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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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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Straightedge-only for Perpendicularity with a Square

There is a square $ABCD$, line $EF$ and point $G$ on a plain. The task is to construct the foot of perpendicular to line $EF$ through point $P$, using only a straightedge (in traditional Euclidean ...
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1 vote
2 answers
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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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3 votes
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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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Is it possible to construct $\cos\frac{\theta}{3}$ from $\{0, 1, \cos\theta\}$ if $\theta$ can be trisected using straightedge and compass?

My attempt: Given $P:=\{0, 1, \cos\theta\}$, clearly we can construct the point $\sin\theta$ on the complex plane since $\sin\theta=\sqrt{1-\cos^2\theta}$, meaning $\sin\theta\in\mathbb{Q}(P)^{py}$. ...
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Could $2^{1/n}$ be straightedge-and-compass constructible for some $n \in \mathbb{N}$ that is NOT a power of 2?

I understand that the square root of an integer is constructible using straightedge and compass; but are they the only constructible radicals? Any hint would be greatly appreciated.
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7 votes
2 answers
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How do you find the center of a cake with just a knife?

Consider an undecorated cylindrical cake and a perfect knife. We want to find the center of the cross-sectional circle. If we can only score the surface of the cake, this reduces to finding the ...
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1 vote
1 answer
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Show $[\mathbb{Q}(\cos\frac{\theta}{3}, \cos\theta): \mathbb{Q}(\cos\theta)]=2$ implies that $\theta$ is trisectable using straightedge and compass.

Show $[\mathbb{Q}(\cos\frac{\theta}{3}, \cos\theta): \mathbb{Q}(\cos\theta)]=2$ implies that $\theta$ is trisectable using straightedge and compass. Proof: $[\mathbb{Q}(\cos\frac{\theta}{3}, \cos\...
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Show that if $[\mathbb Q(\cos\theta,\cos\frac{\theta}{3}):\mathbb Q(\cos \theta)]=1$, then $\theta$ can be trisected using straightedge and compass.

Show that if $[\mathbb Q(\cos\theta,\cos\frac{\theta}{3}):\mathbb Q(\cos \theta)]=1$, then $\theta$ can be trisected using straightedge and compass. Proof: Clearly $\mathbb Q(\cos\theta,\cos\frac{\...
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Proof that there is a point inside an acute-angled triangle so that perpendiculars dropped to the sides will form vertices of equilateral triangle

There is a character limit to the title length and the task may not be clear. So here is a little bit longer task description. Task description: You are given an acute-angled triangle $ABC$. Proof ...
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Which rational angles are constructable?

For which rational angles (I call an angle $\alpha$ rational if $\alpha/\pi \in \mathbb Q$) is $\sin(\alpha)$ algebraic with $[\mathbb Q(\alpha):\mathbb Q]$ being a power of 2? This condition is ...
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Problem in construction by straightedge and compass

I am facing a simple problem in straightedge and compass construction, even for simple kind of real numbers. Rules: A point $P$ in $2$-D plane or $\mathbb{C}$ is constructible by straightedge and ...
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Calculating direction traveling around a circle? (knowing when to move CW vs CCW)

I have a point I am animating that is traveling around a compass. The point should always move in the direction of what would be the shortest path. I have something like this at the moment: ...
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2 votes
3 answers
143 views

Given the red and blue circle, construct with straightedge and compass the yellow one

The problem is from the image: the yellow and the red and tangents, so are the blue and the red and the yellow passes through that black thick dot at the north pole of the blue one. So my attempt: I ...
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If I have the n-polygon regular constructed and the m-polygon regular. How I can construct the lcm(n,m)-polygon regular? And why?

The situation is that I have two regular polygons in the unit circumference, there is a result that there exist one vertex of the n-polygon whose distance for one other vertex of m-polygon is the ...
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9 votes
3 answers
305 views

A Geometric Construction problem that's got me stumped since 1975

Hello, I'm a retired civil engineer from Greece. Ever since I was a student I have really liked maths. I had to solve this problem during my first year at NTUA, back in 1975! I couldn't solve it then ...
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Constructible Numbers with Unit $i$

Recently in my algebraic structures class we discussed constructible numbers: "A real number $\alpha$ if we can construct a line segment of length $|\alpha|$ in a finite number of steps from this ...
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1 answer
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Geometric Construction: Restore the triangle with the only points marked are orthocentre and one vertex

Given obtuse triangle $BAC$ with $\angle A$ obtuse. Let $H$ be the orthocentre. Let $M$ be the midpoint of $BC.$ Define $X$ such that $MB=MC=MX, D\in AH.$ Let $l$ be line $XM.$ Define $k$ as line $XB$ ...
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3 votes
2 answers
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Project a parabolic curve to straight lines.

I am working with Voronoi Diagrams and I'm given a parabolic curve that is described through four parameters: A start point, say $P_0(x_0,y_0)$. An end point, say $P_1(x_1,y_1)$. A focus point, say $...
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Point construction - one point equal to several points?

I have been studying point constructions from ruler and compass in context to field extensions. I follow the proofs without too much trouble, but there is a simple note I do not follow. In A Book of ...
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4 votes
2 answers
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Prove the triangles $LBE$ and $LKD$ are congruent.

I found this problem in a 2019 mathematics competition in Hunan, China. I am a Chinese high school student. There is a triangle $ABC$ which is not equilateral. The point $H$ is the orthocenter of it. ...
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Construct a circle

Circles a and b intersect at M and N. Construct a circle k orthogonal to Circles a and b and touching third given circle c. Problem is supposed to be solved using inversion and/or power of a point. ...
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3 answers
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Multiplication of variables with a compass when a segment of length 1 is not given.

