Question 32
Betal marks $2021$ points on the plane such that no three are collinear, and draws all possible line segments joining these. He then chooses any $1011$ of these line segments, and marks their midpoints. Finally, he chooses a line segment whose midpoint is not marked yet, and challenges Vikram to construct its midpoint using only a straightedge. Can Vikram always complete this challenge?
Note: A straightedge is an infinitely long ruler without markings, which can only be used to draw the line joining any two given distinct points.
Note: A straightedge is an infinitely long ruler without markings, which can only be used to draw the line joining any two given distinct points.