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28 points. Each added line n intersects (n-1) other lines. 7 + 6 + 5 + 4 + 3 + 2 +1 = 28

The formula is P = n(n-1)/2, which here is 8x7/2 = 28

It is possible to have any number of points between 28 and 1 if multiple lines intersect at the same point.

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