Abstract
In 1990, R.Moekel published a famous work about central configurations, in which he proved Conley′s Perpendicular bisector theorem, and the 45° theorem that provide some information on possible shapes of central configuration. In this paper we consider the following problem. Suppose (q₁, m₁), (q₂, m₂),..., (qi, mi) form a convex configuration, can we add a mass and position (qi+1, mi+1) on the boundary to make it a central configuration? The answer is false for i=3, as it follows easily from the Conley′s Perpendicular bisector theorem. Is the answer still negative when i >3? In this paper, we discuss the problem with i=4, and with isosceles trapezoid configuration and some equal masses. In the last section we provide a counter-example for this problem.