I am going to explain a case when a line comes into a face of "3".
Before I begin the explanation, I need to define what I mean when I say a line "comes into" a face. When a line comes into a face, one of its vertices will be connected with a vertex that does not belong to that face.It's too complicated to explain only by text. I will use figures from here.

The star mark indicates the top left vertex of "3". When either one of the edge A or B has a line, which means the vertex is connected with a vertex out of this face, and the other edge is marked with "x", this is called a line is comming into the face of "3". 



These are the examples of a line coming into a face. 


In this figure the line is only contacting a vertex of "3". You cannot
say a line is comming into a face here. It is impossible to contact with a face of "3" at one of its vertices in the first place. 


Now, let's think about how you can draw lines when a line comes into
the face of "3".
Arrows mean one of them will have a line and the other will have "x". 



Since the line comes from outside, it must go around "3".
Either one of two edges, to which the incoming line can extend, will
be marked with "x".
So there are only these two possibilities to draw lines for this case. 

Therefore a rule is defined from the common part of these two cases:
Rule of drawing line to a corner of "3"  

This rule is very powerful. Without knowing this rule it is impossible to solve one puzzle.