Here's a dilemma plenty of us projector owners face: you want to ceiling mount your projector, but there's a ceiling mounted lighting fixture in the way. Jerzy Żygłowicz worked around this problem by making his own flat profile ceiling lamp using parts from affordable IKEA LACK side tables and adding a combo halogen and LED lighting fixtures, permitting his LCD projector beam to reach its intended screen destination...
There's not much you can do in order to remedy a low hanging light fixture with a ceiling mounted projector except to move/remove one or the other. Jerzy opted for something more elaborate: he hacked together a thin ceiling light out of two IKEA LACK coffee table.
His DIY project involved using two different LACK tables. The second one was a smaller one, the LACK side table, which he glued onto the bigger one. The rest of the project required placing mounts for 4 halogen spots. These same holes were used to fix the lamp onto the ceiling. Another hole needs to be drilled in the center, permitting connecting wires from the lights to the electrical system overhead.
We do note this project seems to be a significant fire risk, as halogen lighting and IKEA particle board are poor combinations; even with cooler and smaller LED bulbs, one should take great precaution when wiring their own lighting system.
On top of the 4 halogen spots, he added some LED Dioder strips for ambient lighting. Since the table has a lid, they are perfect for this situation, as the only thing visible is their glow, not the bulbs themselves. He used metal bindings to mount the whole fixture to the ceiling, but before he did this, he painted the fixture to match the ceiling. That way, when it's not in use, it blends into the ceiling.