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.

2
votes
2answers
134 views

Degree of Minimal polynomial of complex number and its components

I've been doing some research on the constructibility of regular polygons, and it led me to come up with the following conjecture: Edit (Sorry I should have imposed much stricter conditions): Let ...
4
votes
1answer
176 views

Find points A and B from their angle bisectors

I need to find points $A$ and $B$ in $ℝ^2$, when I only know three points $X_1, X_2, X_3$ and angle bisectors of the angles $A X_1 B$, $A X_2 B$ and $A X_3 B$. How do I do this geometrically? . If I ...
2
votes
3answers
450 views

Construct right triangle given the sum of legs and the hypotenuse.

I actually made one however with the help of an ellipse. Can the construction be done without using the concept of ellipse? I want another solution since this chapter problem in a book has not yet ...
1
vote
1answer
206 views

Constructible angle definition

While studying a course on field theory, my prof gave the definition of constructible angle as An angle $\theta (0\leq\theta\leq2\pi)$ is said to be constructible, iff the length $\cos\theta$ is ...
1
vote
1answer
79 views

Construction of another regular pentagon

I have a problem that involves construction with the ruler and compass. I want to prove more constructions of the regular pentagon. I finished demonstrating some more constructions, but I couldn’t do ...
-2
votes
4answers
185 views

Construction of a isoceles Triangle [closed]

How can one construct an isosceles triangle with ruler and compass with the following givens the sum of the base and a side the head angle
-1
votes
2answers
584 views

Construct a segment with length of $\frac{2a}{a+b^2}$

Given the following segments how would you construct a segment with length of $\frac{2a}{a+b^2}$? Given the three line segments below, of lengths a, b and 1, respectively: For example if I wanted to ...
0
votes
1answer
22 views

Calculating Path of Motion of Two Uneven Wheels

I had this question when observing something of the same nature with precise measurements. An object has 2 circular ends (objects) joined together by a cylinder. The length of the cylinder is 5.6cm. ...
0
votes
0answers
43 views

Is there a graphic showing where we can and cannot get to utilizing only a compass and straightedge?

I know that we cannot get to the measurement of pi units utilizing a compass and straightedge. I was wondering if there is a graphic that shows points all points that can or cannot be found utilizing ...
1
vote
3answers
170 views

Construct a square of given area

Given are three squares of side lengths $a$, $b$ and $c$ with $a>b>c$. Construct (with compass and straightedge) a square of the area $a^2-b^2-c^2$! I have thought about "cutting" the two ...
0
votes
1answer
262 views

how to construct numbers without compass or a straightedge [closed]

Given arbitrary lines of length a, b, 1. How would we construct stuff like a/b and square root of a without using a compass or straightedge
2
votes
2answers
111 views

Number of Solutions to Apollonius's LLC Problem

I have been practicing my straightedge and compass constructions over the last few days and I'am trying to reproduce the solutions to the ten Apollonius problems (constructing circles which are ...
2
votes
2answers
121 views

Ruler and compass construction [duplicate]

Given the three line segments below, of lengths a, b and 1, respectively: construct the following length using a compass and ruler: $$\frac{1}{\sqrt{b+\sqrt{a}}} \ \ \text{and} \ \ \ \sqrt[4]{a} $$ ...
1
vote
0answers
181 views

quintsect an angle with a compass and a straight edge

I require to construct a reasonably accurate $9^{\circ}$. And quickly. I learn that the quickest way is to make a straight line of $180^{\circ}$, bisect it (to make $90^{\circ}$), bisect that angle (...
2
votes
2answers
205 views

Compass and ruler construction

Given the three line segments below, of lengths a, b and 1, respectively: construct the following length using a compass and ruler: $$\frac{1}{\sqrt{b+\sqrt{a}}} \ \ \text{and} \ \ \ \sqrt[4]{a} $$ ...
0
votes
0answers
174 views

Why does it matter wheter I solve a geometry problem with calculations?

In a task, I was wanted to construct a triangle (e.i. write the steps for construction) if I have been given: The side $a = 6 \operatorname{cm}$ The angle $\alpha = 90°$ The base of the triangle $b = ...
1
vote
2answers
205 views

How to inscribe an ellipse into an isosceles trapezoid?

