Math Is Fun Forum

Al-Allo

Member

Registered: 2012-08-23

Posts: 324

Side-angle-side and side-angle-angle as proved by Euclid in the Elemen

I have small question regarding this proposition : http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI26.html

To prove that one side is equal to another, Euclid assumes that one side is bigger than the other. Finally, when Euclid arrives at a contradiction, he dismisses the assumption about the inequality of sides and considers them equal. What I was wondering is, if we assume that a side is unequal to another one (A is bigger than B) and arrive at a contradiction, shouldn't we also try the inverse, B being bigger than A and assure ourselves that we arrive also at a contradiction to conclude that finally, A is equal to B ? Thank you!

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Side-angle-side and side-angle-angle as proved by Euclid in the Elemen

hi Al-Allo,

He says: "If AB does not equal DE, then one of them is greater. Let AB be greater."

So he chooses to letter the vertices so that this is true.  Therefore he doesn't need to consider the other case.

It is like this:

Two people decide to share their money equally. 

If they have the same already, do nothing as they have achieved their aim.

If not, let A be the one with more and B the other.  Let A's money be a and B's money be b.

Calculate (a-b)/2.  A gives B this amount.

Now A has a - (a-b)/2 = (2a - a + b)/2 = (a + b)/2  and B has b + (a-b)/2 = (2b + a - b)/2 = (a + b)/2 so they have the same.

It doesn't matter what their real names are; we choose A to be the label for the one who has more.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Al-Allo

Member

Registered: 2012-08-23

Posts: 324

Re: Side-angle-side and side-angle-angle as proved by Euclid in the Elemen

Bob did you receive my private message?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Side-angle-side and side-angle-angle as proved by Euclid in the Elemen

Yes and replied.

B

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob