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Hi I have just found the proof that 90 equals 95 and was wondering if I have made some mistake. If so, which step in my proof is not true?

Definitions:
1. $\angle ABC=90^{\circ}$
2. $\angle BCD=95^{\circ}$
3. $|AB|=|CD|$
4. $M:=$ the center of $BC$
5. $N:=$ the center of $AD$
6. $l:=$ a line perpendicular to $BC$ passing through $M$
7. $m:=$ a line perpendicular to $AD$ passing through $N$
8. $S:=$ is the cross-section of $l$ and $m$

Based on definitions 1 through 8 we can draw the following image: 90 = 95

Based on the definitions we can derive the following:
9. $\triangle BSC$ is isosceles (follows from 4, 6 and 8)
10. $\triangle ASD$ is isosceles (follows from 5, 7 and 8)
11. $|BS|=|CS|$ (follows from 9)
12. $|AS|=|DS|$ (follows from 10)
13. $\triangle ABS\cong\triangle DCS$ (follows from 3, 11 and 12)
14. $\angle ABS=\angle DCS$ (follows from 13)
15. $\angle CBS=\angle BCS$ (follows from 9)
16. $\angle ABC=\angle ABS - \angle CBS=\angle DCS-\angle BCS=\angle BCD$ (follows from 14 and 15)
17. $90^{\circ}=95^{\circ}$ (follows from 1, 2 and 16)

Note: point $S$ is indeed lying above $BC$. If however it would be below $BC$, then the 'minus' in step 16 would simply have to be changed into a 'plus'.

Also note: The image is not drawn to scale. It only serves as to provide the reader with a intuitive view of the proof.

Also also note: before posting, I have first investigated if this type of question would be appropriate. Based on How come 32.5 = 31.5? and the following meta Questions about math jokes I have decided to post this question.

Matthijs Wessels
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    This one and its variants are actually fairly well-known: see http://en.wikipedia.org/wiki/Mathematical_fallacy#Geometry for a spoiler. But finding the error is still a fun puzzle. – Vandermonde Oct 22 '10 at 09:37
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    Hav you tried drawing an *accurate* diagram? – Robin Chapman Oct 22 '10 at 09:40
  • @freestorageaccount Wow, I got this one through one of my uncles, but was not able to find anything about it on the internet. I knew there had to be something, but I just did not know the right terms to search for it. I shall forward that link to my uncle ^^. – Matthijs Wessels Oct 22 '10 at 09:42
  • @Robin what is not accurate about it? – Matthijs Wessels Oct 22 '10 at 09:46
  • Shhh, Robin -- don't ruin the fun part. – Vandermonde Oct 22 '10 at 09:56
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    @Robin, no doubt that would help the intuition, but surely the wrongness of the proof has to be independent of the diagram. – LarsH Oct 22 '10 at 10:11
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    @Matthijs: "through one of my uncles" - was it by chance Uncle Shelby?? (http://en.wikipedia.org/wiki/Uncle_Shelby%27s_ABZ_Book) "Or is it correct?" Uhm... yes... 90 = 95, from which we can conclude that pigs fly. – LarsH Oct 22 '10 at 10:14
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    This was actually quite fun to draw and notice the error :) – workmad3 Oct 22 '10 at 11:35
  • As a hint, the diagram provided isn't actually representative of the given constraints. – workmad3 Oct 22 '10 at 11:46
  • About half a year ago I posted this on MathOverflow, but it wasn't received as well as I had expected :D. http://mathoverflow.net/questions/22609/impossible-triangle-proving-that-90-95-closed . Also, my formulation of the question is a bit better here than I did it there. I have also drawn this one on the white board next to the coffee machine at work. Every time I walked by the coffee machine, there would be one or two people standing in front of the white board, coffee in hand, saying things like:"This is not correct!". – Matthijs Wessels Oct 22 '10 at 11:58
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    Or is it correct? – I. J. Kennedy Dec 01 '10 at 16:45

2 Answers2

52

$$ \begin{array}{l} 0\in\mathbb N\\ \forall n\in \mathbb N : n'\in \mathbb N\\ \hline 0'\in\mathbb N\\ \forall n\in \mathbb N : n'\in \mathbb N\\ \hline 0''\in\mathbb N\\ \forall n\in \mathbb N : n'\in \mathbb N\\ \hline 0'''\in\mathbb N\\ \forall n\in \mathbb N : n'\in \mathbb N\\ \hline 0''''\in\mathbb N\\ \forall n\in \mathbb N : n'\neq0\\ \hline 0'''''\neq0\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''\neq0'\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''\neq0''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''\neq0'''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''\neq0''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''\neq0'''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''\neq0''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''\neq0'''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''\neq0''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''\neq0'''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''\neq0''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''\neq0'''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''\neq0''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''\neq0'''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''\neq0''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''\neq0'''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''\neq0''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''\neq0'''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''\neq0''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''\neq0'''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''\neq0''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''\neq0'''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''\neq0''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline \phantom{0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''} \end{array} $$ $$ \begin{array}{l} 0'''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline \phantom{0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''} \end{array} $$ $$ \begin{array}{l} 0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline \phantom{0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''} \end{array} $$ $$ \begin{array}{l} 0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \forall m,n\in \mathbb N:m\neq n\rightarrow m'\neq n' \\ \hline 0'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\neq0''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''\\ \\ \square \end{array} $$

