The planar three-body problem is a mechanical system which consists of three mass points

${m}_{1}$,

${m}_{2}$,

${m}_{3}$ which attract each other according to the Newtonian law. A significant problem is the existence of its periodic solutions. Only a limited number of these solutions is known, including in particular the Lagrangian homographic and homothetic solutions. These ones are considered by the author. Following

*R. Montgomery* [Nonlinearity 11, 363–376 (1998;

Zbl 1076.70503)], a periodic motion of three masses can be characterized by the triplet of winding numbers

$k=({k}_{12},{k}_{23},{k}_{31})$ defining a particular homology class. Fixing

$k$, the action functional can be restricted to the particular set of

$T$-periodic loops. Studying the minimization problem on this set, the author proves that the homographic Lagrangian solutions correspond to the choice

$k=(1,1,1)$ or

$k=(-1,-1,-1)$ (Theorem 1), whilst the homothetic orbits are the minimizers of the action in the case

${k}_{ij}\ne 0$ for all

$(i,j)$ (Theorem 2). It is noticed that the idea to minimize the action on the particular homology class was already introduced by H. Poincaré (1896).