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The Nightstar Zoo • View topic - Triangles

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 Post subject: Triangles
PostPosted: Sat Jun 01, 2002 8:07 am 
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Knight of Daisies, Tulip Slayer
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PostPosted: Sat Jun 01, 2002 10:05 pm 
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Vorpal Bunny Slipper
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Can't be done.

Let's assume that the lengths of the two equal sides are given by a, and the hypotenuse is b. Since this is a right triangle, by the Pythogorean theorem, a^2 + a^2 = b^2. Thus 2a^2 = b^2. If we take the square root of both sides, we get sqrt(2) * a = b, where sqrt is the square root function. Sqrt(2) is an irrational number, so if b is an integer, a can't be, and vice versa.


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PostPosted: Sun Jun 02, 2002 3:13 am 
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PostPosted: Sun Jun 02, 2002 9:53 am 
What's elliptical space?


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PostPosted: Sun Jun 02, 2002 10:23 pm 
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Vorpal Bunny Slipper
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PostPosted: Mon Jun 03, 2002 1:18 am 


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PostPosted: Sat Jun 08, 2002 3:03 am 


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PostPosted: Sat Jun 08, 2002 3:41 am 
The only thing left to prove, at least in my view, is JeremiahSmith's assumption that the right angle is indeed the angle between the two equal sides. I will proceed by contradiction.

First, let us assume that we have an isosceles right triangle ABC, in which sides a and b are equal, and angle B is right. Because opposite angles of equal sides are equal, this means that angle A is also a right angle. Now, as a result of Euclid's fifth postulate, we knwo that the sum of the angles in any triangle is 180°. This means A + B + C = 180, or A + 90 + 90 = 180, which means A = 0. The definition of a triangle states that the points A, B, and C are non-colinear. But if A = 0°, then the points must be colinear, and thus the assumption that ABC is a triangle is violated. Therefore, by contradiction, such a triangle cannot exist in Euclidean geometry. QED.

Now, to describe better what Vorn was talking about, there are fields of geometry in which people decided that Euclid was smoking crack when he came up with his fifth postulate. One such field, spherical geometry, says that the sum of the angles in a triangle can (and indeed always are) greater than 180°. In this geometry, such a triangle can, and does exist.

For a simple example, try a sphere with circumference 4. Hold it out in front of you. Now, place three points on this sphere: One at the topmost point, one at the leftmost point, and one at the point closest to you. Join these three points, and you get, in fact, an equilateral right triangle, with three sides of length 1, and three angles of 90°.

The isosceles right triangle Ogredude is talking about can, in fact, be created on any globe with two points anywhere on the equator and the third point at one of the poles. Notice that the right angle(s) in this triangle is not between the two equal sides, which is how you get around the argument that sqrt(2) is not rational.


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PostPosted: Sat Jun 08, 2002 10:52 am 
sqrt(2) being rational or irrational becomes irrelevant at that point anyway, because the Pythagorean Theorem itself doesn't hold. As Pi mentioned, it's in fact possible to make an equilateral right triangle. The Pythagorean Theorem would need to be reformulated in order to hold on a sphere. I, for one, have no idea where to start, but it seems clear that such a theorem would need to involve the size of the triangle with respect to the size of the sphere.


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PostPosted: Sun Jun 09, 2002 4:40 pm 
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Vorpal Bunny Slipper
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Wow. You guys overanalyzed this to death.


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PostPosted: Sat Aug 17, 2002 1:49 am 


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PostPosted: Sat Aug 17, 2002 7:57 am 
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Knight of Daisies, Tulip Slayer
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PostPosted: Sat Aug 17, 2002 9:33 am 


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PostPosted: Mon Mar 10, 2003 12:27 am 


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PostPosted: Mon Mar 10, 2003 2:18 am 


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PostPosted: Mon Mar 10, 2003 2:29 am 


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PostPosted: Mon Mar 10, 2003 2:34 am 
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Vorpal Bunny Slipper
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Scharr, scharr, verscharr das Gebein, grab es tief unten im Keller ein.
Später dann graben es andere aus, und nennen dein Haus das Knochenhaus.
Scharr, scharr, verscharr das Gebein, leg auch ihre weißen Schädel hinein.
Mit Beton gießt du es aus, das Fundament vom Knochenhaus.
Scharr, scharr, verscharr das Gebein, da ist noch Platz, da paßt noch wer rein.
Hier tobte sich der Teufel aus, unten im Keller im Knochenhaus.


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 Post subject:
PostPosted: Wed Mar 12, 2003 7:39 pm 
A version of mindtrap targetted toward mathematicians and engineers has indeed been suggested and seconded. :twisted:


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PostPosted: Sat Mar 15, 2003 3:32 pm 


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