Questions on the construction of geometrical figures using a limited set of "tools".

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1answer
19 views

Construction of a point with resprect to a triangle

For a given triangle $ABC$, how to construct a point $P$ such that $PA \colon PB \colon PC = 1 \colon 2 \colon4$?
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1answer
32 views

How to embed this circle tangent to the other circles?

I want to construct a circle that would be tangent to the $3$ circles and would have its diameter lie somewhere on the segment $BI$. $EF$ includes the diameters of the $3$ given circles. $EB=BF$. ...
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17answers
2k views

How to create circles and or sections of a circle when the centre is inaccessible

I am doing landscaping and some times I need to create circles or parts of circles that would put the centre of the circle in the neighbours' garden, or there are other obstructions that stop me from ...
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1answer
39 views

A construction of a triangle mapping with a homothety

Given an acute triangle $ABC$ draw a triangle $PQR$ such that $AB=2PQ,BC=2QR,CA=2RP$, and the lines $PQ,QR,RP$ pass through $A,B,C$ respectively. Note $A,B,C,P,Q,R$ are distinct. This is a problem ...
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0answers
24 views

Geometric question (middle line of a parallelogram)

Let $ABCD$ be a parallelogram and $I,J$ are two points such that $AI = ID$ and $BJ = JD$. Show that ($IJ$) is parallel to ($AB$).
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0answers
30 views

Geometrical construction of 7th roots of unity given parabola x^2

Given parabola $x^2$ on plane, how can I construct 7th roots of unity? I was straying with my only idea, that sum of squares of real and imaginary part of roots of 1 equals 1 and belongs to ...
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0answers
16 views

Given a pair of conics, construct (synthetically) their shared tangent lines

There are various well-known ways to construct the common tangents to a pair of circles; this is an easy one. I also just learned that we can use Pascal's Theorem to construct a tangent to a conic at ...
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1answer
20 views

Geometric proof of dot product distributive property

I'm working my way through a text book for fun in order to keep my math brain fresh and came across this simple yet perplexing problem. "Demonstrate geometrically that the dot product is ...
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1answer
33 views

How to prove by induction the constructibility of a line segment of length $\sqrt{n}$?

How to prove the following statement by induction? If a line of unit length is given, then a line of length $\sqrt{n}$ can be constructed with straightedge and compass for each positive integer $n$. ...
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0answers
27 views

Straightedge-Compass Construction vs Origami

I've been looking into these two types of geometric construction and I was wondering- why is origami capable of solving up to cubic equations, when straightedge-compass construction is only capable of ...
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1answer
52 views

Prove : for every integer $n\ge1$, if the regular $n^2$-gon is constructible, then $n$ has no odd prime divisors.

For every integer $n\ge1$, if the regular $n^2$-gon is constructible, then $n$ has no odd prime divisors. I know is has something to do with the fact the output Euler's Totient function on $n^2$ ...
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0answers
29 views

Dummit & Foote: Construction of the regular $17$-gon

In the last step of the exercise where we construct the $17$-gon, I have to draw a circle with a diameter whose endpoints are $(0, 1)$ and $(\eta_1', \eta_4')$ and show that it intersects the positive ...
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1answer
40 views

Is the given triangle unique?

I was reading Polya's How to Solve It when I came across the following problem. Construct a triangle with an angle, the length of altitude through that angle and the perimeter of the triangle given. I ...
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1answer
25 views

Ramanujan (LPS) graph construction

In the original LPS article they mention two constructions for Ramanujan graphs, one with order $q^3$ nodes and the other in order $q$. My question is regarding the second construction (which is ...
2
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2answers
45 views

How to construct the point of intersection of a line and a parabola whose focus and directrix are known?

