A result involving the perpendicular bisectors of a hexagon with opposite sides parallel

Conjecture: The intersections of the adjacent perpendicular bisectors of the sides of a hexagon with opposite sides parallel form a parallelo-hexagon, i.e. hexagon with opposite sides parallel and equal.

Alternative formulation: The respective circumcentres G, H, I, J, K and L of triangles ABC, BCD, CDE, DEF and EFA of a hexagon ABCDEF with opposite sides parallel, form a parallelo-hexagon, i.e. hexagon with opposite sides parallel and equal.

Challenge:
Although the above conjecture lends itself most readily to attack by vector methods or complex algebra, the real challenge is to find a purely geometric proof. Can you find such a proof?

Hint: Click on the Hint button in the sketch for a possible construction which one can use to produce a geometric proof.

Read my 2020 paper related to this result in the International Journal for Mathematical Education in Science & Technology (IJMEST) - the 1st 50 downloads are free and complimentary at this link: A novel proof of a hexagon theorem.