# 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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### Questions about the dot products of a unit cube

I want to find the dot products(It should have 16 dot products.) of a unit cube's diagonal lines. P.S. unit cube is a cube where all the lengths of 12 edges of the cube are 1. It should have 16 dot ...
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### Can you help me on the MPC theory Teminal ingredients?

Can you help me on the MPC theory Teminal ingredients ? more in particulat on the Ellipsoid constrained You can see the picture.
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### How to bisect an arc using only the compass

The Mohr-Mascheroni Theorem (and also here) states that any geometric construction that can be performed by a compass and straightedge can be performed by a compass alone. A key point in the proof of ...
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### Show: No polynomials in $\mathbb{Q}[X]$ satisfy $f(X)^3-f(X)+2=(X^4-7)*g(X)$

Prove: There exist no polynomials $f,g$ in $\mathbb{Q}[X]$ satisfying $$f(X)^3-f(X)+2=(X^4-7)*g(X)$$ As a hint I got: Consider the $4th$ root of $7$. So for $X=7^{1/4}$: $f(7^{1/4})^3-f(7^{1/4})+2=0$. ...
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### Construct a point with given ratio of distances to sides of an angle

We have an angle $\angle AOB$ and two segments of lengths $x$ and $y$. Construct a point $P$ inside $\angle AOB$ such that $$\frac{\mathrm{dist}(P, OA)}{\mathrm{dist}(P,OB)}=\frac{x}y.$$ My idea is ...
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I've done constructing the oval, but now I'm having trouble drawing a square on it. What should I do about this? (The three points given inside the oval are two focuses and the midpoint of the focuses....
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### Circle of a certain radius through a point which cuts a chord of a certain length on a line

I have found the following geometric construction exercise: Given a point $P$, a line $\ell$, a length $m$ and a radius $r$, draw the circumference with radius $r$ which goes through point P and ...
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### Philo's line construction

The Philo line is a line segment defined from an angle and a point inside the angle as the shortest line segment through the point that has its endpoints on the two sides of the angle. There are some ...
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1 vote
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### finding a point that has specific distances from 2 other points and a line using only a compass and straight-edge

Is there a way to find said point, which lies on half-plane, such that it has a distance of n from line L, n-m from point A, and n-2m from point B, with both points A and B located on the same half-...
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### Construct a circle with a specific radius on a sphere, under stereographic projection.

Given three poins $v_1,v_2,v_3$ on the sphere $S^2$, it is obvious that one can construct the circle with radius $|v_1v_2|$ around the point $v_3$, as seen in the following cartoon image: In this ...
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### Prove that if a prime $p = 2^k + 1$, it is a Fermat prime [duplicate]

So, I encountered this issue in the proof of the constructability of a regular n-gon, where it is shown that the n-gon is constructable iff it is a product of $2^k$ and distinct primes $p_i$ such that ...
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### Constructing a circle passing through a complex number $z$, its square roots, and $1$

Let $z\in\Bbb C\setminus\{0,1\}$, and let $\sqrt z$ be a square root of $z$. Then the points $1$, $\sqrt z$, $-\sqrt z$, and $z$ are either concyclic and colinear. This follows from the fact that four ...
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### Constructing fourth collinear point satsfying the mentioned condition

Given three collinear points $A,B,C$ such that $B$ is in between $A$ and $C$ . If $AB = m$ and $BC = n$ construct a point $X$ along $BC$ such that $CX = \frac{BC×AC}{3AB-BC}$ and $C$ lies between $B$ ...
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### Given a triangle construct a point satsfying the mentioned properties.

Given any triangle $ABC$ with $BC$ midpoint being $D$ and also given that $E$ is the midpoint of of $AB$ , construct a point $X$ such that $XB = BD$ along the side BC (not along the $\vec{BC}$ ...
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### Is it possible to find a extension of degree $2^n$ of $\mathbb Q$ which is not constructible?

