Show that it is not possible for the lengths of the
segments of two intersecting chords to be four consecutive
integers

1. Check the picture below.

2. By the well known intersecting secants property: AO*OC=BO*OD

3. Since the segments are assumed to be consecutive integers, AO, OC cannot be the smaller, and neither can AO<BO and OC<OD. 
The only possibility is that the multiplication of the smallest to the largest is equal to the multiplication of the middle 2.

4. For example : 4*5=2*10, we must have 4>2 but 5<10

Here the ordering is 2<4<5<10, similarly assume BO<AO<CO<DO

5. since the segments are of consecutive integers length, let them be 

BO=n, AO=n+1, CO=n+2, DO=n+3

6. AO*CO=BO*DO

so n(n+3)=(n+1)(n+2)
     
     so 0=3 which makes no sense, so no solution with the assumed conditions.