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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Equilateral Triangle by Centroid and Two Points
We're given three random points on the plane: $O,A$ and $B$
Our goal is to construct the equilateral triangle $\triangle XYZ$, such that $O$ is its centroid and points $A$ and $B$ lie on the sides of $...
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A combinatorial problem with coins
I am stuck at a mixture of a combinatorial and maximization problem and don't know how to proceed. Hopefully someone has an idea that can bring me further.
Imagine that we have a sequence of $n$ coins....
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Doubling the cube, field extensions and minimal polynoms
I know the proof of the "Doubling the cube problem". What is used there is the fact that if a number $a$ is constructible then $[\mathbb{Q}(a):\mathbb{Q}]$ is a power of $2$.
I found in a ...
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Construct a perpendicular to a given line from a point not on the line in only three steps
The common compass and straight-edge construction of a perpendicular to a line $\ell$ from a given point $P$ takes $4$ steps:
$\hspace{10pt}$ $1$. Draw a circle centered at $P$. it will cross $\ell$ ...
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Given a rectangle, construct a square with equal area
Given a rectangle, construct a square with equal area. Note: Euclid provides this construction in II.14. I wrote the construction below without looking at Euclid's solution, and request verification,...
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Given a line $l$, a line segment $d$, and a point $O$, construct a circle with center $O$ that cuts off a segment congruent to $d$ on the line $l$.
I found this problem in Hawthorne's Geometry: Euclid and Beyond exercise $2.11$.
Given a line $l$, a line segment $d$, and a point $O$, construct a circle with center $O$ that cuts off a segment ...
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Does Euclid's bisection of an angle fail for $\frac \pi 3$?
Does Euclid's bisection of an angle fail for angles of $\frac \pi 3$ (that is, of equilateral triangles)?
In Book I Prop. 9 Euclid shows how to bisect an angle by constructing an equilateral triangle ...
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Ruler and compass construction of the axis of an ellipse related to a triangle [duplicate]
Let $ABC$ be a triangle. Consider all the six points defined as the opposite of a vertex respect to another.
It is not hard to prove (with affinity) that these 6 points all lie on an ellipse.
Looking ...
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Convert gaussian $XYZ$ into magnetic heading
OK, so I'm stuck, I've tried a combination of what I can find in terms of atan, atan2.
I have a Yost Labs sensor. In the 3D Test Suite it's showing a heading of $\sim234$ magnetic degrees which seems ...
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Show that given two marked points, Euclid can draw a circle centered at one of them and passing through the other.
This is problem 6 of 37th Indian National Mathematical Olympiad - 2023.
Euclid has a tool called cyclos which allows him to do the following:
Given three non-collinear marked points, draw the circle ...
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Let $x$ is the angle of the diagonal lines of a unit cube, then what are the possible values of $\cos x$?
Let $x$ is the angle of the diagonal lines of a unit cube(the cube with edges $1$), then what are the possible values of $\cos x$?
My attempt: Since the dot products of the diagonal lines are $1$ or $...
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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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About the construction of quadrate on oval
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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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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Determine whether or not the following length is convertible with ruler and compass
Determine whether or not the following length is convertible with ruler and compass
$$\sqrt{4+2\sqrt[3]{3}-\sqrt{2}i}.$$
My attempt
Let $z=\sqrt{4+2\sqrt[3]{3}-\sqrt{2}i}$
squaring, we get $z^2=4+2\...
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How to build a protractor without a protractor?
We all know how to use a protractor, it is taught in elementary school. However, I was wondering what type of knowledge is required to build one from scratch.
For instance, was the understanding of $\...
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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 ...
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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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Point on the edge of a triangle
$$\mathrm{I}=\int_{0}^{h_1}\int_{x_L}^{x_U}\Lambda(r^\prime)\frac{e^{-jkR}}{4\pi R}dx dy$$
Here,
$$x_L = \hat{n}\cdot\left(\hat{h_1}\times \overrightarrow{ \ell_2} \right) \left(1-\xi\right)$$
where,
$...
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Constructing a right triangle given $a+b-c$ and $\alpha$
The exercise is to construct a right triangle given $a+b-c$ and $\alpha$.
I know we then have $\beta=90^\circ -\alpha$. I tried to draw the right triangle $\triangle ABC$ and find where I can use $a+...
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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 "...
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How to construct a line perpendicular to a plane
I'm well aware of using the cross product to construct a perpendicular to two vectors. Given a plane, is there a way to construct a line perpendicular to it, using just a straightedge and compass? ...
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Error propagation in compass and straightedge constructions
I was trying to assess the impact of non-idealities on the outcome of a classical geometric construction, performed on paper with actual compass and straightedge.
I was thinking of possible approaches,...
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