Plain TeX: Aligning to an earlier centered line's ends

Hello,

A page has a centered line (via "\centerline{}"):  "Left end of a 
centered line; its right end."

The line below that, "Aligned to L", needs to be left-aligned to "L" of 
the centered line.

The line below that, "Aligned to period.", needs to be right-aligned to 
"." of the centered line.

How exactly to achieve this (create a hbox to the size of the centered 
line, center this hbox, insert the text inside this hbox with hss on the 
right for left-alignment and hss on the left for right alignment)?