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help with problem (2 Viewers)
- Thread starter pl4smaa21
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It is the old Cambridge 3 unit year 12 Ex. 8C Q29 and here is better quality picture
=\textstyle\frac{1}{2}BC\cdot AD=\frac{1}{2}AC\cdot BE)
https://proofwiki.org/wiki/Steiner-Lehmus_Theorem
https://proofwiki.org/wiki/Steiner-Lehmus_Theorem
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There was a post on c. before at https://community.boredofstudies.org/threads/steiner-lehmus-theorem.105751/ but it needs to be revamped due to formatting issues
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aight man thnx i have the old cambridge 3u book cuz i got it at the end of year 10 from school to like preview some of the material ig I posted this becuz it seemed like a bit of a tough problem and I couldn't solve fully so thanks very much.
u know when the proof for c. presented here first appeared? 1880
Journal de mathematiques elementaires et speciales, 1880, p. 538
also attached are 3 articles on c.
Hajja, M., A Short Trigonometric Proof of the Steiner-Lehmus Theorem, Forum Geometricorum Volume 8 (2008) 39–42.
Sauvé, L., The Steiner-Lehmus theorem, Crux Math., 2 (1976) 19–24.
Trigg, C.W., A bibliography of the Steiner-Lehmus theorem, Crux Math., 2 (1976) 191–193.
Journal de mathematiques elementaires et speciales, 1880, p. 538
also attached are 3 articles on c.
Hajja, M., A Short Trigonometric Proof of the Steiner-Lehmus Theorem, Forum Geometricorum Volume 8 (2008) 39–42.
Sauvé, L., The Steiner-Lehmus theorem, Crux Math., 2 (1976) 19–24.
Trigg, C.W., A bibliography of the Steiner-Lehmus theorem, Crux Math., 2 (1976) 191–193.
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damn man u putta lotta effort into this more than I could have hoped for thanks. and yea that's kinda interesting that this small ext. problem has a lotta history
There are at least 80 different ways to do c.
But it remains the subject of modern research, particularly in regards to establishing a direct proof, that being one not relying on reductio ad absurdum.
I attached one such proof, somewhat more modern than the ones i put before. This really raises the bar in the whole scheme of things and here is the abstract for the paper
A direct proof of the Steiner-Lehmus theorem has eluded geometers for over 170 years. The challenge has been that a proof is only considered direct if it does not rely on reductio ad absurdum. Thus, any proof that claims to be direct must show, going back to the axioms, that all of the auxiliary theorems used are also proved directly. In this paper, we give a proof of the Steiner-Lehmus theorem that is guaranteed to be direct. The evidence for this claim is derived from our methodology: we have formalized a constructive axiom set for Euclidean plane geometry in a proof assistant that implements a constructive logic and have built the proof of the Steiner-Lehmus theorem on this constructive foundation.
Kellison, A., A Machine-Checked Direct Proof of the Steiner-Lehmus Theorem, CPP 2022: Proceedings of the 11th ACM SIGPLAN International Conference on Certified Programs and Proofs, Pages 265 - 273
But it remains the subject of modern research, particularly in regards to establishing a direct proof, that being one not relying on reductio ad absurdum.
I attached one such proof, somewhat more modern than the ones i put before. This really raises the bar in the whole scheme of things and here is the abstract for the paper
A direct proof of the Steiner-Lehmus theorem has eluded geometers for over 170 years. The challenge has been that a proof is only considered direct if it does not rely on reductio ad absurdum. Thus, any proof that claims to be direct must show, going back to the axioms, that all of the auxiliary theorems used are also proved directly. In this paper, we give a proof of the Steiner-Lehmus theorem that is guaranteed to be direct. The evidence for this claim is derived from our methodology: we have formalized a constructive axiom set for Euclidean plane geometry in a proof assistant that implements a constructive logic and have built the proof of the Steiner-Lehmus theorem on this constructive foundation.
Kellison, A., A Machine-Checked Direct Proof of the Steiner-Lehmus Theorem, CPP 2022: Proceedings of the 11th ACM SIGPLAN International Conference on Certified Programs and Proofs, Pages 265 - 273
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hmm okay i couldn't have guessed there were over 80 ways to do that wow. So like they rely heavily on proof by contradiction. I only know year 10 math and a bit of year 11 but i do remember that proof by contradiction is taught in Ext 2 right?
yeah. it is
old Cambridge was published from 2000??? Damn this textbook I got is a relic is soo colorful tho esp. when compared to the depressing black and white in the new 4u cambridge textbook lol
joeystarbank
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this problem is easy