Genotyping technologies do not maintain phase? Why exactly does genotyping not maintain phase? I guess I'm more confused on what's the difference between sequencing and genotyping, and as a result how phase is maintained in one, but not the other. This is an image trying to do phasing using genotyping methods.

I understand how phasing works, I just don't quite get what is the difference between genotyping and sequencing, and why one of them doesn't work.
Phasing means that you are able to distinguish the both copies of your target sequence. (Usually you have one from your mother and one from your father). 

If you can do this, depends mainly on the sequencing technology you are using. For example in case of sanger sequencing, you cannot do this, because you (pre)amplify all your copies at once and detect the sequence at the same time at the same location.

In case of ngs you have read information at the end. One read represents the sequence of one original dna molecule. If you detect multiple variants in one read, you can be sure that they are from the same chromosome and thus they are in phase. Now you can go on and try to find other reads that have unambiguous overlaps with these variants on your read. If you find some, you can also say that these variants have their origin on the same chromosome.