3 points  3 lines. 
N points  N lines. 
4 points {A,B,C,D} determine 6 lines {AB,AC,AD, BC, BD, CD} and three additional points {X,Y,Z}. 
The Complete Quadilateral: 4 lines {AB, BC, CD,AD}determine 6 points {A,B, C, D, X,Y} with three additional lines{AC, BD, XY} . 


from a point 
from a line 
Desargue's Spatial Configuration
Pascal: 9 points  7 lines A Point Conic The conic is described by a set of points 
Brianchon: 9 lines  7 points A Line ConicThe conic is described by a set of lines 