# 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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### Construct any regular polygon that has the same area as the sum of $n$ given triangles

Original question: Construct any regular (or similar-scaled to a given) geometric shape that has the same area as given triangle? My idea is application of generalized Pythagora's theorem. Euclid ...
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### Can $\pi$ be approximated by considering polygons with increasing number of sides, but without using circles or trigonometry?

Question in title. Although it should say “regular polygons”, not just “polygons”. When I say "without using circles", I mean without circle constructions. Properties like perimeter and area and ...
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### What's the fastest way to split $45-45-90$ triangle into two parts with equal area (using a straightedge and compass)?

What's the fastest way to split $45-45-90$ triangle into two parts with equal area (using a straightedge and compass)? I know a similar question has been asked before, but I'm asking how to do this ...
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### Straightedge-and-compass construction of the “kissing circles” for three given circles

Let $C_1,C_2,C_3$ be three mutually tangent circles. Call the circle tangent to all of them (that is, intersecting each at one point) and enclosed within the region between them their kissing circle. ...
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### The generality of “neusis plus”

Consider the following progression of facts relating to geometric construction with a limited set of tools: With only a compass and unmarked straightedge, we can only construct numbers that lie in a ...
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### Rationally bracing a rigid regular nonagon

My previous questions on rigid pentagons and heptagons are part of a pet project of mine to make (aka brace) rigid regular polygons using only rational-length sticks. To recap: Hinges can be placed ...
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### Geometry problem - proving atleast three lines are concurrent! [closed]

Each of the given $9$ lines cuts a given square into two quadrilateral,whose areas are in ratio $2:3$. Prove that at least three of these lines pass through the same point. how to approach this ...
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### Proving correctness of a braced regular heptagon from a trigonometric identity

This is a rigid regular heptagon I found on Wikipedia during associated research for my question on rigid pentagons: The accompanying text reads The construction includes two isosceles triangles ...
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### The power of Archimedean spirals: is there an algebraic characterization of Archimedean numbers?

In his famous treatise On spirals, Archimedean used a spiral to square the circle and trisect an angle. There are known algebraic characterizations of the numbers constructed with compass and ...
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### Rigid pentagons and rational solutions of $s^4+s^3+s^2+s+1=y^2$

Gerard 't Hooft, Nobel Prize in Physics laureate, wrote three articles on what he called "Meccano math" (1, 2, 3) – rigid constructions following rules quite similar to my earlier question on doubling ...
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### What is the length the side of of a square inscribed in a triangle?

What is the length of the side of a square inscribed in a triangle? This was inspired by this Numberphile video which showed multiple ways to construct the square with a side on one side of an acute ...
### What is a geometric interpretation of $0x+0y=0$
What is a geometric interpretation of $$0x+0y=0.$$ I would think of this as a plane... Am I right?