Yes, there is a way!
This is not trisecting an angle (which IS impossiible!). This is an inscribed polygon in a circle, and there's a formula in the "Mensuration Formulae" section of the CRC that shows you how.
s = 2R sin (180/n)
where
n is the number of sides of the polygon
R is the radius of the circle
s = the length of each side of the polygon
The polygon we are seeking is an equilateral triangle, so each side will be a a line segment 2R sin (60) in length. Just cut a piece of stick of length s and use it as a chord of the circle to locate the vertices.
Each side of the triangle will be R x 1.73205080757.