Who is asking: Student

Level: Secondary

Question:

A Finite Question

(a) A collection of eight points, no three collinear. If lines are drawn between each pair of these points, how many points of intersection would there be?

(b) what would your answer have been in part (a) if there had been n points to start with?

Hi James

The question needs clarification.

(1) Do we count the original 8 points in the total?

(2) Does the question really call for "lines drawn between points", because that implies that the lines are LINE SEGMENTS. (Otherwise one might say the line JOINS two points.) The difference: any 4 points determine six segments with exactly one intersection point, while they determine 6 lines which determine 3 intersection points unless some lines are parallel.

(2) One certainly wants the points to be "in general position", otherwise there could be instances of 3 or more lines through a single point (and in the infinite line case, the lines could be parallel and not intersect).

If you allow only segments with no three segments through the same point, then the number is n*(n-1)*(n-2)*(n-3)/24, which is 70 when n=8..

Otherwise the best answer is a range of values that would be very hard to determine. For example there are 70 intersection points of segments determined by 8 points placed in general position about a circle, while the number of intersections of the diagonals of a regular octagon (whose eight vertices are evenly spaced about a circle) is only 49 (because each diagonal through the center determines only the center with two other points).

Chris

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