(a) Consider an equilateral triangle, inscribed in a circle of radius a, with a point charge q at each vertex. The electric field is zero (obviously) at the center, but (surprisingly) there are three other points inside the triangle where the field is zero. Where are they? [Answer:r=0.285a—you'll probably need a computer to get it.]
(b) For a regular n-sided polygon there are n points (in addition to the center) where the field is zero. Find their distance from the center for n=4 and n=5. What do you suppose happens as n→∞?