing through these two points such that the given line is tangent to this circle. I have no idea how to ...
2
votes
0answers
45 views

Thread constructions in the Poincaré's disc

I just came across the following image (source) and realised something that should have been obvious to me a while ago: it should be possible to construct the envelopes of curves in the Poincaré disc ...
1
vote
0answers
41 views

Huzita Axiom 6 - Computing the Origami Trisection of an Angle

The Galois theory proof of the improssiblity of angle trisection rests on studying the triple angle formula $\cos 3\theta = 4 \cos^3 \theta - 3 \cos \theta$. Ruler and compass numbers can only be ...
1
vote
1answer
50 views

which equations may be solved with compass and a marked ruler?

Ancient Greeks were not able to trisect a general angle with compass and straightedge: now we know that it is impossible, since we would need to solve a cubic equation while only linear and quadratic ...
0
votes
0answers
43 views

Where does the problem statement say sides are *equal*?

In the book How To Solve It, Part I, chapter/section 19, Pólya's hypothetical teacher poses the following problem to prove to the hypothetical student: Two angles are in different planes but each ...
1
vote
1answer
35 views

Given a $\triangle ABC$ construct another triangle with sides measuring the inverses of the altitudes of $\triangle ABC$

Given a $\triangle ABC$, we want to construct the $\triangle XYZ$ whose sides are the inverses of the altitudes of $\triangle ABC$ . If we denote the altitudes by $h_a,h_b,h_c$ then the sides of ...
3
votes
1answer
66 views

Construction of a triangle given some special points ($O,H,I$)

I'm a newbie in this site. I tried to search if this question was already answered but I'm not sure on how to do it. The problem is: given three distincts points $O,H,I$ namely the circumcenter, the ...
0
votes
3answers
117 views

Construction of an ellipse

Is it possible to construct an ellipse with a line, compasses and a pencil? If yes, how and why is the construction correct?
4
votes
2answers
722 views

Resource for learning straightedge and compass constructions

Does anyone know a good resource for learning about straightedge and compass constructions besides "The Elements?" I tutor geometry to middle-schoolers and high-schoolers and thought that including ...
3
votes
0answers
31 views

how do the conic sections add to the possibilities of geometric construction?

If we are limited by what we can construct with compass and straight edge, then what becomes possible by expanding our toolkit to include all conic sections? The tools would be based on already ...
0
votes
1answer
29 views

smallest set of curves for constructing any real number and angle

If we are limited by what we can construct with compass and straight edge, then what are the fundamental curves required for constructing any real number? In other words, what is the smallest ...
4
votes
1answer
160 views

Geometry: How to find cube root, fourth root, fifth root… and so on?

As we know that square root of a number $n$ can be found by using a compass and a straight edge, given the line of length $n$. What I want to know is how to find cube root, fourth root, fifth root or ...
3
votes
1answer
83 views

Topology proof question?

How to prove that $X_1 := \{(x, y, z) ∈ \mathbb{R}^3 : x^2 + y^2 = 1\} / (S^1 × \{0\} )$ is homeomorphic to the union $X_2$ of two tangent spheres minus two points? What I know: Let $C$ be ...
1
vote
1answer
39 views

How do to derive the following SIMPLE geometric relationship between two points on a plane

Can someone show why: $$x' = L_1 \cos(a_1) + L_2\cos(a_1+a_2)$$ $$y' = L_1 \sin(a_1) + L_2\sin(a_1+a_2)$$ where $L_1$ and $L_2$ are the length of the red lines
4
votes
2answers
212 views

Drawing Euclid?

I decided to study Euclid for fun. I have Oliver Bryne's edition. I also want, as much as possible, to construct the figures myself, to get a deeper understanding. How did people traditionally do ...
2
votes
1answer
69 views

Constructing an example of a infinite set of triangles on the rational line whose union does not contain the interior of a rectangle.

As part of my Topology course, I saw a proof the following proposition (as a consequence of Baire's category theorem): Proposition: Define $\triangledown_{t,h}$ as the interior of an equilateral ...
3
votes
0answers
63 views

Algebraic number of degree four that cannot be constructed with ruler and compass

The real number $b:=\frac{\sqrt{2a}+\sqrt{4\sqrt{a^2-3}-2a}}{2}$, where \begin{equation*} ...
6
votes
2answers
203 views

Geometry construction problem

Given two circles $S_1$ and $S_2$, a line $l_1$, and a length $a$ that is less than the sum of the diameters of the circles, construct a line $l$, parallel to $l_1$, so that the sum of the chords that ...
2
votes
0answers
48 views

Is there any construction method that yields all algebraic numbers?

Compass and Straightedge = rationals and square roots Origami = rationals, square roots and cube roots (I think) How far can we get if we use other tools, like rulers, protractors, pieces of string, ...
0
votes
2answers
67 views

Circle with center point and tangential to lines

I have defined Points all points (3 blue, and one green). All points have the same distance to A point. Yellow lines are bisectors. I have equations of AB and ...
10
votes
4answers
444 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 ...
1
vote
2answers
68 views

Construct a midpoint of two parallel lines with only straightedge

Say I have a large plank of wood that I'm trying to cut in half the long way, but I only have a straightedge (no compass). How can I mark the midpoint between the long edges? (As a note, this isn't a ...
1
vote
1answer
136 views

Ruler-and-Compass metric

Let $\nu \left( x \right) $ be the least number of steps that is required to construct a constructible length $x$, using compass and ruler in the well known fashion. Now, define the distance ...
1
vote
2answers
61 views

Geometric Construction Problem

Given three non-collinear points A, B, and C, construct three circles that are pairwise tangent at these points. Are there any cases where such circles do not exist? I am not sure how to start the ...
1
vote
0answers
59 views

How to calculate the edge sizes of a Goldberg polyhedron?

I want to build a paper model of a Goldberg polyhedron, a Icosahedral G(2,2). But i cant find the formula to calculate the 2 sizes needed for the edges. For a G(4,1) the sizes are here : ...
6
votes
2answers
1k 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 ...
0
votes
2answers
31 views

Find a point C on an infinite line AB which, when connecting two other points M and N, would form congruent angles

On an infinite line $AB$, find a point $C$ such that the rays $CM$ and $CN$ connecting $C$ with two given points $M$ and $N$ situated on the same side of $AB$ would form congruent angles with the ...