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

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2answers
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Can $n$ circles be drawn such that all have a common intersection but no two intersect individually

I was fiddling with plane geometry when a question came into my mind: Can $n$ circles ($n \ge 3$, $n \in \mathbb{N}$) be drawn such that: $C_1 \cap C_2 \cap C_3 \cap \ldots \cap C_n \not = ...
-2
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0answers
23 views

To find/create midpoint, is it easier to bisect a line segment, or double a line segment? With only compass and ruler / straightedge. [closed]

Suppose one wants to find the midpoint of a line segment. Is it generally easier to simply draw two lines of equal length end to end, or is it easier (does it count as less steps) to draw a line and ...
2
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3answers
68 views

Straight Edge - Only Geometric Construction

Given a circle, its diameter and a given point on the diameter, find a procedure to construct a line perpendicular to the diameter using only a straight edge. The perpendicular must pass through ...
3
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2answers
98 views

Construction using a straight edge only

Given a circle, its diameter and a point on the circle, find a procedure to construct a line perpendicular to the diameter using only a straight edge. The perpendicular must pass through the given ...
0
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1answer
35 views

You are given two points and a circle. Construct a circle passing through the given two points and tangent to the given circle. [duplicate]

You are given two points and a circle. Construct a circle passing through the given two points and tangent to the given circle. You are allowed to use a straightedge and a compass.
2
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1answer
83 views

Construct a circle passing through a point $X$, which is externally tangent to two given circles

Given two disjoint circles $S_1$ and $S_2$, and a point $X$ external to both of them, is it possible to find the center of a circle that passes through $X$ and touches $S_1$ and $S_2$ tangentially, ...
0
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1answer
31 views

How to construct a triangle from…?

Two medians and one height (nothing is for the same side!); Outer circle radius, one side and another side's height? Using Compass-and-straightedge construction.
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1answer
24 views

Constructing the asymptotes of a hyperbola by compass and straightedge.

Is it possible to construct the asymptotes of a hyperbola by compass and straightedge? And if so, how to construct them? I have no idea how to approach the first question. It seems it should be ...
0
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0answers
23 views

How can one show algebraically that an angle is constructible?

For example an angle of 30 degrees. I know that geometrically I can obtain the entire 30-60-90 triangle using the standard tools (compass, straightedge and unit length) and by performing iterations. ...
1
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1answer
29 views

A straightedge and compass construction: $\left(\widehat{A},r,b-c\right)$

I am looking for an elegant solution of the following problem: Construct $ABC$ with straightedge and compass, given $\widehat{A},r,b-c$. By taking the lines $AB,AC$ as a skew reference system, ...
2
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4answers
68 views

Construct a triangle given certain lengths related to a bisector

Let $ABC$ be a triangle, and $AD$ the bisector of angle $A$. Write $AB = c$, $AC = b$, $AD = d$, $BD = c'$, $CD = b'$. Using ruler and compass, construct the triangle $ABC$ given the lengths $d$, ...
0
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1answer
45 views

Finding square roots of complex number with ruler and compass

Provide the exact list of steps needed to find, with ruler and compass, the two square roots of a given complex number. (The points $0$ and $1$ are given) I don't really understand what I have to ...
0
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1answer
40 views

Construction of major and minor axes of an ellipse given only 2 focus points($F_1,F_2$) and a point $P$ that is on the ellipse.

Construction of major and minor axes of an ellipse given only 2 focus points $(F_1,F_2)$ and a point $P$ that is on the ellipse. Suppose we define $|F_1P|+|PF_2|:=l$ First I constructed the ...
1
vote
2answers
50 views

Ruler and compass question

Provide the exact list of steps needed to draw, using ruler and compass, a line $M$ through a given point $A$ and parallel to a given line $L$ (given by two points $B$ and $C$ on it). Assume that $A$ ...
2
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3answers
140 views

Construct a parallelogram subject to certain conditions

I am having trouble with the following exercise from Dollon and Gilet's Géométrie plane. Two parallel lines $\Delta$ and $\Delta'$ are given, as well as a point $A$ on $\Delta$ and a point $O$ on ...
2
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1answer
19 views

Accuracy of a Construction

Is there an easy way to find the accuracy of a construction given a straight-edge and compass? For instance setting the point of a compass on an existing line. How do I know how exact that is? Or ...
2
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1answer
78 views

I think I found a method for Squaring the Circle. But I'm not sure if it's valid.

