The question: Let z_{1} and z_{2} be distinct complex numbers. Show that the locus of points z={tz_{1}+(1−t)z_{2},−∞<t<∞}, describes the line through z_{1} and z_{2}. The values $01 give the line segment joining z_{1} and z`2`.

I have an intuitive understanding of why the locus is a line, as it is similar to the description of a line through two points in R^2. However, I'm not sure how to prove this holds in C. Should I be trying to express it as one of the equations for line in C, such as Re(n̄z) = C or w̄z + wz̄ = r?