Let the points on the circle are numbered consequently from 1 to 18, inclusive.

Let call them  , , , . . . , .


Notice that if a rectangle is inscribed in a circle, then its diagonals are DIAMETERS
of the circle. So, the opposite verices of the rectangle are the opposite points on the circle.


Thus, diagonals of our rectangle connect opposite points on the circle.


Of 18 points , , , . . . , , we have 9 pairs with opposite points
(,), (,), . . . , (,).


Now my statement is that the ANSWER to the problem's question is 11 points of 18.


Indeed, among these 11 points, there is at least one pair of opposite points.
Removing this pair, you have remaining 16 points and 11-2 = 9 selected points.
Among these 9 selected points, there is at least one another pair of opposite points.

So, 11 points are ENOUGH, even in the WORST CASE.


From the other side, taking 10 points , , , . . . , , it is clear, that
among them, there is ONLY ONE pair (,) of opposite points to make a diameter/(=a diagonal),
and there are NO points to make another pair/(=another diagonal).



ANSWER.  11 points are needed to choose to GUARANTEE 
         that you will have the four corners of at least one rectangle.

    (1)  This problem is a TWIN to another popular classic problem about 18 socks in a drawer. 
         These 18 socks are of 9 different colors, are separated and lie in disorder.
         How many socks should a person take from a drawer without looking (without seeing) 
         to guarantee that among selected socks there are at least two proper pairs?


    (2)  The condition that the points are uniformly distributed on the circle DOES NOT MATTER
         in this problem, and I did not used it in my solution. What really does matter is that 
         these  18 points are, actually, 9 pair of opposite points.
         The words "uniformly distributed" are used to disguise this fact and to distract 
         attention of a reader from using this conception of opposite points.


    (3)  Interesting, that in the Internet, there is a solution to this problem, 
         generated by the artificial intelligence chatGPT

https://www.quora.com/Eighteen-points-are-equally-spaced-on-a-circle-from-which-a-number-are-chosen-at-random-How-many-must-you-choose-to-guarantee-that-you-will-have-to-the-four-corners-of-at-least-one-rectangle

              +--------------------------------------------------+
              |    AND THIS SOLUTION IS INCORRECT ( !  !  ! )    |
              +--------------------------------------------------+


      It confirms my idea that so called AI in its contemporary status 
      is simply big (VERY BIG, huge) GARBAGE BIN.

      Regarding this so called AI in Math education, it is simply a way 
      to make money without taking any responsibility for a correctness
      of solutions and quality of final product.