Theorem:
If A, B and C are collinear points, then the real numbers x, y, z not all zero can be found such that
x + y + z = 0 xA + yB + zC = 0.
and also the inverse of this thorem is true: If three such numbers not all zero can be ound then the points are collinear.

Conversely
Theorem:
If A, B, C are three given points which are not collinear and we can find three real numbers x, y, z such that x+y+z = 0 and xA + yB + zC = 0,
then we must have x=y=z=0.

Lonicera ”Dropmore Scarlet”