Labels: proof, Pythagoras, Square, Square inside a square

Bleaug converts Fox 306 to Fox 302 by "folding" twice.
I have seen translation, rotation, similarity, etc., but haven't seen anything like this before.
Enjoy and please do respect to human intelligence!
Click here for a SOLUTION.

To see the animation below, you will need any of the following browsers:
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Labels: Area, circle, semi-circle, simple proofs
This is related to Proposition 4 in Book of Lemmas by Archimedes.

 

 

Labels: chords, circle, General Case, proof

Yet again,

try to see the goodness,

see the beauty,

surrounding you.

Forget about the numbers, summations, subscripts.

Leave behind the accounts, stocks, papers, statistics.

Just leave yourselves to the arms of an ocean,

full of love and compassion.

Drift away with the blowing wind...

-- Dervish Fox

 

 

Bob Ryden provided a solution below.  This has been the first attempt to solve this fox. 

His solution has not been confirmed yet, but it is published here to start the discussion.  Wanna say somethin'?  Comment it out!
Labels: 3D, Convexity, Sphere, Tetrahedron, Volume

 

Let the radius of the spheres = 1.
Inside is a tetrahedron whose vertices are the centers
of the spheres. Its edge length = 2
and its volume = (1/8) √3

 
On each face of the tetrahedron, build a triangular prism.
Volume of each prism = (1/2) 2 (√3) 1
Total volume of four prisms = 4√3

 
On each edge of the tetrahedron, build a cylindrical sector.
The angle of the sector = 360 – 90 – 90 – dihedral angle of the tetrahedron
= 360 – 90 – 90 – arccos (1/3) = approx. 109.47°
Length of each cylindrical sector = 2, radius = 1
Total volume of the six sectors

= 6 π (1^2) 2 (109.47 / 360) ≈ 11.46

Finally, there are pieces of the four original spheres that are not covered by any of the above. The four pieces together make one complete sphere, V = (4/3)π.

Total volume is the sum
tetrahedron + 4 prisms + 6 cylindrical sectors + sphere
= approx. 23.52

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Labels: Capitalism sucks, Communism aint no better, Orthogonal, Poligon, proof

Note, the statement should hold in 3-D as well. Don't you agree?

And YES, "plane" is misspelled !

Why?
No, no annoying philosophical stand here.
Just pure simple layyziness.

Labels: Circle inside Square, Pythagoras, Square

 

 

Labels: Capitalism sucks, Center of Gravity, Parallel Lines, Poligon

 

Fox Weber:  Why are you doing this?  What about the human civilization.  What happened to the progress they made on internal combustion engine?

Karl Fox:  And how about the resources.  They were unlimited, right?  Those bourgeois deserve worse than this.

Fox Weber:   But there are thousands of people making a living on this: honest, hardworking people.

Karl Fox:  Oh, you must be talking about the “small people”.

Fox Weber:  You’re a pity old man, lost the cause, and now doodling fancy things. See we lost another potential sponsor, thanks to you Karli!

Karl Fox:  You’re welcome.

 

 

 

We have been delaying this for a while, but Ajit and Vihaan have already solved it. So there is no point of keeping it a secret :)
Note that this may be a general case for Foxes: 296, 297, and 298. It is also related to Fox 301.

Labels: Bisector, chords, circle, General Case, proof, Trigonometry

It would be great if there are purely geometric solutions!!

  
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Labels: Analytic Geometry, proof, similarity of triangles, Solutions, Square, Square inside a square, The truth is out there

 

Two solutions are below:

 

The solution commented out by Julian deserves to be presented here. We often had questions inspired from the "things" we observe around us. Julian's solution is the other way. It uses a very casual observation from real-life for the solution. That's perspective lines converge linearly. See the solution below:by Julian: This one has a simple, intuitive proof. Imagine a long, hollow square-prism. Something like a cardboard tube with a square cross-section. Imagine lines connecting the diagonally-opposite corners of the opposite ends. Due to the symmetry of the tube these four lines will meet at a single point half-way along it.

Now imagine looking through the tube from one end to the other. You see two squares: a large square at the near end, and a smaller square at the far end. By subtly changing the direction the tube is pointing, the far square may move off centre so that the construction looks like the problem diagram. This will not alter the fact that the four lines meet at a single point. QED!

Analytic Geometry by Bob Ryden:

Observe that proving DH implies CG as well (due to rotation).  Neat!

