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

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55 views

Proof of necessary condition for constructibility of a number

I'm reading a proof of the necessary condition for a real number to be constructible, and it seems to leave out a few details that I can't really fill in. This is what I understand so far. We have to ...
0
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0answers
107 views

Application of Compass-and-straightedge construction

Nowadays with computers, Compass-and-straightedge construction doesn't look useful from my point of view. Probably I am just too narrow-minded, So I'm just curious, could anyone tell me any ...
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2answers
233 views

Why only compass and straightedge?

I've read and watched some lectures on euclidean geometry - not so advanced but I've seen the focus on constructions. Two instruments are used, compass and straightedge, I had the following doubts: ...
2
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1answer
105 views

Drawing Euclid?

I decided to study Euclid for fun. I have Oliver Bryne's edition. I also want, as much as possible, to construct the figures myself, to get a deeper understanding. How did people traditionally do ...
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0answers
131 views

Spherical Construction Problem about using a ruler and a compass

I've known the following theorem: Theorem 1: On a plane, if we have both of a primitive ruler and a primitive compass, then we can do the same construction as we can do by using a macro-ruler or a ...
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2answers
2k views

Finding a line perpendicular to a line and passing through the intersection of other two lines

So here is the question as in my text book Find the equation of the line through the intersection of $2x - 3y + 4 = 0 $ and $3x + 4y - 5 = 0$ and perpendicular to $6x - 7y + c = 0$ so I ...
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1answer
71 views

How to draw or construct a brachistochrone

Since the brachistochrone is such a beautiful curve in our planet, I want to build one somewhere around 1.60 m high. I need a quick way to trace the curve on the material to be cut, e.g. a wide sheet ...
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0answers
199 views

Minimal number of moves to construct the challenges (circle packings and regular polygons) in Ancient Greek Geometry?

In the web game Ancient Greek Geometry, there are challenges to construct regular polygons and circle packings using ruler and compass constructions. The game measures the number of line and circles ...
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1answer
82 views

Occurrence of $e$ in intersecting circles.

Consider two identical circles that share a radius such that they intersect. The radii of the circles are $\pi\over 2$. If this new shape sits such that its major axis is horizontal and the shortest ...
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0answers
192 views

Can one trisect $\arccos(6/7)$?

Is this proof correct? Proof: Here $\theta = \arccos(6/7)$. Now to show we can't trisect $\theta$, we show that $\theta/3$ is not constructible by finding the irreducible polynomial in $\mathbb ...
1
vote
1answer
446 views

Equilateral and equiangular polygon

Can we have an equilateral polygon $n \geq 5$, which is not equiangular? Ot does every odd n-gon which is equilateral must be equiangular? Is a construction of an equilateral but not equiangular n-gon ...
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1answer
93 views

Descartes' theorem: Find the midpoint of the $4$th circle

I'm thinking about Descartes' theorem "Wikipedia". I understood how to find the radius with algebra. Now I'm trying to use ruler and compass to find the midpoint of the 4th circle. I thought about ...
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1answer
5k views

How can I construct a square using a compass and straight edge in only 8 moves?

I'm playing this addictive little compass and straight edge game: http://www.sciencevsmagic.net/geo/ I've been able to beat most of the challenges, but I can't construct a square in 8 moves. To ...
2
votes
1answer
225 views

Geometric proof that if n is a non-perfect square, then √n is irrational.

I know there is a geometric proof of the irrationality of √2. I thought maybe this one could be generalized for √n when n is a non-perfect square, but I could not find something like that anywhere. ...
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3answers
571 views

Pair of compasses drawing a square (from children's fiction)

I have read a children's book where alien race of "square people" used a pair of compasses that drafted a perfect square when used. Now I wanted to explain to the child that it is not possible to have ...
5
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0answers
284 views

To construct an ellipse, being a projection of a great circle, given two points on it

I'm looking for a geometric construction which would allow me to draw an ellipse, which is supposed to be an orthographic projection of a great circle of a sphere, given two points on it. The ...
4
votes
2answers
120 views

Construction of triangle

I don't know how to prove or disprove the following problem: How to construct triangle if elements $a$, $b$, $\beta-\gamma$ are given? Is it constructible (if not, how to prove it)? Any help is ...
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2answers
76 views

Are the lengths from this recursive construction a geometric sequence?

