Straightedge+compass angle trisection, given an astroid (= quadricuspid hypocycloid (x^(2/3) + y^(2/3) = 1) = red thing). Draw segment (blue) from center at angle A (= pi/3 in illustration) w.r.t. horizontal, of length 1/4. At its end, draw a circle (green) of radius 3/4. This will cut the astroid in eight places. The cyan, magenta, and yellow radii are at angles -A/3, (2 pi - A)/3, and (4 pi - A)/3. I have yet to work out the significance of the other five intersections.

But where did we get "center" and "horizontal"? (Theological) claim: The cusps are definite, preconstructed points, giving "horizontal", "vertical" and center immediately.