Here it is: Method for constructing a line of length π: Construct a circle labeled A, with a radius of 1. Bisect circle A. Each of the resulting arcs is now length π. Label one arc B. Align one end ...
2
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1answer
50 views

Much used compass and straightedge constructions

I am a editor of wikipedia and would like to know which compass and straightedge constructions deserve a place in the list ...
2
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3answers
38 views

Is my observation correct about geometric constructions?

I have observed that it is possible to construct angles which are multiples of 3 with a ruler and a compass (Angles are in degrees). For example, 135°, 45° etc. can be constructed but Angles like 100° ...
3
votes
2answers
136 views

Construct 60° angle through point, other line in only four compass-and-straightedge steps

PROBLEM Here is a surprisingly intriguing challenge posed on Euclidea, a mobile app for Euclidean constructions: Construct a 60° angle through both a point $P$ and an external (infinite) line ...
2
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3answers
41 views

If $tan^2 \theta = \frac{x}{y}$ how can we construct the angle $\theta$?

If we are given the values of $x$ and $y$ and we know that $\tan^2 \theta = \dfrac{x}{y}$ is it possible for us to construct the angle $\theta$?
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1answer
35 views

Construction of a graph with specific property.

I am trying to find out graphs where eccentricity of every vertex is one except two vertices. And the eccentricity of these two vertices is two. I came to the conclusion that a path graph $P_3$ and a ...
3
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1answer
63 views

Predicting Spirals

I am currently in the process of analyzing a polyspiral, a spiral where each successive length drawn is increased at specified increment at the same angle. *Please note the angles selected are the ...
1
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2answers
47 views

Are all computable numbers constructable after a countable number of steps?

While looking at another question on this site about constructable numbers I started wondering. If you can take a countable number of steps (possibly infinite) can you draw an interval of a length ...
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1answer
51 views

Geometric Construction : Construct a Triangle given 3 heights .. [closed]

Given 3 heights : $h_1=5\mathrm{cm}$ ; $h_2=7\mathrm{cm}$ ; $h_3=8\mathrm{cm}$ ... It is required to draw that triangle using only compass and ruler ! N.B.: It is not allowed to calculate the area ...
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6answers
3k views

Why is not possible to draw this triangle?

Why is it not possible to draw triangle $DEF$ with $EF=5.5cm$,$\angle E=75^0$ and $DE-DF=1.5cm$?(I used this method for ...
0
votes
1answer
27 views

How to build a hexagon according to Poincaré model?

Given a side, I know how to build a hexagon in the euclidean geometry. How can i build it in the hyperbolic geometry according to the Poincaré model? By translating every step using hyperbolic circle ...
6
votes
1answer
165 views

Construct a triangle with its orthocenter and circumcenter on its incircle.

Construct $\triangle ABC$ such that its orthocenter ($H$) and circumcenter ($O$) are on its incircle. I've tried something by inverting everything WRT circumcircle but don't have proper idea... ...
2
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0answers
41 views

A circle can include all but one of n points, but which one can it be?

The answers to the question "Circle enclosing all but one of n points" demonstrate that, given $n$ points, it is possible to construct a circle such that all but one of the points is inside the circle ...
2
votes
2answers
80 views

construct triangle given $b-c$, $r$ and $h_{b}$

As in title: the problem is to construct triangle given difference of sides $b$ and $c$, then in-circle radius $r$, and height $h_{b}$. The problem is from a set of problems exercising various ...
5
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1answer
109 views

Finding $\sqrt{17}$ and $\sqrt{257}$ in the regular $17$-gon and $257$-gon?

