Quote:
In every Pythagorean triangle the following 6 facts are always true:
1. one side is a multiple of 3
2. one side is a multiple of 4
3. one side is a multiple of 5
4. the product of the two legs is always a multiple of 12
5. the area is always a multiple of 6
6. the product of all three sides is always a multiple of 60
endquote
119 120 169:
1, 2, and 3 all apply to 120. 119 is prime, and 160 is 13^2.
--
Engineering is the art of making what you want from things you can get.
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Pythagoras
Started by ●April 9, 2008
Reply by ●April 10, 20082008-04-10
On Wed, 09 Apr 2008 22:41:07 -0400, Jerry Avins <jya@ieee.org> wrote:>Quote: >In every Pythagorean triangle the following 6 facts are always true: > > 1. one side is a multiple of 3 > 2. one side is a multiple of 4 > 3. one side is a multiple of 5 > 4. the product of the two legs is always a multiple of 12 > 5. the area is always a multiple of 6 > 6. the product of all three sides is always a multiple of 60 >endquote > >119 120 169: > >1, 2, and 3 all apply to 120. 119 is prime, and 160 is 13^2.Actually items 4,5 and 6 are redundant and you don't really mean [160 is 13^2].
Reply by ●April 10, 20082008-04-10
On Wed, 09 Apr 2008 22:41:07 -0400, Jerry Avins wrote:> Quote: > In every Pythagorean triangle the following 6 facts are always true: > > 1. one side is a multiple of 3 > 2. one side is a multiple of 4 > 3. one side is a multiple of 5 > 4. the product of the two legs is always a multiple of 12 5. the > area is always a multiple of 6 6. the product of all three sides is > always a multiple of 60 > endquote > > 119 120 169: > > 1, 2, and 3 all apply to 120. 119 is prime, and 160 is 13^2.If a Pythagorean triangle is a right triangle with sides of integer length, AFAIK there are an infinite number of combinations where the sides are relative primes -- you just ran afoul of someone who thinks that a 3-4-5 triangle is the only example of the type. Carpenters are (or were) supposedly good trivia-masters of various such triangles, which is good to know if you need a BIG right angle and all you have is tape measures or strings. -- Tim Wescott Control systems and communications consulting http://www.wescottdesign.com Need to learn how to apply control theory in your embedded system? "Applied Control Theory for Embedded Systems" by Tim Wescott Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html
Reply by ●April 10, 20082008-04-10
On Apr 9, 10:41 pm, Jerry Avins <j...@ieee.org> wrote:> Quote: > In every Pythagorean triangle the following 6 facts are always true: > > 1. one side is a multiple of 3 > 2. one side is a multiple of 4 > 3. one side is a multiple of 5 > 4. the product of the two legs is always a multiple of 12 > 5. the area is always a multiple of 6 > 6. the product of all three sides is always a multiple of 60 > endquote > > 119 120 169: > > 1, 2, and 3 all apply to 120. 119 is prime, and 160 is 13^2. >there are a zillion qualitatively different (not similar or a scaled and rotated copy of each other) Pythagorean Triangles. 5, 12, 13 is the next one after 3, 4, 5. 119, 120, 169 is way down the line. r b-j
Reply by ●April 10, 20082008-04-10
On Apr 10, 2:39 am, robert bristow-johnson <r...@audioimagination.com> wrote:> On Apr 9, 10:41 pm, Jerry Avins <j...@ieee.org> wrote: > > > Quote: > > In every Pythagorean triangle the following 6 facts are always true: > > > 1. one side is a multiple of 3 > > 2. one side is a multiple of 4 > > 3. one side is a multiple of 5 > > 4. the product of the two legs is always a multiple of 12 > > 5. the area is always a multiple of 6 > > 6. the product of all three sides is always a multiple of 60 > > endquote > > > 119 120 169: > > > 1, 2, and 3 all apply to 120. 119 is prime, and 160 is 13^2. > > there are a zillion qualitatively different (not similar or a scaled > and rotated copy of each other) Pythagorean Triangles. > > 5, 12, 13 is the next one after 3, 4, 5. 119, 120, 169 is way down > the line.they're called Pythagorean Triples: http://en.wikipedia.org/wiki/Pythagorean_triple r b-j
Reply by ●April 10, 20082008-04-10
robert bristow-johnson wrote:> > On Apr 9, 10:41 pm, Jerry Avins <j...@ieee.org> wrote: > > Quote: > > In every Pythagorean triangle the following 6 facts are always true: > > > > 1. one side is a multiple of 3 > > 2. one side is a multiple of 4 > > 3. one side is a multiple of 5 > > 4. the product of the two legs is always a multiple of 12 > > 5. the area is always a multiple of 6 > > 6. the product of all three sides is always a multiple of 60 > > endquote > > > > 119 120 169: > > > > 1, 2, and 3 all apply to 120. 119 is prime, and 160 is 13^2. > > > > there are a zillion qualitatively different (not similar or a scaled > and rotated copy of each other) Pythagorean Triangles. > > 5, 12, 13 is the next one after 3, 4, 5. 119, 120, 169 is way down > the line.There's that slippery guy again trying to change the terms of the debate STRAWMAN STRAWMAN STRAWMAN STRAWMAN STRAWMAN STRAWMAN STRAWMAN He didn't say it was about 3,4,5 triangles. In the example he gave Fact 1,2,3 applies to the same side -> 120 -jim ----== Posted via Pronews.Com - Unlimited-Unrestricted-Secure Usenet News==---- http://www.pronews.com The #1 Newsgroup Service in the World! >100,000 Newsgroups ---= - Total Privacy via Encryption =---
