Pappus if Alexandria is a Greek mathematician who is known for his synagouge and pappus's theorem. Pappus's theorem states that if A, B, and C are a set of collinear points, a, b, and c are another set of collinear points, and Ab meets Ba at X, Ac meets Ca at Y, and Bc meets Cb at Z, then points X, Y, and Z are collinear.

Some questions for you to consider:
1. This theorem is considered self-dual, what makes it that way?
2. How do the slopes of the lines affect the points X, Y, and Z?
3. It is possible to only have one point (out of X, Y, and Z) with certain slopes of the lines, but is it possible to create only two points?