Questions tagged [geometric-construction]

Questions on constructing geometrical figures using a limited set of tools. The compass and straightedge are almost always allowed, while other tools like angle trisectors and marked rulers (neusis) may be allowed depending on context.

10
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8answers
283 views

Evaluating $\sum_{k=0}^\infty \left(\frac{1}{5k+1} - \frac{1}{5k+2} - \frac{1}{5k+3} + \frac{1}{5k+4} \right)$

I saw this problem somewhere recently and I was having some difficulty getting started on it. The problem is twofold. The first is to evaluate: $$\sum_{k=0}^\infty \left(\frac{1}{5k+1} - \frac{1}{5k+...
0
votes
1answer
93 views

Alternative compass and straightedge construction

My problem is motivated by software like Geogebra. Suppose that we are allowed to use the following two tools only: given any two points, we can construct a line passing through both of them given ...
1
vote
2answers
67 views

Construct quadrilateral $ABCD$: $AB=AD=4$, $CB=CD=5$ and $BD=6$. Construct incircle of quadrilateral $ABCD$.

Its easy to construct $ABCD$ but how can we construct incircle? As $ABCD$ is a kite is there any method? For my efforts I tried incircle as in triangle will it satisfy the question?
7
votes
4answers
723 views

How to intersect a line with a point off the page using a straight edge and compass (see description)

Its very common in illustration to want to draw a line towards a vanishing point that is off of the page. The specific problem is this: lets say we draw two line segments on a piece of paper lying ...
1
vote
2answers
211 views

If $3-$gon and $5-$gon are constructible, show that $15-$gon is too.

Use the fact that the regular $3-$gon and the regular $5-$gon are constructible to show that the regular $15-$gon is constructible. What is the best way to prove this? I have found a theorem that ...
0
votes
3answers
383 views

How to construct an isosceles triangle given the base angle and height to one side?

I've got this math exercise. It reads: Construct an isosceles triangle given the base angle and height to one side. How many solutions exist? Why? I made a sketch, but still can't figure out how ...
0
votes
1answer
40 views

Given that x is a constructible number, how does one prove that 1/x is constructible?

Given that x is a constructible number, how does one prove that 1/x is constructible? Moreover, what might be one's train of thought when coming up with the proof.
1
vote
2answers
751 views

Proof of construction of $30^{\circ}, 60^{\circ}, 120^{\circ}$ and $135^{\circ}$ angles

What is the proof of constructing $30^{\circ}, 60^{\circ}, 120^{\circ}$ and $135^{\circ}$ angles with ruler and compass? I can prove $90^{\circ}$ by proving that the line joining point of intersection ...
0
votes
1answer
49 views

How to draw/construct the function $\frac{\sqrt{1-x^2}}{x}$

can you help me with some hints how to draw or construct the function $\frac{\sqrt{1-x^2}}{x}$ ($x \in (0,1]$) ? I really have no idea - maybe by using unit circle..?!
1
vote
2answers
42 views

Constructing a line segment through a point so it has a ratio $1:2$

Question: Let $AOB$ be a given angle less than $180^\circ$ and let $P$ be an interior point of the angular region of $\angle AOB$. Show, with proof, how to construct, using only ruler and compass, a ...
2
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0answers
55 views

Given sufficiently large sample of flight paths, is it possible to derive the map?

Background Suppose we do not have a map of the world but we can fly from a random point towards a random direction. The speed are the same for all flights and remain constant throughout each flight....
1
vote
2answers
59 views

Geometric model of real numbers

I'm looking for a way to multiply real numbers using only geometric techniques. Suppose we already know how to halve a real number and multiply a real number with a natural in a geometric way. For ...
10
votes
3answers
2k views

How do you know if a line is straight?

How do you know a line is straight? How can you check in a practical way if something is straight - without assuming that you have a ruler? How do you detect that something is not straight? If you ...
2
votes
0answers
147 views

Relative difficulty of angle trisection and cube roots

It is well-known that arbitrary angles cannot be trisected by using a straightedge and a compass. The proof of this fact is done by reduction to unsolvability of cubic equations with square roots (see ...
8
votes
1answer
234 views

Is is possible to double the cube using compass, straightedge, and angle trisector?

