Wiring new lights

[acursi15]

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

 

[JoeD]

how can I tap into the wire connecting the three way switches

You can't. There is no neutral. There is most likely no constant hot.

 

[Snoonyb]

Interestingly, they used the white, as the hot, without marking it by wrapping with black tape.

The first thing you need is a least one foot of slack in the 12/3, or you'll need two junction boxes and a length of 12/3 too connect them.

Next you'll need some 12/2 to connect a neutral to the hot pair in switch box #2, where the black from the new 12/2 is wire-nutted off.

In doing this, this way, everything is either on, or off.

 

[acursi15]

Thank you.

 

[Snoonyb]

In your case, and I never do this, whoever did the wiring for the three-ways, used the red and black conductors, as travelers, when they should have used either the red or black, (preferably the red) and the white, because that would have left the black as the switched hot conductor.

When there are inconsistencies, repairs become time consuming and expensive.