Question

Let S be a square region (in the plane) of side length 2 inches. Show that among any nine points in S, there are three which are the vertices of a triangle of area ≤ ½ square inch

Answer 1

Let S be a square region (in the plane) of side length 2 inches.

S

In our case if we have 9 points in S, we have at least 3 points, which link them can't cover the square of $\frac{1}{2}$ square inch. It happens because we can't place all 9 points in the way that distance between them was sufficient to cover $\frac{1}{2}$ square inch.