I apologize in advance if my answer will be long, but this is a very interesting question and I'll try to give it the answer it deserves. I will refer not exclusively to notation in geometry but in mathematics in general.
The very short answer is:
it is fundamental to use some notation, or we'd be back to the painful description of equations of early algebra, but use too much and you'll cruelly add a burden to your students.
Now, for the long answer:
Mathematical notation can have on students, especially pre-university or non-maths students (meaning, people that did not choose willingly to attend maths lessons), a double-sided effect:
it can be extremely arousing for us nerds, as it becomes the key to understanding that mathematics is, after all, a way of manipulating concepts as a language per se, and as a language it has its own "alphabet" (forgive the poor wording in terms of linguistics). It is also like giving us new toys to play with, and we then start having fun with them in external contexts - as when we use existential quantifiers in everyday's life.
Furthermore, it speeds up explanation significantly, which means gifted students will be more stimulated;
it can be extremely discouraging for the self-diagnosed-mathematically-impaired, which see in symbols one further obstacle that the teacher/system/Illuminati have made up with the purpose of making this senseless subject even less understandable.
Because of this, as with many things in teaching, it's not intrinsically better to choose to use as much or as little notation notation as possible, but balance is key:
- use too much notation and you'll make things harder for the second group;
- use too little notation and you'll be missing on a very useful tool, and possibly bore the best students to death.
Given all this, let me then give you some, hopefully useful, practical tips for introducing more notation with minimum damage:
- What is fundamental for every student to understand is that no notation has been given to us, perfect and immutable, by some divinity: notation is an agreement between two parts communicating, and different parts (e.g., texts, fields, teachers..) can give to the same symbol different meanings. Mathematics is more alive than one thinks, as well as constantly evolving! Therefore agree explicitly with your students: if you don't mind, from now on I'd use symbol x instead of writing every time [meaning of x]. This will also make the whole teaching feel less an unidirectional flow of pre-compiled stuff from your head to their notebooks;
- A bit at a time: add new symbols progressively, trying to avoid your students an indigestion for receiving too much new information in one go;
- stress out the importance of minor differences, sometimes students tend to quotient over little graphical things thinking they are only quirks while they actually carry a meaning;
- set a "minimum level" of notation and use it in your questions, but encourage your students to use more, for example by providing a little extra credit to those who can also formulate their answers in a concise way rich of notation - potentially, notation-only!