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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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How to construct a point on a semicircle such that one chord is twice the other? [closed]

The Problem: Given a semicircle with diameter $AB$ construct a point $C$ on the semicircle such that $BC = 2AC$. How to construct such point?
Rusurano's user avatar
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6 votes
1 answer
107 views

Permutations of Triangle Centers: Investigating Relationships Between Circumcircle Intersections.

I was solving this problem : Reconstruct the triangle from the points at which the extended bisector, median and altitude drawn from a common vertex intersect the circumscribed circle. I found the ...
pie's user avatar
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2 votes
1 answer
79 views

What arithmetic is possible via compass and straight line?

Given a line segment with length $ a $, a line segment with length $ b $, a compass, and a straightedge (you can only measure line segments with lengths $a $ or $ b $), is it possible to construct a ...
pie's user avatar
  • 5,373
1 vote
2 answers
41 views

Find the Harmonic Mean Using Parabola

beforeAbout an hour ago I was trying to create the harmonic mean using parabola, I haven't managed to get a general answer yet but I got a fairly good answer Is this theorem known in advance? I want ...
زكريا حسناوي's user avatar
2 votes
1 answer
63 views

Can reflect-point-across-point be constructed using points, lines, intersections, and reflect-point-across-line?

The following is a fun problem I thought up while thinking about the role of the compass in ruler and compass constructions. It strictly weakens the ruler and compass system by replacing the compass ...
Geoffrey Sangston's user avatar
3 votes
0 answers
35 views

Why does the path traced by a point reflect on a tangent of a circle create a cardioid

I was messing around in GeoGebra and noticed that this construction creates a shape that, at least, looks like a cardioid. Since both $B$ and $B'$ are equidistant from point $C$, they must be on the ...
Htan's user avatar
  • 31
-4 votes
2 answers
120 views

How can the reals be the set of all points on a number line when there exist non-constructible reals? [closed]

We are given the intuition that the reals form all the numbers on the numberline. However, this intuition wasn't working for me as the existence of non-constructible reals seems to me to imply that ...
Princess Mia's user avatar
  • 2,443
2 votes
0 answers
31 views

3d straightedge and compass

Given a tool that can draw a sphere by given center and a point on it and a surface by given 3 points, is the constructable set of the tool equivalent to the streightedge and compass constructable ...
עמית חי לרמן's user avatar
0 votes
0 answers
28 views

Algebraic varieties associated with (simple) "string" constructions

It is relatively well-known that any arrangement of points that can be constructed with a straightedge and compass can also be constructed with an unstretchable string (of arbitrary length, negligible ...
TLDR's user avatar
  • 131
1 vote
1 answer
38 views

How to construct an hexagon that has sides of two different specific lengths in the order of ababab, opposing sides are parallel

I'm building a three legged sculpture stand. I would like to construct a hexagon in which: Every other side is of equal specific lengths (Lengths of sides: a b a b a b). Opposing sides are parallel. ...
PetaspeedBeaver's user avatar
2 votes
4 answers
178 views

Constructing a circle tangent to another circle and two sides of a triangle

Given the circle tangent to the sides $AB$ and $BC$, I want to construct another circle that is tangent to this circle and also tangent to the sides $AB$ and $AC$. The center of such circle lies on ...
Etemon's user avatar
  • 6,633
0 votes
1 answer
25 views

How to construct $\Delta ABC$, given the angle at $C$ , $\gamma$ the median drawn fom $C$, $t_c$, and the angle at the point $B$, $\beta$? [closed]

I've gotten stuck again, can't seem to figure out what I'm supposed to construct first. $\Delta ABC$ with $\gamma$, $\beta$ and $t_c$ marked. Next to it text reads: $\gamma$ = $\angle ACB$, $\beta$ = $...
WhyWho's user avatar
  • 21
0 votes
1 answer
85 views

How can you construct a right triangle, ABC with a right angle in C, if you are given the hypotenuse, c, and the altitude of the point C? [closed]

How can you construct a right triangle, $\triangle ABC$ with a right angle in $C$, if you are given the hypotenuse, $c$, and the altitude of the point $C$? I know it's very basic but I just can't seem ...
WhyWho's user avatar
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1 vote
2 answers
123 views

How to determine a (compass and straightedge) constructible number of a trigonometric equation?

