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# Show that the points are collinear?....Math Watch

1. (-2a,5a) (0,4a) (6a,a)
2. (Original post by reb0xx)
(-2a,5a) (0,4a) (6a,a)
If you call the points A,B,C, then one way is to show that AB is parallel to BC (i.e. the lines have the same gradient), and since they both go through B, they are the same line, and the points are collinear.
3. If they have same gradient they are collinear.
4. (Original post by marek35)
If they have same gradient they are collinear.
No; that'll mean they are parallel. Points on the line y=2x and y=2x+5 are not collinear. The lines are parallel. As ghostwalker said, if a point on 2 parallel lines share the same point then they are the same line i.e. points are collinear.
5. (Original post by dknt)
No; that'll mean they are parallel. Points on the line y=2x and y=2x+5 are not collinear. The lines are parallel. As ghostwalker said, if a point on 2 parallel lines share the same point then they are the same line i.e. points are collinear.
Well I was saying generally to be collinear they must have the same gradient and lie on the same straight line- which is done by having the same gradient.
6. (Original post by marek35)
Well I was saying generally to be collinear they must have the same gradient and lie on the same straight line- which is done by having the same gradient.
That's a necessary condition but it's not sufficient for 3 points to be collinear. The 2 lines between each pair of points must have the same gradient AND must intersect at a single point (which must be one of the three points we're considering). Your post had the implications going the wrong way.
Same gradient collinear, which is not what you suggested with your original post. However collinear same gradient, which is what I've just said above.
7. maths*
8. (Original post by Farhan.Hanif93)
That's a necessary condition but it's not sufficient for 3 points to be collinear.
It is sufficient. There's no way of having two lines AB and BC which don't both contain the same point (i.e. B), so if you want to show that three points are collinear, all you have to do is show that AB and BC (or AC and BC, or AB and AC) have the same gradient. In general, n points are collinear if n-1 distinct pairs of points are joined with lines of the same gradient.

Showing points are collinear is quite different from showing two lines are the same (in which case you do need to worry about whether they have a common point).

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Updated: December 17, 2010
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