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?
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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 ...
• 5,373
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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 ...
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1 vote
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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 ...
• 2,510
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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 ...
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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 ...
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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 ...
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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 ...
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 ...
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1 vote
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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. ...
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 ...
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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. ...
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1 vote
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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 ...
1 vote
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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 ...
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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 ...
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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 ...
• 7,909
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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 ...
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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 ...
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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 $AB$, ...
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1 vote
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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 ...
• 10.9k
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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 ...
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 ...
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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 ...
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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 ...
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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 ...