slope(first point, second point) = (c+a-b-c)/(b-a) = (a-b)/(b-a) = -1
slope(first point, third point) = (a+b-b-c)/(c-a) = (a-c)/(c-a) = -1
slope(second point, third point) = (a+b-c-a)/(c-b) = (b-c)/(c-b) = -1
If you have 3 points forming 3 lines and the 3 lines are all parallel, then the points have to be collinear.