Euclid's Elements Book 1 Proposition 24

Question

In Euclid's elements book 1 proposition 24, after he establishes that "Again, since DF equals DG, therefore the angle DGF equals the angle DFG. Therefore the angle DFG is greater than the angle EGF" (https://mathcs.clarku.edu/~djoyce/java/elements/bookI/propI24.html)

I'm confused about the justification for the bolded part. After searching online, all sources claim this is the bolded line is justified by the common notion that "the whole is greater than the part." But isn't this is a conclusion drawn from the diagram constructed and not the logical connections? I find that if the diagram is drawn differently, different conclusions can be made.

Thanks for the help!

Answer 1

There are indeed multiple cases here, and Euclid only considers one, as is his habit. See Heath's commentary, pp. 297 - 299.

Answer 2

He uses the facts:

• F is outside $\triangle EDG$

and

• GF projected meets DE projected

along with Proposition 21.

(From https://archive.org/details/EuclidsElementsBooksIIIVolume1Heath/page/n305/mode/2up)