In a previous paper, I proved that two very different constructions of monotone Lagrangian tori are Hamiltonian isotopic inside $\mathbb{CP}^2$ by comparing both of them to a third one called modified Chekanov torus. This modified Chekanov torus has an interesting projection under the standard moment map of $\mathbb{CP}^2$ and motivates a method of construction of (monotone) Lagrangian submanifolds in symplectic toric manifolds. I will explain how this method gives some old and new monotone examples in $\mathbb{CP}^2$ and $\mathbb{CP}^1 \times \mathbb{CP}^1$.
This is joint work with Miguel Abreu (IST, Lisbon).