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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1answer
56 views

Constructing a graph with radius two.

From cycles $C_n$, $n\geq6$, I was trying to form a new graph by adding a single vertex to $C_n$ so that the added vertex has eccentricity two and rest have three. I tried for $C_6$ and $C_7$ as given....
22
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1answer
362 views

How well-studied is origami field theory?

It's well known that angle trisection cannot be done with straightedge and compass alone, as Theorem 1. If $z \in \mathbb C$ is constructible with straightedge and compass from $\mathbb Q$, then $$...
1
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0answers
112 views

Given that $[ABC]$ : Area of small circle = $\frac{3\sqrt3}{4}$ : $\pi$. How many parts of area of small circle is inscribed in large circle? [closed]

In the common region of two circle, $\triangle ABC$ has been drawn with its maximum area such that the proportion of the maximum area of $\triangle ABC$ and the area of small circle is equal to $\frac{...
4
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1answer
65 views

Construct Triangle given bisectors and circumcircle

Suppose we have three concurrent lines $g,h,k$ in the Euclidean plane which meet at a point $P\in g\cap h\cap k.$ Moreover, let $K$ be some circle with center $P$ and some radius $r>0$. I would ...
1
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2answers
70 views

Constructing an equilateral triangle with a given segment as a side, using a compass whose radius is less than the length of the segment

Say I have a line segment $AB$. I have a compass that can only create a circle with some random radius $r$ that is less than the length of $AB$. (I also have a straight-edge to create lines of ...
1
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1answer
76 views

proof for construction of 60 degree angle

I have a construction from the game Euclidea, puzzle 4.2: The puzzle is given point $A$ and line $\overleftrightarrow{BC}$ (just the line -- neither point is given), construct a 60 degree angle with ...
2
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1answer
53 views

Identifying the square $DEFG$ and than finding the value of its perimeter.

Let $ABC$ be a triangle and $DEFG$ be a square, where $D, E$ points are located on $AB$ and $AC$ or their extension line. $F, G$ points are located on $BC$ or the extension of $BC$. The perpendicular ...
0
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1answer
127 views

How to double the circle?

I'm looking for a compass-and-straightedge method to construct a circle that has area twice of the area of another circle, with no prior knowledge of π, without knowledge of the formula for the area ...
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0answers
20 views

Use proven constructions to derive a DFSA.

$$ M1 = < \{A,B,C\}, \{a\}, \{(A, a)\} \to B, (A, a) \to C\}, A, \{B\} >$$ Assume that $T(M1) = {a}$. Use proven constructions to derive a DFSA, $M2$, from $M1$ such that $T(M2) = T(M1)$. My ...
6
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1answer
308 views

Is classical Euclidean geometry Turing complete?

By "classical Euclidean geometry" I don't mean the study of $\mathbb R^2$ with the Euclidean metric, or a model of Hilbert's Grundlagen axioms, or the study of those properties of $\mathbb R^2$ that ...
0
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1answer
38 views

What kind of geometric constructions require marking of a unit length?

Wondering if there is a rule to determine if a geometric construction or question requires marking a unit length. For example, constructing a square root length or product length (a*b) requires unit ...
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0answers
37 views

Show that every point in the interior of one circle is the orthocentre of another triangle inscribed in another circle

Let $C_1$ and $C_2$ be two circles in the plane with radius R and 3R respectively. Show that every point in the interior of $C_2$ is the orthocentre of some triangle inscribed in $C_1$. I gave a ...
0
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1answer
55 views

Question on given compass and ruler construction definition wrt. angle bisection

I am trying to understand chapter 4 in John M. Howie's book called "Fields and Galois Theory" (published by Springer). In chapter 4 regarding geometric constructions he gives the following definition: ...
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3answers
73 views

what geomatric series formula is used?

trying to follow a solution related to geometric series, but not sure what formula is used here. Any pointer is appreciated.Image here, can't embed image yet. I do try to plug in the Sn formula, but ...
2
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0answers
60 views

Find the line that is closest to 4 skew lines

If I have 4 skew lines in $\mathbb{R}^3$, how can I find the line $L_c$, that is closest to all of them? I know that with 3 skew lines, there is always a line that intersects all of them, in fact ...
2
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2answers
86 views

Prove that the following equation has no constructible solutions.

Prove that the following equation has no constructible solution: $\ x^3 - 6x + 2\sqrt{\pi} = 0$ The way I am trying to approach is that: I want to transform the equation into some integer ...
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0answers
119 views

Galois theory: Gauss-Wantzel theorem, proof explanation

I am using Ian Stewart Galois theory book and it says that for $A = $ primitive $p^2$ root of unity $A$ has min poly of $m(t)= 1+ t^p +.....+t^{p(p-1)}$ and so $p(p-1)$ is a power of two. why is ...
1
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1answer
70 views

How does one verify a ruler-compass construction is valid?

