MathLabs

Problem 5

Given n>4n>4 points in the plane, no three collinear, prove that at least (n−32)\binom{n-3}{2} convex quadrilaterals have their vertices among the given points.
Step 1 of 6: Prove the five-point lemma
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Every five points contain a convex quadrilateral\text{Every five points contain a convex quadrilateral}
Detailed analysis

Take five points. If their convex hull has at least four vertices, those vertices form the quadrilateral. Otherwise the hull is a triangle ABCABC, with two points D,ED,E inside it. Two of A,B,CA,B,C lie on the same side of line DEDE; together with D,ED,E they form a convex quadrilateral.