1. The problem statement, all variables and given/known data 1. Suppose A * B * C and A * C * D. a) Prove that no two of A, B, C, D are all equal. b)Prove that A, B, C, D are all on one line. 2. Suppose that A, B, C are points not all on one line. Prove that AB and BC have no points in common except B. 2. Relevant equations Incidence Axioms, Betweenness Axioms (including Pasch Axiom) 3. The attempt at a solution 1a. By definition of A * B * C and A * C * D, we have A != B, A != C, A != D, B != C, and C !=D. So we only have to show that B !=D, and of course that's where I'm stuck. Can I just say that B and D are on opposite sides of C, so they cannot be equal? 1b. I think I did this one correctly. By definition, A, B, and C are all on the same line. Also by definition, D is on the same line as A and C, and therefore on the same line as B. So they are all on the same line. 2. Not really sure how to do this one. Maybe an application of the Pasch axiom? Thanks for any help!