Olympiad Geometry Problem #25: Midpoint, Perpendiculars, Parallels

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Here is a beautiful problem from from 2018 Romania JBMO TST. Only a couple of lines, but it leads to a nontrivial result. Enjoy!
  1. Michael Greenberg
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Sir the solution is very brutal
Any other solution

AmanKumar-loiz
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sir there is something i did now note that if both triangles BED and CFD is moved towards D by distance BC/4 then we get that it becomes homothetic to BGHC so for that setting DMJ is collinear. but if moved back we see that due to congruence the midpoint lies at the same point. so we get CMJ is coliinear. does this work?

ShowriAKBhat