Need a way to do a loopcut- ish thingy on the blue line
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One way is to use vertex bevel (⎈ Ctrl⇧ ShiftB):
Once the triangle fan is converted to quads you can add loopcuts. In the gif I converted the ngon resulting from the bevel to a triangle fan with Poke (⎇ AltP).
As an alternative to bevel, you might also try Inset (I):
Open Specials Menu using W button and select subdivide.
Hit F6 button right after using Subdivide operation to open Subdivide Menu and adjust the number of subdivisions.
Or change the number of subdivisions in Subdivide Operator panel.