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

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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 ...
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2answers
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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), ...
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1answer
79 views
+50

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

$a,b,c$ the sides of the triangles; $A,B,C$ the angles of the triangles; $t_a, t_b, t_c$ the internal bisectors of the angles $A,B,C$. How to construct a triangle with a compass and a ruler(a ...
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14 views

Application of Multivariate Analysis on Straight Lines [closed]

Can anyone suggest to me a reference on the Application of Multivariate Analysis on Straight Lines? The reference should contain the part where we use the concept of orthogonality to formulate ...
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0answers
25 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 ...
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1answer
69 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 ...
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1answer
26 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 $PS$ ...
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2answers
83 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 ...
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1answer
38 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 ...
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2answers
114 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). ...
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17 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 ...
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1answer
37 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 ...
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1answer
32 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 ...
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2answers
2k views

Finding a line perpendicular to a line and passing through the intersection of other two lines

So here is the question as in my text book Find the equation of the line through the intersection of $2x - 3y + 4 = 0 $ and $3x + 4y - 5 = 0$ and perpendicular to $6x - 7y + c = 0$ so I ...
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3answers
48 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 ...
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1answer
36 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 ...
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3answers
85 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 ...
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1answer
56 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 ...
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3answers
177 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 ...
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1answer
34 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 ...
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1answer
105 views

Perpendicular at a defined distance from point on line intersects another line in coordinates?

It approximately looks like the following picture The figure may be rotated at any angle. I know the coordinates of points $A$, $B$, $C$, $D$ and the length of $BF$. $\angle ABD$ and $\angle CBD$ ...
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1answer
55 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 ...
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1answer
52 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, ...
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1answer
42 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 ...
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1answer
72 views

Prove : for every integer $n\ge1$, if the regular $n^2$-gon is constructible, then $n$ has no odd prime divisors.

For every integer $n\ge1$, if the regular $n^2$-gon is constructible, then $n$ has no odd prime divisors. I know is has something to do with the fact the output Euler's Totient function on $n^2$ ...
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1answer
1k 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 ...
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2answers
369 views

Find the angle between two lines using a compass and straight edge.

I've drawn two random, non-parallel, straight lines on a plane. They cross over, forming two angles, $a$ and $b$, where ($a + b + a + b) = 1$ (or $360^\circ$) and $a ≤ b$. (Making $a$ either the acute ...
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2answers
78 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 $h,k$. ...
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1answer
50 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 ...
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92 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. ...
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1answer
37 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: ...
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50 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 ...
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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 ...
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2answers
597 views

What is the difference in radii of two concentric circles given an angle and length of a triangle that is inscribed in the annulus?

In relation to this geometric construction: where D is the center of both circles, if the inner radius (x = length of line segments DA and DE), the angle φ = ∠CAB, and the length Δg of line segment ...
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1answer
53 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 ...
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3answers
96 views

Compass-and-Straightedge Construction [closed]

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 ...
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1answer
26 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 ...
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0answers
43 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 ...
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1answer
44 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 $3.8\ cm$ ...
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0answers
33 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 ...
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59 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.
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2answers
211 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 ...
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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 ...
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93 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 ...
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0answers
49 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 ...
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1answer
82 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?
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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 ...
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2answers
53 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 ...
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1answer
104 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 ...
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125 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 ...