joriki
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    This doesn't seem to have anything to do with the original question—am I missing something? – Isaac Jul 09 '11 at 14:57
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    LOL - a good one! – Jyrki Lahtonen Jul 09 '11 at 15:04
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    Apparently, using your argument and matthijs proof, it follows that arithmetic is inconsistent ;-) – wood Jul 09 '11 at 15:14
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    @Isaac: I think you are. This is a proof from the Peano axioms that $95\neq90$. – joriki Jul 09 '11 at 15:14
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    @jorki: Sure, but that doesn't address the original question of what the mistake is in the geometric proof, nor does it provide any insight related to the original question. – Isaac Jul 09 '11 at 15:16
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    @Isaac: I took the last paragraph of the question, together with Matthijs' last comment under the question, to imply that the question was intended as a joke, so I replied with a joke. Also, note that the question in the title has two parts; I was responding to the second part, to which my answer is quite relevant ;-) – joriki Jul 09 '11 at 15:18
  • Very clever. ${}{}{}$ – davidlowryduda Jul 09 '11 at 15:56
  • @joriki: Well done. (And your answer does address the first sentence of the post.) – Did Jul 09 '11 at 16:16
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    lol, btw. I think there is a mistake in line $0′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′′$ ;) – t.b. Jul 09 '11 at 19:42
  • @joriki: I read your answer yesterday and it still amuses me. I like your sense of humour : ) – Rudy the Reindeer Dec 03 '11 at 08:59
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    -1. This doesn't really answer the question. –  Dec 27 '12 at 23:41
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    @Marvis: It's clearly not intended to answer the question. Do you feel that this sort of humour is out of place on this site? – joriki Dec 28 '12 at 11:16
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    @joriki A short good humored comment is fine with me. But I feel this answer will only add to the OP's confusion, since the OP might not even understand what you are trying to convey and the motivation behind your answer. This is, of course, just my opinion. Given that your answer has received $15$ up-votes compared to just $3$ down-votes, clearly the rest of the site believe such an answer is fine. –  Dec 28 '12 at 19:23
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    @Marvis: Note that I answered almost a year after the question had been properly answered and the proper answer had been accepted; the potential for confusing the OP at this point was probably low. – joriki Mar 11 '13 at 06:02
  • It could have been shorter using induction :). I agree with @claudi85 though: together with my proof, you have proven by contradiction that the Peano axioms are false. – Matthijs Wessels Mar 11 '13 at 13:59
  • a prime is missing in 97th line! right?! –  May 07 '13 at 13:21
  • And also in line 143! –  May 07 '13 at 13:27
  • @Cutie: I'm not sure how you're counting the lines. The way I'm counting the lines, the $97$-th line says $48\ne43$, an it seems to have the right number of primes. Could you please point out the problem more specifically? – joriki May 07 '13 at 13:29
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    If you scroll through the post really fast, it looks like an animation. – Abstract Space Crack Jul 08 '14 at 17:18
  • This is the longest answer on the site. –  Jun 11 '16 at 19:21
16

Your statement 16 depends on the way in which the diagram is drawn to know that ∠DCS = ∠DCB + ∠BCS.

Isaac
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  • How is it false then? PS: use dollar signs to encapsulate math. You wanted to type: \angle DCS=\angle DCB+\angle BCS – Matthijs Wessels Oct 22 '10 at 11:07
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    @Matthis: Carry out the construction in the proof yourself using a geometry set (you can get by with a ruled sheet and a ruler) and you will immediately see what is wrong with the diagram. – Jyotirmoy Bhattacharya Oct 22 '10 at 11:53
  • @Matt: I believe Isaac knows about the latex. The ∠ is an html escape code. Unfortunately, I don't think that exists, though. – Aryabhata Oct 22 '10 at 14:49
  • @Matt, @Matthijs: That's odd—`∠` renders correctly in Chrome... [Apparently, I was supposed to use `∠`, but even that doesn't work in IE6 and IE7](http://stackoverflow.com/questions/709827/displaying-angle-symbol-in-html). – Isaac Oct 22 '10 at 17:30
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    @Matthijs: Specifically, 16 depends on the (false) assertion from the diagram that DS is "between" DA and DC, when it is in fact on the "other side" of DC. – Isaac Oct 22 '10 at 17:33