 

 

Labels: Bisector, chords, circle, proof

 

 

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Labels: Pythagoras, simple proofs, Square, Square inside a square
 

Labels: Bisector, chords, circle, General Case, proof

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Labels: Aint gonna happen, Bisector, chords, circle

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Labels: proof, Rectangle, semi-circle, Solutions, Tangent, Trigonometry

The options are corrected below.
There are about 5 different solutions, that we can post in coming days.

Labels: Rectangle, semi-circle, Tangent, Trigonometry

Not much time to update the blog. But, let's throw this to the public knowledge.
Note: when 3 concentric arcs are given, their center can be identified easily.
(which is a nice exercise and easy by itself).
Let us know if you have a solution.

Labels: 3 concentric circles, equilateral triangle, Geometric Construction

Aaaah, one more detail. More than one equilaterals should be drawn on 3 concentric circles.
Take the figure as it looks and do not go for the "other" equilateral.
Let the simplicity ring over the land, in the morning...
and during the night...

This may be an ancient construction problem, for which we couldn't find a reference.
May be hard or may be easy.   No idea at this time!

 
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Labels: 3 parallel lines, equilateral triangle, Geometric Construction

Commonality of the result may imply the existence of a simply solution, which we don't know.
Good luck!

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Labels: Extremum, Golden Ratio, Semi-circle, Square

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Labels:  Chords, Circle, Square, Square inside circle, Tangent

You can add comments, discuss solution.

We may have a single solution, but not sure if that's the whole story.

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Labels: Angle, Square, Trigonometry

Below is our first geometric construction problem, based on a solution submitted by Bleaug.
We have a few other construction problems in the queue.
For those who are new:
the question asks to draw triangle PQR in any equilateral triangle by using a compass, a ruler (without any numbers on it), and angle α (and a piece of paper too :)
Let us know if you have any questions!

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Labels: Angle, Equilateral Triangle, Geometric Construction

A solution is below...

GEOMETRIC CONSTRUCTION - A Beautiful Greek Antiquity:
(From Weisstein, Eric W. "Geometric Construction." From MathWorld--A Wolfram Web Resource.)
In antiquity, geometric constructions of figures and lengths were restricted to the use of only a straightedge and compass (or in Plato's case, a compass only; a technique now called a Mascheroni construction). Although the term "ruler" is sometimes used instead of "straightedge," the Greek prescription prohibited markings that could be used to make measurements. Furthermore, the "compass" could not even be used to mark off distances by setting it and then "walking" it along, so the compass had to be considered to automatically collapse when not in the process of drawing a circle.

Because of the prominent place Greek geometric constructions held in Euclid's Elements, these constructions are sometimes also known as Euclidean constructions. Such constructions lay at the heart of the geometric problems of antiquity of circle squaring, cube duplication, and angle trisection. The Greeks were unable to solve these problems, but it was not until hundreds of years later that the problems were proved to be actually impossible under the limitations imposed. (Also see Compass and straightedge constructions)

SOLUTION:

Bleaug constructs:
Here is a geometric construction (i.e. compass and ruler) of an inscribed triangle with required property for any angle from 0 to 40°:
1- take any point b on AB and build point c such as angle(Abc)=2α
2- build a inside ABC such as abc is equilateral
3- build O such that angle(Oca)=angle(Oba)=α
4- build P intersection of AO and BC
5- build PQR homothetic to Obc

PQR is such that angle(BPQ)=α, angle (AQR)=2α, angle(CRP)=3α

 
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Labels: 45-45-90 triangle, Circle, Trigonometry

  You can add comments, discuss solutions.

Polar fox: Hey Dervish, listen up! You still believe that the life is beautiful, don't you?
Dervish fox: Most definitely!
Polar fox: Here is my problem: the other day, I killed two little polar bunnies. Where is the beauty in that?
Dervish fox: Did you eat them by yourself?
Polar fox: One was for me; the other was devoured by my cubs in the den.
Dervish fox: How did they feel when they were full? Playfully smiling, right?
Polar fox: Oh, I see where you're going...
Dervish fox: Just like that my friend, death of one may be life for another. Seed decays in the darkness of the soil so that flowers bloom in the sun.
Polar fox: I guess I'll need a heart to believe in what you believe.
Dervish fox: I would start with an eye. Don't just look at it, but try to see...

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Labels: Bisector, Cosine Rule, Law of Sines, Triangle, Trigonometry