In Book III, Proposition 11. The proof uses $|AG|+|GF|>|FA|$, that is $|AG|+|GF|>|FH|$. That is $|AG|>|GH|$, which is impossible. But the proof rely on the following graph, where $F$ is the center of the big circle $ABC$, while $G$ is the center of the small circle $ADE$.
What if the graph is something like this which I think is also possible. Now $F$ is the center of the small circle $ADE$, and $G$ is the center of the big circle $ABC$. Then we cannot have such relationship.
What is wrong of my understanding here?
Sorry about the question. I think I figured it out. If the diagram is the second one, we can extend the segment at the opposite direction. Which is something like:
Then we can still use the $|AF|+|FG|>|AG|$, and since $|AG|=|GH|$, then $|AF|+|FG|>|GH|$. That is $|FD|>|FH|$ which is not possible.