An extension of the theorems of three-tangents and three-normals

Fujiwara et al. [T. Fujiwara, H. Fukuda, A. Kameyama, H. Ozaki, M. Yamada, J. Phys. A 37 (2004) 10571] obtained simple geometrical theorems on the three-body problem with zero-angular momentum. These are the so-called three-tangents and three-normals theorems. We extend these theorems to the planar three-body problem of non-zero angular momentum. We have found that the area of the triangle formed with three tangents (respectively, three normals) is expressed in a simple form using the area of the triangle formed with the three momentum vectors, and the angular momentum (respectively, the time derivative of the moment of inertia) of the system.