In his 1999 review of Edward Tufte's Visual Explanations in the Notices of the AMS (third page), Bill Casselman gives a very pretty proof of the irrationality of the golden mean. More precisely, ...
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2answers
302 views

Importance of construction of polygons

Why did the ancient Greeks give so much importance to the construction of regular polygons with $n$-sides using only ruler and compass and tried to study for what $n$ was such a construction possible? ...
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0answers
119 views

Software to Explain Geometrical Construction

I would like to know about a software that will help me SHOW construction of geometrical figures. Maybe something as simple as constructing a triangle. BUT I need to see a compass and a Straight ...
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0answers
45 views

Geometric calculation: two kneading discs

I have two kneading discs of a screw overlapping with each other at 60 deg. I know the cross section area of one disc and I want to know what will be the overlap area if the other disc is rotated at ...
6
votes
2answers
475 views

Equilateral triangle geometric problem

I have an Equilateral triangle with unknown side $a$. The next thing I do is to make a random point inside the triangle $P$. The distance $|AP|=3$ cm, $|BP|=4$ cm, $|CP|=5$ cm. It is the red ...
3
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2answers
118 views

Apollonius circle

I'm given two points, $A$ and $B$, and two lengths, $b$ and $c$. I need to find the locus of point $C$ such that $BC:AC=b:c$. This link describes Apollonius circle of first type, but I can't seem to ...
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3answers
337 views

Constructing irrational numbers

Which of the following numbers is constructible? 1) $3.14141414\ldots$ 2) $\sqrt{3}$ 3) $5^\frac{1}{4}$ 4) $2^\frac{1}{6}$ Also, Given a segment of length $\pi$, is it possible to construct, ...
2
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2answers
187 views

Doubling the cube neusis

Can anyone explain me in simple maths the neusis construction at http://en.wikipedia.org/wiki/Doubling_the_cube#Using_a_marked_ruler? Why and how does it produce the $\root 3 \of 2$?
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2answers
224 views

Is it fair to say that Kepler's equation involves squaring the circle?

I'm trying to understand the degree to which I'm at sea, anchored, or on firm ground, and how to firm up my understanding as needed. I think Kepler's equation reduces to a task (call it $t$, which ...
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1answer
123 views

How to draw right triangle whose hypotenuse is side of square and whose side is tangent to a circle?

Given a circle whose center (P) is at center of a square, and using pencil, compass, and straight-edge, how can I create a right triangle whose hypotenuse is side of square and whose side is tangent ...
3
votes
1answer
251 views

Finding a circle that touch two other circles and a line

Given two circles $(x1, y1, r1), (x2, y2, r2)$ and a line passing through two points $A(xa, ya)$ and $B(xb, yb)$. How to find a circle $(x3, y3, r3)$ that is tangent to line and two given circles? I ...
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5answers
692 views

Sangaku. How to draw those three circles with only a ruler and a compass?

I found in a book of Sangakus the following problem. Let $R_b$, $R_g$ and $R_r$ the radiuses of the blue, green and red circles $C_b$, $C_g$ and $C_r$. Prove that ...
0
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1answer
273 views

Construct two chords of equal length through points A and B (two arbitrary points INSIDE a circle) that are perpendicular to each other.

Its a construction problem I am having trouble with. I realize I need to use rotations and/or other isometries but I am really stuck. Any help would be really appreciated! Thanks!
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vote
1answer
495 views

Draw a parallelogram given its sides and the angle between diagonals

I'm having trouble with this one: Draw a parallelogram knowing the lengths of its sides and the angle between the diagonals. Bonus points if the answer uses a translation, because that's where this ...
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1answer
39 views

proving that this family of angles, cannot be trisected

Given a field $ \Bbb Q \subset K \subset \Bbb C$. One can prove that $\beta \in \Bbb C$ is constructible over $K$ iff the galois group of the minimal polynomial over $K$, $m_{\beta}(x)\in K[x]$ is a ...
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1answer
124 views

Proving there is a rectangle inside an hexagone.

