  In this paper we introduce new methods to prove the finite cyclicity of some
graphics through a triple nilpotent point of saddle or elliptic type
surrounding a center. After applying a blow-up of the family, yielding a
singular 3-dimensional foliation, this amounts to proving the finite cyclicity
of a family of limit periodic sets of the foliation. The boundary limit
periodic sets of these families were the most challenging, but the new methods
are quite general for treating such graphics. We apply these techniques to
prove the finite cyclicity of the graphic $(I_{14}^1)$, which is part of the
program started in 1994 by Dumortier, Roussarie and Rousseau (and called DRR
program) to show that there exists a uniform upper bound for the number of
limit cycles of a planar quadratic vector field. We also prove the finite
cyclicity of the boundary limit periodic sets in all graphics but one through a
triple nilpotent point at infinity of saddle, elliptic or degenerate type (with
a line of zeros) and surrounding a center, namely the graphics $(I_{6b}^1)$,
$(H_{13}^3)$, and $(DI_{2b})$.
