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

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3
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2answers
60 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 $h,k$. ...
0
votes
1answer
35 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 ...
5
votes
0answers
82 views

Find the intersection of two lines entirely outside the given sheet of paper by straightedge alone

This is a problem from Courant:"Two straight lines entirely outside the given sheet of paper are each given by two pairs of straight lines intersecting at points of the lines outside the paper. ...
0
votes
1answer
33 views

Construction of triangle from side $c$ and heights $h_a, h_b$

I want to construct a triangle $\Delta(A,B,C)$ with given side $c$ and heights $h_a, h_b$. To construct the triangle means to use only ruler and compass. How can I solve this? I started as follows: ...
1
vote
0answers
26 views

How do you visualize ridge, roof and step edges?

I am reading about Canny algorithm in the book Academic Press - Handbook Medical Imaging Processing Analysis where it is written that the algorithm was originally developed for antiasymmetric edges ...
1
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0answers
44 views

Given x,y,w,h can you generate a rainbow box/cuboid with rounded edges?

Given $x$, $y$, $w$, $h$ where $0 \leq x < w$ and $0 \leq y < h$ and $(x, y)=(0, 0)$ is bottom-left and $(x, y)=(w-1, h-1)$ is top-right and they're all integers, can you make a formula that ...
0
votes
1answer
36 views

Online tool for making Geometric Constructions.

There was a website where it tasked you making different geometric shapes using only a compass and straightedge. I've looked for it and I can't find it or even discussion about it. What I do remember ...
1
vote
1answer
23 views

Straightedge-Only for Perpendicularity

Given a triangle ABC and a midpoint M (of the line AB), is it possible to check whether the line CM is perpendicular to AB with a straightedge only? By this, I mean that points can be added ...
5
votes
3answers
84 views

Compass-and-Straightedge Construction

I stumbled upon this question in math class, and I got stuck. The Question: You're are given a circle, and two points. How do you construct a circle that goes through the two points and is tangent to ...
2
votes
0answers
37 views

constructing an equilateral triangle in the Beltrami klein model

I am puzzeling with the following: Using the beltrami klein disk of hyperbolic geometry (see https://en.wikipedia.org/wiki/Beltrami%E2%80%93Klein_model ) (PS not the poincare disk model) and given ...
0
votes
1answer
32 views

Drawing a regular pyramid

I've been asked for help in high school mathematics (some basic stereometry) but I'm not sure how to solve this exercise: Draw a regular triangular pyramid given the lengths of edges $3.8\ cm$ ...
0
votes
0answers
24 views

Constructible points from $\mathbb{Q}\times\mathbb{Q}$

I have recently learned the proof for why you cannot "double" the cube, trisect the angle, and "square" the circle. I understand the whole analysis, assuming that a point is constructible if it is ...
0
votes
0answers
50 views

Find the locus of the centroid of a triangle

Given triangle $ABC$, with points $B(2,3)$ and $C(-2,6)$ and the fact the perimeter is 14, how can we find the locus of its centroid? I do not even know how to begin.
5
votes
2answers
200 views

Angle bisector on a piece of paper?

Let's draw $\overline{AB}$ and $\overline{CD}$ (not parallel) on a piece of paper (rectangular). The intersection of the lines AB and CD is off the paper. Is it possible to construct the section of ...
-1
votes
2answers
42 views

Construct a regular hexagon of specific height?

Is it possible to construct a hexagon of particular height, meaning distance between the faces (not vertices)? I have seen various methods of constructing a hexagon (ruler and compass only) which are ...
6
votes
1answer
81 views

Can eight circles be constructed from three circles?

Given three sufficiently spaced circles in a plane, is it possible, using a straight edge and compass, to construct the eight circles that are tangent to all three given circles?
2
votes
0answers
9 views

Dinamically generate Goldberg polyhedra G(m,n)

In these pages the autor provided a lot of info about some Goldberg polyhedra (http://en.wikipedia.org/wiki/Goldberg_polyhedron): http://dmccooey.com/polyhedra/DualGeodesicIcosahedra.html ...
0
votes
0answers
32 views

To draw a perpendicular on the diameter AB of a circle from an external point P using only a straight-edge.

A perpendicular is to be dropped from external point P on diameter AB I know this question is a duplicate of potato's post, but in potatos post altitudes of triangles were used. But a property of ...
0
votes
2answers
45 views

Constructions of perpendicular in hyperbolic plane

Consider the disc model of hyperbolic plane $\mathbb{D}^2$ and a line $g$ through the origin $(0,0)\in \mathbb{D}\subset\mathbb{C}$, i.e. a diameter of the circle $\partial \mathbb{D}=S^1$. Let ...
5
votes
0answers
106 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
26 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 $$ ...
5
votes
1answer
98 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
28 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
22 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 ...
0
votes
2answers
82 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 ...
8
votes
2answers
75 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 ...
4
votes
3answers
83 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 ...
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 ...
6
votes
1answer
144 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?
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 ...
0
votes
1answer
37 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 ...
2
votes
0answers
34 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
66 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
35 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 ...
2
votes
0answers
47 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 ...
3
votes
3answers
151 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
0answers
50 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
67 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 ...
1
vote
1answer
58 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
46 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
36 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
71 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 ...
3
votes
0answers
35 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
189 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
84 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 ...
0
votes
3answers
142 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?
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
2
votes
1answer
71 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 ...