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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.

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A conjecture involving three parabolas intrinsically bound to any triangle

Given any triangle $\triangle ABC$, we can build the parabola with directrix passing through the side $AB$ and focus in $C$. This curve intersects the other two sides in the points $D$ and $E$. ...
2
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1answer
58 views

An ellipse intrinsically bound to any triangle

Given any triangle $\triangle ABC$, we build the hyperbole with foci in $A$ and $B$ and passing through $C$. The hyperbole always intersects the side of the triangle that is opposite to the vertex ...
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2answers
116 views

A conjecture about the intersections of three hyperboles related to any triangle

Given any triangle $\triangle ABC$, we build the hyperbole with foci in $A$ and $B$ and passing through $C$. Similarly, we can build other two hyperboles, one with foci in $A$ and $C$ and passing ...
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0answers
26 views

Finding the region surrounded by light emitted from the vertices of a triangle through construction

P, Q and R represent the positions of three radio beacons. P is 450 km from Q, Q is 475 km from R, and R is 300 km P. Signals from P have a range of 300 km, Q has a range of 350 km and R has a range ...
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1answer
395 views

How to construct a square equal to a given triangle.

I have a triangle $ABC$ and I want to construct a square of the same area as that of the triangle using ruler and compass. Consider the following image. I first locate the mid-points of $AB$ and $BC$...
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1answer
95 views

Changing the eccentricity of a vertex from $5$ to $4$.

Can anyone help me in changing the eccentricity of any vertex from $5$ to $4$, lying on the outer circle, so that only one vertex is with eccentricity $4$ and rest of the vertices have eccentricity $5$...
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2answers
57 views

Draw a Square Without a Compass, Only a Straightedge — Part Deux

So, I previously asked the question Draw a Square Without a Compass, Only a Straightedge. From the comments and answers, it appears that that question is not solvable. Given that the question I ...
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2answers
166 views

Draw a Square Without a Compass, Only a Straightedge

I remember seeing the following question in an old STEP question: using only a straight-edge and a set of (unmarked) coordinate axes, construct a square. I'm sure I knew how to do it when I was ...
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0answers
149 views

Constructional proof of ellipse property

I came across the fact that the following function defines a family of ellipses with focal distance $f$, parameterized by the value of the function: $$\frac{x-f+\sqrt{(x-f)^2+y^2}}{x+f+\sqrt{(x+f)^2+...
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1answer
33 views

If $a$ and $b$ are constructible numbers and $a ≥ b > 0$, give a geometric proof that $a + b$ and $a - b$ are constructible.

This problem was taken from Joseph Gallian's "Contemporary Abstract Algebra", 8th edition. Chapter 23, Exercise 1, Page 402: If $a$ and $b$ are constructible numbers and $a ≥ b > 0$, give a ...
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2answers
46 views

Construct a perpendicular in confined space?

About 45 years ago, I learned in school how to construct perpendicular lines with straight-edge and compasses, and I remember being taught two techniques (with variations on one). I recall how to ...
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1answer
34 views

On the intersections between ellipses whose foci are the vertices of any triangle

Given any triangle $\triangle ABC$, we can draw two ellipses, one with foci in $A,B$ and passing by $C$, and one with foci in $C,B$ and passing by $A$. We always obtain the points $D,E$, where these ...
1
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1answer
84 views

Construction of a graph with required number of vertices.

I am trying to construct a graph using $K_3$ graph. The graph $G$, obtained by adding vertices to $K_3$, should contain only one vertex with eccentricity two and the rest of the vertices with ...
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0answers
41 views

Squaring the circle, compass-and-straightedge construction

I have to proof that is not possible with compass-and-straightedge to construct a square which has surface equal to a disk. Let $M\subset \mathbb{C}$ with $\{0, 1\} \subset M$ and let $\cal{M}$ the ...
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2answers
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Elliptical version of Pythagoras’ Theorem?

Consider any right triangle $\triangle ABC$. We focus on one side, $AC$, and we take the midpoint $E$ of this side. Then, we draw the circle with center in $E$ and passing by $A,C$. If we take the ...
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2answers
64 views

Construction of arbitrary regular polygons with ruler and compass

A lot of arithmetic geometry has to do and took off with constructions or proofs of non-constructibility of specific figures only with ruler and compass. See e.g. the non-constructibility of the ...
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3answers
77 views

Geometric construction of a triangle, provided an angle, an internal angular bisector of this angle, and the length of the side opposite to this angle [closed]

How can I construct a triangle knowing the following information below? A) An angle. B) the length of the internal bisector for the given angle. C) the length of the side opposite to the given ...
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1answer
22 views

Construct circle internally tangent to a larger circle, and tangent to a point on a chord of the larger circle

Given a larger circle $O$ and its chord $AB$, construct circle $P$ that is internally tangent to $O$ and and tangent to point $C$ on $AB$. The chord and the point on it are completely arbitrary. ...
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0answers
36 views

Constructing or locating a point in a triangle.

