This is a circle _{(citation needed)}.

This is one third of a circle. Dividing circles into thirds is pretty straightforward _{(citation needed)}.

What if we didn't start from the center? Given:

• A circle of radius 1 with a center point at (0, 0)
• An arbitrary point m with y = 0 and 0 < x < 1
• An arbitrary point i on the circle

There is some other point j on the circle such that the three points enclose a 1/3 area. The angle θjm denotes the angle between points m and j. Similarly, the angle θjc denotes the angle from the center to j.

In the following diagram, you can move the points by clicking and dragging. The area is currently (one third would show 0.33)

Unfortunately, the mathematical relationship between these parameters and the position of point j is not readily reducible to a function in terms of θjc. Best form I managed to reduce it to is (c_1-4 being constant factors): c1 = c2*θj + c3*sin(c4*θj). However, computers are fast! We can simply take various guesses and find the best match. Putting all this together, we can smoothly animate the whole thing!