We study the planar three-body problem using the principal axes of inertia frame, determined by one Euler angle in the planar case. Three variables are needed in this frame: two distances R 1 and R 2, related with the principal moments of inertia on the plane of motion of the particles, and an auxiliary angle σ. We also connect these coordinates with the shape sphere of the similarity class of triangles and we give a geometric interpretation of the angle σ. We then write the Hamiltonian and equations of motion in these coordinates to find the symmetries of involution and symmetric periodic orbits. To exemplify this method, we calculate periodic orbits of a three-body problem with two or three equal masses.