Construct a point O in $\Delta ABC$ such that $OA+OC+OB$ value is minimum. Given that it is acute angled. I think $OA=OB=OC$ is the case where we get it's minimum value. But i don't know how to find ...
3
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2answers
93 views

A conjecture about an intrinsic similarity of non-isosceles triangles

Given any non-isosceles triangle $\triangle ABC$, and denoting $AB$ its longest side, the following construction determines the points $DFGE$. In this post is shown that the points $DFCGE$ always ...
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1answer
51 views

A Robbins Pentagon bound to any (non-isosceles) Integer Triangle?

Given any non-isosceles triangle $\triangle ABC$, and denoting $AB$ its longest side, the following construction determines the points $DFGE$ (see this post for details). My conjecture is that if ...
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1answer
41 views

Is there a geometric interpretation for the geometric mean of multiple numbers?

Given a list of $n$ nonnegative real numbers $a_1, a_2, \dots, a_n$, the geometric mean of that list is defined to be $$\sqrt[n]{a_1a_2\cdots a_n}.$$ In the case of $n=2$, there are a few standard "...
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1answer
61 views

A basic relation intrinsic to any (non-isosceles) triangle

Given any non-isosceles triangle $ABC$, let denote with $AB$ its longest sides, and draw the two circles with centers in $A$ and $B$ and passing by $C$. They determine two additional points $E$ and $...
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1answer
65 views

Locus of a vertex of a triangle inside an equilateral one, under an integer constraint

Given an equilateral triangle $ABC$, we choose a point $D$ inside it, determining a new triangle $ADB$. We draw the circles with centers in $A$ and in $B$ passing by $D$, determining the new points $...
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1answer
46 views

Drawing a tangent to a circle at a given point in just 3 ruler-and-compass constructions

A friend of mine recently introduced me to an interesting app called Euclidea that challenges you to complete geometric constructions like the ones we learnt in high school. The app has successfully ...
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0answers
29 views

Subdivide a circle according to a subdivided segment

Given a segment $AB$, subdivided in $m$ sub-segments, I have to draw a circumference $C$, subdivided in $m$ arcs, in such a way that the ratios between the lengths of the sub-segments of $AB$ are the ...
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0answers
52 views

Constraints on conical coffee cup constructions of cardioids & catacaustics

The Mathologer video Times Tables, Mandelbrot and the Heart of Mathematics discusses several relationships. For the n=2 and 3 cases, the cardiod and catacaustic (or nephroid per @Rahul's comment) ...
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2answers
39 views

Given two perpendicular bisectors and a vertex, trace the triangle.

Given two lines $l_1$ and $l_2$ that are not parallel and a point $P$ that is exterior to both of them, trace the triangle with one vertex on $P$ that has $l_1$ and $l_2$ as perpendicular bisectors of ...
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0answers
49 views

Given $\overline{AB}=c$, $\overline{AC}=b$, $m_a$, construct the triangle $ABC$?

Given $\overline{AB}=c$, $\overline{AC}=b$, $m_a$, construct the triangle $ABC$. I did the following: And then used the compass to draw the distances $c,b$: I've drawn the paralelogram: ...
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1answer
53 views

Paraphrasing linear maps as “lines to lines” without coordinates

I taught a Euclidean geometry class last year, which I thought was a dream. It is a first course in proofs, yet is not axiomatic... something with which I'm still wrestling. This summer, I'm trying ...
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3answers
47 views

Geometry: Bisection of an angle and constructing a right angle

When bisecting an angle, some texts suggest that that the arcs drawn from $D$ and $E$ and intersecting at $C$ should be of the same radius as that of the arc drawn from $O$ but it is not necessary. ...
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0answers
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Geometry: Division of a line segment in a given ratio

In geometric construction while dividing a line segment in a given ratio say $m:n$, I have come across the texts that a ray should be drawn making an acute angle with the given line segment. I'm not ...
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1answer
48 views

Model a process of selecting boxes

I am considering the following process: There are $m$ boxes in total and $n$ boxes are red and $0\leq n\leq m$. $m$ and $n$ are all very large. Now we have $K$ (very large) people, where $0\leq K\leq ...
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3answers
71 views

