Central Configurations in the Trapezoidal Four-Body Problems

In this paper we discuss the central configurations of the Trapezoidal four-body Problem. We consider four point masses on the vertices of an isosceles trapezoid with two equal masses $m_1=m_4$ at positions $(\mp 0.5, r_B)$ and $m_2=m_3$ at positions $(\mp \alpha/2, r_A)$. We derive, both analytically and numerically, regions of central configurations in the phase space where it is possible to choose positive masses. It is also shown that in the compliment of these regions no central configurations are possible.