A B727-200 IN CRUISE AT FL240 LOST THE R MAIN LNDG GEAR FLYING DOOR DUE TO INCORRECT INSTALLATION.

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I PICKED UP ACFT X AT ABOUT XA40 WITH THE TASK OF INSTALLING THE R MAIN GEAR FLYING DOOR AND COMPLETING THE INSTALLATION OF THE R INBOARD FLAP FAIRING. THESE ITEMS WERE REMOVED TO FACILITATE THE COMPLETION OF JOB CARD. WHEN INSTALLING THE R MAIN GEAR FLYING DOOR AND TRYING TO LINE UP THE SCREWS FOR THE INBOARD FLAP FAIRING; WE WERE STRUGGLING WITH THE SCREWS FOR THE FLAP FAIRING THAT WERE LOCATED ABOVE THE PARTIALLY INSTALLED GEAR FLYING DOOR. WE FELT IT WOULD EASE THE INSTALLATION OF THE FLAP FAIRING IF WE LOWERED THE GEAR FLYING DOOR DOWN TO ALLOW MORE ROOM TO ALIGN AND INSTALL THE FASTENERS. TO DO THIS WE DETACHED AND MOVED THE BRACE THAT ATTACHES THE FLYING DOOR TO THE GEAR UP AND OUT OF THE WAY. THIS ALLOWED MORE ROOM TO ALIGN THE FASTENERS ON BOTH SIDES OF THE FLAP FAIRING. UPON REINSTALLING THE BRACE THAT ATTACHES THE GEAR TO THE FLYING DOOR; THE BRACE MAY HAVE BEEN ATTACHED INCORRECTLY BELOW THE R MAIN GEAR DRAG BRACE; INADVERTENTLY. THE INSTALLATION OF THE FLYING DOOR AND FLAP FAIRING WAS THEN COMPLETED AND THE APPROPRIATE PAPERWORK SIGNED OFF. WHEN THE ACFT LEFT ZZZ ENRTE TO ZZZ1; THE R GEAR FLYING DOOR DEPARTED THE ACFT AND WAS DISCOVERED MISSING AT THE GATE IN ZZZ1.