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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Constructability of algebraic elements over $\mathbb{Q}$ [duplicate]

I'm currently reading upon the classical straightedge and compass constructions and I came across the following theorem: lf a real number $c$ is constructible, then $c$ is algebraic of degree a power ...
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I'd like to find the center of a circle of fixed radius inscribed within a corner

There are plenty of resources on how to find a circle inscribed within a triangle, with three edges given, however this is not that. I have defined two non-unit vectors on the 2D plane representing ...
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Construct the sizes of the Thales triangle over the comparative parallelogram of a trapezoid

Given are $b, d, e, f$. Can I determine the lengths $q, r$ of the right triangle over the comparative parallelogram, solely by considering symmetry and intercept theorems or angle theorems? If $q$ ...
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How to interpret the ratio $\frac ab=\frac{p^2}{q^2}$ of line segments $a,b,p,q$

Let $a,b,p,q$ line segments (e.g. $a=3\text{cm}$ and so on). How can I interpret or construct the ratio $ \dfrac{a}{b}=\dfrac{p^2}{q^2}$ ? Is this something I can find on a suitable triangle or ...
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Construction by straightedge and compass of a trapezium from the two diagonals and the two non-parallel sides

Let the diagonals and the two non-parallel sides of a trapezium be given: $b,~ d,~ e,~ f$ and, as we know, $a \parallel c$ (the lengths of $a$ and $c$ are unknown). (How to calculate the missing sides ...
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Construction of circular arcs through vertices of a rectangle, tangent to the diagonals?

I know that you can construct the tangent line to a circle, for example, but my question is in the reverse order: If one draws a rectangle, can you use a compass and a straightedge to construct the ...
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Construct triangle $ABC$ given point $C$ and the lines that contain the angle bisectors of angles $A$ and $B$.

The problem is as stated in the title: Construct triangle $ABC$ given point $C$ and the lines that contain the angle bisectors of angles $A$ and $B$. I'm slightly confused about the whole ...
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Construction of segment of given length through an intersection of two circles

"Through an intersection point of two circles, draw a secant such that its segment inside the given disks is congruent to a given length. Hint: Construct a right triangle whose hypotenuse is the ...
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Trapezium from two diagonals and two non-parallel sides

Is it possible to construct (and to calculate) a trapezium from it's two non-parallel sides and it's two diagonals, with other words $b,d,e,f$ are given: I read out the equations system $\begin{...
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Given point $K$ in a triangle, is it possible to construct a line through $K$ that bisects the area of the triangle?

$K$ is an arbitary fixed point inside $\triangle ABC$. Is it possible to construct a line that pass through point $K$ such that area of $\triangle AJH =\triangle ABC/2$
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Is it possible to construct the angle bisector of two lines with only a right-angled ruler?

I have a problem with a special construction. Is it possible to construct the angle bisector of two lines with only a right-angled ruler? Here with a right-angled ruler one can performe the ...
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Construction of a line making equal intercepts on $AC,AB$ on opposite sides of $BC$.

In triangle $ABC$, $D$ is an arbitary point on side $BC$. I need to construct a line that pass through $D$ such that $CF=EB$. (point $F$ lies on the extension of of side $AC$) By Menelaus's theorem $$...
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Use straightedge (no compass) to find the centers of two intersecting circles

We can use the compass and straightedge to find the center of one circle. We have proven we cannot find the center of a circle with straightedge alone (see: http://www.cut-the-knot.org/impossible/...
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Given a point, a line and a circle; how can I find a point on the line having the following property?

First of all: I searched with different search engines and on StackExchange, but did not find a solution to my problem. This may be because its specific description is kind of long. For the same ...
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Construction of a Regular 5-gon

We were given the following true or false question in our Algebra exam Q) The regular $5$-gon is not constructible by using a straight edge and compass. The answer is false but we also had to provide ...
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Given a quadrilateral with 4 equal areas, prove that it is a parallelogram

I have the next quadrilateral with midpoints E F G H. The source of the problem is my class of geometry, I read the book but I don't find anything related to this. I found in Google about Varignon's ...
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For which rational coefficients is $a+b\alpha+c\alpha^2+d\alpha^3+e\alpha^4$ constructible, where $\alpha=3^{1/5}$?