I found this problem in Polya's "How to solve it". It goes as follows Using only a straight edge and a compass, construct the point(s) of intersection of a given line and a parabola whose focus ...
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0answers
15 views

calculate the radius of a sphere-like cloudpoint(not in the origin)

how would you calculate the radius of a piece of a sphere-like(sphere with noise) cloud point matrix(The cloud point represents the surface) which center is not necesarily in the origin. if it was ...
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2answers
58 views

Logic behind a Geometric Construction of regular heptadecagon

I'm reading a Chinese book "Methods of Mathematical Physics" by Wu Chongshi. During introduction of complex analysis, it explains a Geometric Construction of regular heptadecagon. Task: to achieve ...
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0answers
43 views

proof of construction of angle 360/17

One construction for a 17-sided polygon is shown below. ...
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0answers
26 views

To construct a right triangle given the hypotenuse and sum of two legs [duplicate]

NOTE: I want a hint only. A compass and a straightedge construction:Given a hypotenuse and the sum of lengths of the legs,we need to construct a right triangle. MY TRY: From any ray $BE$, ,let ...
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2answers
28 views

Some help needed with a geometry question

What is a formula for all integers n for which a regular polygon with n sides can be constructed using a ruler and compass construction?
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0answers
11 views

Minimum components required to construct a unique polygon

What is the minimum components required to construct a unique n sides polygon? If k is a such number for n sided polygon for any k given components polygon can be constructed by compass and ...
0
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1answer
17 views

construction of a line.

Two non parallel lines l & m are given. For given two angles A & B we have to construct a line n such that it makes angles A & B with lines l & m respectively. Line n intersects l ...
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2answers
27 views

Construction of pentagon with all sides and alternative two angles given.

All sides of a pentagon ABCDE are given. Angles A & C are given. Can such pentagon be created by straightedge and compass?
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1answer
13 views

Geometric Sequence with Normal Distribution Problem

Given: The running time (in seconds) of an algorithm on a data set is approximately normally distributed with mean 3 and variance 0.25. a. What is the probability that the running time of a run ...
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1answer
23 views

Constructing “strange” sets by removing elements: Is there any comparison?

Recently I had a thought about some sets with special properties that are constructed from a starting set $A$ and successively removing elements from it. I am not an expert in set theory but perhaps ...
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0answers
28 views

geometric construction puzzle is my construction right?

I think I solved my own question of hyperbolic geometry (and circle ) construction problem but am not sure if it is correct, it looks that way but can somebody help me with prooving it? My ...
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2answers
80 views

triangle construction given side, angle and median

I can't figure out the solution to this, it looks to me like it doesn't have any solution but I need some proof. problem: Construct a triangle ABC with given $a=6 cm$ $\alpha=75^\circ $ and ...
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2answers
41 views

Construction by compass and straightedge.

l,m,n are three concurrent line concurrent at point A. Given a point B on line l. Is it possible to construct point C on line n such that line m is a median of triangle ABC
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1answer
86 views

What are some alternative ways of describing n-dimensional surfaces using control points other than Bezier surfaces?

I'm interested in problems involving geometric constraints and curve subdivision. I noticed that most of these problems describe the curves/surfaces using the Bezier form. I wanted to know if there ...
10
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1answer
81 views

Using a compass and straightedge, what is the shortest way to divide a line segment into $n$ equal parts?

Sometimes I help my next door neighbor's daughter with her homework. Today she had to trisect a line segment using a compass and straightedge. Admittedly, I had to look this up on the internet, and ...
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0answers
17 views

Open Construction Problems [duplicate]

Are there still any problems in ruler-and-compass construction that remain unsolved? Or have all of the open questions already been answered?
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1answer
78 views

To draw a straight line tangent to two given ellipses

How can I draw a a straight line that touches two ellipses? There are, like for two circles, 4 different solutions. I´m not interested in the analytical solution, but in the geometrical drawing, ...
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1answer
113 views

hyperbolic geometry (and circle ) construction problem

Was thinking about hyperbolic geometry, the Poincare Disk Model and Sweikarts constant and combined them all in a construction puzzle that I was unable to solve. My construction puzzle: Given: A ...
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0answers
61 views

Computing an area with geometric methods

Consider the construction below (it is self-evident, no tricks or misleading drawing) What is the area enclosed by the curve that goes through the points H,J,M,L ? I solved the problem using ...
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1answer
43 views

Geometric construction of 2 soap bubbles meeting

When 2 soap bubbles meet, you get a dividing wall between the soap bubbles. Plateau concluded that the three spheres (2 soap bubbles and the dividing wall) meet at angles of 120. Here's an ...
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1answer
57 views

What are four ways to quadrisect any triangle?

