  We prove Anzis and Tohaneanu conjecture, that is the Dirac-Motzkin conjecture
for supersolvable line arrangements in the projective plane over an arbitrary
field of characteristic zero. Moreover, we show that a divisionally free
arrangements of lines contain at least one double point, that can be regarded
as the Sylvester-Gallai theorem for some free arrangements. This is a corollary
of a general result that if you add a line to a free projective line
arrangement, then that line has to contain at least one double point. Also we
prove some conjectures and one open problems related to supersolvable line
arrangements and the number of double points.
