3 points are drawn on a plane, and inside their triangular region, more points are added such that no 3 are collinear, such that there are n points in total. What is the maximum possible number of line segments one could draw connecting two of these points such that none intersect other than at their endpoints?

(In reply to How Leonhard would solve it by JLo)

The points are non-colinear.  Choose any point as the first and connect it to the three outside points; you have three triangles within the outer one.

Now within each triangle choose an arbitrary point; connect to the vertices of the second-level triangle it's in, for each.

Continue within each newly formed triangle until any triangle is formed with no point inside, but continue with any others.   The result will triangulate all the points.

It's just that I did not know if merely being a triangulation would guarantee the maximum number of lines.

Posted by Charlie on 2006-08-19 00:03:53