  We study the problem of finding pairwise vertex-disjoint triangles in the
randomly perturbed graph model, which is the union of any $n$-vertex graph $G$
satisfying a given minimum degree condition and the binomial random graph
$G(n,p)$. We prove that asymptotically almost surely $G \cup G(n,p)$ contains
at least $\min\{\delta(G), \lfloor n/3 \rfloor\}$ pairwise vertex-disjoint
triangles, provided $p \ge C \log n/n$, where $C$ is a large enough constant.
This is a perturbed version of an old result of Dirac.
  Our result is asymptotically optimal and answers a question of Han, Morris,
and Treglown [RSA, 2021, no. 3, 480--516] in a strong form. We also prove a
stability version of our result, which in the case of pairwise vertex-disjoint
triangles extends a result of Han, Morris, and Treglown [RSA, 2021, no. 3,
480--516]. Together with a result of Balogh, Treglown, and Wagner [CPC, 2019,
no. 2, 159--176] this fully resolves the existence of triangle factors in
randomly perturbed graphs.
  We believe that the methods introduced in this paper are useful for a variety
of related problems: we discuss possible generalisations to clique factors,
cycle factors, and $2$-universality.
