  Denote by $\mathcal{C}^-_{\ell}$ the $3$-uniform hypergraph obtained by
removing one hyperedge from the tight cycle on $\ell$ vertices. It is
conjectured that the Tur\'an density of $\mathcal{C}^-_{5}$ is $1/4$. In this
paper, we make progress toward this conjecture by proving that the Tur\'an
density of $\mathcal{C}^-_{\ell}$ is $1/4$, for every sufficiently large $\ell$
not divisible by $3$. One of the main ingredients of our proof is a
forbidden-subhypergraph characterization of the hypergraphs, for which there
exists a tournament on the same vertex set such that every hyperedge is a
cyclic triangle in this tournament.
  A byproduct of our method is a human-checkable proof for the upper bound on
the maximum number of almost similar triangles in a planar point set, which was
recently proved using the method of flag algebras by Balogh, Clemen, and
Lidick\'y.