Compass and straightedge construction of a square of an arbitrary line segment

If I have some arbitrary length line $AB$ and a unit length line $CD$, how can I construct a line whose length is equal to the square of the length of line $AB$ using a compass and a straightedge?
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427 views

On The Construction Of An Ellipse

You know how when you construct an ellipse, you take a rope, fix it to 2 points, and stretch that rope? When the rope is being stretched, let's call the part of the string attached to the first ...
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2answers
53 views

On The Specifics Deriving The Equation Of Ellipse

I am trying to learn how to derive the equation of an ellipse, from this website (https://people.richland.edu/james/lecture/m116/conics/elldef.html). I am struggling, however, to prove to myself why ...
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1answer
57 views

Creating a Straightedge, Compass and “New Tool” problem.

I've been trying to create a problem that involves similar concepts as straightedge and compass construction. The idea is to add a new tool that allows for a different type of graph to be drawn ...
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1answer
43 views

Constructions: A straight line segment of length pi units. [duplicate]

A line segment of length 22/7 units or 3.14 units can be drawn. But how can a line segment be drawn of exactly pi units?
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1answer
41 views

Largest circle contained in a region delimited by 4 circumferences

I'm working in a region delimited by 4 circumferences, concentric 2 to 2, with opposite centers (shaded area) which vary with respect to a parameter $\alpha$. $$r<(x-a(\alpha))^2+y^2<R$$ $$r&...
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2answers
78 views

How to find the exact value of $\cos(\frac{2\pi}{17})$ with WolframAlpha?

I tried to find the exact value of $\cos(\frac{2\pi}{17})$ with WolframAlpha but only obtained a decimal approximation. Is there any way to find this exact value with WolframAlpha?
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1answer
18 views

Geometric construction to divide a segment

Given a segment AB, I would like to construct using only straightedge and compass, a point C on the segment AB such that $\frac{AC}{CB}$ is equal to $\frac{\phi}{2}$, where $\phi$ is the golden ratio, ...
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5answers
217 views

Where is the hole in this argument asserting the constructibility of all regular polygons?

Some engineers have a so-called "general" method for constructing any (regular) polygon with the classical instruments only, given the length of its side (they may recognise that it appears to be ...
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2answers
52 views

Prove the regular 12-gon is constructible.

Prove, both geometrically and then algebraically, that the regular 12-gon is contructible. I'm pretty stuck on this one and trying to get my head around constructibility, so far I've seen that ...
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1answer
38 views

Constructing a triangle between an angle with an arbitrary centroid.

Let $\overrightarrow {OA}$ and $\overrightarrow {OB}$ be two ray with common end point $O$. Let $G$ be a point lying in the interior of the $\angle AOB$. Construct a $\triangle OCD$ such that the ...
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0answers
15 views

The answer given at 3d-models-of-the-unfoldings-of-the-hypercube

At 3d-models-of-the-unfoldings-of-the-hypercube not all of the 261 graphics show all 8 of the cubic faces that make up each net. I am not yet well versed in Mathematica to be able to extract the ...
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1answer
702 views

Prime Numbers and Architecture

Prime Numbers are widely used in technology and cryptography. They are sometimes also used in small scales such as building gears and evolution of life cycle of Cicada insect. Are they in any way used ...
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3answers
48 views

How/when do we use circle inversions to solve problems?

Given an angle AOB and a point M inside it, construct a segment PQ such that M is the midpoint of PQ P is on side OA Q is on side OB So i've been thinking about this problem and of course the best ...
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2answers
55 views

Constructing a right-angled triangle given the half-perimeter $s$ and an altitude $h_c$

I would like to receive some help about the next problem: Problem: Construct a right-angled triangle given the half-perimeter $s$ and an altitude $h_c$, where $\angle(BCA) = 90°$. What i did: 1) ...
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1answer
44 views

With edge and compass construction, given cubes of volumes $a^3,b^3$, can one construct a cube of volume $a^3+b^3$?

Suppose one has cubes A and B of volumes $r_A^3$ and $r_B^3$. Using only ruler and compass constructions, I need to determine whether it is possible to construct a cube of volume $r_A^3+r_B^3$. I ...
2
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1answer
56 views

How to derive coordinates of the vertices in the unit-distance embedding of the Golomb Graph?

With recent interest in the Hadwiger-Nelson problem on the chromatic number of the plane, thanks to de Grey's theorem that $CNP\ge5$, I've been looking at unit-distance embeddings of various graphs ...