Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
2answers
39 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 ...
0
votes
0answers
12 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 ...
2
votes
0answers
33 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
votes
1answer
30 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 ...
0
votes
0answers
21 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
votes
1answer
34 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: ...
0
votes
3answers
58 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
votes
0answers
42 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
votes
2answers
34 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 ...
0
votes
0answers
35 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
vote
1answer
25 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
vote
1answer
74 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?
0
votes
0answers
34 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 ...
0
votes
1answer
50 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.
-2
votes
1answer
44 views

Geometry - 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. https://www.youtube.com/watch?v=9VceAA2qBPA
-1
votes
2answers
101 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
vote
2answers
97 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
votes
4answers
947 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
votes
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
votes
1answer
126 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
votes
1answer
18 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
votes
2answers
189 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 $...
3
votes
1answer
70 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
votes
2answers
53 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 ...
1
vote
2answers
146 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 ...
0
votes
2answers
59 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 + \...
1
vote
1answer
63 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
votes
2answers
135 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
votes
1answer
95 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
votes
2answers
151 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
vote
0answers
31 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 ...
10
votes
1answer
577 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
votes
1answer
101 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
votes
2answers
76 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
votes
2answers
184 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
votes
0answers
154 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
votes
1answer
44 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
votes
3answers
58 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
vote
1answer
40 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
vote
1answer
86 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
votes
0answers
53 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
votes
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
votes
2answers
89 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 ...
0
votes
3answers
78 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 ...
2
votes
1answer
28 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. ...
0
votes
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
votes
2answers
98 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 ...
1
vote
1answer
52 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 ...
0
votes
1answer
43 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 "...
1
vote
1answer
65 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 $...