Prove Colinearity of Set of n Points

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In summary, the problem is to prove that a set of n distinct points in the plane, where any two points have a third point that is colinear with them, must mean that all n points are colinear. One possible approach is using induction, where you prove the theorem for n=3 and then show that if it holds for n=k-1 points, it also holds for n=k points. However, another proof involves taking a set P that satisfies the conditions but is not a line, and using the fact that there is a minimum non-zero distance between points and lines to reach a contradiction.
  • #1
didi
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Hi,
Can you tell me how to prove the following?

Given a set of n distinct points in the plane such that for any two points in the set there is a third point in the set that is colinear with them (i.e., lies on the same line with them), prove that all the n in the set are colinear (i.e., lie on the same line).
Thank you
didi
 
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  • #2
It seems you can use induction here:
1) Prove that the proposition holds for n=3
2) Try to do the induction step on your own
 
  • #3
arildno said:
2) Try to do the induction step on your own

Suppose the induction hypothesis is that the theorem holds for n = k-1 points, and you want to show that implies it holds for n = k points. In a set of k points, how do you know that there is a subset of k-1 points that satisfies the condition in the statement of the problem? What if that k-th point is the one that holds everything together?
 
  • #4
I've been thinking about this problem and I don't see how you do it with induction. arildno can you explain how the inductive step works because I can't seem to think my way through it.

The proof that I can think goes like this
Say for contradiction you have a point set P that satisfies above but is not a line. Take L to be the set of lines that contain 2 or more points from P (in fact they all contain 3 points). Note that both L and P are finite. Also you can measure distance between a point and a line as the perpendicular distance.

So there are a also a finite amount of distances between points and lines and at least one distance is greater than zero (for every line you have at least one point such that the point is not on the line). Because there are only finitely many there is a minimum non-zero distance. Also I can describe a line by 2 points on that line (although this description is not unique it doesn't have to be). Take point b and line (p,q) such that the point line distance is the minimum. Take r to be the 3rd point on line (p,q). Take c to be the perpendicular projection of point b onto (p,q). Also we can assume the order of the points is p then q then r on line (p,q) (this would be a lot easier with a diagram). Well now either q=c, or q is to the right of c or q is to the left of c.

I will show one case since the rest is the same.
Assume q and r on the same side of c (say right of c). Now call the perp projection of q onto (b,r) d. Now you have two similar triangles (b,c,r) and (q,d,r). But qr is shorter than rc which is shorter than br (since br is the hypotenuse) so qd is shorter than bc. But that means q is closer to line (b,r) than b is to (q,r). But that's a contradiction since we chose distance between b and (q,r) to be of minimum distance.

Again arildno I would love to see the inductive method of solving this.
Thanks, Steven
 
  • #5
Brilliant proof, snoble. Here's a diagram to go with it :smile:
 

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  • #6
colinear points

Thank you all of you for your help

Didi
 

1. How do you define colinearity in a set of points?

Colinearity in a set of points refers to the property of all points lying on the same line. This means that if you connect any two points in the set with a straight line, all the other points in the set will also fall on that line.

2. What is the mathematical proof for colinearity of n points?

The mathematical proof for colinearity of n points is based on the concept of slope. If the slope between any two points in the set is equal, then all the points are colinear. This can be proven using the slope formula (y2-y1)/(x2-x1) and showing that it is equal for all possible combinations of points in the set.

3. Can colinearity be proven for non-linear sets of points?

No, colinearity can only be proven for sets of points that lie on a straight line. If the points are not on a straight line, then they are not colinear.

4. What is the significance of proving colinearity in a set of points?

Proving colinearity in a set of points is important in various mathematical and scientific fields such as geometry, trigonometry, and physics. It allows us to accurately predict the position and movement of objects, as well as solve complex mathematical problems.

5. Are there any real-life applications of proving colinearity in a set of points?

Yes, there are many real-life applications of proving colinearity in a set of points. For example, in surveying, colinearity is used to determine the position of objects and create maps. In physics, it is used to analyze the trajectory of objects in motion. In computer graphics, it is used to create 3D images and animations.

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