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

learn more… | top users | synonyms (1)

2
votes
1answer
49 views

How do I calculate the distance from point A from point B?

I've got this drawing of a circle, and I'd like to know how I can calculate the distance between point A to point B in a straight line. I already have: Radius: 100 Arc length: 78.5 ...
0
votes
0answers
42 views

Constructible decimals?

So I have to figure out if $1.23456$ is constructible. I think that it's not constructible since: I know that this is $\frac{123456}{100000}$ so this goes into $\frac{2^6*3*643}{2^55^5}$ and the ...
1
vote
2answers
48 views

Constructible Solutions

We know that if a cubic equation with a rational coefficients has a constructible root, then the equation has a rational root. Now let; $$x^3-2x+2\sqrt{2}=0$$ Could $\sqrt{2}t$ be a viable ...
0
votes
1answer
23 views

Is it possible to construct an incrementally accurate rectification of a circle?

Exact rectification of a circle (construction of a segment exactly the length of the circumference of a given circle) has been proven impossible. There is a number of rectification constructions that ...
2
votes
2answers
57 views

For every three points on a line, does there exist a triangle such that the three points are the orthocenter, circumcenter and centroid?

The Euler line states that the orthocenter, circumcenter and centroid of a given triangle are on one line. This made me wondering whether the following is true: For every three points on a line ...
0
votes
1answer
31 views

construction of line segment of a length $\sqrt{a^2-b^2+c^2+d^2}$

There are line segments $a, b, c, d$ and $a > b$. I have a question how to construct a line segment of a length $\sqrt{a^2-b^2+c^2+d^2}$. I can use Pythagoras theorem but I don't know how to make ...
3
votes
2answers
54 views

Constructing triangle $\triangle ABC$ given median $AM$ and angles $\angle BAM, \angle CAM$

Constructing triangle $\triangle ABC$ given median $AM$ and angles $\angle BAM, \angle CAM$ I start with the median $AM$. Since $\angle BAM, \angle CAM$ are known I can construct them. So I have ...
0
votes
1answer
55 views

Constructibility of Regular $N$-gon $\implies$ Constructibility of Regular $2N$-gon

I have to prove the following statement: If a regular $n$-gon is constructible, then so is a regular $2n$-gon. My Attempt: 1. Draw a point at the each vertex. 2. Draw a line between each point. ...
1
vote
0answers
54 views

Geometric / Intuitive construction of the rotation axis of a 3D rotation matrix?

I have been looking without success for an intuitive / geometric construction of the rotation axis of a given 3D rotation matrix. To put the problem in more familiar terms, let's assume you have the ...
4
votes
2answers
70 views

Can I square the triangle?

I know I can't construct a square with the same area as a given circle (because $\pi$ is transcendental). Can I construct (ruler and compass) a square with the same area as a given triangle? I think ...
1
vote
2answers
60 views

How to prove three points are collinear when constructing a rectangle

My problem is: Choose a unit segment OI. Then construct a rectangle with base 3 units and height 2 units. I cannot use angle measure. I know I can construct this figure from my unit segment by using ...
2
votes
2answers
65 views

Construction of a square ABCD

There are two nonparallel lines $p,q$ and point $A$, $A \notin p,q $ which lies between lines $p,q$. Construct a square ABCD such that $B \in p$ and $D \in q$. In special case in which $45°$ is angle ...
12
votes
3answers
225 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 ...
0
votes
1answer
45 views

How to find the inradius of orthic triangle?

How to find the inradius of orthic triangle in terms of side lengths or area or circumdiameter of original triangle? The incentre of the orthic triangle is the orthocentre of the original triangle. ...
0
votes
0answers
30 views

Difficult Proof by Construction Question

A teacher introduced this question to me, and am interested in finding the answer, but cannot figure it out. Given the squares of the numbers 1 through 81, can you separate them into three groups with ...
-1
votes
2answers
89 views

Why a segment of length $\sqrt{2}$ can be drawn but a segment of length $\pi$ cannot?

We know that both $\pi$ and $\sqrt{2}$ are irrational. Also, it has been proved that a segment of length $\pi$ can not be drawn whereas a segment of length $\sqrt{2}$ can be drawn. Why is it so, ...
4
votes
2answers
154 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 ...
2
votes
2answers
81 views

How to construct an n-gon by ruler and compass?

Since $\cos[\frac{2\pi}{15}] $ is algebraic and equal to $\frac{1}{8}(1+\sqrt{5}+\sqrt{30-6\sqrt{5}})$ we know that the regular 15-gon is constructible by ruler and compass. Although I know how to ...
4
votes
1answer
122 views

$\sqrt2^{\sqrt3 }$ $<$ $\sqrt3^{\sqrt2 }$ geometrical way [closed]

From the pythagorean theorem we can construct the lengths $\sqrt2$ and $\sqrt3$ Is it possible to construct the lengths $\sqrt2^{\sqrt3 }$ and $\sqrt3^{\sqrt2 }$ in order to show ( geometrically ) ...
0
votes
1answer
25 views

Prove that in a cyclic quadrilateral ABCP where two fo its sides are equal the ratio $ \frac{PA}{PB + PC}= \frac{AC}{BC}$ is constant for any point P

Let ABC be an isosceles triangle where AB = AC. Consider the circumference in which the points A, B and C live. Let P be a point in the chrord of this circunference formed by the points B and C. Prove ...
0
votes
0answers
59 views

Constructing congruent triangles and give a proof

I need to construct from a given triangle $\Delta ABC$ a congruent triangle $\Delta XYZ$ and I know how to do this if I'm allowed to "measure an angle with a compass" and carry that measurement over ...
1
vote
0answers
60 views

Relation between sqrt and ratio in ruler and compass?

