ABCDEFG is a regular heptagon.
I is the intersection of BD and CE, J is the intersection of CF and BE, K is the intersection of CG and BF, L is the intersection of AC and BG.
Show that B,C,I,J,K,L are vertices of a regular heptagon.
An interactive illustration is here.
Can you generalize this result?
Defining a generic n-gon in Geometry Expressions, we observe that the angle BLC is 360/n. Likewise BKC, BJC, BIC. This tells us that BLKJIC are concyclic. (The diagram indicates how this may readily be proved by hand).
Now the angle subtended by chord LK at the circumference of this circle is angle LCK, which is equal to ACG, which is 180/n. Hence LK is the side of a regular ngon inscribed in the circle BLKJIC. Similarly for sides BL, KJ, JI, IC.
One can show in a similar fashion that for given k the intersections of the diagonal from B to the ith point on the ngon with that from C to the i+kth all lie on a regular n-gon of which BD is a diagonal across k points. (The example above has k=1).