Can I split a circuit in the middle of a 3-wire cable?
You can splice into a 3-wire cable just as you'd splice a 2-wire cable, assuming it's not for a three-way switch or some other specialty case. Do it as you've drawn.
You'll need slack. If you don't have it, plan for either using two junction boxes, or splice at an existing box.
For the two-box method, mount two boxes at least 24&quot; apart and cut the cable between. Make your splice in whichever box is most convenient and run a jumper between boxes. Obviously this means you need more /3 cable. Do not run parallel cables as a substitute, which would be illegal.
Be aware that junction boxes must remain accessible. Don't bury them in your walls or ceiling, or cover them with cabinets.