  We introduce a new approach and prove that the maximum number of triangles in
a $C_5$-free graph on $n$ vertices is at most $$(1 + o(1)) \frac{1}{3 \sqrt 2}
n^{3/2}.$$ We also show a connection to $r$-uniform hypergraphs without (Berge)
cycles of length less than six, and estimate their maximum possible size.