My main question is: how can I inscribe an ellipse into an isosceles trapezoid? I want to create an ellipse, whick is tanget to all four sides of the trapeziod (i.e. shares exactly one point with each)...
11
votes
2answers
237 views

Geometric notion of addition for the real projective line

The real projective line $\mathbb{RP}^1 = \mathbb{R} \cup {\infty}$ is usually identified with (or defined as) the set of lines passing through the origin in $\mathbb{R}^2$. Thus, the number $m\in \...
0
votes
2answers
63 views

Are irregular polygons constructible?

I don't know if my question doesn't make sense or it's just too elementary but I can't seem to find something anywhere in internet that guides me to a precise answer, I mean, in my head it's ...
0
votes
1answer
50 views

Geometry/constructions

I want to contruct a trapes and I am finding it hard to find the last point D. 1) Construct a segment AB = 6.8 cm 2) Construct an angle at 67,5 degrees at A. 3) Construct AC = 8 cm 4) CD is ...
2
votes
0answers
54 views

Question regarding constructibility of a point

Given 2 points $ A $ and $ B $ in the complex plane with $ AB=1 $, is it possible to construct with unruled straightedge and compass a point $ C $ on the line $ AB $ such that $ AC \cdot BC^{2}=1 $? ...
2
votes
1answer
298 views

Algebra - Construct circle with radius AB around distinct point C

I'm currently reading Antoine Chambert-Loirs "A field guide to algebra". The very start of the book is dedicated to the topic of Construction with ruler and compass. The most import definition used: ...
1
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0answers
33 views

Constructing Angles [duplicate]

I am a high school student, while solving the 'construction' chapter of my textbook I realized that it is not possible to construct every angle using a compass and a straight edge. why is it not ...
3
votes
4answers
96 views

Prove that BF + CE = BC

In $\triangle ABC$ , $\angle A = 60$ $~BE$ is the bisector of $\angle B$ and $~A-E-C.$ $~CF$ is the bisector of $\angle C$ and $~A-F-B$ Prove that $~BF + CE = BC$
-1
votes
3answers
191 views

Find area of triangle ABD.

Given in $\triangle ABC,~AD$ is the angle bisector of $\angle A $. If area of $\triangle ABC = X$ , prove that area of $\triangle ABD = \dfrac{Xc}{(c+b)}.$ $a=BC$$b=AC$$c=AB$
3
votes
3answers
99 views

Constructing a triangle.

I want a proof that there is one and only one possible unique triangle ABC in which base BC is of length 6 cm,angle B=60 degrees and the sum of other two sides is 9 cm. A proof without use of ...
0
votes
1answer
114 views

Contruction of irrational angles with ruler and compass

I have the following question on constructible numbers. I want to know which angles are constructible using only ruler and compass. I will write my angles always as a multiple of $2\pi$. I already ...
0
votes
3answers
790 views

How to find the height of a tilted rectangle

Suppose we are given two congruent rectangles ABCD and EFGH as shown in the figure, with AB = 8 , AD = 4 and EA = 3. Find the distance of point G from line AB; that is, find X.
1
vote
1answer
63 views

Find length of common chord of nine point circle.

$K$ is any point on side $\overline{BC}$ of triangle $\triangle ABC$ . Find the length of the common chord of the nine-point circle of triangle $\triangle ABK$ and triangle $\triangle AKC$ if $\...
9
votes
1answer
162 views

The smallest parallelogram that contains a convex quadrilateral

I try to find the smallest parallelogram in terms of area that contains a convex quadrilateral(A,B,C,D). I am pretty sure it must be constructed from two neighboring sides of the quadrilateral. But ...
0
votes
1answer
44 views

How to construct $\Delta ABC$ given $a$, height from $A$ on $BC$ and A$B:AC$. [closed]

How to construct $\Delta ABC$ given $a$, height from $A$ on $BC$ and A$B:AC$.
2
votes
4answers
970 views

How to construct a circle tangent to another circle and a line?

How to construct a circle tangent to another circle and a line? Also Find locus of centres of all such circles?
0
votes
0answers
76 views

Geometrical construction of circle and circumscription of a pentagon about it.

How would you draw a circle of a given radius and circumscribe a regular pentagon about it using a scale and a compass. I cannot figure it out. I have checked the internet but cannot find anything ...
6
votes
3answers
368 views

A beautiful geometry problem

Let $PP'$ and $QQ'$ be two parallel lines tangent to a circle of center $C$ and radius $r$ in the points $P$ and $Q$, respectively. $P'Q'$ cuts de circle in $M$ and $N$. Let $Y$ and $X$ be the points ...
1
vote
1answer
561 views

Construction of 3 circles touching each other externally.

