I am trying to prove Pascal's theorem for conics. To do so, I am trying to follow the proof which is given as an exercise in Rey Casse's Projective Geometry. Here is the exercise. 
In this exercise, they use "." as the product of the two line equations. For example, $(AD).(BC)$ means the product of the equations of the lines $AD$ and $BC$.
I have done parts $(a)$ and $(b)$. However, I am stuck at part $(c)$. How do you finish the exercise?
In $(b)$, the line pairs intersect in L and M. Furthermore, the common points in part (b) are A and D, and also the base points of the last pencil in part $(a)$ are also A and D. Is there a link between these two observations (that might help to finish the exercise)?