  This paper is concerned with developing and analyzing convergent
semi-Lagrangian methods for the fully nonlinear elliptic Monge-Amp\`ere
equation on general triangular grids. This is done by establishing an
equivalent (in the viscosity sense) Hamilton-Jacobi-Bellman formulation of the
Monge-Amp\`ere equation. A significant benefit of the reformulation is the
removal of the convexity constraint from the admissible space as convexity
becomes a built-in property of the new formulation. Moreover, this new approach
allows one to tap the wealthy numerical methods, such as semi-Lagrangian
schemes, for Hamilton-Jacobi-Bellman equations to solve Monge-Amp\`ere type
equations. It is proved that the considered numerical methods are monotone,
pointwise consistent and uniformly stable. Consequently, its solutions converge
uniformly to the unique convex viscosity solution of the Monge-Amp\`ere
Dirichlet problem. A super-linearly convergent Howard's algorithm, which is a
Newton type method, is utilized as the nonlinear solver to take advantage of
the monotonicity of the scheme. Numerical experiments are also presented to
gauge the performance of the proposed numerical method and the nonlinear
solver.
