Hello, this question is so much a "how to do it" as it is a "why did they do it that way?"

The problem is to find a condition on three complex numbers $\displaystyle z_1, ~z_2, ~z_3$ showing that they are colinear in the Argand plain.

Obviously the condition is going to be that the slope through any two sets of points must be equal. In the work that I did I used the slope between points $\displaystyle z_1, ~z_3$ and between $\displaystyle z_2, ~z_3$.

The book, however, went in a very screwy direction to my thinking. Their result is that the quantity $\displaystyle \frac{z_1 - z_3}{z_2 - z_3}$ must be real for the points to be colinear. As it happens this condition is exactly the same as mine, just written in a much neater form.

But how the

would you go about doing this problem and say "Oh yeah! I'd get this..." Is there any significance to the fraction $\displaystyle \frac{z_1 - z_3}{z_2 - z_3}$ that I'm not seeing? Why might the book have put the answer in this form?

Thanks!

-Dan