So I found this interesting journal article that I was reading. I understand most of it but the proof of the existence of the orthocenter was not clear to me. Here is a link to the article, http://www.math.siu.edu/kocik/papers/metaptolemiana.pdf

I do not understand the proof of it on the bottom of page 2 (it's in Proposition 1) of the .pdf and the top of page 3. Anyone care to explain the author's meaning? because I honestly cannot figure out why this proves H exists. It's probably really simple and I will look foolish for asking, but I just have to know. Thanks.