Straightedge to draw midpoints
Source: INMO 2021 Problem 3
March 7, 2021
geometryconstructionINMO
Problem Statement
Betal marks points on the plane such that no three are collinear, and draws all possible segments joining these. He then chooses any of these segments, and marks their midpoints. Finally, he chooses a 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. Proposed by Prithwijit De and Sutanay Bhattacharya