It has become obvious that the Shapes category was in way over its head with all sorts of geometrical problems. So, the hard-core Geometry problems will now reside here, where they will fit in much better.

n≥=3 lines lie in the same plane and are concurrent at point O.
If y = mx is the equation of a line passing through the origin O
(where m is the slope), then m_{k} labels the line y = tan(k*180°/n)*x
(for k = 0 to n-1). Note: If n is even and
k = n/2, then line m_{k} is perpendicular to line m_{0}.
Point P (distinct from O) is an arbitrary point
in the plane of the n lines. F_{i} is the foot of the perpendicular from
point P to line m_{i} (for i = 0 to n-1).

Prove that F_{0}F_{1}...F_{n-1} is a regular n-gon.