*There are five points on a flat surface, no two of which are coinciding and no three of which are on a straight line. Prove that four of these points are vertices of a convex quadrilateral.*

This simple theorem has a simple proof but, as the unheard of and complex concept of the so called

*convex hull*is involved, the proof could be unintelligible to non-mathematicians, high school students inclusive. So, here is the simplest proof of mine, in which you can meet the familiar points and lines only.

Let us think about the five points as of two lines and an extra point. Line

**l1**is defined by

**P1**and

**P2**, line

**l2**is defined by

**P3**and

**P4**, and

**P5**is the extra point. There are only 3 cases to consider:

__Case 1.__Neither line intersects the segment between the points on the other line. One can build a convex quadrilateral by connecting the points on

**l1**and

**l2**

**.**

__Case 2.__Each line intersects the segment between the points of the other line. One can build a convex quadrilateral by connecting the points on

**l1**and

**l2**

**.**

__Case 3.__Only one of the lines intersects the segment on the other line. Let us assume without loss of generality that

**l1**intersects the

**P3P4**segment of

**l2**.

**P5**can be in any of the 18 parts of the plane, but fortunately those parts are of only 2 types,

**O**(coming from

**O**ther) and

**S**(coming from

**S**ame).

**O**means there exists a line dividing the plane so that

**P5**is in one half of the plane and a triangle consisting of three of the other points exists in the other half such that when

**P5**is connected to two of its vertices a convex quadrilateral is built (see below the line

**P2P4**allowing us to build the convex quadrilaterals

**P2P3P4P5**and

**P1P2P5P4**).

**S**means there exists a line dividing the plane so that

**P5**is in one half of the plane and a triangle consisting of three of the other points exists in the same half such that when

**P5**is connected to two of its vertices a convex quadrilateral is built (see below the line

**P1P2**allowing us to build the convex quadrilateral

**P1P2P5P4**).