# 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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### 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 ...