# Tagged Questions

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

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### Finding $\sqrt{17}$ and $\sqrt{257}$ in the regular $17$-gon and $257$-gon?

(Edit: I need to revise this question with my original intent. Pls do not answer it yet. Thanks.) Given the regular $n$-gon formed by the $n$-th roots of unity. For some $n$, how do we find ...
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### Drawing a circle tangential to 3 circles (internally to one of them)

The two small circles (in black) are equal in radius, and tangential to the large circle. They also touch each other at the center of the large circle. Now, I want to construct a circle (in orange) ...
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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 ...
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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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### same distance from a point to 2 non-parallel lines

There are 2 nonparallel lines $a,b$ and point $E$ which doesn't belong to any of them and lies anywhere between them. EDIT: Task is to find two couples of points F, G and H, I $\in$ y such that ...
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### How to find the inradius of orthic triangle?

How to find the inradius of orthic triangle in terms of side lengths or area or circumdiameter of original triangle? The incentre of the orthic triangle is the orthocentre of the original triangle. ...
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### Difficult Proof by Construction Question

A teacher introduced this question to me, and am interested in finding the answer, but cannot figure it out. Given the squares of the numbers 1 through 81, can you separate them into three groups with ...
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### Why a segment of length $\sqrt{2}$ can be drawn but a segment of length $\pi$ cannot?

We know that both $\pi$ and $\sqrt{2}$ are irrational. Also, it has been proved that a segment of length $\pi$ can not be drawn whereas a segment of length $\sqrt{2}$ can be drawn. Why is it so, ...
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### Does there exist a tool to construct a perfect sine wave?

For example, a perfect circle can be constructed using a compass and a perfect ellipse can be constructed using two pins and a piece of string, because a circle can be defined as the locus of points ...
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### How to construct an n-gon by ruler and compass?

Since $\cos[\frac{2\pi}{15}]$ is algebraic and equal to $\frac{1}{8}(1+\sqrt{5}+\sqrt{30-6\sqrt{5}})$ we know that the regular 15-gon is constructible by ruler and compass. Although I know how to ...
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### $\sqrt2^{\sqrt3 }$ $<$ $\sqrt3^{\sqrt2 }$ geometrical way [closed]

From the pythagorean theorem we can construct the lengths $\sqrt2$ and $\sqrt3$ Is it possible to construct the lengths $\sqrt2^{\sqrt3 }$ and $\sqrt3^{\sqrt2 }$ in order to show ( geometrically ) ...
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### Prove that in a cyclic quadrilateral ABCP where two fo its sides are equal the ratio $\frac{PA}{PB + PC}= \frac{AC}{BC}$ is constant for any point P

Let ABC be an isosceles triangle where AB = AC. Consider the circumference in which the points A, B and C live. Let P be a point in the chrord of this circunference formed by the points B and C. Prove ...
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### Constructing congruent triangles and give a proof

I need to construct from a given triangle $\Delta ABC$ a congruent triangle $\Delta XYZ$ and I know how to do this if I'm allowed to "measure an angle with a compass" and carry that measurement over ...
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### Relation between sqrt and ratio in ruler and compass?

The construction in most vote reply in Compass-and-straightedge construction of the square root of a given line? uses similar traingles and uses $$\frac{AC}{AD}=\frac{AD}{AB}$$ to compute square root. ...
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### Is it possible to construct an isosceles triangle by using a ruler and without using a pair of compasses?

It is well known that on Euclidean plane one can construct an isosceles triangle on given straight line by using a ruler and a pair of compasses. Also it is possible to construct straight line ...
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### Construct the intersection of a cube by a plane through $3$ points on its edges, no pair of which is on the same face

So this is a rather old problem, but I still cannot find a pure constructive solution to it. Please, do not offer me to write a plane equation, etc. I would be grateful, if you offer a solution by ...
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### Pyramid with unit sides inside a cube

Let $ABCDEFGH$ be a unit cube with base $ABCD$. Let $P$ be the top of the pyramid with base $ABCD$ and all edges of length $1$. One has a standard 2-dimensional projection of this cube on the back ...
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### Can one construct any n-gon if angle trisection is also allowed?

Suppose one is asked to construct a regular n-gon, but with one extra operation allowed in addition to the standard compass-and-straightedge ones: trisecting any angle. Are all n-gons constructible ...
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### construct a triangle with a compass and a ruler, given $a, B, t_a$ [closed]

If $a, b, c$ are the side lengths of a triangle, $A,B,C$ are the opposite internal angles, respectively, and $t_a, t_b, t_c$ are internal bisectors of the angles $A,B,C$, how could I construct a ...
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### Triangle construction procedure

Two lines $L1,L_2$ pass through a common point $O.$ $L_2$ goes through points $P$ and $Q$. How to construct a circle through $P,Q$ to be tangent to $L_1?$ In a particular case, at the tangent ...
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### constructing segments with equal cross ratio

I was puzzeling again and had the following problem: Given: two segments $AD$ and $PS$ on $AD$ there are points $B$ and $C$ so that $AD \gt AC \gt AB$ (so they are in order A, B , C, D ) on $PS$ ...
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### Finding tangents to a circle with a straightedge

There is a geometric construction that I heard years ago and I still haven't figured out why it works despite several attempts. Playing with pen, paper and GeoGebra makes me confident that it does ...
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### Is there a geometric construction for the square root of the volume of a parallelotope?

A parallelotope is the higher dimensional analog of a parallelogram. Now, what I want to know is if there's a way to construct an object with size equal to the square root of the volume of the ...
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### Extension of Isovist concept for a point - to Isovist for a polygon

There is the concept of Isovist/Visibility polygon. They both talking about volume of space visible from a given point in space. My question: What is the algorithamic solution of this problem for a ...
In order to provide and alternative solution to a well-known problem $^{(*)}$ I would like to solve the following sub-problem in the most effective way (i.e. in the least number of steps). ...
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 ...