The context is construction of three circles with different radii so that they touch each other externally using a graduated ruler and a compass. I have done it by drawing a triangle where each side ...
0
votes
1answer
85 views

Epicycloid-alike curve

We knew Epicycloid as a kind of trace curve of a specific point attached to a circle and rolls on another circle. But there's a limit for the ratio R/r of the radius of two circle, R and r, which is ...
1
vote
1answer
2k views

Is it possible to construct a regular heptagon with just compass and straightedge?

Is it possible to construct a regular heptagon (a figure with seven sides) with just compass and straightedge? If so, could you please give me directions for how to do this?
0
votes
2answers
97 views

Construct or prove existence of a certain quadrilateral

I have three questions about a quadrilateral with the following properties: It is convex. It has exactly one pair of congruent opposite sides. It has exactly one pair of congruent opposite angles. It ...
1
vote
2answers
4k views

How to draw a regular pentagon with compass and straightedge

I remember reading that Gauss managed to construct a regular pentagon with just a compass and straightedge, but I don't remember the particulars of how he did this. Could someone help me out and give ...
3
votes
0answers
47 views

Ruler-and-Compass metric: draw a line from any point to separate the area of a triangle?

Given a triangle $ABC$ and a point $X$, is it possible to only use Ruler and Compass, to draw a line $l$ through $X$, such that $l$ will split $ABC$ to two parts with equal area?
0
votes
1answer
364 views

Finding a line segment within an angle based on given midpoint [closed]

Given an angle ∠ABC and an arbitrary point D somewhere within the angle, how would you (using only a compass and straight edge) draw a line segment where one end lies on AB and the other on BC, with D ...
0
votes
2answers
77 views

How to construct the scheme below?

How can one construct the diagram shown below, using just a ruler and a compass, where $ \angle PYX = 2\angle PXY $ and $ \angle PYZ = 2\angle PZY $:
2
votes
0answers
104 views

Straightedge and compass theory in three dimensions

I'm looking for a reference on the theory of straightedge and compass constructions in three dimensions akin to Euclid's Elements in two dimensions. More specifically, I mean a theory of geometric ...
7
votes
3answers
580 views

Graphically locate the axes or foci of an ellipse from 5 arbitrary points on its perimeter.

Five distinct non-collinear points are required to define an ellipse similar to the way that three non-collinear points define a circle and can be used to determine the center point of that circle. I ...
12
votes
1answer
283 views

Construct point on a circle such that the reflection in that point is horiztonal

Let $P$ be a point in the plane outside the unit circle. There is a unique point $Q$ on the circle such that a light ray from $P$ is reflected in the circle at $Q$ and emerges parallel to the $x$-...
0
votes
1answer
50 views

Construct line segment with enpoints on given circles and equal to given line segment

Given a line $p$ and a line segment $\overline{MN}$ on $p$, and two circles $k_1$ and $k_2$, construct a line segment $\overline{AB}$ with endpoints on the circles so that $\overline{AB}$ is parallel ...
2
votes
2answers
370 views

Neusis construction of the 11-gon?

Wikipedia tells me that the 11-gon was found to be neusis constructible in 2014, and the link given doesn't seem to be a crank, but the actual method is behind a paywall. (Interestingly, the page ...
0
votes
2answers
59 views

Creating a circle from 3 points on its circumference when the slope of one line is infinity/undefined?

I have recently run into a problem while trying to get the center of a circle from 3 points $A,B,C$ on its circumference.The equations I used for this require to know the slope values of the lines $\...
2
votes
1answer
546 views

Three-step construction: Euclid's fourth proportional

PROBLEM Straight from Euclid's proposition on the fourth proportional, here is a surprisingly intriguing historical challenge posed on Euclidea, a mobile app for Euclidean constructions. I have a ...
0
votes
1answer
109 views

Construction of equilateral triangle with one vertex given inside an acute angle

If I am given an acute angle in the plane and a vertex $A$ located inside that angle, how would I construct an equilateral triangle $ABC$ such that $B$ is on one side of the angle and $C$ is on the ...