Looking for a math whiz

[Skinny Shooter]

I'm making trivets and need to locate three legs in the circle.
The center is not known. Here is a sample of what I'm working with. The ring is approx 8" across and is made of 3/4" flat stock.



I used a cloth tape measure to figure the OD and then located the legs shown in the photo.
There has to be an easier way. Have thought about using a triangle to find the three points.
thanks.