How to prove that 3 points are collinear

Question

If A, B, C, D are four points such that

(AB) ̅= 2A,

(AC) ̅= 6b -2a,

(AD) ̅= 3b,

how can you prove that B,C,and D are collinear?

Answer 1

Draw and label a diagram!

BC = 6b - 4a

CD = -4b + 2a

BD = 3b - 2a

Use Segment Addition Property to complete.

Answer 2

One way to show that three points are collinear is to show that any two segments are parallel. In this case, if you can show that BC and BD are parallel, it follows that B,C,D are collinear. To that end, we need to show that BC = kBD for some constant k.

BC = AC-AB = (6b-2a)-(2a) = 6b-4a

BD = AD-AB = (3b)-(2a) = 3b-2a

So, BC = 6b-4a = 2(2b-2a) = 2BD as desired.