There is a square, which side length is $2$, To ensure there exists a triangle in the square, with an area less than $0.5$, how many points should you draw in the square at least.
the goal is for all quantification of possible point arrangements, there must exist a triple of points, which triangle area less than $\frac{1}{2}$.
use pigeonhole principle，it's easy to prove the ceiling is 10; I think 9 points is the answer, but i can't prove it. Please give your answer and proof to help me, thank you.