Question

There is a line and five points, which do not belong to this line. Two of these points belong to the first half-plane, and three other points – to the second half-plane. Each pair of points is connected by a segment. How many segments intersect the line?

Answer 1

A segment intersects a line if the ends of this segment belong to different half-planes and doesn’t intersect it, if the ends belong to the same half-plane. Consider the half-plane, which contains 2 points. Each of these points must be connected with all points in the other half-plane to build all segments, which intersect the line. Other half-plane contains 3 points. So, each of 2 points must be connected with 3 points. In this case we get 2*3=6 segments, which intersect the line.