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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Construct any regular polygon that has the same area as the sum of $n$ given triangles

Original question: Construct any regular (or similar-scaled to a given) geometric shape that has the same area as given triangle? My idea is application of generalized Pythagora's theorem. Euclid ...
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Can $\pi$ be approximated by considering polygons with increasing number of sides, but without using circles or trigonometry?

Question in title. Although it should say “regular polygons”, not just “polygons”. When I say "without using circles", I mean without circle constructions. Properties like perimeter and area and ...
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What's the fastest way to split $45-45-90$ triangle into two parts with equal area (using a straightedge and compass)?

What's the fastest way to split $45-45-90$ triangle into two parts with equal area (using a straightedge and compass)? I know a similar question has been asked before, but I'm asking how to do this ...
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Straightedge-and-compass construction of the “kissing circles” for three given circles

Let $C_1,C_2,C_3$ be three mutually tangent circles. Call the circle tangent to all of them (that is, intersecting each at one point) and enclosed within the region between them their kissing circle. ...
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Construct a circle tangent to sides $BC$ and $CD$ and s.t. its meetings with the diagonal $BD$ are tangent points from tangents draw from point $A$

Given square $ABCD$ I want to construct (with ruler and compass) the circle in the interior of the square such that it is tangent to sides $BC$ and $CD$ and such that it's meetings with the diagonal $...
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Is the set of constructible numbers in $\mathbb{C}$ algebraic extension of $\mathbb{Q}$?

Is the set of constructible numbers in $\mathbb{C}$ algebraic extension of $\mathbb{Q}$? I tried assuming that $z=a+ib$ and I am left to show that $ib$ is algebraic over $\mathbb{Q}$ whenever $b$ is. ...
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Divide any given angle into $n$ equal parts using paper folding

I have known that it is possible to trisect an arbitrary angle by Hisashi Abe's method. Of course, it is easy to divide an angle into two or four equal parts using paper folding. My question is ...
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Trisecting an angle with a compass and 2 marks on a ruler

It's well known that there is no possibility to trisect and angle with a compass and a ruler. But there is such a procedure when the ruler has 2 marks on it. See the snippet below. It's not very ...
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how to solve apollonius problem( $CCC$ case) with compass and straight edge

Given three circles, how to construct another circle that is tangent to the three circles with compass and straight edge? Well, this problem can be reduced to ($PCC$ case) by shrinking the smallest ...
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I need an explanation of the end of the problem [closed]

All I need is to know how they get to pi / 6 (what's in the circle) note:sen x =sin x note: it is a single problem that uses trigonometry along with its development [problem][2] ...
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Prove roots of $x^5 - 1$ are constructible

I am trying to show some other result, and by reducing it to this problem I should be able to finish the proof, I am trying to show that $z = e ^ {\frac{2}5 \pi i} $ is constructible (In the sense ...
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Classification of All Constructible Angles

I am familiar with the famous Gauss-Wantzel Theorem which states that a polygon is constructible with compass and straightedge if and only its number of sides is a power of two multiplied by a product ...
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Prove that only fields involving square roots are possible with ruler and compass and find fields that are possible with other conic sections.

Around the 16:40 mark of [1], Mathologer tries to convince us that only "square-rooty" fields can be achieved with ruler and compass. He first shows that any field which involves combining rational ...
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Find $\angle DCF$.

Let $ABC$ an isoscel triangle with $AB=AC$ with $\angle{BAC}=100^{\circ}$ and $D\in (BC)$ s.t. $AC=DC$ and $F$ on $(AB)$ with $DF||AC$. Find $\angle DCF$. I tried a lot of constructions, to ...
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Construction of angle

Let $ABCD$ be the parallelogram and $P$ be the point lie on the perpendicular bisector of $AB$ such that $\angle{PBA}=\angle{PDA}$. What is the trick to construct this?
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How to find the focus of a Cassini oval?

You are given some Cassini oval with foci somewhere on the x-axis. How to construct the foci using a compass and straightedge if only the coordinate axes and the oval are given? The foci are the same ...
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Is it possible to develop this image generator from randomly redrawn detected edges or am I going straight to the wall?

