Why do these points lie on a straight line? I didn't understand why $P_7$, $P_8$, and $P_9$ lie on a straight line in this proposition in Fulton's book.

The corollary:

The definition of $F\cdot G$
 
I need help, maybe I didn't understand what is exactly $F\cdot G$.
Thanks a lot
 A: The intersection cycle basically stores the data of how curves $F$ and $G$ intersect. It keeps the points of intersection, as well as their intersection number at those points. Bezout's theorem tells you that if you have two projective plane curves $F$ and $G$ of degree $n$ and $m$ respectively, then they intersect at $mn$ points counting multiplicity of intersections. In the language of  cycles, the zero cycle  $F\bullet G $ has degree $mn$ (recall that the degree of a zero cycle is just the sum of the coefficients). 
Now if we make the substitutions that the proof of Prop 2 says to make in the proof, we check: 


*

*$F\cap G$ consists of simple points of $F.$ Well $F\cap G$ is where $C$ intersects $Q.$ From $Q\bullet C= P_1 + \cdots + P_6$ we can read off that they intersect precisely at $P_1, \cdots, P_6$, and we are assume they are simple points on $C.$ 

*$H\bullet F \geq G\bullet F.$ This is true since $H\bullet F$ in this case is $P_1 + \cdots + P_9$ and $G\bullet F = P_1 + \cdots + P_6.$ 


So the conditions are satisfied and there is some curve $B$ such that $B\bullet C = P_7 + P_8 + P_9.$ 
Now what does Bezout's theorem tell you about the degree of $B$ ?
