A circle is divided into equal arcs by $n$ diameters. Prove that the bases of the perpendiculars dropped from an arbitrary point $M$ inside the circle to these diameters are vertices of a regular n-gon.
Let the center of the circle be $O$. So, for any feet of the perpendicular $N_i$, $N_j$ and $N_k$ we have that $MN_iON_j$ and $MN_iON_k$ form a cyclic quadrilateral because opposite angles add upto $180$ degrees. But, since ther is a unique circle through $M, N_i, O$ these two circles are coincident and equal. In general, all those points lie on a quadrilateral with diameter $MO$.
I have proved that they all lie on a circle but am unable to prove that the chords or the arcs are equal. How do I proceed? Please help.