I happened upon this paper by Ramanujan in which he gives an approximation for the side length of a square with area nearly equal to that of a given circle. I don't have much experience with ...
1
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1answer
208 views

Inscribe an equilateral triangle inside a triangle

Given a triangle ΔABC, how to draw all possible inscribed equilateral triangles with given side whose vertices lie on different sides of ΔABC?
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0answers
39 views

How do I determine if a complex number $w = f(z_1,\,z_2,\,z_3)$ is a known triangle center?

I've found a number of closely related functions, each of which takes three complex numbers $z_1,\,z_2$, and $z_3$ (which we can consider as the vertices of a triangle) as its arguments, and outputs a ...
1
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1answer
115 views

How to construct a triangle with $BC=7.5$ cm. $\angle ABC$=$60$° and $AC-AB=1.5$ cm.

How to construct a triangle with $BC=7.5$ cm. $\angle ABC$=$60$° and $AC-AB=1.5$ cm. At first I constructed $BC$ then $\angle ABC$ ,but I don't know what to do next. Please help me.
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1answer
369 views

Proof of Construction of regular pentagon by using compass and straightedge.

How to prove that the polygon constructed by the method mentioned in the following link is indeed a regular pentagon? Constructing a Regular Pentagon (Video on YouTube)
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2answers
113 views

A conjecture about the sum of the areas of $3$ triangles built on the sides of any triangle (by means of centroid/orthocenter) [closed]

Given any triangle $\triangle ABC$, let us draw its orthocenter $D$. By means of this point, we can draw three circles with centers in $A,B,C$ and passing through $D$. These circles intersect in the ...
1
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2answers
112 views

Approximation of the quadratic formula with straightedge and compass

Given a directrix and a focus (blue), we can define a parabola as illustrated below. We suppose the parabola intersecting the $x$-axis in correspondence of the red dots. We draw the line ...
9
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4answers
990 views

Proof without words of a simple conjecture about any triangle

Given the midpoint (or centroid) $D$ of any triangle $\triangle ABC$, we build three squares on the three segments connecting $D$ with the three vertices. Then, we consider the centers $K,L,M$ of the ...
22
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2answers
1k views

A conjecture about the sum of the areas of three triangles built on the sides of any given triangle

Given any triangle $\triangle ABC$, and given one of its side, we can draw two lines perpendicular to that side passing through its two vertices. If we do this construction for each side, we obtain ...
9
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1answer
128 views

A simple conjecture (and a question) about three parabolas related to any triangle

Given any triangle, we can build three parabolas, each with focus on one vertex and with directrix the opposing side, as illustrated here: My first conjecture, likely trivial, is that, given any ...
0
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1answer
24 views

Triangle Construction knowing only parts of 2 sides

I have the following triange construction: The values for $\alpha$, $d$, $a$ and $b$ are known: I am trying to calculate the angle $\phi$, depending only on $\alpha$, $d$, $a$ and $b$. To archive ...
9
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2answers
206 views

A conjecture related to any triangle

Given one side $AC$ of any triangle $\triangle ABC$, we can draw the couple of circles with center in $A$ and passing through $C$ and with center in $C$ and passing through $A$, obtaining two points $...
4
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1answer
87 views

A novel (?) construction of the regular pentagon with straightedge and compass

With reference to the triangle $\triangle ABC$ illustrated in the picture below, given the side $AC$, the five points $B,D,E,F,G$, in the conditions discussed here, determine a circle (red). Let us ...
3
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2answers
109 views

Geometric proof of equivalence between two constructs of ellipse

Pretending that we don't know any analytic geometry and trigonometry. Consider the following two constructs of an ellipse, where admittedly the second one is an ad-hoc construct for the ellipse ...
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2answers
161 views

Increasing (or changing) the eccentricity of a vertex in a given graph.

I considered a graph, path $P_8$ and added two more vertices such that eccentricity of two vertices is three and rest of the vertices have eccentricity four, and $P_8$ is induced in the new graph. I ...
1
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2answers
99 views

Show a regular n-sided polygon is constructible, using only ruler and compasses, iff the number $\alpha = 2 \cos(2\pi/n)$ is constructible.

Let $n > 2$. Show that it is possible to construct a regular n-sided polygon, using only ruler and compasses, if and only if the number $\alpha = 2 \cos\left(\frac{2\pi}{n}\right) = \zeta_n + \...
2
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1answer
72 views

An interesting (conjectural) property of any triangle

Given any triangle $\triangle ABC$, we can always build three ellipses, each of them having foci in two of the vertices and passing through the third one, as shown in the following picture: In ...
3
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2answers
166 views

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
106 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 ...
9
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2answers
179 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 ...
1
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0answers
37 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 ...
11
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1answer
1k 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 $BC$...
0
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1answer
104 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 $5$...
2
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2answers
127 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 ...
2
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2answers
299 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 ...
4
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0answers
166 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+...
0
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1answer
129 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 ...
3
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3answers
75 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 ...
1
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1answer
50 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
92 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 ...
0
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0answers
73 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 ...
22
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2answers
2k views

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 ...
3
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2answers
225 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 ...