Making Quadrilaterals
Say we have n points in the plane of which no 3 are collinear. What is the minimum value of n such that it is guaranteed that there will be some 4 of these points that we can join to make a convex quadrilateral?
The answer is 5.
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1 solution
Obviously 4 points is not enough because we can place the fourth point inside the triangle formed by the first three points:
Let's prove that 5 points is enough. Imagine a rubber band around 5 pins that we placed on 5 arbitrary points. Once we release the rubber band it will surround our points and form a convex figure. There are three possibilities:
In that case we can choose any 4 points and they will form a convex quadrilateral.
As we can see we already have a convex quadrilateral.
We have two points inside the triangle. Let's draw a line through them. Since they are inside the triangle there a side that the line doesn't cross. The inner points and points on that side will form a convex quadrilateral.
Thus, among 5 points there will always be four forming a convex quadrilateral.