There is a line and three points (A, B and C), which do not belong to this line. The segment AB intersects the line, and the segment AC doesn’t intersect it. Does the segment BC intersects the line or it doesn’t?
According to fourth Euclid's axiom a line divides the plane into two half-plane. Call these half-plane X and Y. Let the point A belong to the half-plane X. The segment AB intersects the line, thus the point B doesn’t belong to the half-plane X, so it belongs to the half-plane Y. The segment AC doesn’t intersect the line thus the points A and C belongs to the same half-plane – the half-plane X. We get: the point B belongs to the half-plane Y, point C belongs to the half-plane X, so the ends of the segment BC belongs to different half-planes; thus the segment BC intersects the line.