Reply by ●April 10, 20082008-04-10
Tim Wescott wrote:> On Wed, 09 Apr 2008 22:41:07 -0400, Jerry Avins wrote: > > Quote: > > In every Pythagorean triangle the following 6 facts are always true: > > > � � 1. one side is a multiple of 3 > > � � 2. one side is a multiple of 4 > > � � 3. one side is a multiple of 5 > > � � 4. the product of the two legs is always a multiple of 12 > > 5. the area is always a multiple of 6 > > 6. the product of all three sides is always a multiple of 60 > > endquote > > > 119 120 169: > > > 1, 2, and 3 all apply to 120. 119 is prime, and 160 is 13^2. > > If a Pythagorean triangle is a right triangle with sides of integer > length, AFAIK there are an infinite number of combinations where the > sides are relative primes -- you just ran afoul of someone who thinks > that a 3-4-5 triangle is the only example of the type.The sides being relatively prime does not preclude the above conditions to hold. For example (5,12,13) is also relatively prime pythagorean triplet, and the above conditions hold. The conditions certainly look valid, and nobody has provided a counter-example to them yet. Regards, Andor
Reply by ●April 10, 20082008-04-10
Tim Wescott wrote:> On Wed, 09 Apr 2008 22:41:07 -0400, Jerry Avins wrote: > >> Quote: >> In every Pythagorean triangle the following 6 facts are always true: >> >> 1. one side is a multiple of 3 >> 2. one side is a multiple of 4 >> 3. one side is a multiple of 5 >> 4. the product of the two legs is always a multiple of 12. the >> area is always a multiple of 6. the product of all three sides is >> always a multiple of 60 >> endquote >> >> 119 120 169: >> >> 1, 2, and 3 all apply to 120. 119 is prime, and 160 is 13^2. > > If a Pythagorean triangle is a right triangle with sides of integer > length, AFAIK there are an infinite number of combinations where the > sides are relative primes -- you just ran afoul of someone who thinks > that a 3-4-5 triangle is the only example of the type. > > Carpenters are (or were) supposedly good trivia-masters of various such > triangles, which is good to know if you need a BIG right angle and all > you have is tape measures or strings.The post was an email gone astray. Where I wrote 160, I meant 169. The statements are true if you read the first three to mean "at least one and not necessarily a different one", something I hadn't realized until after I posted. It's not just about 4, 3, 5. The statements are from http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/pythag.html#hyplongconsec Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
Reply by ●April 10, 20082008-04-10
robert bristow-johnson wrote:> On Apr 9, 10:41 pm, Jerry Avins <j...@ieee.org> wrote: >> Quote: >> In every Pythagorean triangle the following 6 facts are always true: >> >> 1. one side is a multiple of 3 >> 2. one side is a multiple of 4 >> 3. one side is a multiple of 5 >> 4. the product of the two legs is always a multiple of 12 >> 5. the area is always a multiple of 6 >> 6. the product of all three sides is always a multiple of 60 >> endquote >> >> 119 120 169: >> >> 1, 2, and 3 all apply to 120. 119 is prime, and 160 is 13^2. >> > > there are a zillion qualitatively different (not similar or a scaled > and rotated copy of each other) Pythagorean Triangles. > > 5, 12, 13 is the next one after 3, 4, 5. 119, 120, 169 is way down > the line. > > r b-jIt is third in the set near 45 degrees (i.e., the legs differ by one): 3, 4, 5; 20, 21, 29; 119, 120, 169. The next is 696, 697, 985. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●April 10, 20082008-04-10
On Apr 10, 10:09 am, jim <".sjedgingN0sp"@m...@mwt.net> wrote:> robert bristow-johnson wrote: > > > On Apr 9, 10:41 pm, Jerry Avins <j...@ieee.org> wrote: > > > Quote: > > > In every Pythagorean triangle the following 6 facts are always true: > > > > 1. one side is a multiple of 3 > > > 2. one side is a multiple of 4 > > > 3. one side is a multiple of 5 > > > 4. the product of the two legs is always a multiple of 12 > > > 5. the area is always a multiple of 6 > > > 6. the product of all three sides is always a multiple of 60 > > > endquote > > > > 119 120 169: > > > > 1, 2, and 3 all apply to 120. 119 is prime, and 169 is 13^2. > > > there are a zillion qualitatively different (not similar or a scaled > > and rotated copy of each other) Pythagorean Triangles. > > > 5, 12, 13 is the next one after 3, 4, 5. 119, 120, 169 is way down > > the line. > > There's that slippery guy again trying to change the terms of the debate > > STRAWMAN STRAWMAN STRAWMAN STRAWMAN STRAWMAN STRAWMAN STRAWMAN > > He didn't say it was about 3,4,5 triangles. In the example he gave Fact > 1,2,3 applies to the same side -> 120didn't know i was trying to change the terms of the debate, but i can say i didn't understand it. i think maybe now i do. r b-j
