Our server costs ~$56 per month to run. Please consider donating or becoming a Patron to help keep the site running. Help us gain new members by following us on Twitter and liking our page on Facebook!
Current time: August 23, 2019, 9:07 am

Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Proof by Rearrangement of the Pythagorean Theorem
#1
Proof by Rearrangement of the Pythagorean Theorem
So, what do you think, is the Proof by Rearrangement of the Pythagorean Theorem valid? I've made a SVG animation presenting it here:
http://flatassembler.000webhostapp.com/p...ences.html
At first, it seems to make perfect sense. However, as I was making that animation, it appeared to me that it's actually a form of circular reasoning. Namely, how is it possible to prove that the angles of the c*c square are actually right angles, without appealing to the Pythagorean Theorem itself and the formula for the area of a parallelogram (P=a*b*sin(alpha))? It appears to me that it isn't.
Reply
#2
RE: Proof by Rearrangement of the Pythagorean Theorem
What other angles could they be? The hypotenuses are all the same length, we don’t need to prove that as it’s part of the design of the initial setup. The triangles are all the same, that’s likewise by design. They are all transformed on the plane in the same way.

The only possible shape they can make is a square when all that is taken into account.
Reply
#3
RE: Proof by Rearrangement of the Pythagorean Theorem
I had a hypotenuse once, but I went to the doctor, got a shot for it, and it cleared up.(Note to readers: Nothing to see here, just being silly) We now return you to your regularly scheduled thread already in progress.
Reply
#4
RE: Proof by Rearrangement of the Pythagorean Theorem
"Hypot en use. Hardhat required."

Coffee
[Image: ak_botan_saionji_005.jpg]
Reply
#5
RE: Proof by Rearrangement of the Pythagorean Theorem
(November 22, 2018 at 3:14 am)FlatAssembler Wrote: So, what do you think, is the Proof by Rearrangement of the Pythagorean Theorem valid? I've made a SVG animation presenting it here:
http://flatassembler.000webhostapp.com/p...ences.html
At first, it seems to make perfect sense. However, as I was making that animation, it appeared to me that it's actually a form of circular reasoning. Namely, how is it possible to prove that the angles of the c*c square are actually right angles, without appealing to the Pythagorean Theorem itself and the formula for the area of a parallelogram (P=a*b*sin(alpha))? It appears to me that it isn't.

You just need to take a "look" to see it makes sense, right? Not sure where the necessity to appeal to that formula for the area of a parallelogram comes from.
Reply
#6
RE: Proof by Rearrangement of the Pythagorean Theorem
(November 22, 2018 at 3:14 am)FlatAssembler Wrote: So, what do you think, is the Proof by Rearrangement of the Pythagorean Theorem valid? I've made a SVG animation presenting it here:
http://flatassembler.000webhostapp.com/p...ences.html
At first, it seems to make perfect sense. However, as I was making that animation, it appeared to me that it's actually a form of circular reasoning. Namely, how is it possible to prove that the angles of the c*c square are actually right angles, without appealing to the Pythagorean Theorem itself and the formula for the area of a parallelogram (P=a*b*sin(alpha))? It appears to me that it isn't.

You need to know the angles of a triangle add up to two right angles. That is enough to show that the central figure is a square.

To show the angles of a triangle add up to two right angles requires results on opposite interior and opposite exterior angles for parallel lines.

Those results require the parallel postulate.
Reply
#7
RE: Proof by Rearrangement of the Pythagorean Theorem
Quote: What other angles could they be? The hypotenuses are all the same length, we don’t need to prove that as it’s part of the design of the initial setup. The triangles are all the same, that’s likewise by design. They are all transformed on the plane in the same way.The only possible shape they can make is a square when all that is taken into account.
But the sides being the same length doesn't mean the angles are the same. That's true only for triangles, not for the polygons with four (or more) angles.

As for the other replies, this is not a trolling question, so please don't answer as if it were.

Edit: Sorry, Polymath, I haven't seen your response, which actually makes some sense.
Reply
#8
RE: Proof by Rearrangement of the Pythagorean Theorem
I was being serious, FTR.

Like Tibs said, the triangles are exactly the same and rearranged in the same way. That could only result in a square, not just any rhombus.

Polymath's answer is more rigorous, fair enough.
Reply
#9
RE: Proof by Rearrangement of the Pythagorean Theorem
(November 22, 2018 at 11:35 am)Grandizer Wrote: I was being serious, FTR.

Like Tibs said, the triangles are exactly the same and rearranged in the same way. That could only result in a square, not just any rhombus.

Polymath's answer is more rigorous, fair enough.

Something is required since the result fails for non-Euclidean geometry. And, in fact the point that fails is the angles are not right angles. The lines around the (a+b) squares are no longer straight.
Reply
#10
RE: Proof by Rearrangement of the Pythagorean Theorem
I'd also point out there are other implicit assumptions here. For example, how is the square of side length a+b constructed? Well, we can take a line segment of length a+b, say AB. Do perpendiculars from both ends, giving sides AC and BD, each of length a+b.

iI is an *assumption* that the line CD is perpendicular to both AC and BD and is of length a+b. In non-Euclidean geometry, this line is NOT perpendicular to either AC or BD (the angle is *less* than a right angle) and its length is *more* than a+b. There *are* no squares in Lobachevskian geometry with four equal sides and right angles. They simply don't exist.

So even the original figures have hidden assumptions. To be perfectly rigorous, all these details would need to be addressed.
Reply



Possibly Related Threads...
Thread Author Replies Views Last Post
  The Mathematical Proof Thread Kernel Sohcahtoa 67 7659 July 6, 2018 at 8:37 pm
Last Post: Fireball
  Fundemental theorem of Calculus intuition A Handmaid 19 1866 August 28, 2016 at 12:52 pm
Last Post: Jehanne
  Million Dollar Prize for Math proof of NP problems emilynghiem 6 2382 February 22, 2015 at 12:47 am
Last Post: vorlon13
  "Gödel's ontological proof" proves existence of God Belac Enrobso 41 11369 February 9, 2015 at 3:22 am
Last Post: Alex K
  Godels theorem is invalide for 5 reasons shakuntala 8 2635 December 21, 2014 at 1:04 pm
Last Post: agnesi
  Mathematical proof.. lifesagift 20 4793 September 26, 2014 at 5:01 pm
Last Post: lifesagift
Information My proof for de morgans law LogicMaster 17 3754 May 29, 2014 at 7:55 pm
Last Post: Cyberman
  Mathematician Claims Proof of Connection between Prime Numbers KichigaiNeko 10 5569 September 26, 2012 at 3:18 am
Last Post: Categories+Sheaves
  Need a proof (real analysis) CliveStaples 8 4726 August 2, 2012 at 10:11 pm
Last Post: CliveStaples
  Mathematical proof of the existence of God JudgeDracoAmunRa 20 10568 March 30, 2012 at 11:43 am
Last Post: JudgeDracoAmunRa



Users browsing this thread: 1 Guest(s)