Gravitational and electromagnetic interactions are Hamiltonian systems with forces between pairs of particles. We propose an alternative: Hamiltonian dynamics with triplet interactions between point particles. Our system has a potential energy that depends on the shape of the triangle for each triplet. Similar multibody forces occur in many physical systems, such as polarizable molecules, nucleon interactions, and colloids, but typically are combined with more conventional two-body forces. We focus on potentials that depend only on the triangle perimeter or on its area. The resulting forces point toward a center of the triangle, either the incenter or the orthocenter, respectively. For the planar case, the resulting system has six degrees of freedom but can be reduced to three since it conserves the total momentum and angular momentum. The dynamics often exhibits chaotic motion, but there are a number of special solutions, such as equilateral and isosceles triangles, and perturbations of these can lie on invariant tori. Numerical investigations of several examples show families of such regular trajectories as well as examples of chaotic dynamics.