How to determine if the solutions of $$ \left(\frac{\pi}{12}+\frac{5}{3}x^2-x\sqrt{1-x}\right) = \arcsin x $$ are (compass and straightedge) constructible numbers? I got this question from my students ...
far away high school teacher's user avatar
2 votes
4 answers
701 views

Given a circle, construct three circles tangent to it and tangent to each other

Given a circle, construct three circles tangent to it and tangent to each other. Surprisingly, I can find no reference to this (although there are many similar sounding problems, and Apollonius' ...
SRobertJames's user avatar
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6 votes
4 answers
189 views

Geometric Pattern in Triangle Construction with Squares

I've been exploring some constructions in geometry recently and have been looking at the following algorithm: Start with an arbitrary triangle (red) Construct 3 squares from each of the sides of the ...
Leonidas Lanier's user avatar
1 vote
4 answers
187 views

Construct an inscribed equilateral triangle - proof

Construct an equilateral triangle inscribed in a given circle. Prove the construction using synthetic geometry only. There are two standard constructions for this: Find a $\pi/6$ angle via a right ...
SRobertJames's user avatar
  • 4,410
2 votes
2 answers
97 views

Constructing a quadrilateral with $3$ equal sides given their midpoints

Construct a quadrilateral with three equal sides, given the midpoint of each of these three sides. Also it is given that which of these three midpoints is for the middle side (a side between the two ...
Etemon's user avatar
  • 6,633
4 votes
1 answer
86 views

Ring Structure on the geometric line in synthetic geometry

Synthetic Differential Geometry by Kock opens with the following: The geometric line can, as soon as one chooses two distinct points on it, be made into a commutative ring, with the two points as ...
Mithrandir's user avatar
3 votes
2 answers
153 views

Constructing the orthocenter of a cyclic pentagon

I’d like to construct the following thing in Georgebra: A non-regular cyclic pentagon $ABCDE$ such that the perpendiculars from $A$ to $CD$, $B$ to $DE$... all concur in one point $H$. I managed to ...
PNT's user avatar
  • 4,166
3 votes
3 answers
135 views

What Is the Significance of Geometric Construction?

So, when I was in high school, I learnt several geometric drawings such as bisecting line segments or angles, to constructing a square whose area is the sum of two other squares etc. using a ruler and ...
Della's user avatar
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0 votes
0 answers
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Construct a circle of a given radius such that a ray of a given angle cuts out a given chord length (see diagram)

Given a segment $r$, a segment $c$, and an angle $\angle AOB$, construct (straightedge and compass) a circle whose center is on ray $OA$, whose radius is congruent to $r$, and where the chord created ...
SRobertJames's user avatar
  • 4,410
0 votes
1 answer
53 views

Construction: Given two intersecting lines, find the point where their distance is $P$

Given a segment $R$, a segment $H$, and an angle $\angle ABC$, construct a circle whose center is on ray $BA$, whose radius is congruent to $R$, and where the chord created by ray $BC$ is congruent to ...
SRobertJames's user avatar
  • 4,410
4 votes
1 answer
106 views

Finding more constructible solutions to polynomial equations

Summary: I have two polynomial equations in four variables. How can I find solutions using constructible numbers? Can I use the fact that I know one (or a few closely related) such solutions to find ...
MvG's user avatar
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0 votes
2 answers
44 views

How can I construct a circle given these conditions?