What are four ways to quadrisect any triangle with compass and straightedge? I have a few already: Draw a median and from the midpoint, draw two medians to the remaining sides. Draw a median and ...
2
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1answer
119 views

The Basel Problem and Theodorus' Spiral

I've been trying to find a classical solution to the famous Basel Problem solved by Euler. To those unfamiliar the problem is to find the sum infinite series made up of the reciprocals of square ...
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1answer
47 views

Why do people not use “partially directed” graphs?

Are there structures in use which are a mix of directed and undirected graphs? I.e. the effective edge-set consists of both directed and undirected vertex pairs. In the case that the graph is simple, ...
10
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1answer
343 views

Thinking outside of the box

You want to draw a circle with a 4 inch radius. A trivial task for you and your trusty compass. When you go to grab your compass which has not had much love for a while you find it is rusted shut; ...
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1answer
77 views

A problem of forming equal angles in plane geometry

C and D are two points on the same side of a straight line AB. Find a point X on AB such that angles CXA and DXB are equal.
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5answers
328 views

How to construct three mutually orthogonal circles in stereographic projection?

I'm new to spherical geometry and I enjoy doing ruler-and-compass constructions, so I'm trying to teach myself to do them in stereographic projection. I'm finding it challenging, to put it mildly. ...
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1answer
52 views

How to construct hyperbolically equidistant points on a line?

In Stillwells' "Sources of Hyperbolic Geometry " page 66 figure 3.3 shows an ((incomplete?) construction of hyperbolically equidistant points on a line. I tried to reconstruct the figure but did ...
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1answer
62 views

Constructing image under homography from known information

If $f$ is a homography of the real projective line, $f^2=id$ (is an involution), and $f$ has exactly two fixed points, how can I construct (geometrically) the image of an arbitrary point?
3
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1answer
278 views

Geometrical construction problem (2)

There are given: $[BC],[DE]$, the angles $\phi_1$, $\phi_2$ and a point $A$. Build a isosceles triangle $MAN$ (with$\angle{MAN}$ = $\pi/2$) with the apex of the right angle in $A$ such as ...
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1answer
31 views

Geometrical place of one point

Through one of the intersection points of two given circles, a line is build which intersects 2nd time the 2 circles in $A$ and $B$. Determine the geometrical place of the middle of $AB$. thanks for ...
3
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3answers
223 views

How to build a trapezoid

Build a trapezoid knowing its diagonals, the angle between them, and, also, the sum of $2$ adiacent sides. I appreciate your time and help!
0
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1answer
95 views

what is the most sided sturdy regular n-polygon that can be made with lego?

Was puzzeling with this question: What is the most sided regular n-polygon that can be made with lego? It has to be sturdy (the polygon should stay in shape when pushed around) made with the normal ...
5
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1answer
162 views

Could Euclid have bisected a line segment without his method of superposition?

In Book I Proposition 10 of the Elements, Euclid performs the bisection (i.e. finding a midpoint) of a line segment. In the course of doing so, he uses Book I Proposition 4, the Side-Angle-Side ...
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3answers
96 views

Construct triangle given inradius and circumradius

If we know the inradius $r$ of a triangle and the circumradius $R$ we can find out the distance between the incircle $I$ and the circumcircle $O$: $OI^2 = R^2-2Rr$. Therefore we can draw the incircle ...
5
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1answer
475 views

History of Compass/Straight Edge Construction

I'm interested in learning the origin of compass/straight-edge constructions. In particular, I am interested in the historical interplay between Euclid's axioms for plane geometry, and ...