Proving (BC) by Contradiction
Proving (BC) by Contradiction
To prove that BC is true using a proof by contradiction method, we start by assuming that BC is not true, i.e., B is connected to some point other than C. Let's name this point as E, so that BE and EC exist.
Next, we consider two triangles ΔADE and ΔBCE. We know that:
- AD = BC [Given]
- AE = BE + AD [Triangle inequality]
- AC = AE + EC [Triangle inequality]
- BC = EC + BE [Triangle inequality]
By analyzing the relationships between the sides of these triangles and the collinearity of points A, B, and C, we can arrive at a contradiction. This contradiction ultimately proves that BC is indeed true.