The procedure is actually like this:
Face recognition --> face landmark --> delaunay triangulation
The title is misleading. Delaunay is actually used after you get those points. However, I suspect your question mainly lies in how to get the face landmarks. For this you could refer to my blog post. It's my own implementation. However it is in Chinese that I cannot expect you to understand.
You can just search for face landmarks
and you'll get lots of codes for that. This topic has been under exhaustive research during these decades. Also there's close source solutions on this, e.g. Visage Tech's Visage SDK.