Geometry - Axioms and Point Relations

[steelphantom]

Homework Statement

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.

Homework Equations

Incidence Axioms, Betweenness Axioms (including Pasch Axiom)

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!

 

[mighty2000]

Thank you for your post. Here is my response to your questions:

1a. Your reasoning is correct. Since B and D are on opposite sides of C, they cannot be equal. This can be shown using the Pasch axiom, which states that if a line intersects a triangle, then it intersects at least one of the sides of the triangle. In this case, the line containing A, B, and C intersects the triangle ACD, and therefore must intersect at least one of the sides, which is AC. This means that B and D cannot be equal.

1b. Your reasoning is correct. By definition, A, B, and C are all on the same line, and D is on the same line as A and C. This means that all four points are on the same line.

2. To prove that AB and BC have no points in common except B, we can use the Pasch axiom again. If a line intersects a triangle, then it intersects at least one of the sides of the triangle. In this case, the line containing A, B, and C intersects the triangle ABC, and therefore must intersect at least one of the sides, which is AB or BC. However, since A, B, and C are not all on the same line, the only possible side that the line can intersect is BC. This means that AB and BC have no points in common except B.

I hope this helps. Let me know if you have any further questions. Keep up the good work as a scientist!