# Tagged Questions

Questions on the construction of geometrical figures using a limited set of "tools".

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### Dividing Up A Circular Search Area

BACKSTORY: I need to collect 500 plant samples for strontium analysis. The samples are randomly distributed across a circular area with a radius of 300 kilometers. I have to do this in 30 days, so I ...
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### What types of triangles are constructible?

What types of triangles are constructible? I know that equilateral triangles are easily constructed using compass and straightedge, but what about other types of triangles? Can any other triangles ...
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### What cube roots are constructible using compass and MARKED straightedge?

DUE DILIGENCE: I have reviewed the list of questions possibly related to the one that I pose below, and I find that none of them address my particular question. Is there a formal definition for the ...
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### Is this curve defined by an envelope construction known?

Consider the following construction. Start with the standard envelope construction of a cardioid: on a circle, join each point $\theta$ to $2\theta$. Only, instead of joining with a line, join with ...
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### Constructing a line that passes through $P$

I have recently read a book by Heisuke Hironaka. However, the book is not available on English. The book was basically a biography on his life. Heisuke Hironaka says that his high school teacher had ...
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### Can you construct a rectangle with a given side, equal to a square?

In Euclid's Elements, Book 2, Proposition 14, We are shown how to construct a square from a given rectilinear figure. This allows us to square a rectangle. Is it possible to do the inverse, creating ...
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### Minimal number of steps to construct $\cos(2 \pi /n)$

My question is related to this previous one. I was wondering what is the minimal number of steps $S(a)$ to construct a number $a \in \mathbb R$ that is constructible (as defined here). For instance, ...
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### Is it impossible to construct an equilateral triangle inside a semicircle?

I have made it in a circle(which is very easy)....but I have been unable to make one inside a semicircle....is it not possible to make equilateral triangle inside a semicircle ?... If yes how can we ...
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### Finding the glide reflection using a compass or straightedge

Given two congruent triangles that are not a rotation, translation or reflection of each other; how can I find the glide reflection (the last remaining option) using only compass and straightedge. ...
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### Inscribe square in circle in just seven compass-and-straightedge steps

Problem Here is one of the challenges posed on Euclidea, a mobile app for Euclidean constructions: Given a $\circ O$ centered on point $O$ with a point $A$ on it, inscribe $\square{ABCD}$ within the ...
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### Mascheroni construction

There are two points in one line and a point C which doesn't lie on the line AB. I have to construct a point D such that it is intersection of the line AB and a line which is perpendicular to the line ...
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### What is the simplest way to find $\frac{n}{7}th$ of a line with mathematical proof?

I'd like to know if it is possible to find out the simplest way to get $\frac{1}{7}$, $\frac{2}{7}$, $\frac{3}{7}$, $\frac{4}{7}$, $\frac{5}{7}$ and $\frac{6}{7}$ of a line in 2-dimensional geometry. ...
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### Divide a line segment in the ratio $\sqrt{2}:\sqrt{3}.$

"Divide a line segment in the ratio $\sqrt{2}:\sqrt{3}.$" I have got this problem in a book, but I have no idea how to solve it. Any help will be appreciated.
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### Go from A to D in three equal steps

Given two parallel lines $r$ and $s$, line $p$, perpendicular to both, and points $A$ and $D$ on different sides of $p$ with respect to the parallel lines, how can I prove the existence of two points, ...
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### How do I calculate the distance from point A from point B?

I've got this drawing of a circle, and I'd like to know how I can calculate the distance between point A to point B in a straight line. I already have: Radius: 100 Arc length: 78.5 ...
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### Constructible decimals?

So I have to figure out if $1.23456$ is constructible. I think that it's not constructible since: I know that this is $\frac{123456}{100000}$ so this goes into $\frac{2^6*3*643}{2^55^5}$ and the ...
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### Constructible Solutions

We know that if a cubic equation with a rational coefficients has a constructible root, then the equation has a rational root. Now let; $$x^3-2x+2\sqrt{2}=0$$ Could $\sqrt{2}t$ be a viable ...
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### Is it possible to construct an incrementally accurate rectification of a circle?

Exact rectification of a circle (construction of a segment exactly the length of the circumference of a given circle) has been proven impossible. There is a number of rectification constructions that ...
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### For every three points on a line, does there exist a triangle such that the three points are the orthocenter, circumcenter and centroid?

The Euler line states that the orthocenter, circumcenter and centroid of a given triangle are on one line. This made me wondering whether the following is true: For every three points on a line (...
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### construction of line segment of a length $\sqrt{a^2-b^2+c^2+d^2}$

There are line segments $a, b, c, d$ and $a > b$. I have a question how to construct a line segment of a length $\sqrt{a^2-b^2+c^2+d^2}$. I can use Pythagoras theorem but I don't know how to make ...
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### Constructing triangle $\triangle ABC$ given median $AM$ and angles $\angle BAM, \angle CAM$

Constructing triangle $\triangle ABC$ given median $AM$ and angles $\angle BAM, \angle CAM$ I start with the median $AM$. Since $\angle BAM, \angle CAM$ are known I can construct them. So I have ...
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### Constructibility of Regular $N$-gon $\implies$ Constructibility of Regular $2N$-gon

I have to prove the following statement: If a regular $n$-gon is constructible, then so is a regular $2n$-gon. My Attempt: 1. Draw a point at the each vertex. 2. Draw a line between each point. ...
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### Geometric / Intuitive construction of the rotation axis of a 3D rotation matrix?

I have been looking without success for an intuitive / geometric construction of the rotation axis of a given 3D rotation matrix. To put the problem in more familiar terms, let's assume you have the ...
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### Can I square the triangle?

I know I can't construct a square with the same area as a given circle (because $\pi$ is transcendental). Can I construct (ruler and compass) a square with the same area as a given triangle? I think ...
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### How to prove three points are collinear when constructing a rectangle

My problem is: Choose a unit segment OI. Then construct a rectangle with base 3 units and height 2 units. I cannot use angle measure. I know I can construct this figure from my unit segment by using ...
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### Construction of a square ABCD

There are two nonparallel lines $p,q$ and point $A$, $A \notin p,q$ which lies between lines $p,q$. Construct a square ABCD such that $B \in p$ and $D \in q$. In special case in which $45°$ is angle ...
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### Mapping from Poincare's disk model to UHP

I have a question that : How can I map any point in Poincare's disk model to Upper-half-plane model? I know the function $$f(z) = \frac{z + i}{iz+1}$$ But I want to know the geometric ...
Here is a tricky compass and straightedge construction problem. Given triangle $\triangle ABC$ and point $D$ on segment $\overline{AB}$, construct point $P$ on line $\overleftrightarrow{CD}$ such ...