No four points with pairwise distance 1 can be contained inside a halfdisk of radius 1. An open disk $D$ of radius $1$ in the Euclidean plane is the set of points with distance less than $1$ to the center of the disk. An open half disk  $H$  of radius $1$ is obtained by "cutting" $D$ into two equal sized parts and taking one of them. 
I would like to show that $H$ can not contain 4 points of pairwise minimum distance at least 1. While the statement is intuitively clear, I do not have a good idea as how to tackle the problem. Does anybody have an idea as how to tackle this?
Any pointers would be appreciated. 

Suppose you have four points in the half disk.
Cut the half disk $H$ to three sectors, each opening at an angle $\pi/3$.
By the pigeonhole principle there must be two points in one disk.
Show that the distance between any two points in such a sector is less than one.
