Q:

Determine whether a point is inside triangle in 3D space

I have a 3D triangle that is represented by 3x 3D vertices. 
I'm looking for an algorithm that must cover the following requirements:

a point lies on the same plane as defined by the triangle and
that the point resides within the boundaries of said triangle

for example:

A:

Define basis vectors b=AB, c=AC and n= b x c (vector product), where A, B, C - triangle vertices's coordinates
Represent point P coordinate in this basis, solving linear equation system (unknown t,u,v). Gaussian elimination method is suitable here.
t * b.X + u * c.X + v * n.X = P.X
t * b.Y + u * c.Y + v * n.Y = P.Y
t * b.Z + u * c.Z + v * n.Z = P.Z

Point is 'inside' according to your picture, if 
0 <= t <= 1
0 <= u <= 1
and
t + u <= 1

