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.