3 silver dollar size water stains on ceiling in living room. This is directly below our shower. I suspect it is the shower drain "dognut" or rubber connector

Before you can begin to repair the leak you must first locate it. Some tests should be done before you open up the ceiling. Let's start with the drain. Before you go to bed remove the strainer and put some wet rags in a baggie and stuff them down the drain to seal it off. Now run about a inch or so of water in the shower floor and let it set over night. Make a pencil mark at the water level. Next morning check the mark and your ceiling. Water level down? Ceiling wet? No? Then let the water out and check the ceiling again. Still no wet on the ceiling? If the ceiling stays the same then it's a pretty safe bet the shower drain,trap and drain line's OK and the ceiling won't have to be opened up.
This places the leak in the tile grout. Tile grout shrinks over time and allows water to enter above and outside the shower pan where it will run past the pan and leak out onto the floor. To fix this note where the stream hits the tile when you're taking a shower and regrout the tiles in the area.