My living room does not have any existing lighting, but it does have two three way switches that control the bottom halves of two outlets. Both switches control the bottom halves of both outlets. The wire that connects the two three way switches runs straight across the room through the ceiling. I want to install two recessed lights in line with that wire and then in the middle of those two recessed lights but across several ceiling joists, I want to install a ceiling fan light fixture. So, my question is, how can I tap into the wire connecting the three way switches - going recessed light, ceiling fan, recessed light - so that both switches will control all the lights. I've attached a few pictures. Not sure if they will help. I am comfortable doing the electrical work as long as I know I am connecting the right wires together. Any help would be appreciated.
Left: Rough wiring diagram. Yellow represents white wires. Switch 1 and switch 2 are the three way switches that control the bottom halves of the outlets. Switch 3 controls an outside light.
Right: Junction box behind switches 1 and 3
Left: Same junction box with view of switch 1(left) and 3(right)
Right: Switch 2
Left: Rough wiring diagram. Yellow represents white wires. Switch 1 and switch 2 are the three way switches that control the bottom halves of the outlets. Switch 3 controls an outside light.
Right: Junction box behind switches 1 and 3
Left: Same junction box with view of switch 1(left) and 3(right)
Right: Switch 2