Question 1042421
You need to choose 11 points to guarantee that you will have four corners of a rectangle.  
To see this, the space between any two consecutive points would be {{{20^o}}}, and hence 10 consecutive points, which corresponds 
to 9 {{{20^o}}} spaces, would constitute an entire semi-circle, i.e., {{{9*20^o = 180^o}}}.  
Name the points {{{p[1]}}}, {{{p[2]}}}, {{{p[3]}}},...,{{{p[9]}}}, {{{p[10]}}}

Now add an 11th consecutive point after {{{p[10]}}}, and call it {{{p[11]}}}.

The segment with endpoints {{{p[1]}}} and {{{p[10]}}} will be one diagonal of the rectangle, while the segment with endpoints {{{p[2]}}} and {{{p[11]}}} 
will be the other diagonal of the rectangle formed.