The author links to another blog discussing the related Martinez-Rueda algorithm, which I mention not only because this post ends with "I am considering implementing the Martinez-Rueda algorithm because [... it is more featureful]", but because at the top of it[1] there's a "visual proof" of sorts that I found strikingly concise.
[1]: https://sean.cm/a/polygon-clipping-pt2#annotating-segments