Four-body Central Configurations with Adjacent Equal Masses

TL;DR

This paper proves that convex, non-collinear four-body central configurations with specific equal masses and diagonal lengths must be symmetric and form an isosceles trapezoid, revealing geometric constraints in such configurations.

Contribution

It establishes the symmetry and geometric shape (isosceles trapezoid) of certain four-body central configurations with adjacent equal masses and specific diagonal length conditions.

Findings

Configurations are symmetric and form isosceles trapezoids.

Symmetry holds when diagonals are equal.

Configurations are also isosceles trapezoids under specific length conditions.

Abstract

For any convex non-collinear central configuration of the planar Newtonian 4-body problem with adjacent equal masses $m_1=m_2\neq m_3=m_4$, with equal lengths for the two diagonals, we prove it must possess a symmetry and must be an isosceles trapezoid; furthermore, which is also an isosceles trapezoid when the length between $m_1$ and $ m_4$ equals the length between $m_2$ and $ m_3$.