How can I prove this : Let a regular hexagon ABCDEF and M, N, R and S the respective midpoints of AB, CD, DE and FA: i) Prove that the MNRS is a rectangle. ii) compare the area of MNRS and the area ...
4
votes
2answers
414 views

Construct a triangle given one side, its height and inradius

I've been scratching my head with this problem: "Draw a triangle given one of its sides, the height of that side and the inradius." Now, I can calculate the area and obtain the semiperimeter. From ...
2
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2answers
497 views

Resource for learning straightedge and compass constructions

Does anyone know a good resource for learning about straightedge and compass constructions besides "The Elements?" I tutor geometry to middle-schoolers and high-schoolers and thought that including ...
2
votes
1answer
63 views

Construction of a field

Given the polynomial $$f(x)= x^4-16x^2+4$$ which has $a=\sqrt 3+\sqrt 5$ as one of its roots in $\Bbb C$, can you use $f(x)$ to construct a field $E$ of the form $Q[x]/I $ for some appropriate ideal ...
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3answers
75 views

Prove that altitude² = pq?

The following is a question for my math class. I just cannot figure it out. Given is that: h is the altitude that divides the longest side of this right triangle into p and q. Question: Prove that ...
4
votes
4answers
714 views

Constructing a triangle given three concurrent cevians?

Well, I've been taught how to construct triangles given the $3$ sides, the $3$ angles and etc. This question came up and the first thing I wondered was if the three altitudes (medians, ...
3
votes
3answers
130 views

Construct tangent to a circle

Using a ruler and a compass how can construct a line through a point and tangent to a circle. What I don't want is to eyeball the line by trying to line-up the ruler over the circle. Best if I could ...
2
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4answers
113 views

Construct $\frac{1}{z}$ from $z \in \mathbb{C}$

Suppose $z \in \mathbb{C}$. How can we construct $\frac{1}{z}$ with tools without calculating? My teacher suggested something with a parallel line, but I couldn't figure it out. A unit distance is ...
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3answers
866 views

Construction of a right triangle

It's a high school level question which we can't seem to solve. Here it is: Given 2 lines, one of the length of the hypotenuse and the other with the length of the sum of the 2 legs, construct ...
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2answers
152 views

Construct two circular arcs meeting at a common point

Suppose we are given a ray $\rho_a$ beginning at point $a$ and a ray $\rho_b$ beginning at point $b$. I want to find a circle $C_a$ tangent to $\rho_a$ at point $a$ and another circle $C_b$ tangent to ...
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1answer
128 views

Straightedge Only Construction of Tangents to Circle

Currently, there exists a question regarding straightedge only constructions; however, my specific question pertains something that is not found in that thread, and I do not think it will be answered ...
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0answers
114 views

Ruler-and-Compass metric

Let $\nu \left( x \right) $ be the least number of steps that is required to construct a constructible length $x$, using compass and ruler in the well known fashion. Now, define the distance ...
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1answer
77 views

Contracting an angle (using straightedge and compass)

In my field theory lecture notes I have it that a regular polygon with $n$ sides is constructable iff $\zeta_{n}=\frac{2\pi}{n}$ is constructable. Shouldn't this be $\frac{\pi}{n}$ instead of ...
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1answer
256 views

How can I construct a 2^63-gon with a straightedge and compass?

I entered 2^63 as a stand alone value at WolframAlpha. Among the responses was a factoid that 'A regular 9223372036854775808-gon is constructible with a straightedge and compass.' What is such a ...
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2answers
170 views

Constructing $\sqrt{a}$ for a constructable $0\leq a\in\mathbb{R}$ - Compass and straightedge constructions [duplicate]

Possible Duplicate: Compass-and-straightedge construction of the square root of a given line? I wish to understand how to construct $\sqrt{a}$ for a constructable $0\leq a\in\mathbb{R}$ , ...
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3answers
209 views

What is the cross-ratio of four lines $L_1, L_2, L_3, L_4$ if they are parallel?

What is the cross-ratio of four lines $L_1,L_2,L_3,L_4$ if they are parallel? What is the cross-ratio if $L_4$ is the line at infinity?
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1answer
224 views

Construction of touching circle

My question is: Consider five collinear points $D$, $A$, $C$, $B$ and $E$ such that $DA=AC=a$ , $CB=BE=b$. Let M be the midpoint of $DE$. Let $S_1$ be a circle with center $A$ and radius $a$, Let ...
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2answers
850 views

construct inverse point with respect to the circle by the use of the compass alone

If the given point P lies inside a circle C ,with center O,the circle of radius OP about P intersects C in two points. How to construct point P' inverse to point P with respect to the circle C by ...