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

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3
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1answer
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A straightedge and compass construction: $\left(G,I,Q_a\right)$

Construct $ABC$ with straightedge and compass, given $G,I,Q_a$. $G - $ the intersection point of medians; $I -$ the center inscribed circle; $Q_a -$ point of tangent inscribed circle to the side ...
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3answers
60 views

Constructible $n$-gons

Let $\xi$ be the primitive root of unity. If $n=5$, then the minimal polynomial of $xi$ over rationals would have degree $5-1=2^2$ which is a fermat prime so $5$-gon is constructible. If $n=8$, ...
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0answers
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Cube root of a line

Well this may be simple but I am not getting it. Give a line segment (of length $l$)(and a segment of unit length if you require) how to construct a line of length $l^{1/3}$ with only a straight ...
3
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4answers
107 views

Construct an equilateral triangle with area equal to a given triangle

It is straightforward to construct (straight-edge and compass) an isosceles triangle with area equal to a given triangle $\triangle ABC$, for instance as follows: Construct the line through $A$ ...
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0answers
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Does the collection of algebraic/number-theoretic methods applied to Euclidean Geometry have a name?

I am currently writing an essay on the history of geometry. To educate myself on the subject, I sometimes read the following Wikipedia article on the history of Euclidean Geometry. It seems to me that,...
1
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1answer
27 views

Squaring a line segment

I dont know whether I am being silly or not, but my question is: Given a line segment (say length $l$), how can you draw a line segment of of length $l^2$ using straight edge and compass? I ...
8
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4answers
248 views

Can anyone solve this geometric construction problem?

I remember when I was in high school, one of my all-time favorite books was College Geometry by Nathan Altshiller-Court. Some of its problems kept me wondering for days and even weeks. Now after about ...
4
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0answers
18 views

What makes a geometric construction more or less stable?

As anyone who's actually done geometric construction of n-gons knows, not all construction methods are made equal. Some are very stable (the shape you get is always close to ideal even if you're not ...
5
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1answer
131 views

Is $e$ “constructable” with the appropriate tools?

Of course $e$ cannot be constructed with straightedge and compass. If we allow a marked ruler or a non-rectractable compass (or we use origami...) we can construct numbers like $\sqrt[3]{2}$, but no ...
10
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1answer
79 views

Is there a way to draw a 1 degree angle using only ruler and compass?

There are ways to draw 180, 90, 45, 30, 60, ... degree angles. But is there a way to draw a 1 degree angle? In other words how to divide a circle into 360 equal parts?
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1answer
38 views

Compass only constructions

Solve this using only a compass. Beyond stumped... please help.
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3answers
100 views

The geometric construction of the $90^\circ, 87^\circ, 3^\circ$ triangle

The construction of the $90^\circ, 45^\circ, 45^\circ$ and the $90^\circ, 60^\circ, 30^\circ$ triangles is well known. How can be constructed a triangle with angles $90^\circ, 87^\circ, 3^\circ$ ...
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1answer
26 views

$ax^2+bx+c$ is constructible

If $a$ is constructible, then $\sqrt a$ is constructible. Furthermore, if $a,b,c$ are constructible, then every root of $ax^2+bx+c$ is constructible. I think I know how to prove the first sentence ...
1
vote
1answer
24 views

Line through a given point

Could someone give a proof of "The line through a given point and parallel to a given line can be drawn". I would be able to do this myself but I am so confused on the wording. As in what are we ...
1
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1answer
36 views

Geometry (Locus and constructions)

I want to find the equation for the locus that is at the same distance from the point $(2,3)$ to the line $x=1$. Im not sure if I am right or wrong? Is the locus just the two point at a distance=1 ...
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0answers
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How to construct the circumcenter of a triangle using a compass ONLY.

I just figured out how to find the midpoint between two points using just a compass and no straight edge. A similar approach can be found in this question: Constructing the midpoint of a segment by ...
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0answers
20 views

Construction of a triangle using symmetry

I need to construct a triangle $\Delta \textrm{ABC}$ knowing that $t_a = AS$, $|AS| = 6\, cm$, $|\measuredangle \textrm{BCA}| = 30°$ and $|AB| = 5.5 \,cm$. I've been told that it's possible to do it ...
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0answers
207 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 ...
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1answer
15 views

given two concentric circles construct a particular chord

I am stumped by another Euclidea problem - Euclidea problem 9.8: Given two concentric circles $C_1$ and $C_2$ with radians $r_1$ and $r_2$, with $r_1 < r_2 < 2 r_1 $ Construct the chord $e$ ...
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5answers
92 views

Can we prove that plumb line is vertical to ground?

Using a plumb line to make sure a wall is vertical for instance, is as far as I know one of the most primary tools in the sense that the very-very ancient builders used it as an instrument. I was ...
2
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1answer
28 views

construct triangle given angle and centroid

I am stumped by Euclidea problem 8.11: From a triangle are given angle A and the centroid G Construct the points B and C. Please only a hint
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1answer
558 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 ...
2
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1answer
89 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, ...
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6answers
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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 consruction-http://gradestack.com/CBSE-Class-9th-Complete/Construction/...
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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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3answers
73 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 the ...
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2answers
105 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
36 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.
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1answer
32 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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2answers
3k views

Can a regular heptagon be constructed using a compass, straightedge, and angle trisector?

Euclid has a magical compass with which he can trisect any angle. Together with a regular compass and a straightedge, can he construct a regular heptagon?
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votes
1answer
31 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 ...
4
votes
2answers
179 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 ...
0
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0answers
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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. ...
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2answers
4k views

Trisect unknown angle using pencil, straight edge & compass; Prove validity of technique

This question was posed by my high school geometry teacher, for extra credit: Is it possible, using only a pencil, a straightedge (not a ruler) and a compass to trisect an angle of unknown value? ...
2
votes
3answers
143 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
votes
1answer
39 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
votes
4answers
69 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 $d$, $b-b'...
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vote
2answers
54 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
votes
3answers
477 views

Construction of a triangle, given: side, sum of the other sides and angle between them.

Given: $\overline{AB}$, $\overline{AC}+\overline{BC}$ and $\angle C$. Construct the triangle $\triangle ABC$ using rule and compass.
0
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1answer
56 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
47 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 ...
2
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1answer
55 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 https://en.wikipedia.org/wiki/Compass-and-straightedge_construction#...
2
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1answer
21 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
votes
1answer
80 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 ...
3
votes
2answers
180 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 $\...
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votes
3answers
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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° ...
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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2answers
51 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 ...
1
vote
1answer
36 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
votes
1answer
64 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 ...