Given the angle $∠pOq$ and point $M$, which is not on the ray $p$. Construct a circle with its center on line $q$ that passes through point $M$ and touches line $p$. If I mark the point of tangency on ...
cchris's user avatar
  • 3
0 votes
0 answers
25 views

How to construct isosceles triangle from these given data

How do I construct an isosceles triangle ABC (|AC| = |BC| = a), given $a-v_a$ and the angle $α$?
cchris's user avatar
  • 3
2 votes
2 answers
150 views

Geometric construction of a specific gothic rose window

Christmas is coming up and I wanted to gift my boyfriend some nice laser cut gothic tracery. While browsing "The Power of Form applied to Geometric Tracery" by R. W. Billings, I came upon ...
Sansula's user avatar
  • 53
4 votes
1 answer
156 views

Constructing "Double" circumscribed pentagons.

I want to construct the following thing : A non regular pentagon $ABCDE$ circumscribed about a circle such that if $$A_1 = BC\cap ED, B_1=CD\cap AE...$$ then $A_1B_1C_1D_1E_1$ is also circumscribed ...
PNT's user avatar
  • 4,166
0 votes
0 answers
46 views

Construct a triangle given an angle, the length of internal angle bisector for that angle, and the incircle radius.

I need to write a proof for the following problem. Let theta be an arbitrary angle and suppose the length of the internal angle bisector to that angle and the length of the inradius are given. Perform ...
mathita7100's user avatar
1 vote
1 answer
125 views

Constructing the foci of the conic through five given points, using straightedge and compass [closed]

Given 5 points (every 5 points define a conic) find the foci of their conic using straightedge and compass construction. I thought about this question a while ago and can't solve it. Hypothetically ...
עמית חי לרמן's user avatar
2 votes
4 answers
1k views

Explaining solutions to Pythagorea puzzle 2.16: Drawing a line through a given point parallel to a given line

In this geometry puzzle (Problem 2.16 of the game "Pythagorea"), a 6x6 grid is given. Inside the grid there's a black line and a point A not on the line. The goal is to draw a line through A ...
bob93's user avatar
  • 49
5 votes
0 answers
114 views

There is a compass-like tool that can draw $y=x^2$ on paper. Is there one for $y=x^3$?

Is there a tool that can draw $y=x^3$ on paper? I'm referring to low-tech tools, e.g. not computers. I only know of tools that can draw $y=x^2$. The YouTube video "Conic Sections Compass" ...
Dan's user avatar
  • 23.6k
10 votes
4 answers
441 views

How to draw a parabola using basic equipment?

A straight line can be drawn with a straightedge. A circle can be drawn with a compass. An ellipse can be drawn with string and pins. How can we draw a parabola, using basic equipment? Remarks The ...
Dan's user avatar
  • 23.6k
2 votes
2 answers
148 views

Is there a way to construct a circle through two points and a line using ruler and compass

Given two points and a line in the plane, is it possible to construct a circle tangent to the line which passes through the two points using ruler and compass? For those who don't know, these are the ...
עמית חי לרמן's user avatar
1 vote
1 answer
58 views

Why is the midpoint of a right angled triangle side always on the same height?

it's been a great while since I've had basic geometry, and this seems to be elude me. Here are two images of a triangle $DMC$: How can I prove that the midpoint of $DC$ (point $E$) will always be in ...
ampersander's user avatar
0 votes
0 answers
27 views

straightedge and compass construction in Hartshorne's Geometry: Euclid and Beyond

Do you think that the following straightedge and compass construction as a single step is valid? Given line segment AB and a point C on line L, for a given side of C, find point D such that $AB \cong ...
YAC's user avatar
  • 41
2 votes
2 answers
106 views

Euclidean Geometry - Construction of a Circle Tangent to Given Circle, Line, and Point on Line

This is the original phrasing of the question: "Describe a circle to touch a given circle, and also to touch a given straight line at a given point." A School Geometry H.S. Hall and F.H. ...
procommania's user avatar
1 vote
0 answers
51 views

How do you build a quadrilateral, knowing it's sides, their order, and that this quadrilateral has an inscribed circle?

