Question

A finite set of points in the plane has the property that the triangle formed by any 3 of them has area less than 1.Prove that there is a triangle of area 4 that contains all the points

Answer 1

Explanation:

All of the points must be contained within a circle that will circumscribe an equilateral triangle of area 1. (We refer to this as the inscribed triangle.) Any points on the circle that do not form an equilateral triangle will form a triangle with an area less than 1. Likewise for any points within the circle.

The circle can be circumscribed by an equilateral triangle whose side length is double that of the inscribed triangle. Since the side length is double, the circumscribing triangle will have an area 2² = 4 times that of the inscribed triangle.

The inscribed triangle has an area of 1, so the circumscribing triangle must have an area of 4.