VWO 2001

Consider a triangle and 2 lines that each go through a corner and intersects the opposing segment, such that the areas are as on the attachment.

Find the "?"

Attachments:

In a circle we enscribe a regular $2001$ -gon and inside it a regular $667$ -gon with shared vertices.

Prove that the surface in the $2001$ -gon but not in the $667$ -gon is of the form $k.sin^3\left(\frac{\pi}{2001}\right).cos^3\left(\frac{\pi}{2001}\right)$ with $k$ a positive integer. Find $k$ .