Four-body Central Configurations with Adjacent Equal Masses

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$.