The lines which contain any line segment (e.g. AB and CD) will always intersect (unless they are parallel).
You want to check if the line segments intersect. This is easy to do by finding the point of intersection and checking if it lies on both line segments.
So, using your example, the two lines (on which line segments AB and CD lie on) intersect at P = (15.4434,47.0697). Now, if P lies on AB then the dot product of PA and PB is negative (since the angle between PA and PB is 180° and cos(180°)=-1).
If you compute the dot product of PA and PB, you will get a positive number (thus it does not lie on AB). However, the dot product of PC and PD gives you a negative number (thus it lies on CD).
Thus, you can check if the line segments intersect if the aforementioned dot products (with the point of intersection) are negative.