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Meets from 2004-2005

Feb 2005, Mystery#3: How many different lines contain exactly three points of this square 5 by 5 grid?
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Meet 1 - October 2004
Meet 2 - November 2004
Meet 3 - January 2005
Meet 4 - February 2005
Meet 5 - March 2005
 

Hint: Lines of infinite length, not line segments.