Given a plane alpha and three straight lines AB and AC, BC intersecting it, respectively

Given a plane alpha and three straight lines AB and AC, BC intersecting it, respectively, at points A1, B1, C1. Prove that points A1, B1, C1 belong to one straight line.

The plane is defined by three points. Obviously, three points A, B and C define the plane (ABC). Each of the given lines has two common points with the plane (ABC), which means it belongs to it.

If the straight line AC intersects the plane a, then the planes (ABC) and a intersect along some line c. All common points of the plane (ABC) and a lie on the straight line c. Since points A1, B1, C1 are common points of both planes, these are points of the straight line c.