When to vent new half bathroom plumbing

I am planning the install of a new half bathroom in an existing space in my home. There is currently one full bathroom, that is on the second story of the house. The soil/vent stack runs vertically down from the second floor, has a small horizontal section around grade, and then drops down below grade into the basement. Once in the basement there is a final horizontal run before a 90 down to below the basement foundation, and tie in to the sewer service line.

I am trying to determine the the most efficient way to plumb the new half bathroom, and if additional vents are required. Below is an isometric drawing of the scenario. Red lines are existing plumbing, green dotted lines are just to represent the plane that is the 1st floor grade, and blue is the new plumbing. Since the new plumbing lines (blue) will be less than 6' in length, do they need a vent? I was hoping to make my ties in(s) for the new plumbing horizontal because of a lack of vertical distance between the horizontal run in the basement and 1st floor grade. Tie in fittings would be 45 and Wye, or combo wyes. Install is in the Allegheny County, PA.

Red pipe (existing) is 3". 3D sketch proposed interpretation to Fresh Codemonger