1 Abstract A trigonometric method to simplify the Three-body problem in celestial mechanics. By calculating the centroid of three celestial bodies without and then with motion, it offers a novel approach to determine their dynamic center of mass. Grounded in the Superconducting Field Theory, this method simplifies complex gravitational interactions and dark matter considerations. While applicable within our solar system, its principles may extend to other galaxies, highlighting the potential for broad astronomical applications. This concise methodology underscores the importance of triangulation in understanding celestial mechanics, providing a foundation for future research in the field. Keywords: Three-body problem; N-body problem; Trigonometry. The Double Centroid Approach (Three-body problem) Sergio Pérez Felipe 1, 1 Independent Researcher (Software Engineer, Madrid, Spain) Contents Introduction ........................................................................................................................................................................................ 1 Principles ............................................................................................................................................................................................. 1 1. Center of gravity without motion ................................................................................................................................................... 2 2. Adding motion (centrifugal force) ................................................................................................................................................... 2 3. Calculate the new centroid, or center of mass ............................................................................................................................... 3 References .......................................................................................................................................................................................... 3 Introduction Newton's three-body problem refers to the problem of determining the possible motions of three-point masses that attract each other according to Newton's law of inverse squares. This problem originated from Newton's perturbative studies on the inequalities of lunar motion in the 1740s. Newton's work on the two-body problem, which involved the elliptical orbits of planets around the sun, was highly successful. However, when it came to the three-body problem, finding exact solutions became much more challenging. Leonhard Euler, a Swiss mathematician, made significant contributions to the three-body problem [2]. In 1722, Euler found some narrow solutions to the three-body problem when one of the objects has essentially no mass. Later, in 1760, Euler showed that the three-body problem had an exact solution. He discovered three families of periodic solutions in which the three masses are collinear at each instant. These solutions are known as Euler's collinear solutions and are valid for any mass ratios. Euler's work laid the foundation for further developments in solving the three-body problem. Joseph Louis Lagrange, extended Euler's work on the three-body problem [3]. In 1772, Lagrange found a family of solutions in which the three masses form an equilateral triangle at each instant. These solutions, along with Euler's collinear solutions, are known as the central configurations for the three-body problem. The masses in these solutions move on Keplerian ellipses, and they are valid for any mass ratios. Lagrange's contributions further advanced the understanding of the three-body problem. Important extensions and analyses were done subsequently by Liouville, Laplace, Jacobi, Darboux, Le Verrier, Velde, Hamilton, Poincaré, Birkhoff or E. T. Whittaker, and now me. Principles The main difference with previous approaches is that in this case we take into account the centrifugal force from what would be an equilibrium point without movement. Furthermore, for the calculations of the center of mass of the system, dark matter is taken into account for the calculation of the equilibrium point; This dark matter calculation is an estimate of a change in the gravitational constant as indicated by my first theory (Superconducting Field Theory, 2024) [1]. We will not take into account possible changes in centrifugal force. Fig. 1: Centrifugal force Fcf = mω 2 r = mv 2 /r v = rω ω=2πn