  Let $P \subset \mathbb{R}^m$ be a polytope of dimension $m$ with $n$ facets.
Assume that $P$ is Delzant and Fano. We associate a monotone embedded
Lagrangian $L \subset \mathbb{C}P^{n-1}$ to $P$. As an abstract manifold, the
Lagrangian $L$ fibers over some torus with fiber $\mathcal{R}_P$, where
$\mathcal{R}_P$ is defined by a system of quadrics in $\mathbb{R}P^{n-1}$.
  We find an effective method for computing the Lagrangian quantum cohomology
groups of the mentioned Lagrangians. Then we construct explicitly some rich set
of wide and narrow Lagrangians.
  Our method yields many different monotone Lagrangians with rich topological
properties, including non-trivial Massey products, complicated fundamental
group and complicated singular cohomology ring.
  Interestingly, not only the methods of toric topology can be used to
construct monotone Lagrangians, but the converse is also true: the symplectic
topology of Lagrangians can be used to study the topology of $\mathcal{R}_P$.
General formulas for the rings $H^{*}(\mathcal{R}_P, \mathbb{Z})$,
$H^{*}(\mathcal{R}_P, \mathbb{Z}_2)$ are not known. Since we have a method for
constructing narrow Lagrangians, the spectral sequence of Oh can be used to
study the singular cohomology ring of $\mathcal{R}_P$.
