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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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How to understand the Conway recipe C969qD to construct this polyhedron? Canonicalization then quinto?

The answer to my previous question about the shape below is the Conway notation C969qD. Per the linked viewer in that answer: The specification consists of a ...
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1answer
14 views

Constructing a quadrilateral using the method of translation - Kiselev

Ending the fourteenth chapter of Planimetry is the following construction: Why is this a construction, when the quadrilateral in question is already given.
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Ways of geometrical multiplication

There are at least five ways to multiply two natural numbers $a$ and $b$ given as integer points $A$ and $B$ on the number line by geometrical means. Two of them include counting, the others are ...
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1answer
33 views

If a unit cube can be rounded off using ruler and compasses only, then $r= (3/2\pi)^{1/2}$ is constructible

Show that if a unit cube can be rounded off using ruler and compasses only, then r= $(3/2\pi)^{1/2}$ is constructible. Show that $r$ is transcendental over $\mathbb Q$ I am not that familiar with ...
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How to construct a normal to side $AB$ of acute triangle so that it halves it area?

How to construct a normal to side $AB$ of acute triangle so that it halves it area? So we have $${c\cdot v\over 2} = 2{x\cdot v' \over 2} \implies {v\over v'} = {2x\over c}$$ From similar ...
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1answer
51 views

Drawing a figure in advanced geometry.

I was not able to draw figure for a question. Question was From a point of intersection of two circles, the lines to the centers of similitude bisect the angles between the radii of the circles. ...
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1answer
87 views

Is there a simple perfect squaring of a 1366 by 768 rectangle?

So, a simple perfect squaring of a rectangle is a tiling of that rectangle by squares whose side lengths are all distinct integers. Additionally, not subset of the squares must form a smaller ...
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2answers
58 views

Finding the least possible value of perimeter of $\triangle ABC$ with given ranges of angle

In $\triangle ABC$,$\angle A >2\angle B$ and $\angle C > 90^\circ$. If the length of all side of triangle $\triangle ABC$ are positive integers, then what is the least possible value of ...
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2answers
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Triangular geometric construction with 2 medians and 1 side

Using only a straight edge and compass, geometrically construct a triangle from the three given segments representing the base (side $c$), and the medians to the other two sides ($m_a$ and $m_b$) I'...
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3answers
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Compute $m ( \angle ACD $).

Let $\triangle ABC $ s.t $m (\angle A)=100^{°}, m (\angle B)=20^{°} $. Let $D\in Int (\triangle ABC) $ s.t. $m (\angle BAD)=30^{°} $ and $[BD $ is the bisector of $\angle B $. Compute $m ( \angle ACD ...
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Given three non-overlapping circles, find the triangle of minimum perimeter with one vertex on each circle

G. Polya "Mathematics and plausible reasoning" Chapter 9, problem 2: Three circles in a plane, exterior to each other, are given in position. Find the triangle with minimum perimeter that has one ...
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Are there two non-congruent quadrilaterals with same sets of sides and angles?

Are there two non-congruent quadrilaterals with same sets of sides and angles? It is relatively easy to construct such pentagons. Trying to construct quadrilaterals allows for seemingly multiple ...
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1answer
38 views

Given a compass + straightedge, construct convex quadrilateral tangent to circumcircle

Given a compass, a straightedge, and a convex quadrilateral $ABCD$, construct a point $X$ such that the circumcircle of $\triangle{AXB}$ is tangent to the circumcircle of $\triangle{CXD}$, and the ...
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5answers
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“Given three parallel straight lines. Construct a square three of whose vertices belong to these lines.” Are all three lines required?

Given three parallel straight lines. Construct a square three of whose vertices belong to these lines. What does "belongs" mean in the context of this question? Do the three lines have to be used ...
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1answer
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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....
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299 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 $$...
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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{...
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1answer
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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 ...
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2answers
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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 ...
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1answer
43 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 ...
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Construction of linear space of dimension $n$

1. Let $n$ be an arbitrary cardinal number. Construct a (real or complex) linear space of dimension $n$
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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 ...
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1answer
53 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
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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 ...
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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 ...
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1answer
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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
36 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 ...
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1answer
38 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
67 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 ...
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0answers
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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 ...
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2answers
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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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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 ...
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1answer
35 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 ...
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1answer
103 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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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 ...
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1answer
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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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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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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 ...
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2answers
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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 ...
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956 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 ...
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2answers
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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 ...
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1answer
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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 ...
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1answer
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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 ...
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2answers
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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 $...
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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 ...
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2answers
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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
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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 ...
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2answers
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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 + \...
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1answer
68 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 ...
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2answers
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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 $E$. ...