The construction in most vote reply in Compass-and-straightedge construction of the square root of a given line? uses similar traingles and uses $$\frac{AC}{AD}=\frac{AD}{AB}$$ to compute square root. ...
1
vote
1answer
69 views

Is it possible to construct an isosceles triangle by using a ruler and without using a pair of compasses?

It is well known that on Euclidean plane one can construct an isosceles triangle on given straight line by using a ruler and a pair of compasses. Also it is possible to construct straight line ...
3
votes
1answer
296 views

Construct the intersection of a cube by a plane through $3$ points on its edges, no pair of which is on the same face

So this is a rather old problem, but I still cannot find a pure constructive solution to it. Please, do not offer me to write a plane equation, etc. I would be grateful, if you offer a solution by ...
1
vote
0answers
42 views

Pyramid with unit sides inside a cube

Let $ABCDEFGH$ be a unit cube with base $ABCD$. Let $P$ be the top of the pyramid with base $ABCD$ and all edges of length $1$. One has a standard 2-dimensional projection of this cube on the back ...
3
votes
1answer
63 views

Can one construct any n-gon if angle trisection is also allowed?

Suppose one is asked to construct a regular n-gon, but with one extra operation allowed in addition to the standard compass-and-straightedge ones: trisecting any angle. Are all n-gons constructible ...
4
votes
1answer
189 views

construct a triangle with a compass and a ruler, given $a, B, t_a$ [closed]

If $a, b, c$ are the side lengths of a triangle, $A,B,C$ are the opposite internal angles, respectively, and $t_a, t_b, t_c$ are internal bisectors of the angles $A,B,C$, how could I construct a ...
0
votes
1answer
85 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 ...
0
votes
1answer
45 views

constructing segments with equal cross ratio

I was puzzeling again and had the following problem: Given: two segments $AD$ and $PS$ on $AD$ there are points $B$ and $C$ so that $AD \gt AC \gt AB$ (so they are in order A, B , C, D ) on $PS$ ...
11
votes
2answers
229 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 ...
0
votes
1answer
57 views

Is there a geometric construction for the square root of the volume of a parallelotope?

A parallelotope is the higher dimensional analog of a parallelogram. Now, what I want to know is if there's a way to construct an object with size equal to the square root of the volume of the ...
0
votes
0answers
29 views

Extension of Isovist concept for a point - to Isovist for a polygon

There is the concept of Isovist/Visibility polygon. They both talking about volume of space visible from a given point in space. My question: What is the algorithamic solution of this problem for a ...
7
votes
2answers
162 views

How to find the vertices of a particular ellipse with straightedge and compass?

In order to provide and alternative solution to a well-known problem $^{(*)}$ I would like to solve the following sub-problem in the most effective way (i.e. in the least number of steps). ...
1
vote
1answer
44 views

Mapping from Poincare's disk model to UHP

I have a question that : How can I map any point in Poincare's disk model to Upper-half-plane model? I know the function $$f(z) = \frac{z + i}{iz+1}$$ But I want to know the geometric ...
1
vote
1answer
114 views

Proof of the ultraparallel theorem in the Beltrami Klein model

I was reading (and editing) the proof mentioned at https://en.wikipedia.org/wiki/Ultraparallel_theorem#Proof_in_the_Beltrami-Klein_model and noticed it is not correct. (the ultra parallel theorem is ...
1
vote
3answers
78 views

Easy Compass Construction Problem

Here is a tricky compass and straightedge construction problem. Given triangle $\triangle ABC$ and point $D$ on segment $\overline{AB}$, construct point $P$ on line $\overleftrightarrow{CD}$ such ...
0
votes
1answer
128 views

Constructing a parallelogram according to the given condition

The question #To prove two angles are equal when some angles are supplementary in a parallelogram has been solved. In the process of solving it, I found it is not that easy to draw the corresponding ...
4
votes
3answers
106 views

Degree of minimal polynomial for $\sin (\frac {2 \pi} 7)$

So I was playing around with trying to prove the regular 7-gon is not constructable under qualifier-exam conditions, so I didn't have a book open. I got it down to having (If I didn't make any basic ...
0
votes
1answer
45 views

equation of a cylinder jacket

how would you calculate this? A circular cylinder, height $14$, base radius $2$, has the axis of rotation! What is the equation of the cylinder jacket when the center of the base circle is the ...
3
votes
1answer
90 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 ...
1
vote
1answer
113 views

Constructible real numbers

I'm trying to understand constructible numbers. I know that a real number $r$ is constructible if it can be calculated from 0 and 1 by a finite number of additions, subtractions, multiplications, ...
2
votes
1answer
61 views

Construct with straight edge a parallel to two lines.

It is known that we can't with just a straight edge, given a line and a point out of the line in a plane to construct other line, passing through the point, parallel to the first. I know a proof of ...
1
vote
1answer
118 views

Given a circle, its diameter and an external point, use a straightedge to draw a line through the point and perpendicular to the diameter

Some time back I saw the following problem which originated in Russia: You are given a circle, its diameter and an external point not on the diameter (A, B and P in the diagram below). Using only ...
3
votes
2answers
166 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
71 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 ...
6
votes
0answers
117 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
43 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
38 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
vote
0answers
54 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
507 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 ...