looking at just the end points,
you have the line ab and c is the midpoint of it.
you have the line acb with the segments of ac equal to the segments of cb (equal length).
if d lies on ac, then d must lie between a and c.
that makes the whole line equal to adcb where d is between a and c.
this means that dc + cb = db
your line would look like this:
line: a----------d----------c--------------------b
length of dc d----------c
length of cb c--------------------b
length of db d----------c--------------------b
since the whole is equal to the sum of its parts, the length of db has to be the length of dc plus the length of cb.
the statement that d is on the line ac implies that the point d lies somewhere between a and c.
that's my take.