If point P exists, there would be 4 planar faces, P1...P4.
According to the link:

https://blender.stackexchange.com/questions/46113/how-to-make-all-quads-or-ngons-on-your-mesh-planar-2d

"when a vertex has more than 3 surrounding faces, their planar intersection does not necessarily exist - it would be actually very rate if it did:"

- Brian