Essentially an angle trisector would allow you to construct a root of $4x^3 - 3x - a$ for any constructible $a$ with $|a| \le 1$. By scaling and translation, it would seem this means that using a ...
0
votes
0answers
48 views

Divide a line into $n$ parts using a rusty compass

Using a rusty compass (a compass whose distance between the two needles can't be altered) divide a given line segment ${AB}$ into $n$ equal parts. In fact, a straightedge isn't provided. I know that ...
0
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1answer
263 views

Please Prove Me Wrong (Trisection of a LINE)

I want to preface this by saying I have been trying for a while to prove myself wrong because my results appear to contradict the work of some previous work by people who have studied much more than ...
8
votes
2answers
1k views

Inscribing rhombus in a triangle's angle in only eight compass-and-straightedge steps

EUCLIDEA ANDROID PROBLEM 13.1 Here is a surprisingly intriguing challenge posed on the latest version of Euclidea, a mobile app for Euclidean constructions. The problem used to be considered easy when ...
1
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2answers
180 views

Does this mean that one can construct the cube root of two in three dimensions?

The idea here is to extend to three dimensions what ordinary compass-and-straightedge constructions do in two dimensions. The first thing is to define the tools and rules for their use. For instance,...
0
votes
1answer
67 views

Compass and straight edge [closed]

Is there any method how to grab into compass, by using just compass and straight edge, side length of a cube, which is inside a given sphere, touching this sphere by corners, but this sphere is given ...
3
votes
0answers
78 views

Recognizing integer Pythagorean triples geometrically and whether two points constructed on a line are the same

This is suggested by the question Proof that the sum of the even side and the hypotenuse of a coprime (and positive) Pythagorean triple is a square number and the fact that I really like Euclid's ...
2
votes
3answers
132 views

Geometric construction of a hexagon

How to draw a hexagon with each of its interior angles as 120° using a ruler and a compass? Please help me out.. actually the 6th side I am drawing is becoming slanting in place of being parallel to ...
0
votes
2answers
478 views

Construct tangent through point on circle in 3 steps

I'm stuck in the Euclidea app in level 2.8, which asks: Given a circle and a point on that circle, construct the tangent through that point using only 3 elemental steps. Their FAQ already mentions ...
0
votes
3answers
55 views

How do we find point( b ) on a graph at a specific angle from point( a ). Point( a ) being only value known.

The origin is top-left here because I'm creating this image using SVG. The coordinate system starts in the top left and increases as you move down and/or right. I'm trying to find point b if the ...
2
votes
2answers
35 views

What are good sources for learning to hand draw cartesian and polar functions (and euclidean constructions)?

I would like to get into hand plotting cartesian and polar functions as a hobby. What are good books/websites on this? I will absolutely look at youtube videos but I prefer books/websites because I ...
4
votes
2answers
92 views

can you typically compare lengths in compass and straightedge constructions?

Historically, has it been considered valid in compass and straightedge constructions to, given two line segments, decide which is longer? It isn't hard to see that this problem is equivalent to the ...
0
votes
1answer
93 views

Straight-edge and compass construction of a ray having angle $\theta + \theta'$

Given two complex numbers $z=r\exp(i\theta),z'=r'\exp(i\theta')$, I would like to prove that $zz'$ is constructible using straight-edge and compass. I am stuck on proving the constructability of a ray ...
5
votes
1answer
389 views

Construct a perpendicular to a given line from a given (external) point, using a compass only once

Given a line $AB$ and a point $C$ not on $AB$ it is easy enough to construct a perpendicular line to $AB$ passing through $C$ using two circles as demonstrated in the following picture. Here we pick ...
1
vote
1answer
154 views

Construct the trapezoid $ABCD$ with straightedge and compass

$ABCD -$ is a trapezoid. $AD||BC, AB=CD$. The diagonals of the $AC$ and $BD$ intersect at the point $P$, and the straight lines $AB$ and $CD$ intersect at the point $Q$. Points $O_1$ and $O_2$ are the ...
1
vote
0answers
100 views

Why is it impossible to find the center of a circle with the straight edge? [closed]

The proof in 'What is mathematics' reads: 'there exists a transformation of the plane into itself which has the following properties:a)the given circle is fixed under the transformation b)any straight ...
9
votes
1answer
13k views

Constructing the incenter of a triangle in only six steps

Lately I have become hooked on the game Euclidea. One of the problems gives a triangle and asks you to construct the incenter, or as it is put, "the intersection of angle bisectors." It is stated ...
1
vote
2answers
4k views

Construct parallel line equidistant from two other parallel lines - Euclidea on iOS challenge 2.8

I bet you know this game, Euclidea - if not I recommend it... although I'm stuck on level 2.8 which is different on iOS than other platforms for which there are already walk throughs available. Here ...
2
votes
1answer
86 views

Why is it that if $\alpha$ is a root with minimal polynomial of degree $2^k$ (for some integer $k$), then $\alpha$ is constructible?

