We propose a framework of efficient nonlinear deformable simulation with both fast continuous collision detection and robust collision resolution. We name this new framework Medial IPC as it integrates the merits from medial elastics, for an efficient and versatile reduced simulation, as well as incremental potential contact, for a robust collision and contact resolution. We leverage medial axis transform to construct a kinematic subspace. Instead of resorting to projective dynamics, we use classic hyperelastics to embrace real-world nonlinear materials. A novel reduced continuous collision detection algorithm is presented based on the medial mesh. Thanks to unique geometric properties of medial axis and medial primitives, we derive closedform formulations for identifying between-primitive collision within the