Solution Exercise 4.12.
Definition of Term Used in Mathematics:
Check for Yourself – This is the boring part of the proof, so you can do it on your own time.

Exercise 4.12.
(a) How many cases are there in the proof of the Dual of Axiom 3?

The question is: how many ways can three lines be chosen from four lines?  Hence, four cases.

(b) State the other cases.

(1) Suppose lines AB, BC, and DA are concurrent.
(2) Suppose lines AB, CD, and DA are concurrent.
(3) Suppose lines BC, CD, and DA are concurrent.

(c) Prove at least one of the cases.

Proof (1). Suppose lines AB, BC, and DA are concurrent. By the Dual of Axiom 1, B  is the only point of intersection of AB and BC. Hence B is the point of concurrency for the three lines AB, BC, and DA. Thus B is on DA, which contradicts the initial assumption that A, B, and D are noncollinear.

Proof (2). Suppose lines AB, CD, and DA are concurrent. By the Dual of Axiom 1, D  is the only point of intersection of CD and DA. Hence D is the point of concurrency for the three lines AB, CD, and DA. Thus D is on AB, which contradicts the initial assumption that A, B, and D are noncollinear.

Proof (3). Suppose lines BC, CD, and DA are concurrent. By the Dual of Axiom 1, C  is the only point of intersection of BC and CD. Hence C is the point of concurrency for the three lines BC, CD, and DA. Thus C is on DA, which contradicts the initial assumption that A, C, and D are noncollinear.