I want to build a quadrilateral with sides, for example, $2:3:5:4$, knowing that it also has an inscribed circle (which all 4 sides are tangent to). How can you do it for a general case, using only a ...
danik0011's user avatar
1 vote
0 answers
24 views

In Euclidean geometry, is there a topic of "the shortest construction"?

How do we judge if a certain ruler-and-compass construction is the shortest? For example "the shortest construction of a regular pentagon." Is there a software that can come up with a ...
Maesumi's user avatar
  • 3,702
2 votes
1 answer
239 views

Nondegenerate triangle of zero area. How is it possible?

I read book "A primer of infinitesimal analysis" John Bell. I was confused when I saw example with area under curve. In that example author mentioned about nondegenerate triangle of zero ...
Mike_bb's user avatar
  • 751
0 votes
1 answer
45 views

Rhomboid construction by compass and straight-edge

Consider the following given points: I should construct a Rhomboid $ABCD$ using the facts that $P \in BC$, $Q \in CD$ and $\{M\} = AC \cap BD$.Thus I get I do not really know how to proceed from ...
TheGeekGreek's user avatar
  • 7,909
0 votes
0 answers
27 views

Construction of normals to parabola across external point

I know this is a popular problem for algebra; I am looking for a constructive way to find the normal(s) to a parabola across an external point. The obvious guess is that one needs to find the circle ...
arivero's user avatar
  • 91
29 votes
1 answer
1k views

Group formed on Parabola similarly to how an Elliptic curve forms a group (by drawing lines, circles, intersecting, or taking tangent lines)

There's probably other ways of doing this, but I've found this to be the simplest way (group law) that does indeed work: To add points $A, B \in \{(x, f(x)) : x \in \Bbb{C}\} = G$ where $f$ is any ...
Daniel Donnelly's user avatar
6 votes
3 answers
531 views

Unexpected tangency

I came across an "unexpected tangency" (for lack of a better term) while working out a different construction with GeoGebra. The construction goes like this. Starting with a segment $AB$, ...
kjo's user avatar
  • 14.4k
1 vote
1 answer
75 views

Can a triangle be constructed by right edge and compass when its base, median to the base and the sum of other sides are known?

Can a triangle be constructed by right edge and compass when its base, median and the sum of other sides are known? I found this problem in the book "An introduction to the modern geometry of the ...
sirous's user avatar
  • 10.9k
3 votes
0 answers
47 views

Is there a good resource for constructions using double-edged straightedge?

Double-edged straightedge constructions use only a straightedge, but use the fact that a straightedge has two parallel sides that are a fixed with apart. I've found one obscure paper from the 80s that ...
James Cleveland-Tran's user avatar
0 votes
1 answer
119 views

Appolonius problem: the PPL case

I have a problem to understand the PPL Apollonius problem part as stated here on the page 202. My problem is, where is the diameter and the center of the Thalet circle as given here in the second case ...
user122424's user avatar
  • 3,936
-1 votes
1 answer
131 views

Is it possible to construct a precise regular pentagon with just a straightedge (no compass)? If yes, then how?

Regular polygon = all angles have the same measure AND all sides have equal length. So, is there a possibility to draw a regular pentagon with just a straightedge? (I think you may also call it a ...
Alexander's user avatar
  • 395
5 votes
1 answer
272 views

Using a compass of fixed opening and straightedge, what is the shortest way to find the centroid of $10$ points?

This question is actually mentioned in the OEIS sequence $A157650$. After reviewing the initial terms, I believe that for $n=1..9$, the centroid of $n$ points can be found with the following step ...
Jinyuan's user avatar
  • 240
4 votes
3 answers
195 views

Construction of a circle orthogonal to two given circles and tangential to a given line

Two circles $c$ and $d$ that intersect at points $A$ and $B$ are given. Let $p$ be a line passing through $A$ that intersects circles $c$ and $d$ at points $P_1$ and $P_2$. Construct a circle ...
Katarina's user avatar
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