How can I tell if a point belongs to a certain line?

Question

How can I tell if a point belongs to a certain line?

Examples are appreciated, if possible.

Answer 1

The best way to determine if a point R = (rx, ry) lies on the line connecting points P = (px, py) and Q = (qx, qy) is to check whether the determinant of the matrix

{{qx - px, qy - py}, {rx - px, ry - py}},

namely (qx - px) * (ry - py) - (qy - py) * (rx - px) is close to 0. This solution has several related advantages over the others posted: first, it requires no special case for vertical lines, second, it doesn't divide (usually a slow operation), third, it doesn't trigger bad floating-point behavior when the line is almost, but not quite vertical.