Let be $\alpha=3^{1/5}$, and be $\gamma$ the number $$ \gamma=a+b\alpha+c\alpha^2+d\alpha^3+e\alpha^4 $$ with $a,b,c,d,e \in \mathbb{Q}$ . For which $a,b,c,d,e \in \mathbb{Q}$ is $\gamma$ ...
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Finding the focus of an ellipse with ruler and compass

Given an ellipse E, find his focus with ruler and compass. I tried to generalize the next theorem: Given a circle C find his center. The idea for prove this is with an chord and it's bisector line (...
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Symmedian and orthic triangle properties

I am trying to prove the following lemma that may be useful for junior-level international contests: Given $\Delta ABC$ an acute triangle and $(BE)$ and $(CF)$ its altitudes. Let's consider $(AM)$ ...
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A doubt about what is straightedge and compass construction

The doubt comes from a try. To bisect a segment, I wonder whether I can first choose an arbitrary degree of opening for the compass, then use the compass to measure the segment until I can measure it ...
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Circle of Apollonius; proving a tangent line

Quick recap, since there are several circles of Apollonius: Given a triangle with fixed base and the other two sides in known ratio, the circle of Apollonius (of the first type) gives the possible ...
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Is it possible to construct a given cube?

Is it possible to construct, with unmarked ruler and a compass, a cube whose volume equals to a given sphere? I tried to construct the given cube but I can't see a light of constructing it, maybe it ...
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Construct the vertex of an angle given three points on the plane.

Suppose you are given three points A,B,C on the plane. A is on the bisector of angle XOY, B is on the side OX and C is on the side OY. How can you find the locus of the vertices(find all possile ...
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Constructing a circle which is tangent to a line and circle and passes through a point. [duplicate]

I am trying to solve a geometry problem. as first step I am trying to construct it on Geogebra: To complete the construction I need to draw a circle which passes through the point $O$ and is tangent ...
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Constructing an isosceles trapezoid with a specific decomposition into triangles

A recent question asked about finding the ratio of the bases for the following isosceles trapezoid: That problem has been solved, obtaining a result of $|CD|/|AB|=1-1/\sqrt{2}$. What I'm curious how ...
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Construction of a rhombus with given conditions

given two parallel lines $P$, $Q$ and two points $X$,$Y$ how to construct a rhombus $ABCD$ passing through $X$,$Y$ and opposite sides lie on $P$ and $Q$. I solved a special case of this problem where ...
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Constructibility of regular hexagon with given area

I would like to find the algebraic translation of the following problem: Let $r>0$ be a real number. For some $r$, the regular hexagon with area $\pi r^2$ is not constructible (with straightedge ...
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Construct the angle whose sine is $\frac{3}{2+\sqrt{5}}$

I saw this question in an old trig text book: Construct the angle whose sine is $\frac{3}{2+\sqrt{5}}$. I ask: What solution can people here give ? Is there a solution that does not (like mine) ...
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Is there any solution for converting any irregular shape to any constructible regular polygon without changing area through pure construction?

Is there any solution for converting any irregular shape to any constructible regular polygon without changing area through pure construction?
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if I have a length AB, defined by points A and B, how do I construct a circle with circumference AB with a compass? [closed]

I want to construct a circle with a given linear circumference with compass and straightedge. I know that you cant construct a length ¨pi¨ if the linear distance is ¨1¨, but can you find the radius? I ...
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Construct a triangle given 2 sides and the radius of the inscribed circle (Euclidean geometry) [duplicate]

How do I construct a triangle when I am given 2 sides and the radius of the inscribed circle. Keep in mind the problem must be solved in euclidean geometry so we assume that we don't know trigonometry ...
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To cut a line in two so that the squares of … (geometric construction)

Cut a given straight line so that the sum of the square of one part and twice the square of the other part equals a given size. Given a line of length $L$ which is cut in two pieces, lengths $x$ and $...
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What other angles apart from the right angle can be trisected?

