Assignment:
Q1. Construct a model for incidence geometry that has neither the elliptic, hyperbolic, nor Euclidean parallel properties.
Q2. Consider a finite geometry where the points are interpreted to be the six vertices of a regular octahedron and the lines are sets of exactly two points. Is this interpretation a model of an incidence geometry? How many lines does it have? Which parallel axiom (if any) does it display?
Q3. Given three collinear points A, B, and C, with B between A and C. Let P be a fourth point on the same line. Show that if B is not between A and P, then C is not between A and P.
Q4. Let A and B be two points such that point A lies on line I and B ∉ I. Show that if C ∉ AB- , C ≠ A , B, then B and C are on the same side of I.
Q5. Let γ be a circle in the Euclidean plane with center 0 and let A and B be two points on γ. The segment AB is called a chord of γ: let M be its midpoint. If O≠M, show that the line through O and M is perpendicular to segment AB.
Provide complete and step by step solution for the question and show calculations and use formulas.