How to find two most distant points?

Question

This is a question that I was asked on a job interview some time ago. And I still can\'t figure out sensible answer.

Question is:

you are given set of points

Answer 1

Given a set of points {(x1,y1), (x2,y2) ... (xn,yn)} find 2 most distant points.

My approach:

1). You need a reference point (xa,ya), and it will be:
xa = ( x1 + x2 +...+ xn )/n
ya = ( y1 + y2 +...+ yn )/n

2). Calculate all distance from point (xa,ya) to (x1,y1), (x2,y2),...(xn,yn)
The first "most distant point" (xb,yb) is the one with the maximum distance.

3). Calculate all distance from point (xb,yb) to (x1,y1), (x2,y2),...(xn,yn)
The other "most distant point" (xc,yc) is the one with the maximum distance.

So you got your most distant points (xb,yb) (xc,yc) in O(n)

For example, for points: (0,0), (1,1), (-8, 5)

1). Reference point (xa,ya) = (-2.333, 2)

2). Calculate distances:
from (-2.333, 2) to (0,0) : 3.073
from (-2.333, 2) to (1,1) : 3.480
from (-2.333, 2) to (-8, 5) : 6.411
So the first most distant point is (-8, 5)

3). Calculate distances:
from (-8, 5) to (0,0) : 9.434
from (-8, 5) to (1,1) : 9.849
from (-8, 5) to (-8, 5) : 0
So the other most distant point is (1, 1)