Question 70
Let $P_1, P_2, \dots, P_{2556}$ be distinct interior points of a regular hexagon $ABCDEF$ with side length $1$. Assume that no three points of the set
$$
S = \{A, B, C, D, E, F, P_1, P_2, \dots, P_{2556}\}
$$
are collinear. Show that there is a triangle with area less than $\frac{1}{1700}$ all of whose vertices belong to $S$.
$$
S = \{A, B, C, D, E, F, P_1, P_2, \dots, P_{2556}\}
$$
are collinear. Show that there is a triangle with area less than $\frac{1}{1700}$ all of whose vertices belong to $S$.