Symmetry center of hexagon

Question

How to show that the figure has a symmetry center for instance if we have a convex hexagon where opposite sides are of equal lenght and parallel?

Answer 1

From the given information, the vertices of the given hexagon are $$ p,\quad p+u,\quad p+u+v,\quad p+u+v+w,\quad p+v+w,\quad p+w $$ for some $p,u,v,w\in\mathbb{R}^2$. Then the symmetry center is $$ p + \frac12(u+v+w). $$