The author considers a periodic perturbation of a planar Hamiltonian system of the form \[ \dot{x}=JD_x H(x)+\epsilon g(x,\omega t;\mu),\qquad x\in \mathbb{R}^2. \] It is assumed that the unperturbed system has a one-parameter family of periodic orbits \(q^{\alpha}(t)\) analytic with respect to \(\alpha\). By using a Melnikov-type technique, he proves a criterion for the appearance of Bogdanov-Takens bifurcation points. Moreover, approximate expressions for saddle-node, Hopf and homoclinic bifurcation sets near such points are given. Theoretical results are illustrated with a detailed study of the example \[ x''+x+x^3=\varepsilon\left[(-\delta+x\cos \omega t)x'+\gamma \cos \omega t\right]. \]