We introduce a planner designed to guide robot manipulators in stably placing objects within intricate scenes. Our proposed method reverses the traditional approach to object placement: our planner selects contact points first and then determines a placement pose that solicits the selected points. This is instead of sampling poses, identifying contact points, and evaluating pose quality. Our algorithm facilitates stability-aware object placement planning, imposing no restrictions on object shape, convexity, or mass density homogeneity, while avoiding combinatorial computational complexity. Our proposed stability heuristic enables our planner to find a solution about 20 times faster when compared to the same algorithm not making use of the heuristic and eight times faster than a state-of-the-art method that uses the traditional sample-and-evaluate approach. Our proposed planner is also more successful in finding stable placements than the five other benchmarked algorithms. Derived from first principles and validated in ten real robot experiments, our planner offers a general and scalable method to tackle the problem of object placement planning with rigid objects.