Collinearity
Three or more points are said to be collinear if they all lie on the same straight line.
If A, B and C are collinear then \({m_{AB}} = {m_{BC}}( = {m_{AC}})\) .
If you want to show that three points are collinear, choose two line segments, for example \(AB\) and \(BC\). You then need to establish that they have:
- a common direction (that is, equal gradients)
- a common point (for example, B)
If both of these statements are true then the points are collinear.
Example
Show that \(P(1,4)\), \(Q(4,6)\) and \(R(10,10)\) are collinear.
Solution
For this example we'll use the lines PQ and PR.
\({m_{PQ}} = \frac{{6 - 4}}{{4 - 1}} = \frac{2}{3}\)
\({m_{PR}} = \frac{{10 - 4}}{{10 - 1}} = \frac{6}{9} = \frac{2}{3}\)
The line segments have a common direction (gradients \(= \frac{2}{3}\)) and a common point \((P)\) so \(P\), \(Q\) and \(R\) are collinear.