Aim I would want to procedurally generate totally new images from a set of thousands of images of a given category (simple landscapes for example). Deep-learning won't be used for this purpose. This ...
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How to construct a tangent to a hyperbola that is parallel to a given line? [closed]

You are given a hyperbola $h$, its asymptotes and its foci. You are also given some line $p$. Construct the line(s) tangent to $h$ and parallel to $p$. This problem came up while I was doing ...
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How can I construct a regular 19-gon using two angle trisections?

I'm stuck in how can I construct a regular 19-sided polygon with angle trisection allowed. My interest came from the article "Angle Trisection, the Heptagon, and the Triskaidecagon", published in the ...
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In search of an algebraic number whose degree is a power of 2, yet not constructible [duplicate]

I read on MO & MSE, but still I couldn't understand anything. Hence I am asking a new question explaining the stuffs I know I want to find an algebraic number of degree $4$, which is not ...
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How to show $\sqrt[17]{11}$ is not constructible (through ruler and compass construction)

I understand that for a number to be constructible it must be a membeer of the field of constructible numbers $C$ such that $ 0,1 \in C $ and which is closed under addition, subtraction, division, ...
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How to construct an equilateral triangle on 2 concentric circles

Construct an equilateral triangle with the given vertex so that the other vertices lie on the concentric circles respectively. I constructed the triangle, but I don't know how it works. How does ...
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Is Cairo pentagonal tiling belong to pentagonal tilings type 8?

I am interested in Cairo pentagonal tiling. In following link of wikipedia: https://en.wikipedia.org/wiki/Cairo_pentagonal_tiling It claims "The Cairo pentagonal tiling has two lower symmetry forms ...
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Construct a chord equal to the radius with compass and straight edge.

A circle is given and it has a point inside the circle. How to construct a chord that passes through the point and equal to the radius of the circle with compass and straight edge? I know how to ...
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How to proof that distinct “recursive geometrical constructions” converge to the same object?

It seems like a topological isomorphism problem ... I'm starting with an "intuitive isomorphism recognition" of 3 objects, which are defined by geometric construction. I'm looking for a formalization ...
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Origami-constructible numbers

I apologize if this question has been asked here already; I am wondering whether it is known precisely what is the class of origami-constructible numbers? The class of compass-and-straightedge ...
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How to reconstruct a quadrilateral ABCD only using compass and straight edge?

Reconstruct a quadrilateral ABCD given length of its sides and the length of the midline between the first and third sides (namely all the segments drawn in the given figure) using compass and ...
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Show that no root of the polynomial $x^5 + 21x^4 - 14x^3 + 28x^2 - 7x + 42$ is constructible.

Show that no root of the polynomial $$x^5 + 21x^4 - 14x^3 + 28x^2 - 7x + 42$$ is constructible. Is it enough to say the degree is 5 is not a power of 2?
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Is there a way to determine which angles, $3\theta$, are trisectable into $\theta$? Even when the angle $3\theta$ isn't constructible to begin with?

Is there a way to determine which angles, $3\theta$, are trisectable into $\theta$? Even when the angle $3\theta$ isn't itself constructible to begin with?
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Does angle trisection imply cube root construction and vice versa?

Im curious to know what constructions become possible with the inclusion of angle trisection. Including the construction of heptagons and other polygons, and by how much do the constructible numbers ...
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Can pentagons that are known to tile the plane be ruler and compass construction?

There are 15 types of convex pentagons are known to tile the plane monohedrally. https://en.wikipedia.org/wiki/Pentagonal_tiling I am wondering if all these 15 types pentagons are ruler and compass ...
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Construction of a Simson Line Passing Through a Given Point, and Other Related Questions

Update. Part (a) is somewhat solved (i.e., if I blindly believe the paper I have not yet fully digested). Part (c) is completely solved by timon92. Part (b) is competely solved. There are also two ...
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how to construct an equilateral triangle whose one vertex is given and other two vertices lie on two parallel lines

a vertex and two parallel lines are given. construct an equilateral triangle whose other vertices lie on two parallel lines this is a question from the app euclidia. i tried to make equilateral ...
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Finding an unknown angle (some constructions needed).

Any ideas how I could find the size of $\angle CBD$ in the diagram given that AC=AD, $\angle CAB=6$, $\angle CBA=48$ and $\angle DAC=12$. I think there should be a way to do it with basic geometry ...
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Triangle construction given semiperimeter and radii of inscribed and circumscribed circles.

Given $$ \rho,r,R $$ of a scalene triangle how to construct it geometrically, say when side $c$ is parallel to $x-$ axis... with no Ruler/Compass restriction?
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Constructing a point on a geodesic line in the hyperbolic disc with a given distance from another point

Given a geodesic line $L$ in the Poincaré disc $\{ (x,y) \in \mathbb{R}^2 : |x|^2 + |y|^2 < 1 \}$ with the hyperbolic metric and a point $P \in L$, how can one construct a point $Q \in L$ that has ...
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Constructing an equilateral triangle of a given side length inscribed in a given triangle

I am trying to solve the problem of constructing, with straightedge and compass, an equilateral triangle of given side length $a$ inscribed in a given triangle. I found this post "Inscribe an ...
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Given a Pentagon, Construct a Parallelogram Equal in Area

Euclid claims in I.45 of his Elements to show how to "construct a parallelogram equal to a given rectilinear figure in a given rectilinear angle." In modern terms, he is saying that he will show how ...
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A polynomial relation between the lengths of segments that are constructible with straightedge and compass.

I have read a theorem about the possibility of contructing a segment of given lenght with straightedge and compass. I didn't well understand the theorem, can we use it to deduce the following (...
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Secant chord on intersecting circles construction

"The secant circles $Γ_1(O_1,R_1)$ and $Γ_2(O_2,R_2)$ intersect at points $A$ and $B$. Given a line of lenght $l$, explain how to construct a straight line passing through $A$ intersecting $Γ_1$ and $...
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The generality of “neusis plus”

Consider the following progression of facts relating to geometric construction with a limited set of tools: With only a compass and unmarked straightedge, we can only construct numbers that lie in a ...
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Rationally bracing a rigid regular nonagon

My previous questions on rigid pentagons and heptagons are part of a pet project of mine to make (aka brace) rigid regular polygons using only rational-length sticks. To recap: Hinges can be placed ...
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Geometry problem - proving atleast three lines are concurrent! [closed]

Each of the given $9$ lines cuts a given square into two quadrilateral,whose areas are in ratio $2:3$. Prove that at least three of these lines pass through the same point. how to approach this ...
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Proving correctness of a braced regular heptagon from a trigonometric identity

This is a rigid regular heptagon I found on Wikipedia during associated research for my question on rigid pentagons: The accompanying text reads The construction includes two isosceles triangles ...
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The power of Archimedean spirals: is there an algebraic characterization of Archimedean numbers?

In his famous treatise On spirals, Archimedean used a spiral to square the circle and trisect an angle. There are known algebraic characterizations of the numbers constructed with compass and ...
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Rigid pentagons and rational solutions of $s^4+s^3+s^2+s+1=y^2$

Gerard 't Hooft, Nobel Prize in Physics laureate, wrote three articles on what he called "Meccano math" (1, 2, 3) – rigid constructions following rules quite similar to my earlier question on doubling ...
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What is the length the side of of a square inscribed in a triangle?

What is the length of the side of a square inscribed in a triangle? This was inspired by this Numberphile video which showed multiple ways to construct the square with a side on one side of an acute ...
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What is a geometric interpretation of $0x+0y=0$

What is a geometric interpretation of $$0x+0y=0.$$ I would think of this as a plane... Am I right?
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Is There Any method To construct/Take 1 cm reading to compass without a scale and only using compass.

Hi sorry if I asked this Q but I was wondering was there any method to take 1 cm or any reading to the compass(without the help of ruler) by just using compass itself/properties of circles.Sorry ones ...

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