Tian on Nostr: Assume we have eight points. If at least three had the same parity type, that alone ...

Assume we have eight points. If at least three had the same parity type, that alone would give three different on-grid midpoints. So assume they form four pairs of points, each with a different common parity type. Each pair has an on-grid midpoint. Now assume these midpoints are all the same and call it M. Let A and B be the pair of points that has the same parity type as M. Then the midpoint of A and M is another on-grid point on the line AB.