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.

624 questions
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
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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 ...
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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. ...
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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 ...
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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 ...
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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
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 ...
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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: $$\frac{1}{\sqrt{b+\sqrt{a}}} \ \ \text{and} \ \ \ \sqrt{a}$$ ...
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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 (...
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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: $$\frac{1}{\sqrt{b+\sqrt{a}}} \ \ \text{and} \ \ \ \sqrt{a}$$ ...
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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 ...
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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 ...
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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 $AC \cdot BC^{2}=1$? ...
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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: ...
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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 ...
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 $~BF + CE = BC$
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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$
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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 ...
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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 ...
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.
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$K$ 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 $\... 1answer 162 views The smallest parallelogram that contains a convex quadrilateral I try to find the smallest parallelogram in terms of area that contains a convex quadrilateral(A,B,C,D). I am pretty sure it must be constructed from two neighboring sides of the quadrilateral. But ... 1answer 44 views How to construct$\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$. 4answers 970 views How to construct a circle tangent to another circle and a line? How to construct a circle tangent to another circle and a line? Also Find locus of centres of all such circles? 0answers 76 views Geometrical construction of circle and circumscription of a pentagon about it. How would you draw a circle of a given radius and circumscribe a regular pentagon about it using a scale and a compass. I cannot figure it out. I have checked the internet but cannot find anything ... 3answers 368 views A beautiful geometry problem Let$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$be the points ... 1answer 561 views Construction of 3 circles touching each other externally. The context is construction of three circles with different radii so that they touch each other externally using a graduated ruler and a compass. I have done it by drawing a triangle where each side ... 1answer 85 views Epicycloid-alike curve We knew Epicycloid as a kind of trace curve of a specific point attached to a circle and rolls on another circle. But there's a limit for the ratio R/r of the radius of two circle, R and r, which is ... 1answer 2k views Is it possible to construct a regular heptagon with just compass and straightedge? Is it possible to construct a regular heptagon (a figure with seven sides) with just compass and straightedge? If so, could you please give me directions for how to do this? 2answers 97 views Construct or prove existence of a certain quadrilateral I have three questions about a quadrilateral with the following properties: It is convex. It has exactly one pair of congruent opposite sides. It has exactly one pair of congruent opposite angles. It ... 2answers 4k views How to draw a regular pentagon with compass and straightedge I remember reading that Gauss managed to construct a regular pentagon with just a compass and straightedge, but I don't remember the particulars of how he did this. Could someone help me out and give ... 0answers 47 views Ruler-and-Compass metric: draw a line from any point to separate the area of a triangle? Given a triangle$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$to two parts with equal area? 1answer 364 views Finding a line segment within an angle based on given midpoint [closed] Given an angle ∠ABC and an arbitrary point D somewhere within the angle, how would you (using only a compass and straight edge) draw a line segment where one end lies on AB and the other on BC, with D ... 2answers 77 views How to construct the scheme below? How can one construct the diagram shown below, using just a ruler and a compass, where$ \angle PYX = 2\angle PXY $and$ \angle PYZ = 2\angle PZY $: 0answers 104 views Straightedge and compass theory in three dimensions I'm looking for a reference on the theory of straightedge and compass constructions in three dimensions akin to Euclid's Elements in two dimensions. More specifically, I mean a theory of geometric ... 3answers 580 views Graphically locate the axes or foci of an ellipse from 5 arbitrary points on its perimeter. Five distinct non-collinear points are required to define an ellipse similar to the way that three non-collinear points define a circle and can be used to determine the center point of that circle. I ... 1answer 283 views Construct point on a circle such that the reflection in that point is horiztonal Let$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$-... 1answer 50 views Construct line segment with enpoints on given circles and equal to given line segment Given a line$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}$is parallel ... 2answers 370 views Neusis construction of the 11-gon? Wikipedia tells me that the 11-gon was found to be neusis constructible in 2014, and the link given doesn't seem to be a crank, but the actual method is behind a paywall. (Interestingly, the page ... 2answers 59 views Creating a circle from 3 points on its circumference when the slope of one line is infinity/undefined? I have recently run into a problem while trying to get the center of a circle from 3 points$A,B,C$on its circumference.The equations I used for this require to know the slope values of the lines$\...
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 ...