This paper deals with the implementation of the interior point cutting plane algorithm of Goffin-Haurie-Vial, with a special application to the solution of convex differentiable programming problems. The interior point cutting plane algorithm is closely related to the classical method of Cheney-Goldstein and Kelley, but the cuts are generated from different, more central, points in order to achieve deeper cuts and thereby accelerate convergence. The method is quite general in purpose as it can be applied to a large class of convex differentiable and nondifferentiable optimization problems. The paper focuses on the different stages of a practical implementation and the overall performance of the algorithm. The test problems come from a set of convex geometric programming problems.