Maximum number of points a minimum distance apart in a semicircle of certain radius You have a circle of certain radius $r$. 

I want to put a number of points in either of the semicircles. However, 
no two point can be closer than $r$. 
The points can be put anywhere inside the semicircle, on the straight line, inside area, or on the circumference. There is no relation among the points of the two semicircles. But as you can see, eventually they will be the same.
How do I find the maximum number of points that can be put inside the semicircle?

The answer is five points. Five points can be achieved by placing one at the center of the large circle and four others equally spaced around the circumference of one semicircle (the red points in the picture below). To show that six points is impossible, consider disks of radius $s$ about each of those five points, where $r/\sqrt3 < s < r$. These five smaller disks completely cover the large half-disk; so for any six points in the large half-disk, at least two of them must lie in the same smaller disk. But then those two points are closer than $r$ to each other.

