Something about line

• Two Distinct Point
• Two-point form

so a line should be like

Line { Point1, Point2 }

function check_on_line(l, p)
    #check p on l using Two-point form 
    p1, p2 <- l
    return (x - p1.x) * (p2.y - p1.y) == (y - p1.y) * (p2.x - p1.x)
    

Loop invariant to find all Line and their point_counts

Init: Build a collection(C) for Lines and put first Line with first two Point into it.

1. take a new Point p
2. point_count++ to any Line in C with p on it
3. if p is not on any Line in C, create Lines with p and all former Points
4. put new Lines into C

foreach p in input
  
  on_some_line = False
  
  foreach l in C
    if check_on_line(l, c)
      l.point_count = l.point_count + 1
      on_some_line = True
      
  if not on_some_line
    foreach _p in input before p
       l = build new Line with _p and p
       add l to C
  

Newly Built Line's point_count

It is wrong that newly built Line.point_count = 0. Because the new Line l may cross some Line in C, and the cross point may be the in the input points.

In that case, the new Line should have its point_count = count of former p on it .

Patched not on_some_line block

  if not on_some_line
    foreach _p in input before p
       l = build new Line with _p and p
       
       l.point_count = count of point input before p on l
       
       add l to C

Finding max

Search C to find the max count.