Ok, so how does this sound:

If P, A, B, C are on the same plane, then the vectors PA, PB, PC are on the same plane. Since the all share a common point P, the vectors would look something like so:



Since they are on the same plane, the Parallelogram Law of adding vectors applies. If this is true, then there exists 2 scalars b & c such that:

PA = bPB + cPC

By picking an arbitrary scalar s, any vector parllel to PA would = sPA. So,

sPA = sbPB + scPC

sPA - sbPB - scPC = 0.

And by letting t=-sb and u=-sc,

sPA + tPB + uPC = 0.

Would that work?