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nikolinv
Member since Apr-24-10
May-15-10, 00:52 AM (EST)

"Who can resist?"

 As a truly mathematical lover, i can't resist to temptation. This is my proof of the Pythagorean Theorem:Let D be the foot of the altitude from C of the right triangle ABC, A' be the foot of the altitude from D of the right triangle BCD and B' be the altitude from D of the right triangle ACD. If A'D = a', B'D = A'C = b' and CD = c' we have: Area (ABC)=Area(ACD) + Area(BCD) c·c' / 2 = a·a' / 2 + b·b' / 2 | ·2 c·c' = a·a' +b·b' (1)It is not hard to see that triangles ABC and CDA' are similar, hence: a' = ka b' = kb c' = kcBy substitution in (1) we have:c·kc = a·ka + b·kb | :kand finaly:c² = a² + b².Greetings

alexb
Charter Member
2503 posts
May-16-10, 12:01 PM (EST)

1. "RE: Who can resist?"
In response to message #0

 I have a difficulty making my mind whether this is just a corroborated version of Euclid's VI.31. I am inclined to expand page https://www.cut-the-knot.org/pythagoras/PythagorasBySimilarity.shtmlwith your diagram and proof.Thank you.

nikolinv
Member since Apr-24-10
May-16-10, 03:42 PM (EST)

2. "RE: Who can resist?"
In response to message #1

 Did you mean this?It is obviously related with Euclid's VI.31.

alexb
Charter Member
2503 posts
May-16-10, 03:56 PM (EST)

3. "RE: Who can resist?"
In response to message #2

 This is a good illustration to proof #8. There are 3 or 4 derived from Euclid's more general idea.But no, what I meant is exactly the reference I made.VI.31 tells us that (in your first diagram) ka² + kb² = kc²because such and such triangles are similar and two of them add up to the third. You made it much more specific:aa' + bb' = cc'(for the same reason, i.e. because two triangles add up to the third) and then showed that due to the similaritya' = ka, b' = kb, c' = kcI do not truly know what makes one proof a variant of, or a corroboration on, another, and what makes two proofs independent. Proofs 6, 7, 8 should have been probably one. The reason I have split them is that years ago all I had was just a few proofs. The rest came along over the space of a few years, not all at the same time. At some point it was already hard to make a better system or to change the numbering.