"Suppose that length a and b are given. Construct sqrt(ab) (Not that a segment of length 1 is not given)." I know how to multiply ab together and then take the square root of ab if a length ...
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2 votes
1 answer
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How to construct triangle ABC from these given data

I want to construct triangle $ABC$ and it is given: Segment $BC$ $\angle BAC = \theta$ There is a circle through $B,C$, the orthocenter $H$ of $\triangle ABC$ and the centroid $G$ of $\triangle ABC$. ...
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6 votes
3 answers
368 views

Equilateral triangle and very peculiar inscribed tangent circles

The problem is to find the length of the size of the equilateral triangle below I found one equation: Let $R$ be the radius of the big circle whose red arc touches the two purple circles. Let $A$ be ...
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3 votes
2 answers
93 views

Can we construct this figure in more elementary steps?

The problem is given in image below. We're given a semicircle whose diameter is $AB$ and center $O$ and we want to construct the rectangle triangle $\triangle PCD$ with $CD \parallel AB$, $C$ and $D$ ...
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3 votes
1 answer
112 views

Perpendicular lines in a hexagon should not intersect

The diameter of the circle circumscribed about a regular hexagon is 35 cm. Inside each side of the hexagon we have chosen one point through which we draw a perpendicular line of length 20 cm inside ...
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Question about construction with ruler and compass

Given a circunference $C$ and a line $l$, how can I find a point $A \in l$ such that the tangent lines to the circunference passing through $A$ form an given angle $\beta$? I don't want the answer, ...
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If the inradius is $2$ and the segments determined by the circumference inscribed on one side measure $3$ and $5$, find the area of triangle

For reference: Calculate the area of ​​a triangle if the inradius is $2$ and the segments determined by the circumference inscribed on one side measure $3$ and $5$.(Answer:$\frac{240}{11}$) My ...
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2 votes
2 answers
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What's the value of the area of the triangle $ABC$ below?

For reference: Calculate the area of ​​triangle $ABC$; if $ED = 16$; $AB = 10$ and $D = \angle15^o$(Answer:$20$) My progress: I didn't get much. $\triangle ECD - (15^o, 75^o) \implies EC = 4(\sqrt6-\...
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Rotating circle while moving on a line by a linear speeds relation

I'm searching for a curve identification which comes from the trace of point $P:(0,a+r)$ on circle $C: x^2+(y-a)^2=r^2$ when the circle rotates while its center moves on yAxis toward the cordinate ...
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2 votes
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Origami vs Compass and Ruler construction

So I was inspired by Numberphile to learn about these construction. I find it amazing that folding alone is stronger than compass and ruler because it can solve the problem of trisecting an angle and ...
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1 vote
1 answer
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Why does this construction work? (Explanation)

In the book that I am currently studying, this construction appears (see image) to show that if $a$ and $b$ are constructibles, then $ab^{-1}$ is constructible, but I cannot understand why. I have ...
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1 vote
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Is possible to double the cube with unmarked T-shaped ruler and compass?

As Galois depicted, cube doublication cannot be constructed via unmarked straightedge and compass. Any solution to this problem should make cubic extension or above. I have already known this method ...
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Construct points $C$ and $D$ such that the distance from $C$ to $D$ is $\sqrt{2}$

Suppose that we have points $A$ and $B$, and the distance from $A$ to $B$ is $1$. Construct points $C$ and $D$ such that the distance from $C$ to $D$ is $\sqrt{2}$ So far I have found how to solve ...
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Apollonius's Problem: why PCC can be reduced to PPC

Here is the construction of the Apollonius's Problem PCC by deducing to PPC: https://www.cut-the-knot.org/Curriculum/Geometry/GeoGebra/PCC.shtml But it is not mentioned how it works. I'm trying to ...
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1 vote
2 answers
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What's the measure of the angle $\angle DAC$ in the triangle below?

For reference: In the figure, $AB = 11$, $BC = 5$ and $DE = 3$. Calculate $\angle DAC$. My progress: Extend $BE$ until$I: F\in AD$ Angle chasing: $HBI = \theta$ Can I say that the triangle $ABI$ is ...
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3 votes
1 answer
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How to draw largest possible regular pentagon inside a square?

I have a fixed size square on which I need to draw a regular pentagon to use as a classroom aid. This pentagon should take up as much of the square as possible, and does not need to have any of its ...
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3 votes
0 answers
53 views

Constructing a line through $P$ that meets $\overline{AB}$ and $\overline{AC}$ in points $D$ and $E$ such that $\overline{DB}\cong\overline{EC}$ [duplicate]

Geometry problem in Hironaka Heisuke's book. (I don't think this book is translated into English.) I saw this beautiful problem which the author said he solved it in high school: $P$ is a point ...
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Can you make a Sine wave with a straight edge and compass?

I am trying to make a sine wave with a straight edge and compass, is it even possible?
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Constructing regular $mn$-gon if $m$-, $n$-gons are constructible

I am studying Stillwell's Elements of Algebra. He mentions the following exercise: If $\gcd(m, n) = 1$ and the regular $m$-gon and the regular $n$-gon are constructible, show that the regular $mn$-...
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1 vote
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How do you proceed when constructing a homeomorphism? [closed]

I have maybe quite stupid and quite methodological question: how do you proceed when constructing homeomorphism? I am asking, because I came across many exercises, proofs etc., where some steps would ...
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How to construct a homeomorphism with certain conditions on $\mathbb{R}$?

I am reading the paper "Characterizations of $\beta \mathbb{Q}$ and $\beta \mathbb{R}$ " by Eric Van Douwen. One part of his proof of the Proposition 4 (below) is not clear to me. Suppose $H$...
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Parallel line construction without square root

Given a line defined by two points having rational coordinates, is it possible to construct a parallel line at a given rational distance to the first line, using only $+$, $-$, $*$, $/$, i.e. without $...
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