Finding the Point: Unveiling Euclid's 'Elements' Redux

[murshid_islam]

TL;DR Summary

Question about a proof from Euclid's "Elements" Redux (by Daniel Callahan and John Casey)

The following proof (in the image below) is from the book Euclid's "Elements" Redux (by Daniel Callahan and John Casey). I did not understand why the point D or the $\triangle \rm ABD$ was necessary. (I mean, what was the "point" of D? ) Joking apart, wasn't this sufficient: suppose we have a line segment AB and a point C on AB such that AB=2AC. Let AC = x. Then AB = 2x. Therefore, AB² = (2x)² = 4(x)² = 4(AC)²

 

[anuttarasammyak]

I see Euclid, Elements II 4 at https://web.calstatela.edu/faculty/hmendel/Ancient Mathematics/Euclid/Euclid II/Euclid 2.4/Euclid.2.4.html .
I assume the corollary you refer, though I cannot find it in above linked page, is the special case AG=GB there. I think he, Euclid or the author, says about geometry not algebra of $(2x)^2=4x^2$.

 

[murshid_islam]

I see Euclid, Elements II 4 at https://web.calstatela.edu/faculty/hmendel/Ancient Mathematics/Euclid/Euclid II/Euclid 2.4/Euclid.2.4.html .
I assume the corollary you refer, though I cannot find it in above linked page, is the special case AG=GB there.

Since this is a open textbook, it has been updated several times I guess. So, there are several versions of the book available on the internet. Maybe that's what causes the discrepancies between versions.

I think he, Euclid or the author, says about geometry not algebra of $(2x)^2=4x^2$.

In that case, shouldn't it deal with a square instead of a triangle? I still do not understand the relevance of the triangle and point D.