It is well known that if a number $\lambda$ is constructible, then $\mathbb [Q[\lambda]:\mathbb Q]=2^n$ for some $n.$ Therefore, I am wondering whether there exist a counterexample for the converse? ...
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### Construct a circle orthogonal to another circle and tangent to a given line

This is a though one. It seems like the locus of the centers $B$ of the desired circles all lie in a parabola. How to figure a nice simple way to construct those circles with ruler and compass? I ...
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### Construction of square root of 3

I came up with this construction for $\sqrt{3}$, and want to know if it's valid. Start with a unit segment $AB$. At points $A$ and $B$, draw two circles with radius 1 like so: Then, mark the points ...
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### How to prove that two circles can't have more then 2 intersection points?

I've just understood that any two circles can't have more then 2 points of intersection, drawing several different intersected circles. Is there a way to prove it only by "geometric constructions&...
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### How to find out how many solutions some exercise has?

I'm passing through "geometric constructions" topic and a lot of exercises have the same question in the end of their conditions: How many solutions does the exercise have ? For example, ...
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### Compass-and-straightedge geometric construction problem (see image)

Given circle c with center A and a chord HI, and a point G within segment HI, construct circle d with center C such that the radical axis passes through G and point F lies on both d, line HI and line ...
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### Why does a geometric construction exercise has a unique solution, when two similar triangles are the answers of the exercise?

Exercise: Construct a right triangle by the hypotenuse and the leg with a compass and a ruler without graduations. The solution figures: Note: The text book that I learn math with is on the ...
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### Extruding an implicit/SDF?

Assume we have a convex implicit surface represented as an SDF $f:\mathbb{R}^3\rightarrow\mathbb{R}$, e.g. a sphere or a cube. Given a segment defined by 2 points $p_1, p_2$ I am interested in ...
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### Three dimensional Cairo Pattern

The Cairo pentagonal tiling is an interesting tessellation of the two-dimensional plane by irregular pentagons, which is given by taking two irregular hexagonal tilings, congruent but perpendicular to ...
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### Constructing (x,y) from (x,0) and (0,y) using ruler and compass

Maybe this is a dumb question and I missed the point of something, but... I am reading an old set of notes from when I was in college and doing some of the old homework problems. The first one I am ...
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### Is there a standard way of writing constructible numbers?

Every rational number can be represented by an irreducible fraction with a nonnegative denominator. For expressions involving integers, the four basic operations of arithmetic and square roots, you ...
1 vote
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### Is it possible to draw the center of a given circle using a compass only?

Can you construct the center of a given circle using a compass only. If the answer is yes, how? I've had this question for a while, the only thing I was able to do is find a line containing the ...
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### Construct $\sqrt{9a^2 - 4b^2}$ using compass and ruler, if $a$ and $b$ are given segments [closed]

I really don't know how to do math constructions. The problem is to construct $x$ if $a$ and $b$ are given, which means you can choose the length of them. Construct $x$: $$x = \sqrt{9a^2 - 4b^2}$$ For ...
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### Constructing a tangent line to a point on the curve $y = x^3$ using a compass and straight edge?

There are number of ways of constructing a tangent line to the curve $y = x^2$ using a compass and straight edge. Does anyone know of a way of constructing a tangent line to the curve $y = x^3$ using ...
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### Origami - construction without compass and straightedge

I started reading a book Project Origami by Tom Hull. His first exposition is about constructing equilateral triangle by folding a square paper. I understand the rules as follows: Folding point to ...
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### How do you construct the perpendicular with a square and a straightedge?

There is a square $ABCD$, line $EF$ and point $G$ on a plane. Can you construct the foot of perpendicular to line $EF$ through point $G$, using only a straightedge (in traditional Euclidean ...
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### Inscribe 3 equal circles in a Reuleaux/curved triangle without overlap [closed]

How exactly can I inscribe 3 equal circles into an equilateral Reuleaux triangle like the attached image without any overlap? I have the construction of the Reuleaux triangle down, drawn the altitudes,...
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### Do we extend the geometrically constructible numbers in a 3D space, where lines, circles, spheres and planes can be constructed?

Do we extend the geometrically constructible numbers in a 3D space, where lines, circles, spheres and planes can be constructed? In a 2D plane, we construct lines and circles only with compass and ...
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### Regular pentagon folding a strip

For young students it is an interesting surprise to discover that a knot tied in a strip of paper is a regular pentagon. I'm interested to find a simple, but rigorous, geometrical proof of this "...