Why would it be true that if the minimal polynomial of $\alpha$ has degree equal to a power of $2$ then $\alpha$ is constructible? I came across this fact recently while studying Galois theory on my ...
4
votes
3answers
313 views

How to construct a tangent to a circle between two lines?

Given a circle $P$ between two lines $\ell_1, \ell_2$, we want to find a tangent $AB$ to $P$ such that $A\in\ell_2, B\in\ell_1$ and the midpoint of $AB$ is the tangency point. Is it possible to solve ...
9
votes
4answers
1k views

Why is adding the same as extending a length?

I've come to realize that the more I study some math subjects the more I question some results or ideas that seemed trivial or obvious to me. My question is about the real numbers and their geometric ...
3
votes
1answer
218 views

Construction of spherical regular polygons

Every so often I see spherical geometry/trigonometry questions on this site, and this has piqued my curiosity about geometric constructions on the sphere. Constructions for the equilateral triangle, ...
0
votes
1answer
103 views

Galois Theory: Are separable and normal equivalent?

I have to proof the following Theorem Theorem: Let $\mathbb{Q} \subseteq L$ be a normal field extension with $L \subseteq R$ and $[L:\mathbb{Q}]=2^\lambda$. Then every $\alpha \in L$ is ...
3
votes
3answers
2k views

Why should kids learn how to use a compass and straightedge, and not rely on a drawing program? [closed]

I am curious why it is necessary for people to learn how to use compasses and straightedges in geometry, and not just rely on a drawing program. I have a couple ideas, but I might be missing ...
1
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0answers
45 views

Constructibility of the $\sqrt[n+1]{2} + \sqrt[n+3]{2}$ where $n$ is even

Let $n$ be even. Are there any constructible numbers of the form $\sqrt[n+1]{2} + \sqrt[n+3]{2}$? Attempt My initial hunch is no. Because the $\sqrt[n+1]{2}$ is a root of the polynomial $f(x)=x^{n+1}...
2
votes
1answer
62 views

Counting compass and straightedge constructions

I have found a fun problem concerning compass and straight edge constructions, which I am unable to solve completely at the moment. The problem is the folowing: let $\zeta$ be a primitive elevent ...
2
votes
1answer
69 views

Construct a triangle given its circumscribed circle and the three points on it…

Construct a triangle given its circumscribed circle and the three points on it at which the altitude, the angle bisector and the median, drawn from the same vertex, intersect the circle. The problem ...
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0answers
16 views

Reference for principles regarding construction of geometric figures with fewest steps as possible.

Is there such guiding principle? I have played geometry construction games, and most of the time I do too many steps making it hard to proceed.
2
votes
1answer
42 views

Into a given circle, inscribe a triangle, given the sum of two of its sides and the angle opposite to one of them.

I'm stuck at this problem for hours. This is from a book from Kiselev's geometry.
1
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1answer
69 views

Geometric locus involving tangents of lines and circles that results in an ellipse.

In the left figure below the locus of the center of the circle tangent to a given circle and line is a parabola (yellow "curve"), while in the right is locus of the center of circle tangent to 2 ...
4
votes
1answer
701 views

Construct a triangle, given its angle at the vertex, and the altitude and the median drawn to the base.

The problem is from a book. Also it gives a hint: "Double the median extending it past the base, connect the endpoint the with the vertices at the base, and consider the parallelogram thus formed.". ...
4
votes
1answer
358 views

Can platonic solids be constructed using compass and straightedge?

By moving the concept of geometric construction into three dimensions, could one trace the 3D wireframe of any of the five platonic solids using only a compass and straightedge? If not, what ...
0
votes
2answers
377 views

Find a non-constructible algebraic number of degree $4$ over $\Bbb Q$

I cannot think of a non-constructible algebraic number of degree $4$ over $\Bbb Q$ so far. I wish if I can find such an example. Could some one tell me some such numbers with justification? Also I ...
2
votes
2answers
742 views

Difficult Geometric Construction

I've been trying to construct the following figure geometrically: I've been tearing my hair out all afternoon. Because of the irrational radius lengths of the circles, this problem is (at least to me)...
12
votes
6answers
2k views

Geometry Construction Problems

Recently I've been trying my hand at a few geometrical construction problems using just a straight edge and a compass. So far I have constructed the following: an equilateral triangle a square a ...
2
votes
2answers
134 views

Degree of Minimal polynomial of complex number and its components

I've been doing some research on the constructibility of regular polygons, and it led me to come up with the following conjecture: Edit (Sorry I should have imposed much stricter conditions): Let ...