(Edit: I need to revise this question with my original intent. Pls do not answer it yet. Thanks.) Given the regular $n$-gon formed by the $n$-th roots of unity. For some $n$, how do we find ...
4
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1answer
56 views

Drawing a circle tangential to 3 circles (internally to one of them)

The two small circles (in black) are equal in radius, and tangential to the large circle. They also touch each other at the center of the large circle. Now, I want to construct a circle (in orange) ...
0
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1answer
40 views

Dividing Up A Circular Search Area

BACKSTORY: I need to collect 500 plant samples for strontium analysis. The samples are randomly distributed across a circular area with a radius of 300 kilometers. I have to do this in 30 days, so I ...
3
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2answers
42 views

What types of triangles are constructible?

What types of triangles are constructible? I know that equilateral triangles are easily constructed using compass and straightedge, but what about other types of triangles? Can any other ...
0
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1answer
26 views

What cube roots are constructible using compass and MARKED straightedge?

DUE DILIGENCE: I have reviewed the list of questions possibly related to the one that I pose below, and I find that none of them address my particular question. Is there a formal definition for the ...
3
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0answers
51 views

Is this curve defined by an envelope construction known?

Consider the following construction. Start with the standard envelope construction of a cardioid: on a circle, join each point $\theta$ to $2\theta$. Only, instead of joining with a line, join with ...
2
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2answers
103 views

Constructing a line that passes through $P$

I have recently read a book by Heisuke Hironaka. However, the book is not available on English. The book was basically a biography on his life. Heisuke Hironaka says that his high school teacher had ...
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2answers
104 views

Can you construct a rectangle with a given side, equal to a square?

In Euclid's Elements, Book 2, Proposition 14, We are shown how to construct a square from a given rectilinear figure. This allows us to square a rectangle. Is it possible to do the inverse, creating ...
3
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0answers
49 views

Minimal number of steps to construct $\cos(2 \pi /n)$

My question is related to this previous one. I was wondering what is the minimal number of steps $S(a)$ to construct a number $a \in \mathbb R$ that is constructible (as defined here). For instance, ...
2
votes
1answer
210 views

Is it impossible to construct an equilateral triangle inside a semicircle?

I have made it in a circle(which is very easy)....but I have been unable to make one inside a semicircle....is it not possible to make equilateral triangle inside a semicircle ?... If yes how can we ...
2
votes
1answer
39 views

Finding the glide reflection using a compass or straightedge

Given two congruent triangles that are not a rotation, translation or reflection of each other; how can I find the glide reflection (the last remaining option) using only compass and straightedge. ...
6
votes
2answers
357 views

Inscribe square in circle in just seven compass-and-straightedge steps

Problem Here is one of the challenges posed on Euclidea, a mobile app for Euclidean constructions: Given a $\circ O$ centered on point $O$ with a point $A$ on it, inscribe $\square{ABCD}$ within the ...
4
votes
1answer
82 views

Mascheroni construction

There are two points in one line and a point C which doesn't lie on the line AB. I have to construct a point D such that it is intersection of the line AB and a line which is perpendicular to the line ...
1
vote
4answers
80 views

What is the simplest way to find $\frac{n}{7}th$ of a line with mathematical proof?

I'd like to know if it is possible to find out the simplest way to get $\frac{1}{7}$, $\frac{2}{7}$, $\frac{3}{7}$, $\frac{4}{7}$, $\frac{5}{7}$ and $\frac{6}{7}$ of a line in 2-dimensional geometry. ...
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8answers
326 views

Divide a line segment in the ratio $\sqrt{2}:\sqrt{3}.$

"Divide a line segment in the ratio $\sqrt{2}:\sqrt{3}.$" I have got this problem in a book, but I have no idea how to solve it. Any help will be appreciated.
4
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1answer
82 views

Go from A to D in three equal steps

Given two parallel lines $r$ and $s$, line $p$, perpendicular to both, and points $A$ and $D$ on different sides of $p$ with respect to the parallel lines, how can I prove the existence of two points, ...
2
votes
1answer
36 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
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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
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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
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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 ...