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Subject: "Who can resist?"     Previous Topic | Next Topic
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Conferences The CTK Exchange This and that Topic #927
Reading Topic #927
nikolinv
Member since Apr-24-10
May-15-10, 00:52 AM (EST)
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"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)
cc' / 2 = aa' / 2 + bb' / 2 | 2
cc' = aa' +bb' (1)

It is not hard to see that triangles ABC and CDA' are similar, hence:

a' = ka
b' = kb
c' = kc

By substitution in (1) we have:

ckc = aka + bkb | :k

and finaly:

c = a + b.

Greetings


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

with your diagram and proof.

Thank you.


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nikolinv
Member since Apr-24-10
May-16-10, 03:42 PM (EST)
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2. "RE: Who can resist?"
In response to message #1
 
   Did you mean this?

It is obviously related with Euclid's VI.31.


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alexbadmin
Charter Member
2503 posts
May-16-10, 03:56 PM (EST)
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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 = kc²

because 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 similarity

a' = ka, b' = kb, c' = kc

I 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.


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