How do I measure the leg positions for a three-legged table with a circular top?
An equilateral triangular set of legs is easy for a circular top.

Take a pair of dividers (for those who do not know what I mean, this is just the standard school compass used by kids), set to the radius of the circular top.

Step around the perimeter of the table top using the dividers. There will be EXACTLY 6 such steps until you return to the start point if you had set that distance carefully. Mark the steps you take with a pencil around the perimeter. (Adjust the dividers if you do not return to the start point, then redo the steps.)

Once 6 points are located around the perimeter, choose every other mark. Draw a line from the center of the table top to the corresponding mark on the perimeter.

Finally, with that same compass, draw a circle around the center point. Where that circle crosses the radial lines you have just drawn will be the location of your legs, so choose a reasonable radius here. Stability can be increased if the legs splay out a bit, but not too far as you don't want people tripping over them. Rungs that connect the legs can make them more robust.

The process I described above works because a regular hexagon can be inscribed inside a circle such that the vertices lie on the perimeter of the circle. The length of the edges of that hexagon are the same as the circle radius. 



Theta = 360/12 = 30 degrees, Sin(30) = 1/2  So the chord length is R.

Edit: You often have the center of the circle from when you made the table top. Some circle cutting jigs will actually leave a hole in the center.

But, if you don't have that mark to use, that compass will come again to your aid. Set that tool to mark a distance larger than the radius of the circle. With one point at the perimeter of the circle, draw an arc across the rough center of the top. Now repeat, but from the opposite side of the table. Those arcs will cross at two points. Draw a straight line between them.

Now, rotate the entire process by approximately 90 degrees, drawing a second line between a pair of intersections of circular arcs. The two lines we have drawn will intersect at the center of the circle.