The result position of cylinder is a bit different. I need it to be placed not along one of axis.
I need it to be placed along center line that divide sphere on 2 equal parts. And i also need it to be oriented to normal in that position.

But i see how line-line approach can be helpful. Thank you.
By the way line-line have the same snap issues.

Still i'm interested in orient tool & snaps approach.
I guess the main advantage of orient tool is automatic normal alignment.