  We reach beyond the celebrated theorems of Erd\H{o}s-Ko-Rado and
Hilton-Milner, and, a recent theorem of Han-Kohayakawa, and determine all
maximal intersecting triples systems. It turns out that for each $n\ge7$ there
are exactly 15 pairwise non-isomorphic such systems (and 13 for $n=6$). We
present our result in terms of a hierarchy of Tur\'an numbers $\ex^{(s)}(n,
M_2^{3})$, $s\ge1$, where $M_2^{3}$ is a pair of disjoint triples. Moreover,
owing to our unified approach, we provide short proofs of the above mentioned
results (for triple systems only). The triangle $C_3$ is defined as
$C_3=\{\{x_1,y_3,x_2\},\{x_1,y_2,x_3\}, \{x_2,y_1,x_3\}\}$. Along the way we
show that the largest intersecting triple system $H$ on $n\ge6$ vertices, which
is not a star and is triangle-free, consists of $\max\{10,n\}$ triples. This
facilitates our main proof's philosophy which is to assume that $H$ contains a
copy of the triangle and analyze how the remaining edges of $H$ intersect that
copy.
