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

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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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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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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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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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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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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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53 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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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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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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48 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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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
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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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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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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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41 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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1answer
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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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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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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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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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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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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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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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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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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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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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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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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 ...
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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 $$ ...
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101 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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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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?
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...