Is there a way to characterise the set of all angles $0°<\phi<360°$ such that $\phi$ can be classically trisected? (That is, the trisection can only be done with a finite sequence of straight ...
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Scaling a parallelogram to fit a triangle

How do I, with straightedge and compass, scale the red parallelogram to the green one, given the blue triangle so that both top corners of the green parallelogram touches the triangle?
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Simplifying a radical-trigonometric expression for the hendecagon angle

This question is related to my very first question on this site, on constructing the hendecagon. The Gleason paper I referred to states the following identities, which lead to constructions of a ...
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How to solve this analytical geometry problem?-parable inscribed within a square

This problem appeared on the network, and although it looks simple I am not sure of the result. The polygon $ABCD$ is a square with side $4$ cm and the curve inscribed inside the square is a parabola,...
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Existence of a special pentagonal tiling

Is there a pentagonal tiling composed of only one shape of pentagon so that each pentagon touches exactly 5 other pentagons? Two pentagons are in touch if they share at least one common point. Few ...
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Geometric construction (problem from Swedish 12th grade ‘Student Exam’ from 1932)

The following problem is taken from a Swedish 12th grade ‘Student Exam’ from 1932. Inscribe in a given circle a quadrangle in which de two diagonals have given lengths and, in addition, where the ...
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What is this curved surface constructed inside a rectangular prism?

Given is a rectangular prism with one side $\square ABCD$ and its opposite side $\square EFGH$. Point $J$ lies on $\square ABCD$ some distance $y$ between diagonal $AC$ and edge $AB$, and point $K$ ...
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Constructing a triangle given an angle, the side opposed to this angle, and the median to the given side.

I came across the following problem in my Euclidean Geometry text: Construct a triangle having given an angle, the side opposed to this angle, and the median to the given side. I have tried solving ...
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Construction in Euclidean Geometry

I came across the following problem in my Euclidean Geometry text: Construct a triangle given the ratio of an altitude to the base, the vertical angle (the angle opposite the base), and a median to a ...
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Construct a triangle given side length $b$, the altitude for side $c$, and the angle bisector of $B$.

Construct a triangle given side length $b$, the altitude for side $c$, and the angle bisector of $B$. So far I only found that I can find the angle at $C$ ($\gamma$) by constructing the right ...
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Demonstration of the impossibility to draw a parallel through a point using only a straightedge.

From the responses to this question , it appears to be well know that it is impossible to trace a parallel to a straight line: $\ell$ through a point: $P$, using exclusively a straightedge. ...
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Show that a triangle delimited by parallel lines to the sides of another triangle is similar to said triangle.

I was looking at geometric constructions of similar triangles, and at one point I came across the statement in the question. A triangle delimited by vertices which are intersections of lines parallel ...
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Help with a problem statement: Vertex of a parabolic segment

I found this website related to parabolic segments/sections. I'm interested about the proposition III which states: Proposition-III: Let A be the midpoint of the segment SS'. And let E be the feet of ...
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Constructing triangle $\triangle ABC$ given median to the side $c$ and angles $\alpha$ and $\beta$

Constructing triangle $\triangle ABC$ given median to the side $c$ and angles $\alpha$ and $\beta$ I started with the median. Then I constructed a circle to each side of the median, such that the ...
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Constructing a rhombus $ABCD$ with $A$ a given point, $B$ on a given circle, and $C$ on a given line, such that $\angle A=60^\circ$ [closed]

Given a point $A$, a circle $b$ and a line $c$, construct a rhombus $ABCD$ such that $B \in b$, $C \in c$ and $ \measuredangle A = 60^\circ $. Hello! I saw this geometry problem in a textbook and I ...
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Visual proof for $[\triangle ABC] = rs$?

I'm teaching a high school geometry class and we just got to the part where we proved that the area of a triangle $[\triangle ABC]$ is equal to the inradius $r$ times the semiperimeter $s$. A student ...
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Constructing a Quadrilateral

I am given three sides of length 4.5 and one more side of length 4.2 (dimensionless entities). Also the area is given as 19.575. Now I have been given the task to construct a quadrilateral with these ...
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Construct center of circle without using its interior

I need to find the center of a circle using compass and ruler (or preferably just a straightedge) without using the interior of the circle. In other words, no line or point can be used which is inside ...

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